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(In this "README" file (as well as the "README" files in the "code", "primality-certificates", "unproven-probable-primes", i.e. https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/README.md and https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/primality-certificates/README.md and https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/unproven-probable-primes/README.md), I always offer links to the English Wikipedia (https://en.wikipedia.org/wiki/Main_Page, https://en.wikipedia.org/wiki/English_Wikipedia), The Prime Glossary (https://t5k.org/glossary/) (which is a part of The Prime Pages (https://en.wikipedia.org/wiki/PrimePages, https://www.rieselprime.de/ziki/The_Prime_Pages)), the Prime Wiki (https://www.rieselprime.de/ziki/Main_Page), and the Wolfram MathWorld (https://mathworld.wolfram.com/, https://en.wikipedia.org/wiki/MathWorld), as well as the OEIS (https://oeis.org/, https://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences, https://www.rieselprime.de/ziki/On-Line_Encyclopedia_of_Integer_Sequences) sequences or other references, for the proper nouns, if such page exists, also the factordb (http://factordb.com/, https://www.rieselprime.de/ziki/Factoring_Database) entries for the large primes or probable primes, or factoring status for the N−1 and N+1 for the large primes or probable primes, or factoring status for the numbers of certain forms, also the archived pages (only list the newest available (i.e. neither 4xx nor 5xx) archived pages) for the dead links with the Wayback Machine (https://web.archive.org/, https://en.wikipedia.org/wiki/Wayback_Machine), also, to avoid permanently dead links, I always use both the Wayback Machine (https://web.archive.org/, https://en.wikipedia.org/wiki/Wayback_Machine) and the archive today (https://archive.ph/, https://archive.is/, https://archive.li/, https://archive.vn/, https://archive.fo/, https://archive.md/, https://en.wikipedia.org/wiki/Archive.today) to save all link webpages in this "README" file (as well as all link pages in the "README" files in the "code", "primality-certificates", "unproven-probable-primes", i.e. all link pages in https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/README.md and https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/primality-certificates/README.md and https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/unproven-probable-primes/README.md), including use the archive today (https://archive.ph/, https://archive.is/, https://archive.li/, https://archive.vn/, https://archive.fo/, https://archive.md/, https://en.wikipedia.org/wiki/Archive.today) to save all archived pages for dead links with the wayback machine (https://web.archive.org/, https://en.wikipedia.org/wiki/Wayback_Machine), also, for the ".zip" files (https://en.wikipedia.org/wiki/ZIP_(file_format)) and the ".7z" files (https://en.wikipedia.org/wiki/7z) and the ".lz" files (https://en.wikipedia.org/wiki/Lzip) and the ".gz" files (https://en.wikipedia.org/wiki/Gzip) and the ".xz" files (https://en.wikipedia.org/wiki/XZ_Utils) and the ".exe" files (https://en.wikipedia.org/wiki/.exe) and the ".dll" files (https://en.wikipedia.org/wiki/Dynamic-link_library), I always use both the Wayback Machine (https://web.archive.org/, https://en.wikipedia.org/wiki/Wayback_Machine) and the megalodon (https://megalodon.jp/, https://en.wikipedia.org/wiki/Megalodon_(website)) to save (since these files cannot be saved with the archive today (https://archive.ph/, https://archive.is/, https://archive.li/, https://archive.vn/, https://archive.fo/, https://archive.md/, https://en.wikipedia.org/wiki/Archive.today)), also, for the link webpages with programs (such as the "Prime checkers" and the "Integer factorizers" and the "Prime generators" and the "Base converters" and the "Expression generators" link webpages in the bottom of this "README" file), I always use the ghost archive (https://ghostarchive.org/, https://en.wikipedia.org/wiki/Ghost_Archive) as well as the Wayback Machine (https://web.archive.org/, https://en.wikipedia.org/wiki/Wayback_Machine) and the archive today (https://archive.ph/, https://archive.is/, https://archive.li/, https://archive.vn/, https://archive.fo/, https://archive.md/, https://en.wikipedia.org/wiki/Archive.today) to save (since in the archived pages of these webpages with the archive today (https://archive.ph/, https://archive.is/, https://archive.li/, https://archive.vn/, https://archive.fo/, https://archive.md/, https://en.wikipedia.org/wiki/Archive.today) or the megalodon (https://megalodon.jp/, https://en.wikipedia.org/wiki/Megalodon_(website)), the programs will not work), unfortunately, as of Feb. 2024, Google has removed links to its cached pages from Google search results (known informally as "Google cache" (https://web.archive.org/web/20240127100249/https://support.google.com/websearch/answer/1687222, https://en.wikipedia.org/wiki/Google_cache)), see https://arstechnica.com/gadgets/2024/02/google-search-kills-off-cached-webpages/, but you can still use time travel (http://timetravel.mementoweb.org/) to find the archive pages, also, I did not only use the Wayback Machine (https://web.archive.org/, https://en.wikipedia.org/wiki/Wayback_Machine) to save but also use another archive service since some webpages have been excluded from the Wayback Machine (see https://wiki.archiveteam.org/index.php/List_of_websites_excluded_from_the_Wayback_Machine and https://wiki.archiveteam.org/index.php/List_of_websites_excluded_from_the_Wayback_Machine/Partial_exclusions and https://wiki.archiveteam.org/index.php/List_of_websites_excluded_from_the_Wayback_Machine/Former_exclusions) (also, before Apr. 2017, the Wayback Machine has respected the robots exclusion standard (i.e. robots.txt) (https://www.robotstxt.org/, https://datatracker.ietf.org/doc/html/rfc9309, https://en.wikipedia.org/wiki/Robots.txt) in determining if a website would be crawled (or if already crawled, if its archives would be publicly viewable). Website owners had the option to opt-out of Wayback Machine through the use of robots.txt. It applied robots.txt rules retroactively; if a site blocked the Wayback Machine, any previously archived pages from the domain were immediately rendered unavailable as well. However, as of Apr. 2017, the Wayback Machine changed the policy to now require an explicit exclusion request to remove it from the Wayback Machine, see https://help.archive.org/help/using-the-wayback-machine/), for more web archive sites (https://en.wikipedia.org/wiki/Web_archiving), see https://en.wikipedia.org/wiki/Wikipedia:List_of_web_archives_on_Wikipedia and https://en.wikipedia.org/wiki/Help:Archiving_a_source, also, for the cached copy of the pdf files (https://en.wikipedia.org/wiki/PDF) references see https://github.com/xayahrainie4793/pdf-files-cached-copy (you can click https://github.com/xayahrainie4793/pdf-files-cached-copy/archive/refs/heads/main.zip to download all these pdf files by one click), and for the cached copy of the prime programs (including: The primality (or probable primality) testing (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/index.html) programs (https://www.rieselprime.de/ziki/Primality_testing_program), the sieving (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.primegrid.com/stats_psp_sieve.php, https://www.primegrid.com/stats_pps_sieve.php, https://www.primegrid.com/stats_sr5_sieve.php, https://www.primegrid.com/stats_cw_sieve.php, https://www.primegrid.com/sieving/rsp/) programs (https://www.rieselprime.de/ziki/Sieving_program), the integer factoring (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) programs (https://www.rieselprime.de/ziki/Factoring_program)) see https://github.com/xayahrainie4793/prime-programs-cached-copy (you can click https://github.com/xayahrainie4793/prime-programs-cached-copy/archive/refs/heads/main.zip to download all these programs by one click), also not only webpage references, but also ".zip" files (https://en.wikipedia.org/wiki/ZIP_(file_format)) and ".7z" files (https://en.wikipedia.org/wiki/7z) and ".lz" files (https://en.wikipedia.org/wiki/Lzip) and ".gz" files (https://en.wikipedia.org/wiki/Gzip) and ".xz" files (https://en.wikipedia.org/wiki/XZ_Utils) and ".exe" files (https://en.wikipedia.org/wiki/.exe) and ".dll" files (https://en.wikipedia.org/wiki/Dynamic-link_library), which you can download (https://en.wikipedia.org/wiki/Download), also you can download the C (https://en.wikipedia.org/wiki/C_(programming_language)) program files ".c" and ".h" (https://en.wikipedia.org/wiki/Include_directive), also C++ (https://en.wikipedia.org/wiki/C%2B%2B) program files ".cpp", also the ".ini" files (https://en.wikipedia.org/wiki/INI_file) and the text files (https://en.wikipedia.org/wiki/Text_file) and the html files (https://en.wikipedia.org/wiki/HTML), of course, you can also click https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/archive/refs/heads/main.zip to download all files in this GitHub (https://en.wikipedia.org/wiki/GitHub, https://github.com/) page by one click)

(Note: I do not offer links to the OEIS Wiki (https://oeis.org/wiki/Main_Page) or the PrimeGrid Wiki (http://primegrid.wikia.com/wiki/PrimeGrid_Wiki) or the Rechenkraft Wiki (https://www.rechenkraft.net/wiki/Willkommen_beim_Verein_Rechenkraft.net_e.V.), with only two exceptions: the page "OEIS sequences needing factors" (https://oeis.org/wiki/OEIS_sequences_needing_factors) in the OEIS Wiki and the page "Five or Bust" (https://www.rechenkraft.net/wiki/Five_or_Bust) in the Rechenkraft Wiki, since fewer people contribute the OEIS Wiki and the PrimeGrid Wiki, and most things in the Rechenkraft Wiki are out of date)

A Prime Game:

Write down a multidigit prime number (i.e. a prime number > 10), and I can always strike out 0 or more digits to get a prime in this list:

{11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}

e.g.

  • Write down the prime 149 → I can strike out the digit 4, to get the prime 19
  • Write down the prime 439 → I can strike out the digit 9, to get the prime 43
  • Write down the prime 857 → I can strike out zero digits, to get the prime 857
  • Write down the prime 2081 → I can strike out the digit 0, to get the prime 281
  • Write down the largest known double Mersenne prime 170141183460469231731687303715884105727 (2127−1) → I can strike out all digits except the third-leftmost 1 and the second-rightmost 3, to get the prime 13 (also I can choose to strike out all digits except the second-leftmost 4 and the third-rightmost 7, to get the prime 47)
  • Write down the largest known Fermat prime 65537 → I can strike out the 6 and the 3, to get the prime 557 (also I can choose to strike out the 6 and two 5's, to get the prime 37) (also I can choose to strike out two 5's and the 3, to get the prime 67) (also I can choose to strike out the 6, one 5, and the 7, to get the prime 53)
  • Write down the famous repunit prime 1111111111111111111 (with 19 1's) → I can strike out 17 1's, to get the prime 11
  • Write down the prime 1000000000000000000000000000000000000000000000000000000000007 (which is the next prime after 1060) → I can strike out all 0's, to get the prime 17
  • Write down the prime 95801 → I can strike out the 9, to get the prime 5801
  • Write down the prime 946969 → I can strike out the first 9 and two 6's, to get the prime 499
  • Write down the prime 90000000581 → I can strike out five 0's, the 5, and the 8, to get the prime 9001
  • Write down the prime 8555555555555555555551 → I can strike out the 8 and nine 5's, to get the prime 555555555551

These primes are called minimal primes.

Let b > 1 be a natural number (https://en.wikipedia.org/wiki/Natural_number, https://www.rieselprime.de/ziki/Natural_number, https://mathworld.wolfram.com/NaturalNumber.html). A minimal prime base b is a prime number (https://en.wikipedia.org/wiki/Prime_number, https://t5k.org/glossary/xpage/Prime.html, https://www.rieselprime.de/ziki/Prime, https://mathworld.wolfram.com/PrimeNumber.html, https://www.numbersaplenty.com/set/prime_number/, http://www.numericana.com/answer/primes.htm#definition, http://irvinemclean.com/maths/pfaq2.htm, https://oeis.org/A000040, https://t5k.org/lists/small/1000.txt, https://t5k.org/lists/small/10000.txt, https://t5k.org/lists/small/100000.txt, https://t5k.org/lists/small/millions/) greater than (https://en.wikipedia.org/wiki/Greater_than, https://mathworld.wolfram.com/Greater.html) b whose base-b (i.e. the positional numeral system (https://en.wikipedia.org/wiki/Positional_numeral_system) with radix (https://en.wikipedia.org/wiki/Radix, https://t5k.org/glossary/xpage/Radix.html, https://www.rieselprime.de/ziki/Base, https://mathworld.wolfram.com/Radix.html) b) representation has no proper subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html) which is also a prime number greater than b.

For example, 857 is a minimal prime in decimal (base b = 10) because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a minimal prime in decimal (base b = 10) (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a minimal prime in decimal (base b = 10) even though a subset of the digits can form the shorter prime 19 > 10 by changing the order.

Now we extend minimal primes to bases b other than 10.

The minimal elements (https://en.wikipedia.org/wiki/Minimal_element) (https://mathworld.wolfram.com/MaximalElement.html for maximal element, the dual of minimal element, unfortunately there is no article "minimal element" in mathworld, a minimal element of a set (https://en.wikipedia.org/wiki/Set_(mathematics), https://mathworld.wolfram.com/Set.html) under a partial ordering binary relation (https://en.wikipedia.org/wiki/Binary_relation, https://mathworld.wolfram.com/BinaryRelation.html) is a maximal element of the same set under its converse relation (https://en.wikipedia.org/wiki/Converse_relation), a converse relation of a partial ordering relation must also be a partial ordering relation) of the prime numbers (https://en.wikipedia.org/wiki/Prime_number, https://t5k.org/glossary/xpage/Prime.html, https://www.rieselprime.de/ziki/Prime, https://mathworld.wolfram.com/PrimeNumber.html, https://www.numbersaplenty.com/set/prime_number/, http://www.numericana.com/answer/primes.htm#definition, http://irvinemclean.com/maths/pfaq2.htm, https://oeis.org/A000040, https://t5k.org/lists/small/1000.txt, https://t5k.org/lists/small/10000.txt, https://t5k.org/lists/small/100000.txt, https://t5k.org/lists/small/millions/) which are > b written in the positional numeral system (https://en.wikipedia.org/wiki/Positional_numeral_system) with radix (https://en.wikipedia.org/wiki/Radix, https://t5k.org/glossary/xpage/Radix.html, https://www.rieselprime.de/ziki/Base, https://mathworld.wolfram.com/Radix.html) b, as digit (https://en.wikipedia.org/wiki/Numerical_digit, https://www.rieselprime.de/ziki/Digit, https://mathworld.wolfram.com/Digit.html) strings (https://en.wikipedia.org/wiki/String_(computer_science), https://mathworld.wolfram.com/String.html) under the subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html) ordering (https://en.wikipedia.org/wiki/Partially_ordered_set, https://mathworld.wolfram.com/PartialOrder.html, https://mathworld.wolfram.com/PartiallyOrderedSet.html), for 2 ≤ b ≤ 36 (I stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the symbols 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 Arabic numerals (https://en.wikipedia.org/wiki/Arabic_numerals, https://mathworld.wolfram.com/ArabicNumeral.html) and the 26 Latin letters (https://en.wikipedia.org/wiki/Latin_alphabet, https://en.wikipedia.org/wiki/ISO_basic_Latin_alphabet), also the 36 characters in Morse code (https://en.wikipedia.org/wiki/Morse_code) (see https://upload.wikimedia.org/wikipedia/commons/b/b5/International_Morse_Code.svg), also the 36 characters in NATO phonetic alphabet (https://en.wikipedia.org/wiki/NATO_phonetic_alphabet, http://www.nato.int/cps/en/natohq/declassified_136216.htm, https://web.archive.org/web/20190626195301/https://www.icao.int/secretariat/PostalHistory/annex_10_aeronautical_telecommunications.htm) (see https://upload.wikimedia.org/wikipedia/commons/e/e0/FAA_Phonetic_and_Morse_Chart2.svg), also the 36 characters in international maritime signal flags (https://en.wikipedia.org/wiki/International_maritime_signal_flags, http://www.quadibloc.com/other/flaint.htm)), i.e. bases 2 ≤ b ≤ 36 are case-insensitive (https://en.wikipedia.org/wiki/Case-insensitive) alphanumeric (https://en.wikipedia.org/wiki/Alphanumericals) numeral system using ASCII (https://en.wikipedia.org/wiki/ASCII) characters (https://en.wikipedia.org/wiki/Character_(computing)), while bases b > 36 are not, references: http://www.tonymarston.net/php-mysql/converter.html, https://www.dcode.fr/base-36-cipher, http://www.urticator.net/essay/5/567.html, http://www.urticator.net/essay/6/624.html, https://docs.python.org/3/library/functions.html#int, https://numpy.org/doc/stable/reference/generated/numpy.base_repr.html, https://reference.wolfram.com/language/ref/BaseForm.html, https://support.microsoft.com/en-us/office/base-function-2ef61411-aee9-4f29-a811-1c42456c6342, https://www.cut-the-knot.org/recurrence/word_primes.shtml, https://oeis.org/A072922, https://oeis.org/A073421, https://oeis.org/A002488 (the Alonso del Arte comment in Jul 01 2012), https://en.wikipedia.org/wiki/Base36, https://web.archive.org/web/20150320103231/https://en.wikipedia.org/wiki/Base_36, https://fr.wikipedia.org/wiki/Syst%C3%A8me_%C3%A0_base_36 (in French), https://zh.wikipedia.org/wiki/%E4%B8%89%E5%8D%81%E5%85%AD%E8%BF%9B%E5%88%B6 (in Chinese), https://ja.wikipedia.org/wiki/%E4%B8%89%E5%8D%81%E5%85%AD%E9%80%B2%E6%B3%95 (in Japanese), https://baseconvert.com/, https://baseconvert.com/high-precision, https://www.calculand.com/unit-converter/zahlen.php?og=Base+2-36&ug=1, http://www.unitconversion.org/unit_converter/numbers.html, http://www.unitconversion.org/unit_converter/numbers-ex.html, http://extraconversion.com/base-number, https://web.archive.org/web/20190629223750/http://thedevtoolkit.com/tools/base_conversion, http://www.kwuntung.net/hkunit/base/base.php (in Chinese), https://linesegment.web.fc2.com/application/math/numbers/RadixConversion.html (in Japanese), also https://jpbenney.blogspot.com/2021/08/pentatrigesimal.html and https://jpbenney.blogspot.com/2021/08/pentatrigesimal-periods.html and https://jpbenney.blogspot.com/2021/08/can-you-recognise-this-list.html (although they use base b = 35 instead of base b = 36, since they do not use the Latin letter O as a digit), also see https://t5k.org/notes/words.html for the English words which are prime numbers when viewed as a number base 36 (also see https://oeis.org/A038842), also, the pangrams (https://en.wikipedia.org/wiki/Pangram, http://clagnut.com/blog/2380/, http://rinkworks.com/words/pangrams.shtml, https://wordsmith.org/pangram/) together with the digits 0−9 will be pandigital numbers (https://en.wikipedia.org/wiki/Pandigital_number, https://mathworld.wolfram.com/PandigitalNumber.html, https://oeis.org/A171102, https://oeis.org/A050288, https://oeis.org/A049363, https://oeis.org/A185122, https://arxiv.org/pdf/2403.20304.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_441.pdf)) in base b = 36, e.g. the famous pangram "The quick brown fox jumps over the lazy dog" (https://en.wikipedia.org/wiki/The_quick_brown_fox_jumps_over_the_lazy_dog) together with the digits 0−9 is a pandigital number in base b = 36, see http://factordb.com/index.php?showid=1100000002639212386&base=36 (although this number is not prime (it is divisible by 2), see http://factordb.com/index.php?id=1100000002639212386&open=ecm for its prime factorization), and the shorter pangram "Waltz, bad nymph, for quick jigs vex" together with the digits 0−9 is a pandigital number in base b = 36, see http://factordb.com/index.php?showid=1100000007048545780&base=36 (although this number is not prime (it is divisible by 3), see http://factordb.com/index.php?id=1100000007048545780&open=ecm for its prime factorization), also, the palindromes (https://en.wikipedia.org/wiki/Palindrome) will be palindromic numbers (https://en.wikipedia.org/wiki/Palindromic_number, https://en.wikipedia.org/wiki/Palindromic_prime, https://t5k.org/glossary/xpage/PalindromicPrime.html, https://mathworld.wolfram.com/PalindromicNumber.html, https://mathworld.wolfram.com/PalindromicPrime.html, https://www.numbersaplenty.com/set/palindromic_number/, https://t5k.org/top20/page.php?id=53, https://t5k.org/primes/search.php?Comment=Palindrome&OnList=all&Number=1000000&Style=HTML, https://stdkmd.net/nrr/abbba.htm, https://stdkmd.net/nrr/aabaa.htm, https://stdkmd.net/nrr/prime/prime_nrpl.htm, https://stdkmd.net/nrr/prime/prime_nrpl.txt, https://stdkmd.net/nrr/prime/prime_n2pl.htm, https://stdkmd.net/nrr/prime/prime_n2pl.txt, https://stdkmd.net/nrr/prime/prime_n3pl.htm, https://stdkmd.net/nrr/prime/prime_n3pl.txt, https://stdkmd.net/nrr/prime/prime_pd.htm, https://stdkmd.net/nrr/prime/prime_pd.txt, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#pdprime, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#nrpprime, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#pdprp, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#nrpprp, https://www.asahi-net.or.jp/~KC2H-MSM/mathland/aba/index.htm, http://factordb.com/tables.php?open=3, http://factordb.com/tables.php?open=5, http://factordb.com/tables.php?open=6, https://oeis.org/A002113, https://oeis.org/A002385) in base b = 36, e.g. the longest known palindrome word "saippuakivikauppias" is a palindromic number in base b = 36, see http://factordb.com/index.php?showid=1100000003910734942&base=36 (although this number is not prime (it is divisible by 2), see http://factordb.com/index.php?id=1100000003910734942&open=ecm for its prime factorization), and Demetri Martin's Palindrome in https://web.archive.org/web/20150923005512/http://classes.yale.edu/fractals/panorama/Literature/Martin/MartinPalindrome.html is a palindromic number in base b = 36, see http://factordb.com/index.php?showid=1100000004709774731&base=36 (although this number is not prime (it is divisible by 23), see http://factordb.com/index.php?id=1100000004709774731&open=ecm for its prime factorization), (you can try to convert these numbers to base b = 36: 133, 391, 417, 853, 1030, 15238, 35665, 36825, 599609, 620303, 630308, 739172, 957182, 1329077, 1334693, 19353617, 43427410, 816958261, 1421722899, 23508730562, you will get something interesting! A similar example is the fraction 1480479987/1679615 in base b = 36, see https://www.reddit.com/r/funny/comments/1dhog4/wolfram_alpha_gives_up/) also, the digits in bases 2 ≤ b ≤ 36 can use either sixteen-segment display (https://en.wikipedia.org/wiki/Sixteen-segment_display) or fourteen-segment display (https://en.wikipedia.org/wiki/Fourteen-segment_display) to show, see https://upload.wikimedia.org/wikipedia/commons/5/5b/Sixteen-segment_display_0-9_A-Z.gif and https://upload.wikimedia.org/wikipedia/commons/b/b8/Arabic_number_on_a_14_segement_display.gif and https://upload.wikimedia.org/wikipedia/commons/6/62/Latin_alphabet_on_a_14_segement_display.gif), using upper case letters (https://en.wikipedia.org/wiki/Upper-case_letter) A−Z to represent digit values 10 to 35 (A represents digit value 10, B represents digit value 11, C represents digit value 12, D represents digit value 13, E represents digit value 14, F represents digit value 15, G represents digit value 16, H represents digit value 17, I represents digit value 18, J represents digit value 19, K represents digit value 20, L represents digit value 21, M represents digit value 22, N represents digit value 23, O represents digit value 24, P represents digit value 25, Q represents digit value 26, R represents digit value 27, S represents digit value 28, T represents digit value 29, U represents digit value 30, V represents digit value 31, W represents digit value 32, X represents digit value 33, Y represents digit value 34, Z represents digit value 35). (note: the number 36 also has number theory significances, although 24 is the most important number in number theory (see https://sites.google.com/view/24-important-number-theory), but the next half of 24 (i.e. 12) numbers after 24 immediately have four perfect powers (https://oeis.org/A001597, https://en.wikipedia.org/wiki/Perfect_power, https://mathworld.wolfram.com/PerfectPower.html, https://www.numbersaplenty.com/set/perfect_power/), i.e. 25 = 52, 27 = 33, 32 = 25, 36 = 62, and the main problem in this project in perfect power bases b are more interesting since a large minimal prime in base b = mr can be written as a base m form, and both the top definitely primes page (https://t5k.org/primes/lists/all.txt) and the generalized Proth/Riesel primes page (https://pzktupel.de/Primetables/TableProthGen.php, https://pzktupel.de/Primetables/TableRieselGen.php) convert the perfect power bases (i.e. b = mr with r > 1) to their "ground bases" (https://oeis.org/A052410) (i.e. b = m), i.e. the bases are normalized, e.g. it converts the prime 2805222×252805222+1 to 2805222×55610444+1 (i.e. converts base 25 = 52 to base 5) (see https://t5k.org/primes/page.php?id=129893 for the entry of this prime in the top definitely primes page), and it converts the prime 2622×121810960−1 to 2622×111621920−1 (i.e. converts base 121 = 112 to base 11) (see https://t5k.org/primes/page.php?id=119929 for the entry of this prime in the top definitely primes page), see https://www.mersenneforum.org/showpost.php?p=121374&postcount=1 and https://www.mersenneforum.org/showpost.php?p=656659&postcount=1 and https://www.mersenneforum.org/showpost.php?p=643173&postcount=9, and 36 is exactly 24 + (half of 24), besides, the number 36 also has other number theory properties, e.g. it is the smallest perfect power (https://oeis.org/A001597, https://en.wikipedia.org/wiki/Perfect_power, https://mathworld.wolfram.com/PerfectPower.html, https://www.numbersaplenty.com/set/perfect_power/) which is not prime power (https://oeis.org/A246655, https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html), it is the smallest square triangular number (https://en.wikipedia.org/wiki/Square_triangular_number, https://mathworld.wolfram.com/SquareTriangularNumber.html, https://oeis.org/A001110) (i.e. a number which is both square (https://en.wikipedia.org/wiki/Square_number, https://www.rieselprime.de/ziki/Square_number, https://mathworld.wolfram.com/SquareNumber.html, https://www.numbersaplenty.com/set/square_number/, https://oeis.org/A000290) and triangular (https://en.wikipedia.org/wiki/Triangular_number, https://mathworld.wolfram.com/TriangularNumber.html, https://www.numbersaplenty.com/set/triangular_number/, https://oeis.org/A000217)) greater than 1, it is the smallest number greater than 1 which is neither prime power (https://oeis.org/A246655, https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html) nor squarefree (https://en.wikipedia.org/wiki/Square-free_integer, https://mathworld.wolfram.com/Squarefree.html, https://oeis.org/A005117), it is a highly composite number (https://en.wikipedia.org/wiki/Highly_composite_number, https://mathworld.wolfram.com/HighlyCompositeNumber.html, https://www.numbersaplenty.com/set/highly_composite_number/, https://oeis.org/A002182) (highly composite numbers are more suitable for the stopping base b since it is more convenient for the fractions (https://en.wikipedia.org/wiki/Fraction, https://mathworld.wolfram.com/Fraction.html) with small denominators, since all fractions with denominators dividing (https://en.wikipedia.org/wiki/Divides, https://t5k.org/glossary/xpage/Divides.html, https://t5k.org/glossary/xpage/Divisor.html, https://www.rieselprime.de/ziki/Factor, https://mathworld.wolfram.com/Divides.html, https://mathworld.wolfram.com/Divisor.html, https://mathworld.wolfram.com/Divisible.html, http://www.numericana.com/answer/primes.htm#divisor) the base b have only one digit after the radix point (https://en.wikipedia.org/wiki/Radix_point), this is the generalization of repeating decimal (https://en.wikipedia.org/wiki/Repeating_decimal, https://t5k.org/glossary/xpage/PeriodOfADecimal.html, https://mathworld.wolfram.com/RepeatingDecimal.html) to other bases b, also base b+k will have more same congruence for small modulos (https://en.wikipedia.org/wiki/Integers_modulo_n, https://mathworld.wolfram.com/Mod.html), thus have more same properties in the main problem in this project since they are congruent mod many small numbers (thus no need to repeat them, and stop at such base b is better), and for the case for base b = 36, it is divisible by 2 and 3, and the number one less than 36 is the product of the next two primes (5 and 7), it can approximate many fractions well for its size), it is a highly abundant number (https://en.wikipedia.org/wiki/Highly_abundant_number, https://oeis.org/A002093), it is a superabundant number (https://en.wikipedia.org/wiki/Superabundant_number, https://mathworld.wolfram.com/SuperabundantNumber.html, https://www.numbersaplenty.com/set/superabundant_number/, https://oeis.org/A004394), all even perfect numbers (https://en.wikipedia.org/wiki/Perfect_number, https://en.wikipedia.org/wiki/List_of_Mersenne_primes_and_perfect_numbers, https://t5k.org/glossary/xpage/PerfectNumber.html, https://www.rieselprime.de/ziki/Perfect_number, https://mathworld.wolfram.com/PerfectNumber.html, https://mathworld.wolfram.com/EvenPerfectNumber.html, https://www.numbersaplenty.com/set/perfect_number/, https://www.numericana.com/answer/numbers.htm#perfect, https://t5k.org/notes/proofs/EvenPerfect.html, https://t5k.org/notes/proofs/Theorem3.html, https://oeis.org/A000396) except 6 end with the digit S in base b = 36 (the first twelve perfect numbers (of course all are even, since currently there are no known odd perfect numbers) written in base b = 36 are 6, S, DS, 69S, JZ3LS, 3Y26PDS, 1R4ZG4XS, HIO94MA0BJLS, 49GANX6QRHCA3LSAR3NSUI9S, 2C0BVYDSTGQL620GUGC18745PAZSVNZIPDS, 21O3YNZFCCMBFGSF6MZB3QSTL75SELZYD2D9DTAPDS, SPN7SJDRBBR9EKYVZLYMKAIYODC0J4OYKTWFTY3W9V1MZQSXS), and if there exists an odd perfect number (https://web.archive.org/web/20120426061657/http://oddperfect.org/, https://www.lirmm.fr/~ochem/opn/, https://en.wikipedia.org/wiki/Perfect_number, https://en.wikipedia.org/wiki/List_of_Mersenne_primes_and_perfect_numbers, https://t5k.org/glossary/xpage/PerfectNumber.html, https://www.rieselprime.de/ziki/Perfect_number, https://mathworld.wolfram.com/PerfectNumber.html, https://mathworld.wolfram.com/OddPerfectNumber.html, https://maths-people.anu.edu.au/~brent/pub/pub116.html, https://maths-people.anu.edu.au/~brent/pub/pub100.html, https://maths-people.anu.edu.au/~brent/pub/pub106.html, https://maths-people.anu.edu.au/~brent/pd/rpb116a.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_398.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb116.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_399.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb116p.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_400.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb100a.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_401.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_402.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb100s.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_403.pdf), https://www.lirmm.fr/~ochem/opn/opn.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_404.pdf), https://www.lirmm.fr/~ochem/opn/opnf.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_437.pdf), https://math.colgate.edu/~integers/x79/x79.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_438.pdf), https://www.lirmm.fr/~ochem/opn/opn_slide.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_439.pdf), https://www.ams.org/journals/mcom/2007-76-260/S0025-5718-07-02033-9/S0025-5718-07-02033-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_440.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb106i.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_405.pdf)), then it must be end with one of the digits {1,9,D,P} in base b = 36, but whether there exists an odd perfect number is a famous open problem (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://t5k.org/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, http://www.numericana.com/answer/open.htm, https://t5k.org/notes/conjectures/), and currently it is only know that there is no odd perfect number ≤ 101500, also all records for highest point of trajectory before reaching 1 in the sequence of the Collatz conjecture (https://en.wikipedia.org/wiki/Collatz_conjecture, https://mathworld.wolfram.com/CollatzProblem.html, http://www.numericana.com/answer/open.htm#collatz, http://www.ericr.nl/wondrous/index.html, http://www.ericr.nl/wondrous/delrecs.html, http://www.ericr.nl/wondrous/pathrecs.html, http://www.ericr.nl/wondrous/comprecs.html, http://www.ericr.nl/wondrous/glidrecs.html, http://www.ericr.nl/wondrous/residues.html, http://www.ericr.nl/wondrous/classrec.html, http://www.ericr.nl/wondrous/strengths.html, http://www.ericr.nl/wondrous/progress.html, http://www.ericr.nl/wondrous/techpage.html, http://www.ericr.nl/wondrous/showsteps.html, https://pcbarina.fit.vutbr.cz/, https://pcbarina.fit.vutbr.cz/path-records.htm, http://sweet.ua.pt/tos/3x_plus_1.html, http://www.rechenkraft.net/yoyo/y_status_col.php, https://oeis.org/A006370, https://oeis.org/A070165, https://oeis.org/A006577, https://oeis.org/A006877, https://oeis.org/A006878, https://oeis.org/A025586, https://oeis.org/A006884, https://oeis.org/A006885) except 1 and 2 end with the digit G in base b = 36 (the first twenty record numbers written in base b = 36 are 1, 2, G, 1G, 4G, 74G, A4G, UDG, W1G, 5DAG, RDAG, 41YQG, 4URDG, G55MG, TUQSG, 1RBMDG, 201Q4G, 9T814G, PZ884G, 151S9DG, also, the famous "sequence of the Collatz conjecture" starting with the number R (27 in decimal) (i.e. the sequence https://oeis.org/A008884, which gives the record number 74G (9232 in decimal)) written in base b = 36 is R, 2A, 15, 3G, 1Q, V, 2M, 1B, 3Y, 1Z, 5Y, 2Z, 8Y, 4H, DG, 6Q, 3D, A4, 52, 2J, 7M, 3T, BG, 5Q, 2V, 8M, 4B, CY, 6H, JG, 9Q, 4V, EM, 7B, LY, AZ, WY, GH, 1DG, OQ, CD, 114, IK, 9A, 4N, DY, 6Z, KY, AH, VG, FQ, 7V, NM, BT, ZG, HQ, 8V, QM, DB, 13Y, JZ, 1NY, TZ, 2HY, 18Z, 3QY, 1VH, 5MG, 2T8, 1EM, PB, 23Y, 11Z, 35Y, 1KZ, 4QY, 2DH, 74G, 3K8, 1S4, W2, G1, 1C4, O2, C1, 104, I2, 91, R4, DK, 6S, 3E, 1P, 54, 2K, 1A, N, 1Y, Z, 2Y, 1H, 4G, 28, 14, K, A, 5, G, 8, 4, 2, 1), also 36 is the number of nonzero numbers (which you can place a bet, you cannot place a bet on the zero numbers) in a roulette (https://en.wikipedia.org/wiki/Roulette), also the famous 36 officers problem (https://en.wikipedia.org/wiki/Thirty-six_officers_problem, https://mathworld.wolfram.com/36OfficerProblem.html) (36 is the only number beside 4 such that this problem has no solutions), also 36 is the largest number n2 (n2 is the total number of squares in an n×n board) such that it is possible to color all n2 squares in an n×n board with black and white with no squares (with all four vertices on the n2 squares in the n×n board) having all four vertices all black or all white (the total number of squares with all four vertices on the n2 squares in the n×n board is n2×(n2−1)/12 (https://oeis.org/A002415), e.g. the total number of squares with all four vertices on the 36 squares in the 6×6 board is 105), also 36 is the total number of chemical elements (https://en.wikipedia.org/wiki/Chemical_element) in the first 4 periods (https://en.wikipedia.org/wiki/Period_(periodic_table)) of the periodic table (https://en.wikipedia.org/wiki/Periodic_table), and 4 is the smallest n such that the chemical elements in the first n periods of the periodic table contain chemical elements in all 18 groups (https://en.wikipedia.org/wiki/Group_(periodic_table)), i.e. the chemical elements in the first 4 periods are exactly the chemical elements with atomic numbers (https://en.wikipedia.org/wiki/Atomic_number) ≤ 36 (the atomic number is also related to number theory, e.g. there are 92 "atomic elements" for the look-and-say sequence (https://en.wikipedia.org/wiki/Look-and-say_sequence, https://mathworld.wolfram.com/LookandSaySequence.html, http://www.se16.info/js/looknsay.htm, https://oeis.org/A001155, https://oeis.org/A005150, https://oeis.org/A006751, https://oeis.org/A006715, https://oeis.org/A049064, https://oeis.org/A001387, https://oeis.org/A001388, https://oeis.org/A001389, https://oeis.org/A045918) in bases b ≥ 4 (although in base b = 2 there are only 13 "atomic elements" and its Conway's constant (the limit of the ratio of two consecutive terms of the look-and-say sequence with any start number except 111, including the limit of the ratio of two consecutive terms in the sequences https://oeis.org/A001609 and https://oeis.org/A049194) is the root of x3x2−1 (i.e. the number https://oeis.org/A092526), see https://www.math.uni-bielefeld.de/~sillke/SEQUENCES/series001 and https://www.nathanieljohnston.com/2010/11/the-binary-look-and-say-sequence/ and https://arxiv.org/pdf/2004.06414.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_442.pdf), and in base b = 3 there are only 27 "atomic elements" and its Conway's constant (the limit of the ratio of two consecutive terms of the look-and-say sequence with any start number except 22, including the limit of the ratio of two consecutive terms in the sequence https://oeis.org/A046639) is the root of x3x−1 (i.e. the number https://oeis.org/A060006), see https://www.nathanieljohnston.com/2011/01/further-variants-of-the-look-and-say-sequence/) (note that the base b "Look and Say" number LS(n,b) is small when n is a large minimal prime in base b, e.g. for the largest minimal prime in decimal (base b = 10), i.e. 5000000000000000000000000000027, its "Look and Say" number LS(5000000000000000000000000000027,10) is only 152801217, since it can be read as "one 5, twenty-eight 0's, one 2, one 7"), and John Conway named these 92 "atomic elements" after the 92 naturally-occurring chemical elements up to uranium, see https://mathworld.wolfram.com/CosmologicalTheorem.html and http://www.se16.info/js/lands2.htm and https://www.nathanieljohnston.com/2010/10/a-derivation-of-conways-degree-71-look-and-say-polynomial/ and https://web.archive.org/web/20000820121520/http://www.mathsoft.com/asolve/constant/cnwy/cnwy4.gif and https://web.archive.org/web/20000820121758/http://www.mathsoft.com/asolve/constant/cnwy/cnwy5.gif and https://oeis.org/A119566 (note that the lowest polynomial with Conway's constant (the limit of the ratio of two consecutive terms of the look-and-say sequence with any start number except 22, including the limit of the ratio of two consecutive terms in the sequences https://oeis.org/A022471 and https://oeis.org/A005341) (https://oeis.org/A014715) as a root (this polynomial is called Conway's polynomial) has degree 71, thus it has 72 (= double of 36) coefficients (including 0), see https://oeis.org/A137275, and the 71 roots of this polynomial have 36 different complex conjugates (https://en.wikipedia.org/wiki/Complex_conjugate, https://mathworld.wolfram.com/ComplexConjugate.html), since Conway's constant is the only one real root of this polynomial, and a complex number is the complex conjugate of itself if and only if it is a real number), also http://factordb.com/index.php?showid=1100000004405258711&base=36 is a large number (although this number is not prime, see http://factordb.com/index.php?id=1100000004405258711&open=ecm for its prime factorization), which when written in base b = 36, is the concatenation of the atomic numbers (https://en.wikipedia.org/wiki/Atomic_number) and the chemical symbol (https://en.wikipedia.org/wiki/Chemical_symbol) of all currently known chemical elements, i.e. the chemical elements with atomic numbers ≤ 118), also the next number 37 is the smallest irregular prime (https://en.wikipedia.org/wiki/Irregular_prime, https://t5k.org/glossary/xpage/Regular.html, https://mathworld.wolfram.com/IrregularPrime.html, https://t5k.org/top20/page.php?id=26, https://t5k.org/primes/search.php?Comment=^Irregular&OnList=all&Number=1000000&Style=HTML, https://arxiv.org/pdf/0912.2121.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_237.pdf), https://math.dartmouth.edu/~carlp/irreg.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_239.pdf), https://www.ams.org/journals/mcom/1978-32-142/S0025-5718-1978-0491465-4/S0025-5718-1978-0491465-4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_240.pdf), https://www.ams.org/journals/mcom/1974-28-126/S0025-5718-1974-0347727-0/S0025-5718-1974-0347727-0.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_294.pdf), https://www.ams.org/journals/mcom/1975-29-129/S0025-5718-1975-0376606-9/S0025-5718-1975-0376606-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_295.pdf), https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1197511-5/S0025-5718-1993-1197511-5.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_300.pdf), https://oeis.org/A000928) (thus, all prime number ≤ 36 are regular, and 36 is the largest number satisfying this property), also the previous number 35 is the smallest number n > 1 such that gcd(n, bnb) = 1 for some b (see https://oeis.org/A121707 and https://oeis.org/A321487 and https://oeis.org/A267999 and https://oeis.org/A306097) and the smallest semiprime (https://en.wikipedia.org/wiki/Semiprime, https://t5k.org/glossary/xpage/Semiprime.html, https://mathworld.wolfram.com/Semiprime.html, https://www.numbersaplenty.com/set/semiprime/, https://oeis.org/A001358) which is not in the range of the primary pretenders (i.e. is the smallest n such that bn == b mod n, for some b) (see https://oeis.org/A000790 and https://oeis.org/A108574) and likely to be the largest semiprime (https://en.wikipedia.org/wiki/Semiprime, https://t5k.org/glossary/xpage/Semiprime.html, https://mathworld.wolfram.com/Semiprime.html, https://www.numbersaplenty.com/set/semiprime/, https://oeis.org/A001358) n such that bn+1 == 1 mod n for every b coprime to n (see https://oeis.org/A208728) and the smallest non-perfect power composite number coprime to 6 (i.e. not divisible by either 2 or 3) and the smallest quasi-Carmichael number (see https://oeis.org/A257750) and the smallest composite squarefree (https://en.wikipedia.org/wiki/Square-free_integer, https://mathworld.wolfram.com/Squarefree.html, https://oeis.org/A005117) number k such that k2−1 is divisible by p−1 and p+1, where p are all the prime factors of k (see https://oeis.org/A306685) (all of these properties mean that the number 35+1 = 36 has many divisors), but these are not the main reason, the main reason is base b = 36 is the maximum base for which it is possible to write the numbers with the symbols 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 Arabic numerals (https://en.wikipedia.org/wiki/Arabic_numerals, https://mathworld.wolfram.com/ArabicNumeral.html) and the 26 Latin letters (https://en.wikipedia.org/wiki/Latin_alphabet, https://en.wikipedia.org/wiki/ISO_basic_Latin_alphabet)), i.e. bases 2 ≤ b ≤ 36 are case-insensitive (https://en.wikipedia.org/wiki/Case-insensitive) alphanumeric (https://en.wikipedia.org/wiki/Alphanumericals) numeral system using ASCII (https://en.wikipedia.org/wiki/ASCII) characters (https://en.wikipedia.org/wiki/Character_(computing)), while bases b > 36 are not)

A string (https://en.wikipedia.org/wiki/String_(computer_science), https://mathworld.wolfram.com/String.html) x is a subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html) of another string y, if x can be obtained from y by deleting zero or more of the characters (https://en.wikipedia.org/wiki/Character_(computing)) (in this project, digits (https://en.wikipedia.org/wiki/Numerical_digit, https://www.rieselprime.de/ziki/Digit, https://mathworld.wolfram.com/Digit.html)) in y. For example, 514 is a subsequence of 352148, "STRING" is a subsequence of "MEISTERSINGER". In contrast, 758 is not a subsequence of 378259, "ABC" is not a subsequence of "CBACACBA", since the characters (in this project, digits) must be in the same order. The empty string (https://en.wikipedia.org/wiki/Empty_string) 𝜆 is a subsequence of every string. There are 2n subsequences of a string with length n, e.g. the subsequences of 123456 are (totally 26 = 64 subsequences):

𝜆, 1, 2, 3, 4, 5, 6, 12, 13, 14, 15, 16, 23, 24, 25, 26, 34, 35, 36, 45, 46, 56, 123, 124, 125, 126, 134, 135, 136, 145, 146, 156, 234, 235, 236, 245, 246, 256, 345, 346, 356, 456, 1234, 1235, 1236, 1245, 1246, 1256, 1345, 1346, 1356, 1456, 2345, 2346, 2356, 2456, 3456, 12345, 12346, 12356, 12456, 13456, 23456, 123456

"The set of strings ordered by subsequence" is a partially ordered set (https://en.wikipedia.org/wiki/Partially_ordered_set, https://mathworld.wolfram.com/PartialOrder.html, https://mathworld.wolfram.com/PartiallyOrderedSet.html), since this binary relation (https://en.wikipedia.org/wiki/Binary_relation, https://mathworld.wolfram.com/BinaryRelation.html) is reflexive (https://en.wikipedia.org/wiki/Reflexive_relation, https://mathworld.wolfram.com/Reflexive.html), antisymmetric (https://en.wikipedia.org/wiki/Antisymmetric_relation), and transitive (https://en.wikipedia.org/wiki/Transitive_relation), and thus we can draw its Hasse diagram (https://en.wikipedia.org/wiki/Hasse_diagram, https://mathworld.wolfram.com/HasseDiagram.html) and find its greatest element (https://en.wikipedia.org/wiki/Greatest_element), least element (https://en.wikipedia.org/wiki/Least_element), maximal elements (https://en.wikipedia.org/wiki/Maximal_element, https://mathworld.wolfram.com/MaximalElement.html), and minimal elements (https://en.wikipedia.org/wiki/Minimal_element), however, the greatest element and least element may not exist, and for an infinite set (such as the set of the "prime numbers > b" strings in base b (for a given base b ≥ 2), for the proofs for that there are infinitely many primes, see https://en.wikipedia.org/wiki/Euclid%27s_theorem, https://mathworld.wolfram.com/EuclidsTheorems.html, http://www.numericana.com/answer/primes.htm#euclid, https://t5k.org/notes/proofs/infinite/, https://t5k.org/notes/proofs/infinite/euclids.html, https://t5k.org/notes/proofs/infinite/topproof.html, https://t5k.org/notes/proofs/infinite/goldbach.html, https://t5k.org/notes/proofs/infinite/kummers.html, https://t5k.org/notes/proofs/infinite/Saidak.html)), the maximal elements also may not exist, thus we are only interested on finding the minimal elements of these sets, and we define "minimal set" of a set as the set of the minimal elements of this set, under a given partially ordered binary relation (this binary relation is "is a subsequence of" in this project))

Two strings x and y are comparable (https://en.wikipedia.org/wiki/Comparability, https://mathworld.wolfram.com/ComparableElements.html) if either x is a subsequence of y, or y is a subsequence of x. A surprising result from formal language theory (https://en.wikipedia.org/wiki/Formal_language_theory) is that every set of pairwise incomparable strings is finite (https://en.wikipedia.org/wiki/Finite_set, https://mathworld.wolfram.com/FiniteSet.html) (which is proved by M. Lothaire), i.e. there are no infinite (https://en.wikipedia.org/wiki/Infinite_set, https://t5k.org/glossary/xpage/Infinite.html, https://mathworld.wolfram.com/InfiniteSet.html) antichains (https://en.wikipedia.org/wiki/Antichain, https://mathworld.wolfram.com/Antichain.html) for the subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html) ordering (https://en.wikipedia.org/wiki/Partially_ordered_set, https://mathworld.wolfram.com/PartialOrder.html, https://mathworld.wolfram.com/PartiallyOrderedSet.html).

By the theorem that there are no infinite (https://en.wikipedia.org/wiki/Infinite_set, https://t5k.org/glossary/xpage/Infinite.html, https://mathworld.wolfram.com/InfiniteSet.html) antichains (https://en.wikipedia.org/wiki/Antichain, https://mathworld.wolfram.com/Antichain.html) (i.e. a subset of a partially ordered set such that any two distinct elements in the subset are incomparable (https://en.wikipedia.org/wiki/Comparability, https://mathworld.wolfram.com/ComparableElements.html)) for the subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html) ordering (https://en.wikipedia.org/wiki/Partially_ordered_set, https://mathworld.wolfram.com/PartialOrder.html, https://mathworld.wolfram.com/PartiallyOrderedSet.html) (i.e. the set of the minimal elements of any set under the subsequence ordering must be finite, even if this set is infinite, such as the set of the "prime numbers > b" strings in base b (for a given base b ≥ 2), for the proofs for that there are infinitely many primes, see https://en.wikipedia.org/wiki/Euclid%27s_theorem, https://mathworld.wolfram.com/EuclidsTheorems.html, http://www.numericana.com/answer/primes.htm#euclid, https://t5k.org/notes/proofs/infinite/, https://t5k.org/notes/proofs/infinite/euclids.html, https://t5k.org/notes/proofs/infinite/topproof.html, https://t5k.org/notes/proofs/infinite/goldbach.html, https://t5k.org/notes/proofs/infinite/kummers.html, https://t5k.org/notes/proofs/infinite/Saidak.html), there must be only finitely such minimal elements in every base b.

In this project, we want to find the set of the minimal strings of the "prime number > b" digit strings in bases 2 ≤ b ≤ 36, since decimal (base 10) is not special in mathematics, there is no reason to only find this set in decimal (base 10), also, finding this set in decimal (base 10) is too easy to be researched in an article (only harder than bases 2, 3, 4, 6), thus it is necessary to research this set in other bases b.

Equivalently, a string x in a set of strings S is a minimal string if and only if any proper subsequence of x (subsequence of x which is unequal to x, like proper subset (https://en.wikipedia.org/wiki/Proper_subset, https://mathworld.wolfram.com/ProperSubset.html)) is not in S.

The minimal set M(L) of a language (https://en.wikipedia.org/wiki/Formal_language, https://mathworld.wolfram.com/FormalLanguage.html) L is interesting, this is because it allows us to compute two natural and related languages, defined as follows:

  • sub(L) = {x ∈ Σ* : there exists yL such that x is a subsequence of y}
  • sup(L) = {x ∈ Σ* : there exists yL such that y is a subsequence of x}

An amazing fact is that sub(L) and sup(L) are always regular. This follows from the classical theorem that every set of pairwise incomparable strings is finite, for the proof see https://www.sciencedirect.com/science/article/pii/S0021980069801110 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_329.pdf).

Although the minimal set M(L) is necessary finite even for infinite set L, but computing (https://en.wikipedia.org/wiki/Computing) the minimal set M(L) is undecidable (https://en.wikipedia.org/wiki/Undecidable_problem, https://mathworld.wolfram.com/Undecidable.html) in general and can be very difficult to compute even for simple languages, and can lead to some strange behaviour ...

  • The minimal set of the primes > 7 in base b = 7 has 71 elements, but the largest of which has only 17 digits.
  • The minimal set of the primes > 5 in base b = 5 has only 22 elements, but the largest of which has 96 digits!

And ...

  • The minimal set of the primes > 10 in base b = 10 has 77 elements, but the largest of which has only 31 digits.
  • The minimal set of the primes > 12 in base b = 12 has 106 elements, but the largest of which has only 42 digits.
  • The minimal set of the primes > 8 in base b = 8 has only 75 elements, but the largest of which has 221 digits!

Also, more strange ...

  • The minimal set of the primes > 15 in base b = 15 has 1284 elements, but the largest of which has only 157 digits.
  • The minimal set of the primes > 9 in base b = 9 has only 151 elements, but the largest of which has 1161 digits!
  • The minimal set of the primes > 18 in base b = 18 has only 549 elements, but the largest of which has 6271 digits!
  • The minimal set of the primes > 14 in base b = 14 has only 650 elements, but the largest of which has 19699 digits!

And ...

  • The minimal set of the primes > 20 in base b = 20 has 3314 elements, and the largest of which also has 6271 digits.
  • The minimal set of the primes > 24 in base b = 24 has 3409 elements, and the largest of which has 8134 digits.

And the finales ...

  • The minimal set of the primes > 11 in base b = 11 has only 1068 elements, but the largest of which has 62669 digits! (technically, probable primality tests were used to show this (which have a very small chance of making an error) because all known primality tests run far too slowly to run on a number of this size)
  • The minimal set of the primes > 16 in base b = 16 has only 2347 elements, but the largest of which has 116139 digits! (technically, probable primality tests were used to show this (which have a very small chance of making an error) because all known primality tests run far too slowly to run on a number of this size)
  • The minimal set of the primes > 13 in base b = 13 has only 3197 elements, but the largest of which has 592199 digits! (technically, probable primality tests were used to show this (which have a very small chance of making an error) because all known primality tests run far too slowly to run on a number of this size)

In this project, we will find the minimal set of the language (https://en.wikipedia.org/wiki/Formal_language, https://mathworld.wolfram.com/FormalLanguage.html) of base (https://en.wikipedia.org/wiki/Radix, https://t5k.org/glossary/xpage/Radix.html, https://www.rieselprime.de/ziki/Base, https://mathworld.wolfram.com/Radix.html) b representations (https://en.wikipedia.org/wiki/Representation_(mathematics)) of the prime numbers (https://en.wikipedia.org/wiki/Prime_number, https://t5k.org/glossary/xpage/Prime.html, https://www.rieselprime.de/ziki/Prime, https://mathworld.wolfram.com/PrimeNumber.html, https://www.numbersaplenty.com/set/prime_number/, http://www.numericana.com/answer/primes.htm#definition, http://irvinemclean.com/maths/pfaq2.htm, https://oeis.org/A000040, https://t5k.org/lists/small/1000.txt, https://t5k.org/lists/small/10000.txt, https://t5k.org/lists/small/100000.txt, https://t5k.org/lists/small/millions/) which are > b, and the language of base-b representations of the prime numbers which are > b are strings (https://en.wikipedia.org/wiki/String_(computer_science), https://mathworld.wolfram.com/String.html) of symbols (https://en.wikipedia.org/wiki/Symbol) over the alphabet (https://en.wikipedia.org/wiki/Alphabet_(formal_languages)) Σb = {0, 1, ..., b−1} (the set of the base b digits (https://en.wikipedia.org/wiki/Numerical_digit, https://www.rieselprime.de/ziki/Digit, https://mathworld.wolfram.com/Digit.html)).

In this project, we will find the sets of the minimal elements of these 35 sets under the subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html) ordering (https://en.wikipedia.org/wiki/Partially_ordered_set, https://mathworld.wolfram.com/PartialOrder.html, https://mathworld.wolfram.com/PartiallyOrderedSet.html):

bthe base b representations of the prime numbers (https://en.wikipedia.org/wiki/Prime_number, https://t5k.org/glossary/xpage/Prime.html, https://www.rieselprime.de/ziki/Prime, https://mathworld.wolfram.com/PrimeNumber.html, https://www.numbersaplenty.com/set/prime_number/, http://www.numericana.com/answer/primes.htm#definition, http://irvinemclean.com/maths/pfaq2.htm, https://oeis.org/A000040, https://t5k.org/lists/small/1000.txt, https://t5k.org/lists/small/10000.txt, https://t5k.org/lists/small/100000.txt, https://t5k.org/lists/small/millions/) > b (only list the first 1000 elements in the sets)
211, 101, 111, 1011, 1101, 10001, 10011, 10111, 11101, 11111, 100101, 101001, 101011, 101111, 110101, 111011, 111101, 1000011, 1000111, 1001001, 1001111, 1010011, 1011001, 1100001, 1100101, 1100111, 1101011, 1101101, 1110001, 1111111, 10000011, 10001001, 10001011, 10010101, 10010111, 10011101, 10100011, 10100111, 10101101, 10110011, 10110101, 10111111, 11000001, 11000101, 11000111, 11010011, 11011111, 11100011, 11100101, 11101001, 11101111, 11110001, 11111011, 100000001, 100000111, 100001101, 100001111, 100010101, 100011001, 100011011, 100100101, 100110011, 100110111, 100111001, 100111101, 101001011, 101010001, 101011011, 101011101, 101100001, 101100111, 101101111, 101110101, 101111011, 101111111, 110000101, 110001101, 110010001, 110011001, 110100011, 110100101, 110101111, 110110001, 110110111, 110111011, 111000001, 111001001, 111001101, 111001111, 111010011, 111011111, 111100111, 111101011, 111110011, 111110111, 111111101, 1000001001, 1000001011, 1000011101, 1000100011, 1000101101, 1000110011, 1000111001, 1000111011, 1001000001, 1001001011, 1001010001, 1001010111, 1001011001, 1001011111, 1001100101, 1001101001, 1001101011, 1001110111, 1010000001, 1010000011, 1010000111, 1010001101, 1010010011, 1010010101, 1010100001, 1010100101, 1010101011, 1010110011, 1010111101, 1011000101, 1011001111, 1011010111, 1011011101, 1011100011, 1011100111, 1011101111, 1011110101, 1011111001, 1100000001, 1100000101, 1100010011, 1100011101, 1100101001, 1100101011, 1100110101, 1100110111, 1100111011, 1100111101, 1101000111, 1101010101, 1101011001, 1101011011, 1101011111, 1101101101, 1101110001, 1101110011, 1101110111, 1110001011, 1110001111, 1110010111, 1110100001, 1110101001, 1110101101, 1110110011, 1110111001, 1111000111, 1111001011, 1111010001, 1111010111, 1111011111, 1111100101, 1111110001, 1111110101, 1111111011, 1111111101, 10000000111, 10000001001, 10000001111, 10000011001, 10000011011, 10000100101, 10000100111, 10000101101, 10000111111, 10001000011, 10001000101, 10001001001, 10001001111, 10001010101, 10001011101, 10001100011, 10001101001, 10001111111, 10010000001, 10010001011, 10010010011, 10010011101, 10010100011, 10010101001, 10010110001, 10010111101, 10011000001, 10011000111, 10011001101, 10011001111, 10011010101, 10011100001, 10011101011, 10011111101, 10011111111, 10100000011, 10100001001, 10100001011, 10100010001, 10100010101, 10100010111, 10100011011, 10100100111, 10100101001, 10100101111, 10101010001, 10101010111, 10101011101, 10101100101, 10101110111, 10110000001, 10110001111, 10110010011, 10110010101, 10110011001, 10110011111, 10110100111, 10110101011, 10110101101, 10110110011, 10110111111, 10111001001, 10111001011, 10111001111, 10111010001, 10111010101, 10111011011, 10111100111, 10111110011, 10111111011, 11000000111, 11000001101, 11000010001, 11000010111, 11000011111, 11000100011, 11000101011, 11000101111, 11000111101, 11001000001, 11001000111, 11001001001, 11001001101, 11001010011, 11001010101, 11001011011, 11001100101, 11001111001, 11001111111, 11010000011, 11010000101, 11010011101, 11010100001, 11010100011, 11010101101, 11010111001, 11010111011, 11011000101, 11011001101, 11011010011, 11011011001, 11011011111, 11011110001, 11011110111, 11011111011, 11011111101, 11100001001, 11100010011, 11100011111, 11100100111, 11100110111, 11101000101, 11101001011, 11101001111, 11101010001, 11101010101, 11101010111, 11101100001, 11101101101, 11101110011, 11101111001, 11110001011, 11110001101, 11110011101, 11110011111, 11110110101, 11110111011, 11111000011, 11111001001, 11111001101, 11111001111, 11111010011, 11111011011, 11111100001, 11111101011, 11111101101, 11111110111, 100000000101, 100000001111, 100000010101, 100000100001, 100000100011, 100000100111, 100000101001, 100000110011, 100000111111, 100001000001, 100001010001, 100001010011, 100001011001, 100001011101, 100001011111, 100001101001, 100001110001, 100010000011, 100010011011, 100010011111, 100010100101, 100010101101, 100010111101, 100010111111, 100011000011, 100011001011, 100011011011, 100011011101, 100011100001, 100011101001, 100011101111, 100011110101, 100011111001, 100100000101, 100100000111, 100100011101, 100100100011, 100100100101, 100100101011, 100100101111, 100100110101, 100101000011, 100101001001, 100101001101, 100101001111, 100101010101, 100101011001, 100101011111, 100101101011, 100101110001, 100101110111, 100110000101, 100110001001, 100110001111, 100110011011, 100110100011, 100110101001, 100110101101, 100111000111, 100111011001, 100111100011, 100111101011, 100111101111, 100111110101, 100111110111, 100111111101, 101000010011, 101000011111, 101000100001, 101000110001, 101000111001, 101000111101, 101001001001, 101001010111, 101001100001, 101001100011, 101001100111, 101001101111, 101001110101, 101001111011, 101001111111, 101010000001, 101010000101, 101010001011, 101010010011, 101010010111, 101010011001, 101010011111, 101010101001, 101010101011, 101010110101, 101010111101, 101011000001, 101011001111, 101011011001, 101011100101, 101011100111, 101011101101, 101011110001, 101011110011, 101100000011, 101100010001, 101100010101, 101100011011, 101100100011, 101100101001, 101100101101, 101100111111, 101101000111, 101101010001, 101101010111, 101101011101, 101101100101, 101101101111, 101101111011, 101110001001, 101110001101, 101110010011, 101110011001, 101110011011, 101110110111, 101110111001, 101111000011, 101111001011, 101111001111, 101111011101, 101111100001, 101111101001, 101111110101, 101111111011, 110000000111, 110000001011, 110000010001, 110000100101, 110000101111, 110000110001, 110001000001, 110001011011, 110001011111, 110001100001, 110001101101, 110001110011, 110001110111, 110010000011, 110010001001, 110010010001, 110010010101, 110010011101, 110010110011, 110010110101, 110010111001, 110010111011, 110011000111, 110011100011, 110011100101, 110011101011, 110011110001, 110011110111, 110011111011, 110100000001, 110100000011, 110100001111, 110100010011, 110100011111, 110100100001, 110100101011, 110100101101, 110100111101, 110100111111, 110101001111, 110101010101, 110101101001, 110101111001, 110110000001, 110110000101, 110110000111, 110110001011, 110110001101, 110110100011, 110110101011, 110110110111, 110110111101, 110111000111, 110111001001, 110111001101, 110111010011, 110111010101, 110111011011, 110111100101, 110111100111, 110111110011, 110111111101, 110111111111, 111000001001, 111000010111, 111000011101, 111000100001, 111000100111, 111000101111, 111000110101, 111000111011, 111001001011, 111001010111, 111001011001, 111001011101, 111001101011, 111001110001, 111001110101, 111001111101, 111010000111, 111010001111, 111010010101, 111010011011, 111010110001, 111010110111, 111010111001, 111011000011, 111011010001, 111011010101, 111011011011, 111011101101, 111011101111, 111011111001, 111100000111, 111100001011, 111100001101, 111100010111, 111100100101, 111100101001, 111100110001, 111101000011, 111101000111, 111101001101, 111101001111, 111101010011, 111101011001, 111101011011, 111101100111, 111101101011, 111101111111, 111110010101, 111110100001, 111110100011, 111110100111, 111110101101, 111110110011, 111110110101, 111110111011, 111111010001, 111111010011, 111111011001, 111111101001, 111111101111, 111111111011, 111111111101, 1000000000011, 1000000001111, 1000000011111, 1000000100001, 1000000100101, 1000000101011, 1000000111001, 1000000111101, 1000000111111, 1000001010001, 1000001101001, 1000001110011, 1000001111001, 1000001111011, 1000010000101, 1000010000111, 1000010010001, 1000010010011, 1000010011101, 1000010100011, 1000010100101, 1000010101111, 1000010110001, 1000010111011, 1000011000001, 1000011001001, 1000011100111, 1000011110001, 1000011110011, 1000011111101, 1000100000101, 1000100001011, 1000100010101, 1000100100111, 1000100101101, 1000100111001, 1000101000101, 1000101000111, 1000101011001, 1000101011111, 1000101100011, 1000101101001, 1000101101111, 1000110000001, 1000110000011, 1000110001101, 1000110011011, 1000110100001, 1000110100101, 1000110100111, 1000110101011, 1000111000011, 1000111000101, 1000111010001, 1000111010111, 1000111100111, 1000111101111, 1000111110101, 1000111111011, 1001000001101, 1001000011101, 1001000011111, 1001000100011, 1001000101001, 1001000101011, 1001000110001, 1001000110111, 1001001000001, 1001001000111, 1001001010011, 1001001011111, 1001001110001, 1001001110011, 1001001111001, 1001001111101, 1001010001111, 1001010010111, 1001010101111, 1001010110011, 1001010110101, 1001010111001, 1001010111111, 1001011000001, 1001011001101, 1001011010001, 1001011011111, 1001011111101, 1001100000111, 1001100001101, 1001100011001, 1001100100111, 1001100101101, 1001100110111, 1001101000011, 1001101000101, 1001101001001, 1001101001111, 1001101010111, 1001101011101, 1001101100111, 1001101101001, 1001101101101, 1001101111011, 1001110000001, 1001110000111, 1001110001011, 1001110010001, 1001110010011, 1001110011101, 1001110011111, 1001110101111, 1001110111011, 1001111000011, 1001111010101, 1001111011001, 1001111011111, 1001111101011, 1001111101101, 1001111110011, 1001111111001, 1001111111111, 1010000011011, 1010000100001, 1010000101111, 1010000110011, 1010000111011, 1010001000101, 1010001001101, 1010001011001, 1010001101011, 1010001101111, 1010001110001, 1010001110101, 1010010001101, 1010010011001, 1010010011111, 1010010100001, 1010010110001, 1010010110111, 1010010111101, 1010011001011, 1010011010101, 1010011100011, 1010011100111, 1010100000101, 1010100001011, 1010100010001, 1010100010111, 1010100011111, 1010100100101, 1010100101001, 1010100101011, 1010100110111, 1010100111101, 1010101000001, 1010101000011, 1010101001001, 1010101011111, 1010101100101, 1010101100111, 1010101101011, 1010101111101, 1010101111111, 1010110000011, 1010110001111, 1010110010001, 1010110010111, 1010110011011, 1010110110101, 1010110111011, 1010111000001, 1010111000101, 1010111001101, 1010111010111, 1010111110111, 1011000000111, 1011000001001, 1011000001111, 1011000010011, 1011000010101, 1011000011001, 1011000011011, 1011000100101, 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611, 15, 21, 25, 31, 35, 45, 51, 101, 105, 111, 115, 125, 135, 141, 151, 155, 201, 211, 215, 225, 241, 245, 251, 255, 301, 305, 331, 335, 345, 351, 405, 411, 421, 431, 435, 445, 455, 501, 515, 521, 525, 531, 551, 1011, 1015, 1021, 1025, 1035, 1041, 1055, 1105, 1115, 1125, 1131, 1141, 1145, 1151, 1205, 1231, 1235, 1241, 1245, 1311, 1321, 1335, 1341, 1345, 1355, 1411, 1421, 1431, 1435, 1445, 1501, 1505, 1521, 1535, 1541, 1555, 2001, 2011, 2015, 2025, 2041, 2045, 2051, 2055, 2115, 2131, 2135, 2151, 2155, 2205, 2225, 2231, 2301, 2311, 2325, 2335, 2345, 2351, 2401, 2415, 2425, 2435, 2441, 2451, 2501, 2505, 2511, 2531, 2545, 2551, 2555, 3005, 3015, 3021, 3041, 3045, 3055, 3111, 3125, 3141, 3155, 3211, 3221, 3231, 3235, 3251, 3301, 3305, 3321, 3325, 3351, 3405, 3425, 3431, 3445, 3451, 3455, 3501, 3515, 3541, 3545, 3551, 3555, 4021, 4025, 4031, 4035, 4111, 4115, 4131, 4145, 4201, 4205, 4215, 4225, 4251, 4255, 4305, 4315, 4331, 4341, 4401, 4405, 4415, 4421, 4435, 4441, 4451, 4505, 4511, 4525, 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23225, 23231, 23251, 23255, 23315, 23321, 23335, 23341, 23405, 23411, 23435, 23445, 23521, 23545, 24001, 24005, 24011, 24015, 24021, 24055, 24111, 24131, 24141, 24155, 24201, 24205, 24215, 24221, 24231, 24245, 24251, 24311, 24325, 24331, 24345, 24411, 24421, 24425, 24435, 24451, 24501, 24511, 24535, 24555, 25001, 25005, 25031, 25041, 25045, 25101, 25115, 25131, 25141, 25151, 25225, 25235, 25241, 25255, 25321, 25325, 25335, 25405, 25411, 25425, 25451, 25455, 25501, 25515, 25541, 25545, 30001, 30031, 30035, 30045, 30051, 30055, 30105, 30111, 30131, 30135, 30211, 30245, 30305, 30311, 30315, 30325, 30335, 30341, 30351, 30425, 30431, 30441, 30505, 30515, 30535, 30541, 30551, 31011, 31035, 31041, 31045, 31055, 31121, 31125, 31131, 31201, 31241, 31255, 31305, 31311, 31325, 31331, 31345, 31351, 31405, 31415, 31421, 31435, 31441, 31455, 31505, 31521, 32011, 32025, 32031, 32045, 32101, 32111, 32125, 32155, 32205, 32225, 32245, 32251, 32321, 32331, 32335, 32345, 32355, 32425, 32431, 32445, 32511, 32521, 32525, 32531, 32535, 33015, 33021, 33041, 33051, 33115, 33131, 33141, 33151, 33221, 33245, 33251, 33255, 33305, 33311, 33321, 33331, 33345, 33355, 33415, 33435, 33505, 33511, 33521, 33525, 33555, 34011, 34051, 34055, 34101, 34105, 34115, 34121, 34141, 34145, 34211, 34301, 34315, 34325, 34345, 34411, 34421, 34435, 34455, 34501, 34505, 34515, 34531, 34541, 34555, 35001, 35005, 35031, 35041, 35051, 35055, 35105, 35111, 35125, 35131, 35155, 35215, 35231, 35301, 35305, 35315, 35335, 35341, 35351, 35401, 35411, 35455, 35505, 35531, 35535, 35551, 40005, 40021, 40041, 40111, 40115, 40121, 40125, 40205, 40225, 40235, 40241, 40305, 40315, 40325, 40351, 40405, 40431, 40435, 40525, 40535, 40545, 40555, 41011, 41021, 41025, 41031, 41051, 41101, 41105, 41111, 41121, 41155, 41205, 41211, 41215, 41245, 41251, 41255, 41315, 41321, 41331, 41335, 41421, 41431, 41441, 41445, 41501, 41515, 42011, 42035, 42041, 42051, 42055, 42101, 42105, 42111, 42125, 42151, 42201, 42205, 42221, 42235, 42245, 42321, 42325, 42331, 42341, 42431, 42435, 42451, 42505, 42515, 42525, 42541, 42551, 43011, 43015, 43025, 43031, 43041, 43045, 43055, 43101, 43115, 43121, 43145, 43155, 43231, 43235, 43255, 43321, 43405, 43415, 43451, 43455, 43525, 43541, 43551, 43555, 44005, 44031, 44041, 44051, 44105, 44111, 44125, 44145, 44201, 44215, 44221, 44235, 44251, 44311, 44325, 44405, 44411, 44415, 44431, 44441, 44445, 44501, 44531, 44545, 44555, 45005, 45011, 45021, 45035, 45055, 45101, 45115, 45125, 45135, 45145, 45201, 45211, 45225, 45235, 45241, 45251, 45301, 45311, 45325, 45341, 45421, 45431, 45505, 45511, 45541, 45545, 50001, 50015, 50105, 50121, 50151, 50155, 50201, 50215, 50225, 50231, 50241, 50245, 50315, 50331, 50351, 50421, 50445, 50455, 50501, 50521, 50531, 50545, 50551, 51005, 51011, 51021, 51035, 51101, 51105, 51145, 51151, 51215, 51221, 51235, 51241, 51255, 51331, 51335, 51341, 51345, 51401, 51425, 51435, 51445, 51451, 51511, 51535, 51551, 51555, 52005, 52055, 52101, 52115, 52121, 52131, 52135, 52145, 52155, 52211, 52221, 52225, 52245, 52255, 52311, 52331, 52335, 52401, 52421, 52435, 52515, 52525, 52545, 52555, 53001, 53035, 53051, 53121, 53135, 53145, 53211, 53215, 53221, 53231, 53245, 53301, 53311, 53315, 53325, 53415, 53441, 53455, 53501, 53521, 53535, 53541, 54005, 54011, 54041, 54121, 54151, 54201, 54225, 54255, 54305, 54311, 54341, 54345, 54355, 54401, 54415, 54431, 54445, 54455, 54505, 54521, 54525, 54535, 54541, 54555, 55001, 55021, 55025, 55035, 55045, 55051, 55111, 55115, 55141, 55211, 55215, 55225, 55301, 55305, 55321, 55331, 55335, 55351, 55355, 55421, 55431, 55435, 55501, 55521, 55525, 55531, 100021, 100025, 100105, 100115, 100125, 100145, 100205, 100231, 100241, 100245, 100251, 100255, 100325, 100335, 100355, 100411, 100421, 100425, ...
714, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 104, 113, 115, 124, 131, 133, 142, 146, 155, 166, 203, 205, 212, 214, 221, 241, 245, 254, 256, 302, 304, 313, 322, 326, 335, 344, 346, 362, 364, 401, 403, 421, 436, 443, 445, 452, 461, 463, 506, 515, 524, 533, 535, 544, 551, 553, 566, 616, 623, 625, 632, 652, 661, 1004, 1006, 1013, 1022, 1033, 1042, 1051, 1055, 1064, 1105, 1112, 1123, 1136, 1141, 1154, 1156, 1165, 1202, 1211, 1222, 1226, 1231, 1235, 1253, 1264, 1301, 1312, 1316, 1325, 1343, 1345, 1402, 1411, 1424, 1433, 1442, 1444, 1453, 1466, 1505, 1514, 1516, 1525, 1534, 1541, 1543, 1561, 1604, 1606, 1613, 1622, 1631, 1633, 1651, 1655, 1664, 2005, 2021, 2032, 2045, 2056, 2065, 2104, 2111, 2122, 2131, 2135, 2146, 2153, 2203, 2216, 2234, 2236, 2252, 2254, 2261, 2263, 2306, 2326, 2333, 2335, 2342, 2362, 2366, 2401, 2405, 2434, 2441, 2452, 2465, 2506, 2513, 2522, 2531, 2551, 2555, 2564, 2603, 2614, 2623, 2641, 2645, 2654, 2656, 3002, 3004, 3013, 3026, 3031, 3044, 3046, 3055, 3112, 3116, 3121, 3125, 3134, 3143, 3154, 3163, 3202, 3233, 3235, 3251, 3262, 3305, 3314, 3323, 3334, 3352, 3356, 3365, 3404, 3406, 3415, 3433, 3446, 3503, 3505, 3512, 3521, 3523, 3532, 3536, 3541, 3545, 3563, 3565, 3604, 3653, 3662, 4001, 4012, 4036, 4052, 4102, 4106, 4111, 4115, 4124, 4135, 4142, 4144, 4153, 4201, 4214, 4216, 4223, 4225, 4232, 4241, 4256, 4304, 4315, 4333, 4342, 4346, 4355, 4366, 4403, 4414, 4421, 4441, 4445, 4454, 4456, 4463, 4502, 4504, 4513, 4526, 4555, 4564, 4601, 4603, 4636, 4643, 4645, 4661, 5006, 5011, 5024, 5035, 5044, 5053, 5062, 5116, 5125, 5132, 5134, 5152, 5165, 5213, 5224, 5246, 5266, 5305, 5312, 5314, 5321, 5323, 5336, 5354, 5363, 5402, 5426, 5431, 5453, 5455, 5516, 5525, 5536, 5545, 5552, 5554, 5561, 5602, 5611, 5624, 5626, 5642, 5662, 6005, 6014, 6032, 6034, 6041, 6043, 6056, 6104, 6106, 6131, 6133, 6142, 6146, 6151, 6164, 6205, 6232, 6265, 6302, 6311, 6322, 6344, 6346, 6353, 6364, 6416, 6421, 6425, 6436, 6445, 6454, 6461, 6506, 6511, 6542, 6551, 6553, 6562, 6566, 6605, 6625, 6634, 6641, 6643, 6652, 6656, 6665, 10013, 10022, 10031, 10051, 10055, 10064, 10112, 10123, 10132, 10136, 10204, 10231, 10244, 10255, 10262, 10301, 10303, 10312, 10343, 10361, 10363, 10415, 10426, 10433, 10451, 10501, 10514, 10516, 10523, 10534, 10543, 10552, 10556, 10561, 10565, 10604, 10615, 10622, 10624, 10633, 10646, 10651, 10664, 11005, 11012, 11032, 11045, 11063, 11065, 11104, 11111, 11113, 11135, 11155, 11162, 11201, 11212, 11221, 11225, 11252, 11263, 11306, 11315, 11324, 11335, 11351, 11366, 11416, 11423, 11432, 11441, 11443, 11513, 11515, 11531, 11542, 11546, 11566, 11603, 11614, 11632, 11641, 11656, 11663, 12002, 12031, 12044, 12046, 12101, 12136, 12143, 12145, 12163, 12202, 12206, 12224, 12233, 12244, 12251, 12262, 12323, 12325, 12332, 12334, 12352, 12422, 12424, 12433, 12442, 12451, 12455, 12464, 12466, 12514, 12521, 12536, 12541, 12554, 12556, 12611, 12613, 12635, 12644, 13003, 13025, 13036, 13043, 13045, 13052, 13054, 13115, 13126, 13144, 13153, 13166, 13201, 13205, 13214, 13216, 13225, 13241, 13243, 13261, 13304, 13306, 13322, 13342, 13351, 13355, 13364, 13405, 13414, 13423, 13445, 13463, 13465, 13502, 13522, 13531, 13535, 13546, 13562, 13603, 13612, 13621, 13652, 13661, 13663, 14006, 14026, 14033, 14042, 14066, 14101, 14114, 14134, 14141, 14143, 14156, 14206, 14213, 14224, 14251, 14255, 14264, 14266, 14303, 14312, 14314, 14332, 14336, 14365, 14426, 14444, 14446, 14453, 14462, 14501, 14503, 14512, 14543, 14545, 14554, 14606, 14615, 14633, 14635, 14644, 14662, 15014, 15016, 15023, 15032, 15052, 15056, 15061, 15115, 15151, 15164, 15203, 15205, 15221, 15223, 15236, 15241, 15254, 15263, 15265, 15311, 15313, 15326, 15335, 15346, 15421, 15434, 15436, 15452, 15463, 15502, 15515, 15542, 15551, 15566, 15614, 15616, 15643, 15652, 15656, 15665, 16004, 16031, 16033, 16046, 16066, 16105, 16112, 16114, 16121, 16154, 16156, 16204, 16213, 16235, 16246, 16255, 16264, 16321, 16343, 16345, 16352, 16361, 16363, 16402, 16411, 16424, 16433, 16451, 16466, 16523, 16525, 16534, 16541, 16565, 16606, 16642, 16646, 16651, 16655, 16664, 16666, 20014, 20021, 20041, 20113, 20126, 20135, 20153, 20203, 20212, 20225, 20243, 20245, 20252, 20261, 20302, 20311, 20324, 20326, 20333, 20353, 20362, 20401, 20405, 20414, 20416, 20432, 20434, 20456, 20504, 20515, 20542, 20546, 20555, 20603, 20605, 20614, 20623, 20632, 21002, 21011, 21031, 21035, 21046, 21062, 21103, 21121, 21145, 21152, 21154, 21161, 21224, 21242, 21251, 21253, 21305, 21314, 21323, 21343, 21356, 21406, 21413, 21455, 21464, 21503, 21512, 21523, 21532, 21536, 21541, 21556, 21565, 21602, 21604, 21613, 21644, 21653, 21655, 21662, 22016, 22021, 22025, 22043, 22045, 22054, 22061, 22126, 22135, 22144, 22151, 22162, 22205, 22252, 22304, 22306, 22315, 22322, 22324, 22331, 22333, 22346, 22366, 22405, 22412, 22423, 22436, 22445, 22504, 22511, 22513, 22522, 22564, 22601, 22612, 22625, 22634, 22643, 22654, 22663, 23011, 23015, 23024, 23026, 23035, 23042, 23051, 23053, 23066, 23101, 23123, 23132, 23161, 23165, 23213, 23233, 23303, 23312, 23341, 23345, 23402, 23413, 23422, 23426, 23435, 23455, 23464, 23503, 23516, 23521, 23534, 23552, 23563, 23606, 23611, 23624, 23635, 23653, 23666, 24032, 24034, 24041, 24052, 24061, 24065, 24106, 24133, 24146, 24155, 24164, 24166, 24205, 24221, 24236, 24241, 24254, 24263, 24302, 24311, 24322, 24331, 24344, 24353, 24355, 24364, 24403, 24412, 24425, 24436, 24502, 24511, 24542, 24544, 24601, 24605, 24616, 24632, 25004, 25015, 25042, 25046, 25051, 25064, 25103, 25105, 25114, 25121, 25145, 25156, 25204, 25231, 25253, 25262, 25264, 25312, 25321, 25334, 25336, 25352, 25354, 25363, 25406, 25426, 25433, 25466, 25501, 25523, 25525, 25541, 25543, 25556, 25615, 25622, 25624, 25631, 25642, 25664, 26003, 26012, 26014, 26032, 26054, 26065, 26102, 26111, 26153, 26155, 26201, 26203, 26212, 26216, 26225, 26234, 26245, 26254, 26261, 26306, 26315, 26326, 26344, 26351, 26401, 26416, 26432, 26465, 26504, 26522, 26531, 26533, 26564, 26605, 26632, 26645, 26654, 30004, 30011, 30013, 30022, 30035, 30046, 30055, 30062, 30101, 30143, 30163, 30206, 30211, 30226, 30242, 30244, 30266, 30301, 30325, 30361, 30415, 30424, 30446, 30503, 30512, 30514, 30541, 30545, 30554, 30556, 30602, 30613, 30626, 30635, 30644, 30655, 30662, 31001, 31003, 31016, 31021, 31036, 31043, 31052, 31061, 31063, 31111, 31115, 31135, 31162, 31166, 31205, 31234, 31241, 31252, 31261, 31265, 31306, 31313, 31333, 31342, 31346, 31366, 31414, 31421, 31423, 31465, 31502, 31535, 31544, 31553, 31601, 31616, 31636, 31645, 31652, 31654, 31661, 32015, 32024, 32042, 32053, 32062, 32066, 32114, ...
813, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 103, 107, 111, 117, 123, 131, 141, 145, 147, 153, 155, 161, 177, 203, 211, 213, 225, 227, 235, 243, 247, 255, 263, 265, 277, 301, 305, 307, 323, 337, 343, 345, 351, 357, 361, 373, 401, 407, 415, 417, 425, 431, 433, 445, 463, 467, 471, 475, 513, 521, 533, 535, 541, 547, 557, 565, 573, 577, 605, 615, 621, 631, 643, 645, 657, 661, 667, 673, 701, 711, 715, 717, 723, 737, 747, 753, 763, 767, 775, 1011, 1013, 1035, 1043, 1055, 1063, 1071, 1073, 1101, 1113, 1121, 1127, 1131, 1137, 1145, 1151, 1153, 1167, 1201, 1203, 1207, 1215, 1223, 1225, 1241, 1245, 1253, 1263, 1275, 1305, 1317, 1327, 1335, 1343, 1347, 1357, 1365, 1371, 1401, 1405, 1423, 1435, 1451, 1453, 1465, 1467, 1473, 1475, 1507, 1525, 1531, 1533, 1537, 1555, 1561, 1563, 1567, 1613, 1617, 1627, 1641, 1651, 1655, 1663, 1671, 1707, 1713, 1721, 1727, 1737, 1745, 1761, 1765, 1773, 1775, 2007, 2011, 2017, 2031, 2033, 2045, 2047, 2055, 2077, 2103, 2105, 2111, 2117, 2125, 2135, 2143, 2151, 2177, 2201, 2213, 2223, 2235, 2243, 2251, 2261, 2275, 2301, 2307, 2315, 2317, 2325, 2341, 2353, 2375, 2377, 2403, 2411, 2413, 2421, 2425, 2427, 2433, 2447, 2451, 2457, 2521, 2527, 2535, 2545, 2567, 2601, 2617, 2623, 2625, 2631, 2637, 2647, 2653, 2655, 2663, 2677, 2711, 2713, 2717, 2721, 2725, 2733, 2747, 2763, 2773, 3007, 3015, 3021, 3027, 3037, 3043, 3053, 3057, 3075, 3101, 3107, 3111, 3115, 3123, 3125, 3133, 3145, 3171, 3177, 3203, 3205, 3235, 3241, 3243, 3255, 3271, 3273, 3305, 3315, 3323, 3331, 3337, 3361, 3367, 3373, 3375, 3411, 3423, 3437, 3447, 3467, 3505, 3513, 3517, 3521, 3525, 3527, 3541, 3555, 3563, 3571, 3613, 3615, 3635, 3637, 3665, 3673, 3703, 3711, 3715, 3717, 3723, 3733, 3741, 3753, 3755, 3767, 4005, 4017, 4025, 4041, 4043, 4047, 4051, 4063, 4077, 4101, 4121, 4123, 4131, 4135, 4137, 4151, 4161, 4203, 4233, 4237, 4245, 4255, 4275, 4277, 4303, 4313, 4333, 4335, 4341, 4351, 4357, 4365, 4371, 4405, 4407, 4435, 4443, 4445, 4453, 4457, 4465, 4503, 4511, 4515, 4517, 4525, 4531, 4537, 4553, 4561, 4567, 4605, 4611, 4617, 4633, 4643, 4651, 4655, 4707, 4731, 4743, 4753, 4757, 4765, 4767, 4775, 5023, 5037, 5041, 5061, 5071, 5075, 5111, 5127, 5141, 5143, 5147, 5157, 5165, 5173, 5177, 5201, 5205, 5213, 5223, 5227, 5231, 5237, 5251, 5253, 5265, 5275, 5301, 5317, 5331, 5345, 5347, 5355, 5361, 5363, 5403, 5421, 5425, 5433, 5443, 5451, 5455, 5477, 5507, 5521, 5527, 5535, 5545, 5557, 5573, 5611, 5615, 5623, 5631, 5633, 5667, 5671, 5703, 5713, 5717, 5735, 5741, 5751, 5765, 5773, 6007, 6013, 6021, 6045, 6057, 6061, 6101, 6133, 6137, 6141, 6155, 6163, 6167, 6203, 6211, 6221, 6225, 6235, 6263, 6265, 6271, 6273, 6307, 6343, 6345, 6353, 6361, 6367, 6373, 6401, 6403, 6417, 6423, 6437, 6441, 6453, 6455, 6475, 6477, 6517, 6525, 6551, 6571, 6601, 6605, 6607, 6613, 6615, 6643, 6653, 6667, 6675, 6707, 6711, 6715, 6723, 6725, 6733, 6745, 6747, 6763, 6775, 6777, 7011, 7027, 7035, 7041, 7047, 7057, 7065, 7073, 7113, 7127, 7131, 7135, 7153, 7161, 7165, 7175, 7207, 7217, 7225, 7233, 7261, 7267, 7271, 7303, 7321, 7325, 7333, 7355, 7357, 7371, 7407, 7413, 7415, 7427, 7445, 7451, 7461, 7503, 7507, 7515, 7517, 7523, 7531, 7533, 7547, 7553, 7577, 7625, 7641, 7643, 7647, 7655, 7663, 7665, 7673, 7721, 7723, 7731, 7751, 7757, 7773, 7775, 10003, 10017, 10037, 10041, 10045, 10053, 10071, 10075, 10077, 10121, 10151, 10163, 10171, 10173, 10205, 10207, 10221, 10223, 10235, 10243, 10245, 10257, 10261, 10273, 10301, 10311, 10347, 10361, 10363, 10375, 10405, 10413, 10425, 10447, 10455, 10471, 10505, 10507, 10531, 10537, 10543, 10551, 10557, 10601, 10603, 10615, 10633, 10641, 10645, 10647, 10653, 10703, 10705, 10721, 10727, 10747, 10757, 10765, 10773, 11015, 11035, 11037, 11043, 11051, 11053, 11061, 11067, 11101, 11107, 11123, 11137, 11161, 11163, 11171, 11175, 11217, 11227, 11257, 11263, 11265, 11271, 11277, 11301, 11315, 11321, 11337, 11375, 11407, 11415, 11431, 11447, 11455, 11467, 11503, 11505, 11511, 11517, 11527, 11535, 11547, 11551, 11555, 11573, 11601, 11607, 11613, 11621, 11623, 11635, 11637, 11657, 11673, 11703, 11725, 11731, 11737, 11753, 11755, 11763, 11771, 11777, 12033, 12041, 12057, 12063, 12073, 12105, 12115, 12131, 12153, 12157, 12161, 12165, 12215, 12231, 12237, 12241, 12261, 12267, 12275, 12313, 12325, 12343, 12347, 12405, 12413, 12421, 12427, 12437, 12445, 12451, 12453, 12467, 12475, 12501, 12503, 12511, 12537, 12545, 12547, 12553, 12575, 12577, 12603, 12617, 12621, 12627, 12633, 12665, 12673, 12701, 12705, 12715, 12727, 12767, 13007, 13011, 13017, 13023, 13025, 13031, 13033, 13045, 13063, 13071, 13075, 13105, 13117, 13125, 13151, 13155, 13157, 13165, 13223, 13227, 13237, 13251, 13257, 13265, 13275, 13303, 13317, 13323, 13331, 13333, 13341, 13345, 13353, 13355, 13367, 13371, 13411, 13417, 13443, 13447, 13463, 13501, 13535, 13543, 13567, 13573, 13615, 13625, 13633, 13637, 13645, 13663, 13671, 13677, 13711, 13713, 13725, 13741, 13751, 13763, 13765, 13777, 14007, 14023, 14035, 14065, 14067, 14073, 14103, 14111, 14115, 14125, 14147, 14161, 14167, 14175, 14177, 14205, 14217, 14233, 14235, 14247, 14255, 14263, 14271, 14301, 14307, 14321, 14327, 14331, 14337, 14345, 14353, 14365, 14375, 14425, 14433, 14461, 14463, 14505, 14511, 14521, 14533, 14571, 14601, 14623, 14627, 14631, 14643, 14651, 14653, 14661, 14665, 14707, 14717, 14733, 14755, 14775, 15003, 15005, 15021, 15027, 15041, 15043, 15055, 15057, 15065, 15077, 15115, 15121, 15151, 15153, 15173, 15175, 15207, 15211, 15223, 15247, 15253, 15255, 15261, 15271, 15311, 15317, 15325, 15327, 15343, 15363, 15373, 15377, 15405, 15443, 15445, 15457, 15461, 15467, 15473, 15501, 15507, 15517, 15525, 15531, 15545, 15553, 15563, 15577, 15603, 15621, 15635, 15647, 15677, 15705, 15721, 15727, 15731, 15757, 15767, 16011, 16023, 16031, 16047, 16053, 16055, 16063, 16075, 16105, 16113, 16117, 16125, 16163, 16201, 16213, 16215, 16231, 16243, 16245, 16265, 16267, 16311, 16341, 16363, 16371, 16411, 16433, 16441, 16443, 16465, 16471, 16477, 16501, 16513, 16523, 16535, 16543, 16551, 16561, 16565, 16573, 16575, 16607, 16611, 16625, 16631, 16637, 16645, 16647, 16663, 16667, 16705, 16727, 16733, 16741, 16765, 16771, 17001, 17007, 17013, 17023, 17027, 17045, 17053, 17057, 17075, 17111, 17115, 17117, 17155, 17161, 17211, 17217, 17225, 17241, 17255, 17273, 17301, 17305, 17307, 17313, 17335, 17343, 17357, 17367, 17375, 17401, 17415, ...
912, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 102, 108, 117, 122, 124, 128, 131, 135, 151, 155, 162, 164, 175, 177, 184, 201, 205, 212, 218, 221, 232, 234, 238, 241, 254, 267, 272, 274, 278, 285, 287, 308, 315, 322, 328, 331, 337, 342, 344, 355, 371, 375, 377, 382, 407, 414, 425, 427, 432, 438, 447, 454, 461, 465, 472, 481, 485, 504, 515, 517, 528, 531, 537, 542, 548, 557, 562, 564, 568, 582, 601, 605, 614, 618, 625, 638, 641, 661, 667, 678, 685, 702, 704, 711, 722, 728, 735, 737, 744, 751, 755, 757, 771, 782, 784, 788, 805, 812, 814, 827, 832, 838, 847, 858, 867, 878, 887, 1004, 1011, 1015, 1024, 1031, 1035, 1044, 1048, 1064, 1075, 1088, 1101, 1112, 1114, 1118, 1121, 1132, 1147, 1152, 1154, 1158, 1174, 1178, 1181, 1185, 1217, 1222, 1231, 1242, 1251, 1255, 1262, 1268, 1284, 1288, 1305, 1312, 1321, 1327, 1341, 1345, 1352, 1354, 1365, 1367, 1374, 1385, 1387, 1408, 1411, 1417, 1437, 1442, 1444, 1448, 1455, 1462, 1471, 1477, 1484, 1518, 1521, 1532, 1541, 1552, 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2A47, 2A52, 2A54, 2A65, 2A69, 2A87, 2AA7, 3008, 300A, 3013, 3019, 3024, 3026, 3031, 3051, 3053, 3059, 3073, 3079, 308A, 3091, 3097, 30A8, 3112, 3114, 3118, 3123, 3136, 313A, 3141, 3158, 317A, 3189, 3194, 3196, 31A5, 31A7, 3206, 3208, 3217, 3222, 3224, 3233, 3235, 3244, 324A, 3257, 3284, 3293, 3295, 32A4, 3301, 3307, 3316, 3332, 3338, 3349, 335A, 3361, 3378, 3383, 3387, 3392, 3398, 3404, 3406, 3415, 3428, 3433, 3437, 3439, 3442, 3464, 3466, 3477, 3482, 3497, 34A4, 34AA, 3505, 3521, 3536, 3538, 3541, 3547, 3549, 3554, 355A, 3569, 3574, 3585, 3596, 3602, 3604, 360A, 3613, 362A, 3637, 3659, 3662, 3664, 3668, 3673, 3675, 3686, 368A, 36A2, 371A, 3729, 3734, 3745, 3758, 3763, 3772, 3783, 3785, 3789, 3794, 37A1, 37A7, 3806, 3808, 3811, 3824, 382A, 3835, 3839, 3844, 3846, 3855, 3857, 3871, 3882, 388A, 38A6, 38AA, 3905, 3916, 3918, 3923, 3929, 3934, 395A, 3965, 3978, 3981, 3989, 3998, 39A5, 3A06, 3A22, 3A26, 3A28, 3A31, 3A53, 3A64, 3A6A, 3A71, 3A86, 3A91, 3A97, 3AAA, 4009, 4021, 4025, 4052, 4058, 4063, 4069, 4076, 4081, 4085, 4087, 4098, 40A3, 40A7, 40A9, 4104, 4124, 412A, 4131, 4135, 4151, 4153, 4157, 4168, 416A, 4175, 4179, 41A2, 41A8, 4203, 4207, 4214, 4223, 4252, 4267, 4269, 4274, 4278, 427A, 4283, 4285, 4294, 42A7, 4302, 4306, 4313, 4322, 4328, 4346, 434A, 4351, 4357, 4384, 4388, 4395, 43A4, 43AA, 4405, 4412, 4418, 4429, 4432, 4438, 443A, 4445, 4449, 4454, 4456, 4465, 4467, 4481, 4487, 44A5, 44A9, 450A, 4522, 4548, 4553, 4571, 4575, 4591, 4599, 45A4, 45A8, 4603, 4616, 4621, 4627, 4636, 4638, 4647, 4658, 4665, 4674, 4676, 4685, 4692, 46A3, 4702, 4724, 4726, 472A, 4737, 4742, 4746, 4753, 476A, 4779, 4784, 478A, 4791, 4797, 47A6, 4807, 4809, 4818, 4823, 4829, 4834, 4841, 4847, 4856, 4861, 4863, 4869, 4874, 487A, 4889, 4896, 4908, 4913, 4933, 4935, 4951, 4955, 4962, 4971, 4999, 49A6, 4A12, 4A16, 4A18, 4A27, 4A32, 4A34, 4A3A, 4A43, 4A5A, 4A67, 4A78, 4A94, 4AA9, 5004, 5006, 5017, 5022, 5031, 5033, 5042, 5044, 504A, 5059, 5071, 5075, 5097, 5099, 5103, 5105, 5114, 5116, 5125, 5143, 5147, 5149, 5152, 515A, 5174, 517A, 5185, 5187, 5198, 5202, 520A, 5213, 5219, 5246, 5248, 5257, 5259, 5264, 5268, 5273, 5279, 5286, 5291, 5295, 52A6, 5301, 5309, 531A, 5323, 5336, 5347, 5356, 5378, 5383, 5394, 539A, 53A1, 5411, 5419, 5435, 5444, 544A, 5462, 5466, 5468, 5473, 5482, 548A, 5495, 5499, 54A4, 5521, 5534, 5543, 5545, 5556, 5565, 5567, 5581, 5583, 559A, 5611, 5628, 5633, 5648, 5664, 566A, 5671, 5688, 5691, 5697, 5699, 56A8, 5705, 5714, 571A, 5725, 5732, 5736, 5741, 5743, 5752, 5754, 5765, 5769, 5774, 577A, 5781, 5792, 5796, 57A9, 5815, 5819, 5824, 5842, 5846, 5853, 5859, 5862, 586A, 5873, 5886, 5891, 5895, 58A8, 5909, 5912, 5914, 5941, 5945, 5967, 5972, 5978, 5989, 599A, 5A02, 5A08, 5A11, 5A13, 5A17, 5A33, 5A39, 5A4A, 5A57, 5A62, 5A66, 5A77, 5A79, ...
1211, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 105, 107, 111, 117, 11B, 125, 12B, 131, 13B, 141, 145, 147, 157, 167, 16B, 171, 175, 17B, 181, 18B, 195, 19B, 1A5, 1A7, 1B1, 1B5, 1B7, 205, 217, 21B, 221, 225, 237, 241, 24B, 251, 255, 25B, 267, 271, 277, 27B, 285, 291, 295, 2A1, 2AB, 2B1, 2BB, 301, 307, 30B, 315, 321, 325, 327, 32B, 33B, 347, 34B, 357, 35B, 365, 375, 377, 391, 397, 3A5, 3AB, 3B5, 3B7, 401, 40B, 415, 41B, 421, 427, 431, 435, 437, 447, 455, 457, 45B, 465, 46B, 471, 481, 485, 48B, 497, 4A5, 4B1, 4BB, 507, 511, 517, 51B, 527, 531, 535, 541, 545, 557, 565, 575, 577, 585, 587, 58B, 591, 59B, 5B1, 5B5, 5B7, 5BB, 611, 615, 617, 61B, 637, 63B, 647, 655, 661, 665, 66B, 675, 687, 68B, 695, 69B, 6A7, 6B1, 701, 705, 70B, 711, 71B, 721, 727, 735, 737, 745, 747, 751, 767, 76B, 771, 775, 77B, 785, 791, 797, 7A1, 7BB, 801, 80B, 817, 825, 82B, 835, 841, 851, 855, 85B, 865, 867, 871, 881, 88B, 8A5, 8A7, 8AB, 8B5, 8B7, 901, 905, 907, 90B, 91B, 921, 927, 955, 95B, 965, 971, 987, 995, 9A7, 9AB, 9B1, 9B5, 9BB, A07, A0B, A11, A17, A27, A35, A37, A3B, A41, A45, A4B, A5B, A6B, A77, A87, A91, A95, A9B, AA7, AAB, AB7, ABB, B11, B15, B1B, B21, B25, B2B, B31, B37, B45, B61, B67, B6B, B71, B91, B95, B97, BA5, BB5, BB7, 1005, 1011, 1017, 1021, 1027, 1041, 1047, 104B, 1051, 1061, 106B, 107B, 1087, 109B, 10B1, 10B7, 10BB, 1101, 1105, 1107, 1115, 1125, 112B, 1135, 114B, 1151, 1165, 1167, 1185, 118B, 1197, 11A1, 11A5, 11A7, 11AB, 11B7, 1201, 120B, 1211, 121B, 1231, 123B, 1245, 1255, 1257, 125B, 1261, 126B, 127B, 1281, 1295, 1297, 12A1, 12A5, 12A7, 12B5, 1301, 1317, 1337, 133B, 1345, 1351, 1365, 1367, 136B, 1377, 138B, 1391, 1395, 13A1, 13A7, 13B1, 13B5, 1405, 1407, 1425, 142B, 1431, 1437, 143B, 1445, 1457, 1461, 1465, 1467, 1471, 1475, 147B, 148B, 1495, 149B, 14B1, 14B5, 14BB, 150B, 1517, 1521, 1525, 1547, 1561, 156B, 1577, 157B, 1585, 1587, 1591, 15AB, 15BB, 1601, 1615, 1621, 1625, 1635, 1647, 1655, 1657, 165B, 1667, 1671, 1677, 167B, 1681, 1685, 168B, 1697, 169B, 16A1, 16A7, 16B5, 16B7, 1705, 1711, 1715, 1727, 1735, 1745, 1747, 1751, 1755, 1757, 176B, 1781, 1785, 178B, 1797, 17A1, 17A5, 17BB, 1807, 1815, 181B, 1825, 1831, 183B, 184B, 1861, 1865, 186B, 1875, 1877, 189B, 18A1, 18AB, 18B7, 18BB, 1911, 1915, 1921, 1931, 1937, 1947, 194B, 1955, 1971, 197B, 1981, 1995, 19B7, 19BB, 1A01, 1A11, 1A17, 1A1B, 1A2B, 1A35, 1A41, 1A45, 1A51, 1A6B, 1A71, 1A75, 1A77, 1A87, 1AAB, 1AB1, 1AB7, 1B01, 1B07, 1B0B, 1B15, 1B17, 1B27, 1B2B, 1B3B, 1B41, 1B4B, 1B51, 1B65, 1B67, 1B7B, 1B85, 1BA1, 1BB5, 2001, 2005, 2007, 200B, 2011, 202B, 2037, 2047, 2051, 205B, 2061, 2065, 206B, 2071, 2077, 2085, 2087, 2097, 20A5, 20A7, 20B5, 2107, 2111, 2115, 211B, 2127, 2131, 2137, 214B, 215B, 2161, 2165, 2177, 2181, 2185, 2191, 219B, 21A7, 21B1, 21B7, 2215, 221B, 2221, 222B, 2241, 2245, 224B, 2265, 2267, 2275, 2287, 228B, 2291, 229B, 22B1, 22B5, 2301, 2317, 231B, 2325, 2327, 232B, 2335, 2337, 2347, 234B, 2367, 2385, 2395, 2397, 239B, 23A5, 23AB, 23B1, 23B7, 2415, 2417, 2421, 2435, 243B, 244B, 2451, 2457, 2467, 247B, 2481, 2485, 248B, 24A1, 24A5, 24A7, 2501, 2521, 252B, 2535, 2537, 2545, 2547, 2555, 2557, 2565, 256B, 2571, 257B, 2581, 258B, 2595, 25A1, 2607, 2615, 2617, 2625, 2631, 2637, 2645, 265B, 2665, 2675, 2685, 2687, 26A1, 26A7, 26AB, 26B5, 26BB, 2715, 2717, 2725, 2737, 2741, 2745, 2747, 274B, 276B, 2771, 2781, 2787, 279B, 27A7, 27B1, 27B7, 2811, 2825, 2827, 282B, 2835, 2837, 2841, 2847, 2855, 285B, 286B, 287B, 2895, 2897, 28A1, 28A5, 28BB, 2907, 2927, 292B, 2931, 2935, 293B, 2941, 2951, 2955, 2967, 2991, 299B, 29A5, 29B5, 2A07, 2A11, 2A1B, 2A2B, 2A31, 2A35, 2A3B, 2A47, 2A51, 2A5B, 2A61, 2A65, 2A77, 2A81, 2A87, 2A8B, 2A95, 2A97, 2AA5, 2AA7, 2ABB, 2B0B, 2B17, 2B31, 2B35, 2B3B, 2B4B, 2B51, 2B57, 2B61, 2B67, 2B8B, 2B95, 2BA7, 2BAB, 2BB7, 3005, 3011, 3021, 3037, 303B, 3041, 3045, 3065, 3075, 307B, 3081, 3095, 309B, 30A5, 30B7, 3105, 3117, 311B, 3145, 314B, 3155, 315B, 3167, 3171, 3175, 3177, 3187, 3191, 3195, 3197, 31A1, 31BB, 3205, 3207, 320B, 3225, 3227, 322B, 323B, 3241, 3247, 324B, 3271, 3277, 3281, 3285, 3291, 329B, 3307, 331B, 3321, 3327, 332B, 3331, 3335, 3337, 3345, 3357, 3361, 3365, 3371, 337B, 3385, 33A1, 33A5, 33A7, 33B1, 3417, 341B, 3427, 3435, 343B, 3445, 3451, 3457, 3467, 346B, 3475, 3477, 3481, 3485, 348B, 3491, 349B, 34A1, 34B5, 34BB, 3517, 351B, 352B, 3541, 3565, 356B, 3587, 358B, 35A5, 35B1, 35B7, 35BB, 3605, 3617, 3621, 3627, 3635, 3637, 3645, 3655, 3661, 366B, 3671, 367B, 3687, 3697, 36A5, 3705, 3707, 370B, 3717, 3721, 3725, 3731, 3747, 3755, 375B, 3765, 3767, 3771, 377B, 378B, 3791, 379B, 37A5, 37AB, 37B5, 3801, 3807, 3815, 381B, 3821, 3827, 3831, 3837, 3845, 3851, 3871, 3877, 3895, 3897, 38B1, 38B5, 3901, 390B, 3935, 3941, 3957, 395B, 3961, 396B, 3975, 3977, 3981, 3985, 399B, 39A7, 39B7, 3A11, 3A25, 3A2B, 3A31, 3A41, 3A47, 3A55, 3A57, 3A65, 3A67, 3A71, 3A7B, 3A91, 3A95, 3AB5, 3AB7, 3B0B, 3B11, 3B1B, 3B21, 3B2B, 3B47, 3B4B, 3B51, 3B55, 3B61, 3B75, 3B7B, 3B85, 3B87, 3B97, 3BAB, 3BB7, 3BBB, 4005, 402B, 4031, 403B, 4041, 4047, 404B, 4055, 405B, 4067, 4071, 4075, 4085, 408B, 4097, 40A7, 40AB, 4101, 4111, 411B, 413B, 4145, 4155, 415B, 4161, 417B, 4187, 41A1, 41AB, 41B5, 4207, 420B, 4211, 4217, 4225, 4231, 4237, 423B, 4245, 426B, 4281, 428B, 4291, 42A1, 42AB, 42B1, 4305, 4307, 4321, 4341, 4357, 4361, 4375, 438B, 4395, 4397, 43B1, 43B5, 43BB, 4401, 440B, 4417, 4425, 442B, 4435, 4441, 4445, 444B, 4451, 445B, 4461, 4471, 4475, 447B, 4485, 4487, 4497, 449B, 44B1, 4507, 450B, 4515, 4531, 4535, 4541, 4547, 454B, 4557, 455B, 4571, 4577, 457B, 4591, 45A1, 45A5, 45A7, 4611, 4615, 4635, 463B, 4645, 4655, 4665, 4677, 4681, 4685, 4687, 468B, 46A5, 46AB, 46BB, 4707, 4711, 4715, 4725, 4727, ...
1314, 16, 1A, 23, 25, 2B, 32, 34, 38, 41, 47, 49, 52, 56, 58, 61, 65, 6B, 76, 7A, 7C, 83, 85, 89, 9A, A1, A7, A9, B6, B8, C1, C7, CB, 104, 10A, 10C, 119, 11B, 122, 124, 133, 142, 146, 148, 14C, 155, 157, 164, 16A, 173, 179, 17B, 184, 188, 18A, 197, 1A8, 1AC, 1B1, 1B5, 1C6, 1CC, 209, 20B, 212, 218, 223, 229, 232, 236, 23C, 247, 24B, 256, 263, 265, 272, 274, 27A, 281, 287, 292, 296, 298, 29C, 2AB, 2B6, 2BA, 2C5, 2C9, 302, 311, 313, 328, 331, 33B, 344, 34A, 34C, 355, 362, 368, 371, 373, 379, 382, 386, 388, 397, 3A4, 3A6, 3AA, 3B3, 3B9, 3BB, 3CA, 401, 407, 412, 41C, 427, 434, 43C, 445, 44B, 452, 45A, 463, 467, 472, 476, 487, 494, 4A3, 4A5, 4B2, 4B4, 4B8, 4BA, 4C7, 508, 50C, 511, 515, 526, 52A, 52C, 533, 54A, 551, 559, 566, 571, 575, 57B, 584, 595, 599, 5A2, 5A8, 5B3, 5B9, 5C8, 5CC, 605, 607, 614, 616, 61C, 629, 62B, 638, 63A, 643, 658, 65C, 661, 665, 66B, 674, 67C, 685, 68B, 6A7, 6A9, 6B6, 6C1, 6CB, 704, 70A, 715, 724, 728, 731, 737, 739, 742, 751, 75B, 773, 775, 779, 782, 784, 78A, 791, 793, 797, 7A6, 7A8, 7B1, 809, 812, 818, 823, 838, 845, 856, 85A, 85C, 863, 869, 874, 878, 87A, 883, 892, 89C, 8A1, 8A5, 8A7, 8AB, 8B4, 8C3, 902, 90A, 919, 922, 926, 92C, 937, 93B, 946, 94A, 95B, 962, 968, 96A, 971, 977, 979, 982, 98C, 9A6, 9AC, 9B3, 9B5, A03, A07, A09, A16, A25, A27, A34, A3C, A45, A4B, A54, A69, A72, A76, A78, A87, A94, AA3, AAB, AC1, B02, B08, B0C, B11, B15, B17, B24, B33, B39, B42, B57, B59, B6C, B71, B8A, B93, B9B, BA4, BA8, BAA, BB1, BB9, BC2, BCC, C01, C0B, C1C, C29, C32, C41, C43, C47, C49, C56, C65, C67, C7A, C7C, C85, C89, C8B, C98, CA3, CB8, 1006, 100A, 1013, 101B, 1031, 1033, 1037, 1042, 1055, 1057, 105B, 1066, 106C, 1075, 1079, 1088, 108A, 10A6, 10AC, 10B1, 10B7, 10BB, 10C4, 1105, 110B, 1112, 1114, 111A, 1121, 1127, 1136, 113C, 1145, 1156, 115A, 1163, 1172, 117A, 1183, 1187, 11A7, 11BC, 11C9, 1204, 1208, 1211, 1213, 1219, 1235, 1244, 1246, 1259, 1264, 1268, 1277, 1288, 1295, 1297, 129B, 12A6, 12AC, 12B5, 12B9, 12BB, 12C2, 12C8, 1303, 1307, 1309, 1312, 131C, 1321, 132B, 1336, 133A, 134B, 1358, 1367, 1369, 1372, 1376, 1378, 138B, 139C, 13A3, 13A9, 13B4, 13BA, 13C1, 1406, 1411, 141B, 1424, 142A, 1435, 1442, 1451, 1462, 1466, 146C, 1475, 1477, 1499, 149B, 14A8, 14B3, 14B7, 14C8, 14CC, 1507, 1516, 151C, 152B, 1532, 1538, 1552, 155C, 1561, 1574, 1594, 1598, 159A, 15A9, 15B2, 15B6, 15C5, 15CB, 1606, 160A, 1615, 1631, 1633, 1637, 1639, 1648, 166A, 166C, 1675, 167B, 1684, 1688, 1691, 1693, 16A2, 16A6, 16B5, 16B7, 16C4, 16C6, 1709, 170B, 1721, 1727, 1741, 1754, 175C, 1763, 1765, 1769, 176B, 1787, 1792, 17A1, 17A7, 17B4, 17B6, 17BA, 17C3, 17C5, 17CB, 1808, 180A, 1819, 1826, 1828, 1835, 1846, 184C, 1853, 1859, 1864, 186A, 1873, 1886, 1895, 1897, 189B, 18AC, 18B5, 18B9, 18C4, 1901, 1909, 1912, 1918, 1934, 193A, 193C, 1949, 195A, 1961, 1967, 197C, 1981, 198B, 199C, 19A3, 19A5, 19B2, 19C3, 19C7, 1A02, 1A17, 1A1B, 1A24, 1A26, 1A2A, 1A33, 1A35, 1A44, 1A48, 1A62, 1A7B, 1A8A, 1A8C, 1A93, 1A99, 1AA2, 1AA4, 1AAA, 1AC6, 1AC8, 1B01, 1B14, 1B1A, 1B29, 1B2B, 1B34, 1B43, 1B56, 1B58, 1B5C, 1B65, 1B76, 1B7A, 1B7C, 1B94, 1BB2, 1BBC, 1BC5, 1BC7, 1C04, 1C06, 1C13, 1C15, 1C22, 1C28, 1C2A, 1C37, 1C39, 1C46, 1C4C, 1C57, 1C7B, 1C88, 1C8A, 1C97, 1CA2, 1CA8, 1CB5, 1CCA, 2003, 2012, 2021, 2023, 2038, 2041, 2045, 204B, 2054, 2069, 206B, 2078, 2089, 2092, 2096, 2098, 209C, 20BA, 20BC, 20CB, 2104, 2117, 2122, 2128, 2131, 2146, 2159, 215B, 2162, 2168, 216A, 2173, 2179, 2186, 218C, 219B, 21AA, 21C2, 21C4, 21CA, 2201, 2216, 2221, 223C, 2243, 2245, 2249, 2252, 2254, 2263, 2267, 2278, 229C, 22A9, 22B2, 22C1, 2302, 2308, 2315, 2324, 2326, 232A, 2333, 233B, 2344, 2351, 2353, 2357, 2368, 2371, 2377, 237B, 2384, 2386, 2393, 2395, 23A8, 23B7, 23C2, 2407, 240B, 2414, 2423, 2425, 242B, 2434, 243A, 245C, 2465, 2476, 247A, 2485, 2492, 249A, 24A9, 24C1, 24C5, 24C7, 24CB, 2519, 2528, 2531, 2533, 2546, 254C, 2555, 2566, 2573, 2584, 2588, 25AC, 25B5, 25BB, 25C4, 25CC, 2605, 2609, 260B, 261A, 2623, 2627, 2629, 2632, 264B, 2654, 2656, 265A, 2672, 2674, 2678, 2687, 2689, 2692, 2696, 26B6, 26BC, 26C5, 26C9, 2704, 2711, 2737, 274A, 274C, 2755, 2759, 275B, 2762, 2764, 2771, 2782, 2788, 278C, 2797, 27A4, 27AA, 27C4, 27C8, 27CA, 2803, 2827, 282B, 2836, 2843, 2849, 2852, 285A, 2863, 2872, 2876, 287C, 2881, 2887, 288B, 2894, 2896, 28A3, 28A5, 28B8, 28C1, 2908, 290C, 291B, 292C, 2951, 2957, 2971, 2975, 298A, 2995, 299B, 29A2, 29A8, 29B9, 29C2, 29C8, 2A05, 2A07, 2A14, 2A23, 2A2B, 2A38, 2A3A, 2A47, 2A52, 2A61, 2A6B, 2A89, 2A8B, 2A92, 2A9A, 2AA3, 2AA7, 2AB2, 2AC7, 2B04, 2B0A, 2B13, 2B15, 2B1B, 2B28, 2B37, 2B39, 2B46, 2B4C, 2B55, 2B5B, 2B66, 2B6C, 2B79, 2B82, 2B84, 2B8A, 2B93, 2B99, 2BA6, 2BB1, 2BCC, 2C05, 2C21, 2C23, 2C38, 2C3C, 2C47, 2C54, 2C78, 2C83, 2C98, 2C9C, 2CA1, 2CAB, 2CB4, 2CB6, 2CBC, 2CC3, 3008, 3013, 3022, 3037, 304A, 3053, 3055, 3064, 306A, 3077, 3079, 3086, 3088, 3091, 309B, 30AC, 30B3, 3101, 3103, 3116, 3118, 3125, 3127, 3134, 314B, 3152, 3154, 3158, 3163, 3176, 317C, 3185, 3187, 3196, 31A9, 31B4, 31B8, 31C1, 3215, 3217, 3224, 3226, 322C, 3233, 3239, 3242, 324A, 3253, 3257, 3266, 326C, 3277, 3286, 328A, 329B, 32AA, 32B7, 3305, 330B, 331A, 3323, 3325, 3341, 3349, 3361, 336B, 3374, 3385, 3389, 338B, 3394, 33A1, 33A9, 33B2, 33B6, 33BC, 3413, 3424, 3431, 3433, 3442, 344C, 3451, 3464, 3466, 347B, 3499, 34B1, 34B7, 34CA, 3512, 3518, 351A, 3532, 3536, 353C, 3541, 354B, 3556, 3563, 3569, 3572, 357A, 3581, 3587, 3589, 3596, 3598, 35A7, 35AB, 35B4, 35BA, 35BC, 35CB, 3602, 3613, 3628, 362C, 3635, 364C, 3653, 365B, 3664, 3668, 3673, 3677, 3688, 3691, 3695, 36A6, 36B5, 36B9, 36BB, 3712, 3716, 3734, 373A, 3743, 3752, 3761, 3772, 3778, 377C, 3781, 3785, 379A, 37A3, 37B2, 37BA, 37C3, 37C7, 3806, 3808, 3817, ...
1413, 15, 19, 21, 23, 29, 2D, 31, 35, 3B, 43, 45, 4B, 51, 53, 59, 5D, 65, 6D, 73, 75, 79, 7B, 81, 91, 95, 9B, 9D, A9, AB, B3, B9, BD, C5, CB, CD, D9, DB, 101, 103, 111, 11D, 123, 125, 129, 131, 133, 13D, 145, 14B, 153, 155, 15B, 161, 163, 16D, 17D, 183, 185, 189, 199, 1A1, 1AB, 1AD, 1B3, 1B9, 1C3, 1C9, 1D1, 1D5, 1DB, 205, 209, 213, 21D, 221, 22B, 22D, 235, 239, 241, 249, 24D, 251, 255, 263, 26B, 271, 279, 27D, 285, 293, 295, 2A9, 2B1, 2BB, 2C3, 2C9, 2CB, 2D3, 2DD, 305, 30B, 30D, 315, 31B, 321, 323, 331, 33B, 33D, 343, 349, 351, 353, 361, 365, 36B, 375, 381, 389, 395, 39D, 3A5, 3AB, 3B1, 3B9, 3C1, 3C5, 3CD, 3D3, 403, 40D, 41B, 41D, 429, 42B, 431, 433, 43D, 44D, 453, 455, 459, 469, 46D, 471, 475, 48B, 491, 499, 4A5, 4AD, 4B3, 4B9, 4C1, 4D1, 4D5, 4DB, 503, 50B, 513, 521, 525, 52B, 52D, 539, 53B, 543, 54D, 551, 55B, 55D, 565, 579, 57D, 581, 585, 58B, 593, 59B, 5A3, 5A9, 5C3, 5C5, 5D1, 5D9, 605, 60B, 613, 61B, 629, 62D, 635, 63B, 63D, 645, 653, 65D, 673, 675, 679, 681, 683, 689, 68D, 691, 695, 6A3, 6A5, 6AB, 6D3, 6D9, 701, 709, 71D, 729, 739, 73D, 741, 745, 74B, 755, 759, 75B, 763, 771, 77B, 77D, 783, 785, 789, 791, 79D, 7AB, 7B5, 7C3, 7C9, 7CD, 7D5, 7DD, 803, 80B, 811, 821, 825, 82B, 82D, 833, 839, 83B, 843, 84D, 865, 86B, 871, 873, 88D, 893, 895, 8A1, 8AD, 8B1, 8BB, 8C5, 8CB, 8D3, 8D9, 90D, 915, 919, 91B, 929, 935, 943, 94B, 95D, 96D, 975, 979, 97B, 981, 983, 98D, 99B, 9A3, 9A9, 9BD, 9C1, 9D3, 9D5, A0D, A15, A1D, A25, A29, A2B, A31, A39, A41, A4B, A4D, A59, A69, A75, A7B, A89, A8B, A91, A93, A9D, AAB, AAD, AC1, AC3, AC9, ACD, AD1, ADB, B05, B19, B35, B39, B41, B49, B5B, B5D, B63, B6B, B7D, B81, B85, B8D, B95, B9B, BA1, BAD, BB1, BC9, BD1, BD3, BD9, BDD, C05, C15, C1B, C21, C23, C29, C2D, C35, C43, C49, C51, C61, C65, C6B, C79, C83, C89, C8D, CAB, CC1, CCB, CD5, CD9, D01, D03, D09, D23, D31, D33, D45, D4D, D53, D61, D71, D7B, D7D, D83, D8B, D93, D99, D9D, DA1, DA5, DAB, DB5, DB9, DBB, DC3, DCD, DD1, DDB, 1005, 1009, 1019, 1025, 1033, 1035, 103B, 1041, 1043, 1055, 1065, 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1775, 1781, 1783, 178D, 1791, 179B, 17A3, 17A5, 17B1, 17B3, 17BD, 17C5, 17CD, 1811, 181B, 181D, 1829, 1833, 1839, 1845, 1859, 1861, 186D, 187B, 187D, 1893, 1899, 189D, 18A5, 18AB, 18C1, 18C3, 18CD, 18DD, 1905, 1909, 190B, 1911, 192B, 192D, 193B, 1943, 1955, 195D, 1965, 196B, 1981, 1993, 1995, 1999, 19A1, 19A3, 19A9, 19B1, 19BB, 19C3, 19D1, 19DD, 1A13, 1A15, 1A1B, 1A21, 1A35, 1A3D, 1A59, 1A5D, 1A61, 1A65, 1A6B, 1A6D, 1A7B, 1A81, 1A91, 1AB3, 1ABD, 1AC5, 1AD3, 1B03, 1B09, 1B15, 1B23, 1B25, 1B29, 1B31, 1B39, 1B41, 1B4B, 1B4D, 1B53, 1B63, 1B69, 1B71, 1B75, 1B7B, 1B7D, 1B89, 1B8B, 1B9D, 1BAB, 1BB5, 1BC9, 1BCD, 1BD5, 1C03, 1C05, 1C0B, 1C13, 1C19, 1C39, 1C41, 1C51, 1C55, 1C5D, 1C69, 1C73, 1C81, 1C95, 1C99, 1C9B, 1CA1, 1CBB, 1CC9, 1CD1, 1CD3, 1D05, 1D0B, 1D13, 1D23, 1D2D, 1D3D, 1D43, 1D65, 1D6B, 1D73, 1D79, 1D83, 1D89, 1D8D, 1D91, 1D9D, 1DA5, 1DA9, 1DAB, 1DB3, 1DCB, 1DD3, 1DD5, 1DD9, 200D, 2011, 2015, 2023, 2025, 202B, 2031, 204D, 2055, 205B, 2061, 2069, 2075, 2099, 20AB, 20AD, 20B5, 20B9, 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2835, 283B, 2849, 2851, 2853, 286B, 2875, 2889, 2895, 289B, 28AB, 28B1, 28B3, 28B9, 28C5, 28CD, 28D5, 28D9, 2901, 2923, 2933, 293D, 2941, 294D, 2959, 295B, 296D, 2971, 2985, 29A1, 29B5, 29BB, 29CD, 2A03, 2A09, 2A0B, 2A21, 2A25, 2A2B, 2A2D, 2A39, 2A43, 2A4D, 2A55, 2A5B, 2A65, 2A69, 2A71, 2A73, 2A7D, 2A81, 2A8D, 2A93, 2A99, 2AA1, 2AA3, 2AB1, 2AB5, 2AC5, 2AD9, 2ADD, 2B05, 2B1B, 2B21, 2B29, 2B31, 2B35, 2B3D, 2B43, 2B53, 2B59, 2B5D, 2B6D, 2B7B, 2B81, 2B83, 2BA5, 2BA9, 2BC5, 2BCB, 2BD3, 2C01, 2C0D, 2C1D, 2C25, 2C29, 2C2B, 2C31, 2C45, 2C4B, 2C59, 2C63, 2C69, 2C6D, 2C7B, 2C7D, 2C8B, ...
1512, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 102, 104, 108, 10E, 111, 11B, 122, 128, 12E, 131, 137, 13B, 13D, 148, 157, 15B, 15D, 162, 171, 177, 182, 184, 188, 18E, 197, 19D, 1A4, 1A8, 1AE, 1B7, 1BB, 1C4, 1CE, 1D1, 1DB, 1DD, 1E4, 1E8, 1EE, 207, 20B, 20D, 212, 21E, 227, 22B, 234, 238, 23E, 24B, 24D, 261, 267, 272, 278, 27E, 281, 287, 292, 298, 29E, 2A1, 2A7, 2AD, 2B2, 2B4, 2C1, 2CB, 2CD, 2D2, 2D8, 2DE, 2E1, 2ED, 302, 308, 311, 31B, 324, 32E, 337, 33D, 344, 348, 351, 357, 35B, 364, 368, 377, 382, 38E, 391, 39B, 39D, 3A2, 3A4, 3AE, 3BD, 3C2, 3C4, 3C8, 3D7, 3DB, 3DD, 3E2, 407, 40B, 414, 41E, 427, 42B, 432, 438, 447, 44B, 452, 458, 461, 467, 474, 478, 47E, 481, 48B, 48D, 494, 49E, 4A1, 4AB, 4AD, 4B4, 4C7, 4CB, 4CD, 4D2, 4D8, 4DE, 4E7, 4ED, 504, 51B, 51D, 528, 531, 53B, 542, 548, 551, 55D, 562, 568, 56E, 571, 577, 584, 58E, 5A2, 5A4, 5A8, 5AE, 5B1, 5B7, 5BB, 5BD, 5C2, 5CE, 5D1, 5D7, 60B, 612, 618, 621, 634, 63E, 64D, 652, 654, 658, 65E, 667, 66B, 66D, 674, 681, 68B, 68D, 692, 694, 698, 69E, 6AB, 6B8, 6C1, 6CD, 6D4, 6D8, 6DE, 6E7, 6EB, 704, 708, 717, 71B, 722, 724, 728, 72E, 731, 737, 742, 757, 75D, 762, 764, 77D, 782, 784, 78E, 79B, 79D, 7A8, 7B1, 7B7, 7BD, 7C4, 7D7, 7DD, 7E2, 7E4, 801, 80B, 818, 821, 832, 841, 847, 84B, 84D, 852, 854, 85E, 86B, 872, 878, 88B, 88D, 89E, 8A1, 8B8, 8BE, 8C7, 8CD, 8D2, 8D4, 8D8, 8E1, 8E7, 902, 904, 90E, 91D, 928, 92E, 93B, 93D, 942, 944, 94E, 95B, 95D, 96E, 971, 977, 97B, 97D, 988, 991, 9A4, 9BD, 9C2, 9C8, 9D1, 9E2, 9E4, 9E8, A01, A12, A14, A18, A21, A27, A2D, A32, A3E, A41, A58, A5E, A61, A67, A6B, A72, A81, A87, A8B, A8D, A94, A98, A9E, AAB, AB2, AB8, AC7, ACB, AD2, ADE, AE7, AED, B02, B1D, B31, B3B, B44, B48, B4E, B51, B57, B6E, B7B, B7D, B8E, B97, B9B, BA8, BB7, BC2, BC4, BC8, BD1, BD7, BDD, BE2, BE4, BE8, BEE, C07, C0B, C0D, C14, C1E, C21, C2B, C34, C38, C47, C52, C5E, C61, C67, C6B, C6D, C7E, C8D, C92, C98, CA1, CA7, CAB, CBE, CC7, CD2, CD8, CDE, CE7, D02, D0E, D1D, D22, D28, D2E, D31, D4E, D51, D5B, D64, D68, D77, D7B, D84, D91, D97, DA4, DA8, DAE, DC4, DCE, DD1, DE2, E0D, E12, E14, E21, E27, E2B, E38, E3E, E47, E4B, E54, E6B, E6D, E72, E74, E81, E9E, EA1, EA7, EAD, EB4, EB8, EBE, EC1, ECD, ED2, EDE, EE1, EEB, EED, 100E, 1011, 1022, 1028, 103D, 104E, 1057, 105B, 105D, 1062, 1064, 107B, 1084, 1091, 1097, 10A2, 10A4, 10A8, 10AE, 10B1, 10B7, 10C2, 10C4, 10D1, 10DB, 10DD, 10E8, 1107, 110D, 1112, 1118, 1121, 1127, 112D, 113E, 114B, 114D, 1152, 1161, 1167, 116B, 1174, 117E, 1187, 118D, 1194, 11AB, 11B2, 11B4, 11BE, 11CD, 11D2, 11D8, 11EB, 11ED, 1208, 1217, 121B, 121D, 1228, 1237, 123B, 1244, 1257, 125B, 1262, 1264, 1268, 126E, 1271, 127D, 1282, 1297, 12AE, 12BB, 12BD, 12C2, 12C8, 12CE, 12D1, 12D7, 12EE, 1301, 1307, 1318, 131E, 132B, 132D, 1334, 1341, 1352, 1354, 1358, 135E, 136D, 1372, 1374, 1387, 13A1, 13AB, 13B2, 13B4, 13BE, 13C1, 13CB, 13CD, 13D8, 13DE, 13E1, 13EB, 13ED, 1408, 140E, 1417, 1437, 1442, 1444, 144E, 1457, 145D, 1468, 147B, 1482, 148E, 149B, 149D, 14B1, 14B7, 14BB, 14C2, 14C8, 14DB, 14DD, 14E8, 1507, 150D, 1512, 1514, 1518, 1532, 1534, 1541, 1547, 1558, 1561, 1567, 156D, 1581, 1592, 1594, 1598, 159E, 15A1, 15A7, 15AD, 15B8, 15BE, 15CB, 15D8, 15EB, 15ED, 1604, 1608, 161B, 1624, 163D, 1642, 1644, 1648, 164E, 1651, 165D, 1662, 1671, 1691, 169B, 16A2, 16AE, 16BD, 16C4, 16CE, 16DB, 16DD, 16E2, 16E8, 1701, 1707, 1712, 1714, 1718, 1727, 172D, 1734, 1738, 173E, 1741, 174B, 174D, 175E, 176B, 1774, 1787, 178B, 1792, 179E, 17A1, 17A7, 17AD, 17B4, 17D2, 17D8, 17E7, 17EB, 1804, 180E, 1817, 1824, 1837, 183B, 183D, 1842, 185B, 1868, 186E, 1871, 1882, 1888, 188E, 189D, 18A8, 18B7, 18BB, 18DB, 18E2, 18E8, 18EE, 1907, 190D, 1912, 1914, 1921, 1927, 192B, 192D, 1934, 194B, 1952, 1954, 1958, 196B, 196D, 1972, 197E, 1981, 1987, 198B, 19A7, 19AD, 19B4, 19B8, 19C1, 19CB, 19ED, 1A0E, 1A11, 1A17, 1A1B, 1A1D, 1A22, 1A24, 1A2E, 1A3D, 1A44, 1A48, 1A51, 1A5B, 1A62, 1A77, 1A7B, 1A7D, 1A84, 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FD5, FDB, FEA, FF1, FF7, FFD, FGE, G0D, G0F, G12, G18, G1A, G1G, G25, G2F, G34, G3G, G4B, G5C, G5E, G63, G67, G78, G7G, G96, G9A, G9C, G9G, GA5, GA7, GB2, GB6, GC3, GDG, GE9, GEF, GFA, GG7, GGD, 1006, 1011, 1013, 1017, 101D, 1024, 102A, 1033, 1035, 1039, 1046, 104C, 1051, 1055, 105B, 105D, 1066, 1068, 1077, 1082, 108A, 109B, 109F, 10A4, 10AG, 10B1, 10B7, 10BD, 10C2, 10DD, 10E2, 10EG, 10F3, 10FB, 10G4, 10GC, 1107, 1118, 111C, 111E, 1121, 1138, 1143, 1149, 114B, 115A, 115G, 1165, 1172, 117C, 1189, 118D, 11A9, 11AF, 11B4, 11BA, 11C1, 11C7, 11CB, 11CD, 11D8, 11DE, 11E1, 11E3, 11E9, 11FE, 11G3, 11G5, 11G9, 120A, 120C, 120G, 121B, 121D, 1222, 1226, 123F, 1244, 124A, 124E, 1255, 125F, 127D, 128C, 128E, 1293, 1297, 1299, 129D, 129F, 12A8, 12B5, 12BB, 12BF, 12C6, 12CG, 12D5, 12E8, 12EC, 12EE, 12F3, 12GG, 1303, 130B, 1314, 131A, 131G, 1327, 132D, 1338, 133C, 1341, 1343, 1349, 134D, 1352, 1354, 135E, 135G, 136F, 1374, 1387, 138B, 1396, 13A3, 13BE, 13C3, 13D6, 13DA, 13EB, 13F2, 13F8, 13FC, 13G1, 13GF, 1404, 140A, 1413, 1415, 141F, 142A, 1431, 143B, 143D, 1446, 144E, 1459, 1462, 1479, 147B, 147F, 1486, 148C, 148G, 1497, 14A8, 14B1, 14B7, 14BD, 14BF, 14C4, 14CE, 14D9, 14DB, 14E4, 14EA, 14EG, 14F5, 14FD, 14G2, 14GC, 1501, 1503, 1509, 150F, 1514, 151E, 1525, 153C, 1541, 1556, 1558, 1569, 156D, 1574, 157E, 159A, 15A1, 15B2, 15B6, 15B8, 15C1, 15C7, 15C9, 15CF, 15D2, 15E3, 15EB, 15F6, 15G7, 1606, 160C, 160E, 1619, 161F, 1628, 162A, 1633, 1635, 163B, 1644, 1651, 1655, 166C, 166E, 167D, 167F, 1688, 168A, 1693, 16A6, 16AA, 16AC, 16AG, 16B7, 16C6, 16CC, 16D1, 16D3, 16DF, 16EE, 16F5, 16F9, 16FF, 170B, 170D, 1716, 1718, 171E, 1721, 1727, 172D, 1734, 173A, 173E, 1749, 174F, 1756, 1761, 1765, 1772, 177E, 1787, 179E, 17A3, 17AF, 17B4, 17B6, 17CB, 17D2, 17E3, 17ED, 17F2, 17FG, 17G3, 17G5, 17GB, 1804, 180C, 1811, 1815, 181B, 1837, 1844, 184E, 184G, 185B, 1864, 1866, 1875, 1877, 1888, 189F, 18AG, 18B5, 18C4, 18D5, 18DB, 18DD, 18EE, 18F1, 18F7, 18F9, 18G2, 18GA, 1903, 1909, 190F, 1916, 191A, 191G, 1921, 192B, 192D, 1938, 193C, 1941, 1947, 1949, 1954, 1958, 1965, 1976, 197A, 197G, 1992, 1996, 199E, 19A3, 19A7, 19AF, 19B2, 19BG, 19C5, 19C9, 19D6, 19E1, 19E5, 19E7, 19G3, 19G7, 1A0E, 1A13, 1A19, 1A24, 1A2G, 1A3D, 1A42, 1A46, 1A48, 1A4C, 1A5D, 1A62, 1A6E, 1A75, 1A7B, 1A7F, 1A8A, 1A8C, 1A97, 1AB3, ...
1811, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 10D, 115, 117, 11B, 11H, 127, 12D, 131, 135, 13B, 141, 145, 14D, 155, 157, 15H, 161, 167, 16B, 16H, 177, 17B, 17D, 17H, 18B, 191, 195, 19D, 19H, 1A5, 1AH, 1B1, 1C1, 1C7, 1CH, 1D5, 1DB, 1DD, 1E1, 1EB, 1EH, 1F5, 1F7, 1FD, 1G1, 1G5, 1G7, 1H1, 1HB, 1HD, 1HH, 205, 20B, 20D, 217, 21B, 21H, 227, 22H, 237, 23H, 247, 24D, 251, 255, 25D, 261, 265, 26D, 26H, 27D, 285, 28H, 291, 29B, 29D, 29H, 2A1, 2AB, 2B7, 2BB, 2BD, 2BH, 2CD, 2CH, 2D1, 2D5, 2E7, 2EB, 2F1, 2FB, 2G1, 2G5, 2GB, 2GH, 2HD, 2HH, 305, 30B, 311, 317, 321, 325, 32B, 32D, 335, 337, 33D, 345, 347, 34H, 351, 357, 367, 36B, 36D, 36H, 375, 37B, 381, 387, 38D, 39H, 3A1, 3AB, 3B1, 3BB, 3BH, 3C5, 3CD, 3D7, 3DB, 3DH, 3E5, 3E7, 3ED, 3F7, 3FH, 3GH, 3H1, 3H5, 3HB, 3HD, 401, 405, 407, 40B, 415, 417, 41D, 43B, 43H, 445, 44D, 45D, 465, 471, 475, 477, 47B, 47H, 487, 48B, 48D, 491, 49D, 4A5, 4A7, 4AB, 4AD, 4AH, 4B5, 4BH, 4CB, 4D1, 4DD, 4E1, 4E5, 4EB, 4F1, 4F5, 4FD, 4FH, 4GD, 4GH, 4H5, 4H7, 4HB, 4HH, 501, 507, 50H, 521, 527, 52B, 52D, 541, 545, 547, 54H, 55B, 55D, 565, 56D, 571, 577, 57D, 58D, 591, 595, 597, 5A1, 5AB, 5B5, 5BD, 5CB, 5D7, 5DD, 5DH, 5E1, 5E5, 5E7, 5EH, 5FB, 5FH, 5G5, 5H5, 5H7, 605, 607, 61B, 61H, 627, 62D, 62H, 631, 635, 63D, 641, 64B, 64D, 655, 661, 66B, 66H, 67B, 67D, 67H, 681, 68B, 695, 697, 6A5, 6A7, 6AD, 6AH, 6B1, 6BB, 6C1, 6D1, 6E7, 6EB, 6EH, 6F7, 6G5, 6G7, 6GB, 6H1, 6HH, 701, 705, 70D, 711, 717, 71B, 725, 727, 73B, 73H, 741, 747, 74B, 74H, 75D, 761, 765, 767, 76D, 76H, 775, 77H, 785, 78B, 797, 79B, 79H, 7AB, 7B1, 7B7, 7BB, 7D1, 7E1, 7EB, 7F1, 7F5, 7FB, 7FD, 7G1, 7H5, 7HH, 801, 80H, 817, 81B, 825, 831, 83B, 83D, 83H, 847, 84D, 851, 855, 857, 85B, 85H, 867, 86B, 86D, 871, 87B, 87D, 885, 88D, 88H, 89D, 8A5, 8AH, 8B1, 8B7, 8BB, 8BD, 8CB, 8D7, 8DB, 8DH, 8E7, 8ED, 8EH, 8FH, 8G7, 8GH, 8H5, 8HB, 901, 90B, 915, 921, 925, 92B, 92H, 931, 94B, 94D, 955, 95D, 95H, 96D, 96H, 977, 981, 987, 991, 995, 99B, 9AD, 9B5, 9B7, 9C5, 9DD, 9DH, 9E1, 9ED, 9F1, 9F5, 9FH, 9G5, 9GD, 9GH, 9H7, A0B, A0D, A0H, A11, A1D, A35, A37, A3D, A41, A47, A4B, A4H, A51, A5D, A5H, A6B, A6D, A75, A77, A85, A87, A95, A9B, AAD, ABB, AC1, AC5, AC7, ACB, ACD, ADH, AE7, AF1, AF7, AFH, AG1, AG5, AGB, AGD, AH1, AHB, AHD, B07, B0H, B11, B1B, B27, B2D, B2H, B35, B3D, B41, B47, B55, B5H, B61, B65, B71, B77, B7B, B81, B8B, B91, B97, B9D, BAH, BB5, BB7, BBH, BCD, BCH, BD5, BE5, BE7, BEH, BFD, BFH, BG1, BGB, BH7, BHB, C01, C11, C15, C1B, C1D, C1H, C25, C27, C31, C35, C47, C5B, C65, C67, C6B, C6H, C75, C77, C7D, C8H, C91, C97, CA5, CAB, CB5, CB7, CBD, CC7, CD5, CD7, CDB, CDH, CED, CEH, CF1, CG1, CH7, CHH, D05, D07, D0H, D11, D1B, D1D, D25, D2B, D2D, D35, D37, D3H, D45, D4D, D67, D6H, D71, D7B, D81, D87, D8H, D9H, DA5, DAH, DBB, DBD, DCD, DD1, DD5, DDB, DDH, DEH, DF1, DFB, DG7, DGD, DGH, DH1, DH5, E0B, E0D, E17, E1D, E2B, E31, E37, E3D, E4D, E5B, E5D, E5H, E65, E67, E6D, E71, E7B, E7H, E8B, E95, EA5, EA7, EAD, EAH, EBH, EC7, EDD, EDH, EE1, EE5, EEB, EED, EF7, EFB, EG7, F01, F0B, F0H, F1B, F27, F2D, F35, F3H, F41, F45, F4B, F51, F57, F5H, F61, F65, F71, F77, F7D, F7H, F85, F87, F8H, F91, F9H, FAB, FB1, FC1, FC5, FCB, FD5, FD7, FDD, FE1, FE7, FFH, FG5, FH1, FH5, FHD, G05, G0D, G17, G27, G2B, G2D, G2H, G45, G4H, G55, G57, G65, G6B, G6H, G7D, G85, G91, G95, GAH, GB5, GBB, GBH, GC7, GCD, GCH, GD1, GDD, GE1, GE5, GE7, GED, GFH, GG5, GG7, GGB, GHB, GHD, GHH, H0B, H0D, H11, H15, H2D, H31, H37, H3B, H41, H4B, H67, H75, H77, H7D, H7H, H81, H85, H87, H8H, H9D, HA1, HA5, HAD, HB5, HBB, HCD, HCH, HD1, HD7, HF1, HF5, HFD, HG5, HGB, HGH, HH7, HHD, 1007, 100B, 100H, 1011, 1017, 101B, 101H, 1021, 102B, 102D, 103B, 103H, 1051, 1055, 105H, 106D, 1085, 108B, 109D, 109H, 10AH, 10B7, 10BD, 10BH, 10C5, 10D1, 10D7, 10DD, 10E5, 10E7, 10EH, 10FB, 10G1, 10GB, 10GD, 10H5, 10HD, 1107, 110H, 1125, 1127, 112B, 1131, 1137, 113B, 1141, 1151, 115B, 115H, 1165, 1167, 116D, 1175, 117H, 1181, 118B, 118H, 1195, 119B, 11A1, 11A7, 11AH, 11B5, 11B7, 11BD, 11C1, 11C7, 11CH, 11D7, 11ED, 11F1, 11G5, 11G7, 11H7, 11HB, 1201, 120B, 1225, 122D, 123D, 123H, 1241, 124B, 124H, 1251, 1257, 125B, 126B, 1271, 127D, 128D, 129B, 129H, 12A1, 12AD, 12B1, 12BB, 12BD, 12C5, 12C7, 12CD, 12D5, 12E1, 12E5, 12FB, 12FD, 12GB, 12GD, 12H5, 12H7, 12HH, 1311, 1315, 1317, 131B, 1321, 132H, 1335, 133B, 133D, 1347, 1355, 135D, 135H, 1365, 137H, 1381, 138B, 138D, 1391, 1395, 139B, 139H, 13A7, 13AD, 13AH, 13BB, 13BH, 13C7, 13D1, 13D5, 13E1, 13ED, 13F5, 13GB, 13GH, 13HB, 13HH, 1401, 1415, 141D, 142D, 1435, 143B, 1447, 144B, 144D, 1451, 145B, 1461, 1467, 146B, 146H, 148B, 1497, 149H, 14A1, 14AD, 14B5, 14B7, 14C5, 14C7, 14D7, 14ED, 14FD, 14G1, 14GH, 14HH, 1505, 1507, 1517, 151B, 151H, 1521, 152B, 1531, 153B, 153H, 1545, 154D, 154H, 1555, 1557, 155H, 1561, 156D, 156H, 1575, 157B, 157D, 1587, 158B, 1597, 15A7, 15AB, 15AH, 15C1, 15C5, 15CD, 15D1, 15D5, 15DD, 15DH, 15ED, 15F1, 15F5, 15G1, 15GD, 15GH, 15H1, 160D, 160H, 1625, 162B, 162H, 163B, 1645, 1651, 1657, 165B, 165D, 165H, 166H, 1675, 167H, 1687, 168D, 168H, 169B, 169D, 16A7, 16C1, ...
1914, 1A, 1C, 1I, 23, 25, 29, 2F, 32, 34, 3A, 3E, 3G, 43, 47, 4D, 52, 56, 58, 5C, 5E, 5I, 6D, 6H, 74, 76, 7G, 7I, 85, 8B, 8F, 92, 98, 9A, A1, A3, A7, A9, B2, BE, BI, C1, C5, CB, CD, D4, DA, DG, E3, E5, EB, EF, EH, F8, G3, G7, G9, GD, H8, HE, I5, I7, IB, IH, 106, 10C, 10I, 113, 119, 11H, 122, 12A, 131, 133, 13D, 13F, 142, 146, 14C, 151, 155, 157, 15B, 164, 16C, 16G, 175, 179, 17F, 188, 18A, 199, 19F, 1A6, 1AC, 1AI, 1B1, 1B7, 1BH, 1C4, 1CA, 1CC, 1CI, 1D5, 1D9, 1DB, 1E4, 1EE, 1EG, 1F1, 1F7, 1FD, 1FF, 1G8, 1GC, 1GI, 1H7, 1HH, 1I6, 1IG, 205, 20B, 20H, 212, 21A, 21G, 221, 229, 22D, 238, 23I, 24B, 24D, 254, 256, 25A, 25C, 263, 26H, 272, 274, 278, 283, 287, 289, 28D, 29E, 29I, 2A7, 2AH, 2B6, 2BA, 2BG, 2C3, 2CH, 2D2, 2D8, 2DE, 2E3, 2E9, 2F2, 2F6, 2FC, 2FE, 2G5, 2G7, 2GD, 2H4, 2H6, 2HG, 2HI, 2I5, 304, 308, 30A, 30E, 311, 317, 31F, 322, 328, 33B, 33D, 344, 34C, 353, 359, 35F, 364, 36G, 371, 377, 37D, 37F, 382, 38E, 395, 3A4, 3A6, 3AA, 3AG, 3AI, 3B5, 3B9, 3BB, 3BF, 3C8, 3CA, 3CG, 3EC, 3EI, 3F5, 3FD, 3GC, 3H3, 3HH, 3I2, 3I4, 3I8, 3IE, 403, 407, 409, 40F, 418, 41I, 421, 425, 427, 42B, 42H, 43A, 443, 44B, 454, 45A, 45E, 461, 469, 46D, 472, 476, 481, 485, 48B, 48D, 48H, 494, 496, 49C, 4A3, 4B4, 4BA, 4BE, 4BG, 4D2, 4D6, 4D8, 4DI, 4EB, 4ED, 4F4, 4FC, 4FI, 4G5, 4GB, 4HA, 4HG, 4I1, 4I3, 4IF, 506, 50I, 517, 524, 52I, 535, 539, 53B, 53F, 53H, 548, 551, 557, 55D, 56C, 56E, 57B, 57D, 58G, 593, 59B, 59H, 5A2, 5A4, 5A8, 5AG, 5B3, 5BD, 5BF, 5C6, 5D1, 5DB, 5DH, 5EA, 5EC, 5EG, 5EI, 5F9, 5G2, 5G4, 5H1, 5H3, 5H9, 5HD, 5HF, 5I6, 5IE, 60D, 61I, 623, 629, 62H, 63E, 63G, 641, 649, 656, 658, 65C, 661, 667, 66D, 66H, 67A, 67C, 68F, 692, 694, 69A, 69E, 6A1, 6AF, 6B2, 6B6, 6B8, 6BE, 6BI, 6C5, 6CH, 6D4, 6DA, 6E5, 6E9, 6EF, 6F8, 6FG, 6G3, 6G7, 6HE, 6ID, 704, 70C, 70G, 713, 715, 71B, 72E, 737, 739, 746, 74E, 74I, 75B, 766, 76G, 76I, 773, 77B, 77H, 784, 788, 78A, 78E, 791, 799, 79D, 79F, 7A2, 7AC, 7AE, 7B5, 7BD, 7BH, 7CC, 7D3, 7DF, 7DH, 7E4, 7E8, 7EA, 7F7, 7G2, 7G6, 7GC, 7H1, 7H7, 7HB, 7IA, 7II, 809, 80F, 812, 81A, 821, 82D, 838, 83C, 83I, 845, 847, 85G, 85I, 869, 86H, 872, 87G, 881, 889, 892, 898, 8A1, 8A5, 8AB, 8BC, 8C3, 8C5, 8D2, 8E9, 8ED, 8EF, 8F8, 8FE, 8FI, 8GB, 8GH, 8H6, 8HA, 8HI, 902, 904, 908, 90A, 913, 92C, 92E, 931, 937, 93D, 93H, 944, 946, 94I, 953, 95F, 95H, 968, 96A, 977, 979, 986, 98C, 99D, 9AA, 9AI, 9B3, 9B5, 9B9, 9BB, 9CE, 9D3, 9DF, 9E2, 9EC, 9EE, 9EI, 9F5, 9F7, 9FD, 9G4, 9G6, 9GI, 9H9, 9HB, 9I2, 9IG, A03, A07, A0D, A12, A18, A1E, A2B, A34, A36, A3A, A45, A4B, A4F, A54, A5E, A63, A69, A6F, A7I, A85, A87, A8H, A9C, A9G, AA3, AB2, AB4, ABE, AC9, ACD, ACF, AD6, AE1, AE5, AED, AFC, AFG, AG3, AG5, AG9, AGF, AGH, AHA, AHE, AIF, B0I, B1B, B1D, B1H, B24, B2A, B2C, B2I, B42, B44, B4A, B57, B5D, B66, B68, B6E, B77, B84, B86, B8A, B8G, B9B, B9F, B9H, BAG, BC2, BCC, BCI, BD1, BDB, BDD, BE4, BE6, BEG, BF3, BF5, BFF, BFH, BG8, BGE, BH3, BIE, C05, C07, C0H, C16, C1C, C23, C32, C38, C41, C4D, C4F, C5E, C61, C65, C6B, C6H, C7G, C7I, C89, C94, C9A, C9E, C9G, CA1, CB6, CB8, CC1, CC7, CD4, CDC, CDI, CE5, CF4, CG1, CG3, CG7, CGD, CGF, CH2, CH8, CHI, CI5, CIH, D0A, D19, D1B, D1H, D22, D31, D39, D4E, D4I, D51, D55, D5B, D5D, D66, D6A, D75, D8G, D97, D9D, DA6, DB1, DB7, DBH, DCA, DCC, DCG, DD3, DDB, DDH, DE8, DEA, DEE, DF9, DFF, DG2, DG6, DGC, DGE, DH5, DH7, DI4, DIG, E05, E14, E18, E1E, E27, E29, E2F, E32, E38, E4H, E54, E5I, E63, E6B, E72, E7A, E83, E92, E96, E98, E9C, EAH, EBA, EBG, EBI, ECF, ED2, ED8, EE3, EED, EF8, EFC, EH4, EHA, EHG, EI3, EIB, EIH, F02, F04, F0G, F13, F17, F19, F1F, F2I, F35, F37, F3B, F4A, F4C, F4G, F59, F5B, F5H, F62, F79, F7F, F82, F86, F8E, F95, FAI, FBF, FBH, FC4, FC8, FCA, FCE, FCG, FD7, FE2, FE8, FEC, FF1, FFB, FFH, FGI, FH3, FH5, FHB, G03, G07, G0F, G16, G1C, G1I, G27, G2D, G36, G3A, G3G, G3I, G45, G49, G4F, G4H, G58, G5A, G67, G6D, G7E, G7I, G8B, G96, GAF, GB2, GC3, GC7, GD6, GDE, GE1, GE5, GEB, GF6, GFC, GFI, GG9, GGB, GH2, GHE, GI3, GID, GIF, H06, H0E, H17, H1H, H33, H35, H39, H3H, H44, H48, H4G, H5F, H66, H6C, H6I, H71, H77, H7H, H8A, H8C, H93, H99, H9F, HA2, HAA, HAG, HB7, HBD, HBF, HC2, HC8, HCE, HD5, HDD, HEI, HF5, HG8, HGA, HH9, HHD, HI2, HIC, I14, I1C, I2B, I2F, I2H, I38, I3E, I3G, I43, I47, I56, I5E, I67, I76, I83, I89, I8B, I94, I9A, IA1, IA3, IAD, IAF, IB2, IBC, IC7, ICB, IDG, IDI, IEF, IEH, IF8, IFA, IG1, IH2, IH6, IH8, IHC, II1, IIH, 1004, 100A, 100C, 1015, 1022, 102A, 102E, 1031, 104C, 104E, 1055, 1057, 105D, 105H, 1064, 106A, 106I, 1075, 1079, 1082, 1088, 108G, 1099, 109D, 10A8, 10B1, 10BB, 10CG, 10D3, 10DF, 10E2, 10E4, 10F7, 10FF, 10GE, 10H5, 10HB, 10I6, 10IA, 10IC, 10II, 1109, 110H, 1114, 1118, 111E, 1136, 1141, 114B, 114D, 1156, 115G, 115I, 116F, 116H, 117G, 1192, 11A1, 11A7, 11B4, 11C3, 11C9, 11CB, 11DA, 11DE, 11E1, 11E3, 11ED, 11F2, 11FC, 11FI, 11G5, 11GD, 11GH, 11H4, 11H6, 11HG, 11HI, 11IB, 11IF, 1202, 1208, 120A, 1213, 1217, 1222, 1231, 1235, 123B, 124C, 124G, 1255, 125B, 125F, 1264, 1268, 1273, 1279, 127D, 1288, 1291, 1295, 1297, 12AI, 12B3, 12C8, 12CE, 12D1, 12DD, 12E6, 12F1, 12F7, 12FB, 12FD, 12FH, 12GG, 12H3, 12HF, 12I4, 12IA, 12IE, 1307, 1309, 1312, 132D, 133A, ...
2013, 19, 1B, 1H, 21, 23, 27, 2D, 2J, 31, 37, 3B, 3D, 3J, 43, 49, 4H, 51, 53, 57, 59, 5D, 67, 6B, 6H, 6J, 79, 7B, 7H, 83, 87, 8D, 8J, 91, 9B, 9D, 9H, 9J, AB, B3, B7, B9, BD, BJ, C1, CB, CH, D3, D9, DB, DH, E1, E3, ED, F7, FB, FD, FH, GB, GH, H7, H9, HD, HJ, I7, ID, IJ, J3, J9, JH, 101, 109, 10J, 111, 11B, 11D, 11J, 123, 129, 12H, 131, 133, 137, 13J, 147, 14B, 14J, 153, 159, 161, 163, 171, 177, 17H, 183, 189, 18B, 18H, 197, 19D, 19J, 1A1, 1A7, 1AD, 1AH, 1AJ, 1BB, 1C1, 1C3, 1C7, 1CD, 1CJ, 1D1, 1DD, 1DH, 1E3, 1EB, 1F1, 1F9, 1FJ, 1G7, 1GD, 1GJ, 1H3, 1HB, 1HH, 1I1, 1I9, 1ID, 1J7, 1JH, 209, 20B, 211, 213, 217, 219, 21J, 22D, 22H, 22J, 233, 23H, 241, 243, 247, 257, 25B, 25J, 269, 26H, 271, 277, 27D, 287, 28B, 28H, 293, 29B, 29H, 2A9, 2AD, 2AJ, 2B1, 2BB, 2BD, 2BJ, 2C9, 2CB, 2D1, 2D3, 2D9, 2E7, 2EB, 2ED, 2EH, 2F3, 2F9, 2FH, 2G3, 2G9, 2HB, 2HD, 2I3, 2IB, 2J1, 2J7, 2JD, 301, 30D, 30H, 313, 319, 31B, 31H, 329, 32J, 33H, 33J, 343, 349, 34B, 34H, 351, 353, 357, 35J, 361, 367, 381, 387, 38D, 391, 39J, 3A9, 3B3, 3B7, 3B9, 3BD, 3BJ, 3C7, 3CB, 3CD, 3CJ, 3DB, 3E1, 3E3, 3E7, 3E9, 3ED, 3EJ, 3FB, 3G3, 3GB, 3H3, 3H9, 3HD, 3HJ, 3I7, 3IB, 3IJ, 3J3, 3JH, 401, 407, 409, 40D, 40J, 411, 417, 41H, 42H, 433, 437, 439, 44D, 44H, 44J, 459, 461, 463, 46D, 471, 477, 47D, 47J, 48H, 493, 497, 499, 4A1, 4AB, 4B3, 4BB, 4C7, 4D1, 4D7, 4DB, 4DD, 4DH, 4DJ, 4E9, 4F1, 4F7, 4FD, 4GB, 4GD, 4H9, 4HB, 4ID, 4IJ, 4J7, 4JD, 4JH, 4JJ, 503, 50B, 50H, 517, 519, 51J, 52D, 533, 539, 541, 543, 547, 549, 54J, 55B, 55D, 569, 56B, 56H, 571, 573, 57D, 581, 58J, 5A3, 5A7, 5AD, 5B1, 5BH, 5BJ, 5C3, 5CB, 5D7, 5D9, 5DD, 5E1, 5E7, 5ED, 5EH, 5F9, 5FB, 5GD, 5GJ, 5H1, 5H7, 5HB, 5HH, 5IB, 5IH, 5J1, 5J3, 5J9, 5JD, 5JJ, 60B, 60H, 613, 61H, 621, 627, 62J, 637, 63D, 63H, 653, 661, 66B, 66J, 673, 679, 67B, 67H, 68J, 69B, 69D, 6A9, 6AH, 6B1, 6BD, 6C7, 6CH, 6CJ, 6D3, 6DB, 6DH, 6E3, 6E7, 6E9, 6ED, 6EJ, 6F7, 6FB, 6FD, 6FJ, 6G9, 6GB, 6H1, 6H9, 6HD, 6I7, 6IH, 6J9, 6JB, 6JH, 701, 703, 70J, 71D, 71H, 723, 72B, 72H, 731, 73J, 747, 74H, 753, 759, 75H, 767, 76J, 77D, 77H, 783, 789, 78B, 79J, 7A1, 7AB, 7AJ, 7B3, 7BH, 7C1, 7C9, 7D1, 7D7, 7DJ, 7E3, 7E9, 7F9, 7FJ, 7G1, 7GH, 7I3, 7I7, 7I9, 7J1, 7J7, 7JB, 803, 809, 80H, 811, 819, 82B, 82D, 82H, 82J, 83B, 84J, 851, 857, 85D, 85J, 863, 869, 86B, 873, 877, 87J, 881, 88B, 88D, 899, 89B, 8A7, 8AD, 8BD, 8C9, 8CH, 8D1, 8D3, 8D7, 8D9, 8EB, 8EJ, 8FB, 8FH, 8G7, 8G9, 8GD, 8GJ, 8H1, 8H7, 8HH, 8HJ, 8IB, 8J1, 8J3, 8JD, 907, 90D, 90H, 913, 91B, 91H, 923, 92J, 93B, 93D, 93H, 94B, 94H, 951, 959, 95J, 967, 96D, 96J, 981, 987, 989, 98J, 99D, 99H, 9A3, 9B1, 9B3, 9BD, 9C7, 9CB, 9CD, 9D3, 9DH, 9E1, 9E9, 9F7, 9FB, 9FH, 9FJ, 9G3, 9G9, 9GB, 9H3, 9H7, 9I7, 9J9, A01, A03, A07, A0D, A0J, A11, A17, A29, A2B, A2H, A3D, A3J, A4B, A4D, A4J, A5B, A67, A69, A6D, A6J, A7D, A7H, A7J, A8H, AA1, AAB, AAH, AAJ, AB9, ABB, AC1, AC3, ACD, ACJ, AD1, ADB, ADD, AE3, AE9, AEH, AG7, AGH, AGJ, AH9, AHH, AI3, AID, AJB, AJH, B09, B11, B13, B21, B27, B2B, B2H, B33, B41, B43, B4D, B57, B5D, B5H, B5J, B63, B77, B79, B81, B87, B93, B9B, B9H, BA3, BB1, BBH, BBJ, BC3, BC9, BCB, BCH, BD3, BDD, BDJ, BEB, BF3, BG1, BG3, BG9, BGD, BHB, BHJ, BJ3, BJ7, BJ9, BJD, BJJ, C01, C0D, C0H, C1B, C31, C3B, C3H, C49, C53, C59, C5J, C6B, C6D, C6H, C73, C7B, C7H, C87, C89, C8D, C97, C9D, C9J, CA3, CA9, CAB, CB1, CB3, CBJ, CCB, CCJ, CDH, CE1, CE7, CEJ, CF1, CF7, CFD, CFJ, CH7, CHD, CI7, CIB, CIJ, CJ9, CJH, D09, D17, D1B, D1D, D1H, D31, D3D, D3J, D41, D4H, D53, D59, D63, D6D, D77, D7B, D91, D97, D9D, D9J, DA7, DAD, DAH, DAJ, DBB, DBH, DC1, DC3, DC9, DDB, DDH, DDJ, DE3, DF1, DF3, DF7, DFJ, DG1, DG7, DGB, DHH, DI3, DI9, DID, DJ1, DJB, E13, E1J, E21, E27, E2B, E2D, E2H, E2J, E39, E43, E49, E4D, E51, E5B, E5H, E6H, E71, E73, E79, E8J, E93, E9B, EA1, EA7, EAD, EB1, EB7, EBJ, EC3, EC9, ECB, ECH, ED1, ED7, ED9, EDJ, EE1, EEH, EF3, EG3, EG7, EGJ, EHD, EJ1, EJ7, F07, F0B, F19, F1H, F23, F27, F2D, F37, F3D, F3J, F49, F4B, F51, F5D, F61, F6B, F6D, F73, F7B, F83, F8D, F9H, F9J, FA3, FAB, FAH, FB1, FB9, FC7, FCH, FD3, FD9, FDB, FDH, FE7, FEJ, FF1, FFB, FFH, FG3, FG9, FGH, FH3, FHD, FHJ, FI1, FI7, FID, FIJ, FJ9, FJH, G11, G17, G29, G2B, G39, G3D, G41, G4B, G61, G69, G77, G7B, G7D, G83, G89, G8B, G8H, G91, G9J, GA7, GAJ, GBH, GCD, GCJ, GD1, GDD, GDJ, GE9, GEB, GF1, GF3, GF9, GFJ, GGD, GGH, GI1, GI3, GIJ, GJ1, GJB, GJD, H03, H13, H17, H19, H1D, H21, H2H, H33, H39, H3B, H43, H4J, H57, H5B, H5H, H77, H79, H7J, H81, H87, H8B, H8H, H93, H9B, H9H, HA1, HAD, HAJ, HB7, HBJ, HC3, HCH, HD9, HDJ, HF3, HF9, HG1, HG7, HG9, HHB, HHJ, HIH, HJ7, HJD, I07, I0B, I0D, I0J, I19, I1H, I23, I27, I2D, I43, I4H, I57, I59, I61, I6B, I6D, I79, I7B, I89, I9D, IAB, IAH, IBD, ICB, ICH, ICJ, IDH, IE1, IE7, IE9, IEJ, IF7, IFH, IG3, IG9, IGH, IH1, IH7, IH9, IHJ, II1, IID, IIH, IJ3, IJ9, IJB, J03, J07, J11, J1J, J23, J29, J39, J3D, J41, J47, J4B, J4J, J53, J5H, J63, J67, J71, J7D, J7H, J7J, J99, J9D, JAH, JB3, JB9, JC1, JCD, JD7, JDD, JDH, JDJ, JE3, JF1, JF7, JFJ, JG7, JGD, JGH, JH9, JHB, JI3, JJD, 1009, ...
2112, 18, 1A, 1G, 1K, 21, 25, 2B, 2H, 2J, 34, 38, 3A, 3G, 3K, 45, 4D, 4H, 4J, 52, 54, 58, 61, 65, 6B, 6D, 72, 74, 7A, 7G, 7K, 85, 8B, 8D, 92, 94, 98, 9A, A1, AD, AH, AJ, B2, B8, BA, BK, C5, CB, CH, CJ, D4, D8, DA, DK, ED, EH, EJ, F2, FG, G1, GB, GD, GH, H2, HA, HG, I1, I5, IB, IJ, J2, JA, JK, K1, KB, KD, KJ, 102, 108, 10G, 10K, 111, 115, 11H, 124, 128, 12G, 12K, 135, 13H, 13J, 14G, 151, 15B, 15H, 162, 164, 16A, 16K, 175, 17B, 17D, 17J, 184, 188, 18A, 191, 19B, 19D, 19H, 1A2, 1A8, 1AA, 1B1, 1B5, 1BB, 1BJ, 1C8, 1CG, 1D5, 1DD, 1DJ, 1E4, 1E8, 1EG, 1F1, 1F5, 1FD, 1FH, 1GA, 1GK, 1HB, 1HD, 1I2, 1I4, 1I8, 1IA, 1IK, 1JD, 1JH, 1JJ, 1K2, 1KG, 1KK, 201, 205, 214, 218, 21G, 225, 22D, 22H, 232, 238, 241, 245, 24B, 24H, 254, 25A, 261, 265, 26B, 26D, 272, 274, 27A, 27K, 281, 28B, 28D, 28J, 29G, 29K, 2A1, 2A5, 2AB, 2AH, 2B4, 2BA, 2BG, 2CH, 2CJ, 2D8, 2DG, 2E5, 2EB, 2EH, 2F4, 2FG, 2FK, 2G5, 2GB, 2GD, 2GJ, 2HA, 2HK, 2IH, 2IJ, 2J2, 2J8, 2JA, 2JG, 2JK, 2K1, 2K5, 2KH, 2KJ, 304, 31H, 322, 328, 32G, 33D, 342, 34G, 34K, 351, 355, 35B, 35J, 362, 364, 36A, 371, 37B, 37D, 37H, 37J, 382, 388, 38K, 39B, 39J, 3AA, 3AG, 3AK, 3B5, 3BD, 3BH, 3C4, 3C8, 3D1, 3D5, 3DB, 3DD, 3DH, 3E2, 3E4, 3EA, 3EK, 3FJ, 3G4, 3G8, 3GA, 3HD, 3HH, 3HJ, 3I8, 3IK, 3J1, 3JB, 3JJ, 3K4, 3KA, 3KG, 40D, 40J, 412, 414, 41G, 425, 42H, 434, 43K, 44D, 44J, 452, 454, 458, 45A, 45K, 46B, 46H, 472, 47K, 481, 48H, 48J, 49K, 4A5, 4AD, 4AJ, 4B2, 4B4, 4B8, 4BG, 4C1, 4CB, 4CD, 4D2, 4DG, 4E5, 4EB, 4F2, 4F4, 4F8, 4FA, 4FK, 4GB, 4GD, 4H8, 4HA, 4HG, 4HK, 4I1, 4IB, 4IJ, 4JG, 4KJ, 502, 508, 50G, 51B, 51D, 51H, 524, 52K, 531, 535, 53D, 53J, 544, 548, 54K, 551, 562, 568, 56A, 56G, 56K, 575, 57J, 584, 588, 58A, 58G, 58K, 595, 59H, 5A2, 5A8, 5B1, 5B5, 5BB, 5C2, 5CA, 5CG, 5CK, 5E4, 5F1, 5FB, 5FJ, 5G2, 5G8, 5GA, 5GG, 5HH, 5I8, 5IA, 5J5, 5JD, 5JH, 5K8, 601, 60B, 60D, 60H, 614, 61A, 61G, 61K, 621, 625, 62B, 62J, 632, 634, 63A, 63K, 641, 64B, 64J, 652, 65G, 665, 66H, 66J, 674, 678, 67A, 685, 68J, 692, 698, 69G, 6A1, 6A5, 6B2, 6BA, 6BK, 6C5, 6CB, 6CJ, 6D8, 6DK, 6ED, 6EH, 6F2, 6F8, 6FA, 6GH, 6GJ, 6H8, 6HG, 6HK, 6ID, 6IH, 6J4, 6JG, 6K1, 6KD, 6KH, 702, 711, 71B, 71D, 728, 73D, 73H, 73J, 74A, 74G, 74K, 75B, 75H, 764, 768, 76G, 77H, 77J, 782, 784, 78G, 7A2, 7A4, 7AA, 7AG, 7B1, 7B5, 7BB, 7BD, 7C4, 7C8, 7CK, 7D1, 7DB, 7DD, 7E8, 7EA, 7F5, 7FB, 7GA, 7H5, 7HD, 7HH, 7HJ, 7I2, 7I4, 7J5, 7JD, 7K4, 7KA, 7KK, 801, 805, 80B, 80D, 80J, 818, 81A, 821, 82B, 82D, 832, 83G, 841, 845, 84B, 84J, 854, 85A, 865, 86H, 86J, 872, 87G, 881, 885, 88D, 892, 89A, 89G, 8A1, 8B2, 8B8, 8BA, 8BK, 8CD, 8CH, 8D2, 8DK, 8E1, 8EB, 8F4, 8F8, 8FA, 8FK, 8GD, 8GH, 8H4, 8I1, 8I5, 8IB, 8ID, 8IH, 8J2, 8J4, 8JG, 8JK, 8KJ, 90K, 91B, 91D, 91H, 922, 928, 92A, 92G, 93H, 93J, 944, 94K, 955, 95H, 95J, 964, 96G, 97B, 97D, 97H, 982, 98G, 98K, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9C8, 9CA, 9CK, 9D1, 9DB, 9DH, 9DJ, 9E8, 9EA, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9I2, 9IA, 9IG, 9J5, 9K2, 9K8, 9KK, A0B, A0D, A1A, A1G, A1K, A25, A2B, A38, A3A, A3K, A4D, A4J, A52, A54, A58, A6B, A6D, A74, A7A, A85, A8D, A8J, A94, AA1, AAH, AAJ, AB2, AB8, ABA, ABG, AC1, ACB, ACH, AD8, ADK, AEH, AEJ, AF4, AF8, AG5, AGD, AHG, AHK, AI1, AI5, AIB, AID, AJ4, AJ8, AK1, B0A, B0K, B15, B1H, B2A, B2G, B35, B3H, B3J, B42, B48, B4G, B51, B5B, B5D, B5H, B6A, B6G, B71, B75, B7B, B7D, B82, B84, B8K, B9B, B9J, BAG, BAK, BB5, BBH, BBJ, BC4, BCA, BCG, BE2, BE8, BF1, BF5, BFD, BG2, BGA, BH1, BHJ, BI2, BI4, BI8, BJB, BK2, BK8, BKA, C05, C0B, C0H, C1A, C1K, C2D, C2H, C45, C4B, C4H, C52, C5A, C5G, C5K, C61, C6D, C6J, C72, C74, C7A, C8B, C8H, C8J, C92, C9K, CA1, CA5, CAH, CAJ, CB4, CB8, CCD, CCJ, CD4, CD8, CDG, CE5, CFG, CGB, CGD, CGJ, CH2, CH4, CH8, CHA, CHK, CID, CIJ, CJ2, CJA, CJK, CK5, D04, D08, D0A, D0G, D24, D28, D2G, D35, D3B, D3H, D44, D4A, D51, D55, D5B, D5D, D5J, D62, D68, D6A, D6K, D71, D7H, D82, D91, D95, D9H, DAA, DBH, DC2, DD1, DD5, DE2, DEA, DEG, DEK, DF5, DFJ, DG4, DGA, DGK, DH1, DHB, DI2, DIA, DIK, DJ1, DJB, DJJ, DKA, DKK, E12, E14, E18, E1G, E21, E25, E2D, E3A, E3K, E45, E4B, E4D, E4J, E58, E5K, E61, E6B, E6H, E72, E78, E7G, E81, E8B, E8H, E8J, E94, E9A, E9G, EA5, EAD, EBG, EC1, ED2, ED4, EE1, EE5, EED, EF2, EGB, EGJ, EHG, EHK, EI1, EIB, EIH, EIJ, EJ4, EJ8, EK5, EKD, F04, F11, F1H, F22, F24, F2G, F31, F3B, F3D, F42, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA2, FA4, FA8, FAG, FBB, FBH, FC2, FC4, FCG, FDB, FDJ, FE2, FE8, FFH, FFJ, FG8, FGA, FGG, FGK, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G01, G0D, G12, G25, G2B, G32, G38, G3A, G4B, G4J, G5G, G65, G6B, G74, G78, G7A, G7G, G85, G8D, G8J, G92, G98, GAH, GBA, GBK, GC1, GCD, GD2, GD4, GDK, GE1, GEJ, GG1, GGJ, GH4, GHK, GIH, GJ2, GJ4, GK1, GK5, GKB, GKD, H02, H0A, H0K, H15, H1B, H1J, H22, H28, H2A, H2K, H31, H3D, H3H, H42, H48, H4A, H51, H55, H5J, H6G, H6K, H75, H84, H88, H8G, H91, H95, H9D, H9H, HAA, HAG, HAK, HBD, HC4, HC8, HCA, HDJ, HE2, HF5, HFB, HFH, HG8, HGK, HHD, HHJ, HI2, HI4, HI8, HJ5, HJB, HK2, HKA, HKG, HKK, I0B, I0D, I14, I2D, I38, ...
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3512, 16, 18, 1C, 1I, 1O, 1Q, 1W, 21, 23, 29, 2D, 2J, 2R, 2V, 2X, 32, 34, 38, 3M, 3Q, 3W, 3Y, 49, 4B, 4H, 4N, 4R, 4X, 54, 56, 5G, 5I, 5M, 5O, 61, 6D, 6H, 6J, 6N, 6T, 6V, 76, 7C, 7I, 7O, 7Q, 7W, 81, 83, 8D, 8R, 8V, 8X, 92, 9G, 9M, 9W, 9Y, A3, A9, AH, AN, AT, AX, B4, BC, BG, BO, BY, C1, CB, CD, CJ, CN, CT, D2, D6, D8, DC, DO, DW, E1, E9, ED, EJ, EV, EX, FG, FM, FW, G3, G9, GB, GH, GR, GX, H4, H6, HC, HI, HM, HO, I1, IB, ID, IH, IN, IT, IV, J8, JC, JI, JQ, K1, K9, KJ, KR, KX, L4, L8, LG, LM, LQ, LY, M3, MH, MR, N4, N6, NG, NI, NM, NO, NY, OD, OH, OJ, ON, P2, P6, P8, PC, PW, Q1, Q9, QJ, QR, QV, R2, R8, RM, RQ, RW, S3, SB, SH, ST, SX, T4, T6, TG, TI, TO, TY, U1, UB, UD, UJ, V2, V6, V8, VC, VI, VO, VW, W3, W9, WV, WX, X8, XG, XQ, XW, Y3, YB, YN, YR, YX, 104, 106, 10C, 10O, 10Y, 11H, 11J, 11N, 11T, 11V, 122, 126, 128, 12C, 12O, 12Q, 12W, 13V, 142, 148, 14G, 14Y, 159, 15N, 15R, 15T, 15X, 164, 16C, 16G, 16I, 16O, 171, 17B, 17D, 17H, 17J, 17N, 17T, 186, 18I, 18Q, 193, 199, 19D, 19J, 19R, 19V, 1A4, 1A8, 1AM, 1AQ, 1AW, 1AY, 1B3, 1B9, 1BB, 1BH, 1BR, 1CC, 1CI, 1CM, 1CO, 1DD, 1DH, 1DJ, 1DT, 1E6, 1E8, 1EI, 1EQ, 1EW, 1F3, 1F9, 1FR, 1FX, 1G2, 1G4, 1GG, 1GQ, 1H3, 1HB, 1HR, 1I6, 1IC, 1IG, 1II, 1IM, 1IO, 1IY, 1JB, 1JH, 1JN, 1K6, 1K8, 1KO, 1KQ, 1LD, 1LJ, 1LR, 1LX, 1M2, 1M4, 1M8, 1MG, 1MM, 1MW, 1MY, 1N9, 1NN, 1NX, 1O4, 1OG, 1OI, 1OM, 1OO, 1OY, 1PB, 1PD, 1PT, 1PV, 1Q2, 1Q6, 1Q8, 1QI, 1QQ, 1R9, 1RX, 1S2, 1S8, 1SG, 1SW, 1SY, 1T3, 1TB, 1TR, 1TT, 1TX, 1U6, 1UC, 1UI, 1UM, 1UY, 1V1, 1VN, 1VT, 1VV, 1W2, 1W6, 1WC, 1WQ, 1WW, 1X1, 1X3, 1X9, 1XD, 1XJ, 1XV, 1Y2, 1Y8, 1YM, 1YQ, 1YW, 209, 20H, 20N, 20R, 21I, 221, 22B, 22J, 22N, 22T, 22V, 232, 23O, 241, 243, 24J, 24R, 24V, 258, 25M, 25W, 25Y, 263, 26B, 26H, 26N, 26R, 26T, 26X, 274, 27C, 27G, 27I, 27O, 27Y, 281, 28B, 28J, 28N, 292, 29C, 29O, 29Q, 29W, 2A1, 2A3, 2AJ, 2AX, 2B2, 2B8, 2BG, 2BM, 2BQ, 2C9, 2CH, 2CR, 2CX, 2D4, 2DC, 2DM, 2DY, 2ED, 2EH, 2EN, 2ET, 2EV, 2FO, 2FQ, 2G1, 2G9, 2GD, 2GR, 2GV, 2H4, 2HG, 2HM, 2HY, 2I3, 2I9, 2IT, 2J4, 2J6, 2JM, 2KD, 2KH, 2KJ, 2KV, 2L2, 2L6, 2LI, 2LO, 2LW, 2M1, 2M9, 2MV, 2MX, 2N2, 2N4, 2NG, 2O9, 2OB, 2OH, 2ON, 2OT, 2OX, 2P4, 2P6, 2PI, 2PM, 2PY, 2Q1, 2QB, 2QD, 2QT, 2QV, 2RC, 2RI, 2S3, 2SJ, 2SR, 2SV, 2SX, 2T2, 2T4, 2TQ, 2TY, 2UB, 2UH, 2UR, 2UT, 2UX, 2V4, 2V6, 2VC, 2VM, 2VO, 2W1, 2WB, 2WD, 2WN, 2X2, 2X8, 2XC, 2XI, 2XQ, 2XW, 2Y3, 2YJ, 2YV, 2YX, 302, 30G, 30M, 30Q, 30Y, 319, 31H, 31N, 31T, 32G, 32M, 32O, 32Y, 33D, 33H, 33N, 346, 348, 34I, 34W, 351, 353, 35D, 35R, 35V, 364, 36M, 36Q, 36W, 36Y, 373, 379, 37B, 37N, 37R, 38C, 38Y, 39B, 39D, 39H, 39N, 39T, 39V, 3A2, 3AO, 3AQ, 3AW, 3BD, 3BJ, 3BV, 3BX, 3C4, 3CG, 3CW, 3CY, 3D3, 3D9, 3DN, 3DR, 3DT, 3EC, 3F1, 3FB, 3FH, 3FJ, 3FT, 3FV, 3G6, 3G8, 3GI, 3GO, 3GQ, 3H1, 3H3, 3HD, 3HJ, 3HR, 3IM, 3IW, 3IY, 3J9, 3JH, 3JN, 3JX, 3KG, 3KM, 3KY, 3LB, 3LD, 3LV, 3M2, 3M6, 3MC, 3MI, 3N1, 3N3, 3ND, 3NR, 3NX, 3O2, 3O4, 3O8, 3OW, 3OY, 3PB, 3PH, 3PX, 3Q6, 3QC, 3QI, 3R1, 3RH, 3RJ, 3RN, 3RT, 3RV, 3S2, 3S8, 3SI, 3SO, 3T1, 3TD, 3TV, 3TX, 3U4, 3U8, 3UQ, 3UY, 3VN, 3VR, 3VT, 3VX, 3W4, 3W6, 3WI, 3WM, 3X1, 3XV, 3Y6, 3YC, 3YO, 403, 409, 40J, 40V, 40X, 412, 418, 41G, 41M, 41W, 41Y, 423, 42H, 42N, 42T, 42X, 434, 436, 43G, 43I, 43Y, 44B, 44J, 452, 456, 45C, 45O, 45Q, 45W, 463, 469, 472, 478, 47M, 47Q, 47Y, 489, 48H, 48T, 49C, 49G, 49I, 49M, 4AB, 4AN, 4AT, 4AV, 4BC, 4BI, 4BO, 4C3, 4CD, 4CR, 4CV, 4DQ, 4DW, 4E3, 4E9, 4EH, 4EN, 4ER, 4ET, 4F6, 4FC, 4FG, 4FI, 4FO, 4GB, 4GH, 4GJ, 4GN, 4H6, 4H8, 4HC, 4HO, 4HQ, 4HW, 4I1, 4IR, 4IX, 4J4, 4J8, 4JG, 4JQ, 4KN, 4L4, 4L6, 4LC, 4LG, 4LI, 4LM, 4LO, 4LY, 4MD, 4MJ, 4MN, 4MV, 4N6, 4NC, 4NW, 4O1, 4O3, 4O9, 4P4, 4P8, 4PG, 4PQ, 4PW, 4Q3, 4QB, 4QH, 4QT, 4QX, 4R4, 4R6, 4RC, 4RG, 4RM, 4RO, 4RY, 4S1, 4SH, 4SN, 4T8, 4TC, 4TO, 4U3, 4UV, 4V2, 4VM, 4VQ, 4W9, 4WH, 4WN, 4WR, 4WX, 4XC, 4XI, 4XO, 4XY, 4Y1, 4YB, 4YN, 4YV, 506, 508, 50I, 50Q, 513, 51D, 522, 524, 528, 52G, 52M, 52Q, 52Y, 53H, 53R, 53X, 544, 546, 54C, 54M, 54Y, 551, 55B, 55H, 55N, 55T, 562, 568, 56I, 56O, 56Q, 56W, 573, 579, 57J, 57R, 58G, 58M, 599, 59B, 59T, 59X, 5A6, 5AG, 5BB, 5BJ, 5C2, 5C6, 5C8, 5CI, 5CO, 5CQ, 5CW, 5D1, 5DJ, 5DR, 5E4, 5EM, 5F3, 5F9, 5FB, 5FN, 5FT, 5G4, 5G6, 5GG, 5GI, 5GO, 5GY, 5HD, 5HH, 5I6, 5I8, 5IO, 5IQ, 5J1, 5J3, 5JD, 5JX, 5K2, 5K4, 5K8, 5KG, 5KW, 5L3, 5L9, 5LB, 5LN, 5M4, 5MC, 5MG, 5MM, 5NH, 5NJ, 5NT, 5NV, 5O2, 5O6, 5OC, 5OI, 5OQ, 5OW, 5P1, 5PD, 5PJ, 5PR, 5Q4, 5Q8, 5QM, 5QY, 5R9, 5RX, 5S4, 5SG, 5SM, 5SO, 5TB, 5TJ, 5U2, 5UC, 5UI, 5UW, 5V1, 5V3, 5V9, 5VJ, 5VR, 5VX, 5W2, 5W8, 5X3, 5XH, 5XR, 5XT, 5Y6, 5YG, 5YI, 5YY, 601, 60J, 618, 61Q, 61W, 62D, 62V, 632, 634, 63M, 63Q, 63W, 63Y, 649, 64H, 64R, 64X, 654, 65C, 65G, 65M, 65O, 65Y, 661, 66D, 66H, 66N, 66T, 66V, 678, 67C, 67Q, 689, 68D, 68J, 694, 698, 69G, 69M, 69Q, 69Y, 6A3, 6AH, 6AN, 6AR, 6B6, 6BI, 6BM, 6BO, 6CJ, 6CN, 6DC, 6DI, 6DO, 6E1, 6ED, 6ER, 6EX, 6F2, 6F4, 6F8, 6FQ, 6FW, 6G9, 6GH, 6GN, 6GR, 6H4, 6H6, 6HI, 6ID, 6IT, 6IV, 6J2, 6JO, ...
3611, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 101, 105, 107, 10B, 10N, 10P, 10V, 11T, 11Z, 125, 12D, 12V, 135, 13J, 13N, 13P, 13T, 13Z, 147, 14B, 14D, 14J, 14V, 155, 157, 15B, 15D, 15H, 15N, 15Z, 16B, 16J, 16V, 171, 175, 17B, 17J, 17N, 17V, 17Z, 18D, 18H, 18N, 18P, 18T, 18Z, 191, 197, 19H, 1A1, 1A7, 1AB, 1AD, 1B1, 1B5, 1B7, 1BH, 1BT, 1BV, 1C5, 1CD, 1CJ, 1CP, 1CV, 1DD, 1DJ, 1DN, 1DP, 1E1, 1EB, 1EN, 1EV, 1FB, 1FP, 1FV, 1FZ, 1G1, 1G5, 1G7, 1GH, 1GT, 1GZ, 1H5, 1HN, 1HP, 1I5, 1I7, 1IT, 1IZ, 1J7, 1JD, 1JH, 1JJ, 1JN, 1JV, 1K1, 1KB, 1KD, 1KN, 1L1, 1LB, 1LH, 1LT, 1LV, 1LZ, 1M1, 1MB, 1MN, 1MP, 1N5, 1N7, 1ND, 1NH, 1NJ, 1NT, 1O1, 1OJ, 1P7, 1PB, 1PH, 1PP, 1Q5, 1Q7, 1QB, 1QJ, 1QZ, 1R1, 1R5, 1RD, 1RJ, 1RP, 1RT, 1S5, 1S7, 1ST, 1SZ, 1T1, 1T7, 1TB, 1TH, 1TV, 1U1, 1U5, 1U7, 1UD, 1UH, 1UN, 1UZ, 1V5, 1VB, 1VP, 1VT, 1VZ, 1WB, 1WJ, 1WP, 1WT, 1XJ, 1Y1, 1YB, 1YJ, 1YN, 1YT, 1YV, 1Z1, 1ZN, 1ZZ, 201, 20H, 20P, 20T, 215, 21J, 21T, 21V, 21Z, 227, 22D, 22J, 22N, 22P, 22T, 22Z, 237, 23B, 23D, 23J, 23T, 23V, 245, 24D, 24H, 24V, 255, 25H, 25J, 25P, 25T, 25V, 26B, 26P, 26T, 26Z, 277, 27D, 27H, 27Z, 287, 28H, 28N, 28T, 291, 29B, 29N, 2A1, 2A5, 2AB, 2AH, 2AJ, 2BB, 2BD, 2BN, 2BV, 2BZ, 2CD, 2CH, 2CP, 2D1, 2D7, 2DJ, 2DN, 2DT, 2ED, 2EN, 2EP, 2F5, 2FV, 2FZ, 2G1, 2GD, 2GJ, 2GN, 2GZ, 2H5, 2HD, 2HH, 2HP, 2IB, 2ID, 2IH, 2IJ, 2IV, 2JN, 2JP, 2JV, 2K1, 2K7, 2KB, 2KH, 2KJ, 2KV, 2KZ, 2LB, 2LD, 2LN, 2LP, 2M5, 2M7, 2MN, 2MT, 2ND, 2NT, 2O1, 2O5, 2O7, 2OB, 2OD, 2OZ, 2P7, 2PJ, 2PP, 2PZ, 2Q1, 2Q5, 2QB, 2QD, 2QJ, 2QT, 2QV, 2R7, 2RH, 2RJ, 2RT, 2S7, 2SD, 2SH, 2SN, 2SV, 2T1, 2T7, 2TN, 2TZ, 2U1, 2U5, 2UJ, 2UP, 2UT, 2V1, 2VB, 2VJ, 2VP, 2VV, 2WH, 2WN, 2WP, 2WZ, 2XD, 2XH, 2XN, 2Y5, 2Y7, 2YH, 2YV, 2YZ, 2Z1, 2ZB, 2ZP, 2ZT, 301, 30J, 30N, 30T, 30V, 30Z, 315, 317, 31J, 31N, 327, 32T, 335, 337, 33B, 33H, 33N, 33P, 33V, 34H, 34J, 34P, 355, 35B, 35N, 35P, 35V, 367, 36N, 36P, 36T, 36Z, 37D, 37H, 37J, 381, 38P, 38Z, 395, 397, 39H, 39J, 39T, 39V, 3A5, 3AB, 3AD, 3AN, 3AP, 3AZ, 3B5, 3BD, 3C7, 3CH, 3CJ, 3CT, 3D1, 3D7, 3DH, 3DZ, 3E5, 3EH, 3ET, 3EV, 3FD, 3FJ, 3FN, 3FT, 3FZ, 3GH, 3GJ, 3GT, 3H7, 3HD, 3HH, 3HJ, 3HN, 3IB, 3ID, 3IP, 3IV, 3JB, 3JJ, 3JP, 3JV, 3KD, 3KT, 3KV, 3KZ, 3L5, 3L7, 3LD, 3LJ, 3LT, 3LZ, 3MB, 3MN, 3N5, 3N7, 3ND, 3NH, 3NZ, 3O7, 3OV, 3OZ, 3P1, 3P5, 3PB, 3PD, 3PP, 3PT, 3Q7, 3R1, 3RB, 3RH, 3RT, 3S7, 3SD, 3SN, 3SZ, 3T1, 3T5, 3TB, 3TJ, 3TP, 3TZ, 3U1, 3U5, 3UJ, 3UP, 3UV, 3UZ, 3V5, 3V7, 3VH, 3VJ, 3VZ, 3WB, 3WJ, 3X1, 3X5, 3XB, 3XN, 3XP, 3XV, 3Y1, 3Y7, 3YZ, 3Z5, 3ZJ, 3ZN, 3ZV, 405, 40D, 40P, 417, 41B, 41D, 41H, 425, 42H, 42N, 42P, 435, 43B, 43H, 43V, 445, 44J, 44N, 45H, 45N, 45T, 45Z, 467, 46D, 46H, 46J, 46V, 471, 475, 477, 47D, 47Z, 485, 487, 48B, 48T, 48V, 48Z, 49B, 49D, 49J, 49N, 4AD, 4AJ, 4AP, 4AT, 4B1, 4BB, 4C7, 4CN, 4CP, 4CV, 4CZ, 4D1, 4D5, 4D7, 4DH, 4DV, 4E1, 4E5, 4ED, 4EN, 4ET, 4FD, 4FH, 4FJ, 4FP, 4GJ, 4GN, 4GV, 4H5, 4HB, 4HH, 4HP, 4HV, 4I7, 4IB, 4IH, 4IJ, 4IP, 4IT, 4IZ, 4J1, 4JB, 4JD, 4JT, 4JZ, 4KJ, 4KN, 4KZ, 4LD, 4M5, 4MB, 4MV, 4MZ, 4NH, 4NP, 4NV, 4NZ, 4O5, 4OJ, 4OP, 4OV, 4P5, 4P7, 4PH, 4PT, 4Q1, 4QB, 4QD, 4QN, 4QV, 4R7, 4RH, 4S5, 4S7, 4SB, 4SJ, 4SP, 4ST, 4T1, 4TJ, 4TT, 4TZ, 4U5, 4U7, 4UD, 4UN, 4UZ, 4V1, 4VB, 4VH, 4VN, 4VT, 4W1, 4W7, 4WH, 4WN, 4WP, 4WV, 4X1, 4X7, 4XH, 4XP, 4YD, 4YJ, 4Z5, 4Z7, 4ZP, 4ZT, 501, 50B, 515, 51D, 51V, 51Z, 521, 52B, 52H, 52J, 52P, 52T, 53B, 53J, 53V, 54D, 54T, 54Z, 551, 55D, 55J, 55T, 55V, 565, 567, 56D, 56N, 571, 575, 57T, 57V, 58B, 58D, 58N, 58P, 58Z, 59J, 59N, 59P, 59T, 5A1, 5AH, 5AN, 5AT, 5AV, 5B7, 5BN, 5BV, 5BZ, 5C5, 5CZ, 5D1, 5DB, 5DD, 5DJ, 5DN, 5DT, 5DZ, 5E7, 5ED, 5EH, 5ET, 5EZ, 5F7, 5FJ, 5FN, 5G1, 5GD, 5GN, 5HB, 5HH, 5HT, 5HZ, 5I1, 5IN, 5IV, 5JD, 5JN, 5JT, 5K7, 5KB, 5KD, 5KJ, 5KT, 5L1, 5L7, 5LB, 5LH, 5MB, 5MP, 5MZ, 5N1, 5ND, 5NN, 5NP, 5O5, 5O7, 5OP, 5PD, 5PV, 5Q1, 5QH, 5QZ, 5R5, 5R7, 5RP, 5RT, 5RZ, 5S1, 5SB, 5SJ, 5ST, 5SZ, 5T5, 5TD, 5TH, 5TN, 5TP, 5TZ, 5U1, 5UD, 5UH, 5UN, 5UT, 5UV, 5V7, 5VB, 5VP, 5W7, 5WB, 5WH, 5X1, 5X5, 5XD, 5XJ, 5XN, 5XV, 5XZ, 5YD, 5YJ, 5YN, 5Z1, 5ZD, 5ZH, 5ZJ, 60D, 60H, 615, 61B, 61H, 61T, 625, 62J, 62P, 62T, 62V, 62Z, 63H, 63N, 63Z, 647, 64D, 64H, 64T, 64V, 657, 661, 66H, 66J, 66P, 67B, ...

The OEIS sequences for the base b representations of the prime numbers (of course, also includes the primes ≤ b) are: https://oeis.org/A004676 (b = 2), https://oeis.org/A001363 (b = 3), https://oeis.org/A004678 (b = 4), https://oeis.org/A004679 (b = 5), https://oeis.org/A004680 (b = 6), https://oeis.org/A004681 (b = 7), https://oeis.org/A004682 (b = 8), https://oeis.org/A004683 (b = 9), https://oeis.org/A000040 (b = 10), there are no OEIS sequences for b > 10 since OEIS disallows the alpha digits and only allows decimal characters (i.e. the 10 Arabic numerals (https://en.wikipedia.org/wiki/Arabic_numerals, https://mathworld.wolfram.com/ArabicNumeral.html)), see https://oeis.org/wiki/Disallowed_sequences#Sequences_of_rational_integers_with_digits_other_than_0_to_9, but there is a short OEIS sequence https://oeis.org/A004684 for b = 11, which stops exactly before the first term with a nondecimal character (i.e. the prime 43, which is written "3A" in base b = 11) and stops with the prime 41, which is written "38" in base b = 11.

Some bases 2 ≤ b ≤ 36 are notable: (fortunately, all of these bases except b = 32 and b = 36 are solved, except the primality proving for the probable prime 5762668 in base b = 11 and the probable primes C523755C, 8032017111, 95197420, A3592197A in base b = 13 and the probable primes DB32234, 472785DD, 3116137AF in base b = 16 and the probable prime I024608D in base b = 30)

Prime numbers (https://en.wikipedia.org/wiki/Prime_number, https://t5k.org/glossary/xpage/Prime.html, https://www.rieselprime.de/ziki/Prime, https://mathworld.wolfram.com/PrimeNumber.html, https://www.numbersaplenty.com/set/prime_number/, http://www.numericana.com/answer/primes.htm#definition, http://irvinemclean.com/maths/pfaq2.htm, https://oeis.org/A000040, https://t5k.org/lists/small/1000.txt, https://t5k.org/lists/small/10000.txt, https://t5k.org/lists/small/100000.txt, https://t5k.org/lists/small/millions/) are central in number theory (https://en.wikipedia.org/wiki/Number_theory, https://www.rieselprime.de/ziki/Number_theory, https://mathworld.wolfram.com/NumberTheory.html) because of the fundamental theorem of arithmetic (https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic, https://t5k.org/glossary/xpage/FundamentalTheorem.html, https://mathworld.wolfram.com/FundamentalTheoremofArithmetic.html, http://www.numericana.com/answer/primes.htm#fta, http://irvinemclean.com/maths/pfaq1.htm): every natural number (https://en.wikipedia.org/wiki/Natural_number, https://www.rieselprime.de/ziki/Natural_number, https://mathworld.wolfram.com/NaturalNumber.html) greater than (https://en.wikipedia.org/wiki/Greater_than, https://mathworld.wolfram.com/Greater.html) 1 is either a prime itself or can be factorized (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) as a product (https://en.wikipedia.org/wiki/Product_(mathematics), https://mathworld.wolfram.com/Product.html) of primes that is unique up to (https://en.wikipedia.org/wiki/Up_to) their order (sociology (https://en.wikipedia.org/wiki/Sociology) is applied psychology, psychology (https://en.wikipedia.org/wiki/Psychology) is applied biology, biology (https://en.wikipedia.org/wiki/Biology) is applied chemistry, chemistry (https://en.wikipedia.org/wiki/Chemistry) is applied physics, physics (https://en.wikipedia.org/wiki/Physics) is applied mathematics, the basics of mathematics (https://en.wikipedia.org/wiki/Mathematics, https://www.rieselprime.de/ziki/Mathematics, https://mathworld.wolfram.com/Mathematics.html) is the numbers, the basics of the numbers (https://en.wikipedia.org/wiki/Number, https://www.rieselprime.de/ziki/Number, https://mathworld.wolfram.com/Number.html) is the natural numbers, the researching of the natural numbers (https://en.wikipedia.org/wiki/Natural_number, https://www.rieselprime.de/ziki/Natural_number, https://mathworld.wolfram.com/NaturalNumber.html) is number theory (https://en.wikipedia.org/wiki/Number_theory, https://www.rieselprime.de/ziki/Number_theory, https://mathworld.wolfram.com/NumberTheory.html)). Also, for a completely multiplicative function (https://en.wikipedia.org/wiki/Completely_multiplicative_function, https://t5k.org/glossary/xpage/CompletelyMultiplicative.html, https://mathworld.wolfram.com/CompletelyMultiplicativeFunction.html, http://www.numericana.com/answer/numbers.htm#totally) f(x) (i.e. an arithmetic function (https://en.wikipedia.org/wiki/Arithmetic_function, https://mathworld.wolfram.com/ArithmeticFunction.html) (i.e. a function (https://en.wikipedia.org/wiki/Function_(mathematics), https://mathworld.wolfram.com/Function.html) whose domain (https://en.wikipedia.org/wiki/Domain_of_a_function, https://mathworld.wolfram.com/Domain.html) is the natural numbers (https://en.wikipedia.org/wiki/Natural_number, https://www.rieselprime.de/ziki/Natural_number, https://mathworld.wolfram.com/NaturalNumber.html)), such that f(1) = 1 and f(x×y) = f(xf(y) holds for all positive integers x and y), all f(n) are completely determined by f(p) with prime p (i.e. a completely multiplicative function is completely determined by its values at the prime numbers). Also many functions in number theory are highly related to prime numbers, such as Liouville function (https://en.wikipedia.org/wiki/Liouville_function, https://mathworld.wolfram.com/LiouvilleFunction.html, https://oeis.org/A008836), Möbius function (https://en.wikipedia.org/wiki/M%C3%B6bius_function, https://mathworld.wolfram.com/MoebiusFunction.html, http://www.numericana.com/answer/numbers.htm#moebius, https://oeis.org/A008683), Euler's totient function (https://en.wikipedia.org/wiki/Euler%27s_totient_function, https://t5k.org/glossary/xpage/EulersPhi.html, https://mathworld.wolfram.com/TotientFunction.html, http://www.numericana.com/answer/modular.htm#phi, http://www.javascripter.net/math/calculators/eulertotientfunction.htm, https://oeis.org/A000010), Carmichael function (https://en.wikipedia.org/wiki/Carmichael_function, https://mathworld.wolfram.com/CarmichaelFunction.html, http://www.numericana.com/answer/modular.htm#lambda, https://oeis.org/A002322), Dedekind psi function (https://en.wikipedia.org/wiki/Dedekind_psi_function, https://mathworld.wolfram.com/DedekindFunction.html, https://oeis.org/A001615), and divisor function (https://en.wikipedia.org/wiki/Divisor_function, https://t5k.org/glossary/xpage/SigmaFunction.html, https://mathworld.wolfram.com/DivisorFunction.html, http://www.javascripter.net/math/calculators/divisorscalculator.htm, https://oeis.org/A000203) (all of them are multiplicative functions (https://en.wikipedia.org/wiki/Multiplicative_function, https://t5k.org/glossary/xpage/MultiplicativeFunction.html, https://mathworld.wolfram.com/MultiplicativeFunction.html, http://www.numericana.com/answer/numbers.htm#multiplicative), although only Liouville function is a completely multiplicative function (https://en.wikipedia.org/wiki/Completely_multiplicative_function, https://t5k.org/glossary/xpage/CompletelyMultiplicative.html, https://mathworld.wolfram.com/CompletelyMultiplicativeFunction.html, http://www.numericana.com/answer/numbers.htm#totally)). Also, see https://t5k.org/ (The Prime Pages, https://en.wikipedia.org/wiki/PrimePages, https://www.rieselprime.de/ziki/The_Prime_Pages) and https://www.primegrid.com/ (Primegrid, https://en.wikipedia.org/wiki/PrimeGrid, https://www.rieselprime.de/ziki/PrimeGrid, https://t5k.org/bios/page.php?id=950) and http://www.numericana.com/answer/primes.htm (the set of the primes) (warning: the related link "The n−1 and n+1 primality tests by Curtis Bright, INTP (2013-10-09)" in this article is wrong, the correct link is http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/) and http://www.numericana.com/answer/factoring.htm (factoring into primes).

additionmultiplication
subtractiondivision
01
negationreciprocal
the set {1}the set of the prime numbers
less thandivides
1 + 1 + 1 + ... + 1 with n 1'sthe prime factorization of n (e.g. 360 = 23 × 32 × 5)

Addition (https://en.wikipedia.org/wiki/Addition, https://www.rieselprime.de/ziki/Addition, https://mathworld.wolfram.com/Addition.html) and multiplication (https://en.wikipedia.org/wiki/Multiplication, https://www.rieselprime.de/ziki/Multiplication, https://mathworld.wolfram.com/Multiplication.html) are the basic operations of arithmetic (https://en.wikipedia.org/wiki/Arithmetic, https://www.rieselprime.de/ziki/Arithmetic, https://mathworld.wolfram.com/Arithmetic.html) (which is also the basics of mathematics (https://en.wikipedia.org/wiki/Mathematics, https://www.rieselprime.de/ziki/Mathematics, https://mathworld.wolfram.com/Mathematics.html)). In the addition operation, the identity element (https://en.wikipedia.org/wiki/Identity_element, https://mathworld.wolfram.com/IdentityElement.html) is 0, and all natural numbers > 0 can be written as the sum of many 1's, and the number 1 cannot be broken up; in the multiplication operation, the identity element is 1, and all natural numbers > 1 can be written as the product of many prime numbers, and the prime numbers cannot be broken up. Also, primes are the natural numbers n > 1 such that if n divides (https://en.wikipedia.org/wiki/Divides, https://t5k.org/glossary/xpage/Divides.html, https://t5k.org/glossary/xpage/Divisor.html, https://www.rieselprime.de/ziki/Factor, https://mathworld.wolfram.com/Divides.html, https://mathworld.wolfram.com/Divisor.html, https://mathworld.wolfram.com/Divisible.html, http://www.numericana.com/answer/primes.htm#divisor) x×y (x and y are natural numbers), then n divides either x or y (or both). Also, prime numbers are the natural numbers n such that the ring (https://en.wikipedia.org/wiki/Ring_(mathematics), https://mathworld.wolfram.com/Ring.html) of integers modulo n (https://en.wikipedia.org/wiki/Integers_modulo_n, https://mathworld.wolfram.com/Mod.html) (i.e. the ring Zn) is a field (https://en.wikipedia.org/wiki/Field_(mathematics), https://mathworld.wolfram.com/Field.html) (also is a domain (https://en.wikipedia.org/wiki/Domain_(ring_theory), https://mathworld.wolfram.com/Domain.html), also is an integral domain (https://en.wikipedia.org/wiki/Integral_domain, https://mathworld.wolfram.com/IntegralDomain.html), also is a division ring (https://en.wikipedia.org/wiki/Division_ring), also has no zero divisors (https://en.wikipedia.org/wiki/Zero_divisor, https://mathworld.wolfram.com/ZeroDivisor.html) other than 0 (for the special case that n = 1, it is the zero ring (https://en.wikipedia.org/wiki/Zero_ring, https://mathworld.wolfram.com/TrivialRing.html))). Also, see https://t5k.org/ (The Prime Pages, https://en.wikipedia.org/wiki/PrimePages, https://www.rieselprime.de/ziki/The_Prime_Pages) and https://www.primegrid.com/ (Primegrid, https://en.wikipedia.org/wiki/PrimeGrid, https://www.rieselprime.de/ziki/PrimeGrid, https://t5k.org/bios/page.php?id=950) and http://www.numericana.com/answer/primes.htm (the set of the primes) and http://www.numericana.com/answer/factoring.htm (factoring into primes). Besides, "the set of the minimal elements of the base b representations of the prime numbers > b under the subsequence ordering" to "the set of the prime numbers (except b itself) digit strings with length > 1 in base b" to "the partially ordered binary relation by subsequence" is "the set of the prime numbers" to "the set of the integers > 1" to "the partially ordered binary relation by divisibility" (and indeed, the "> 1" in "the prime numbers (except b itself) digit strings with length > 1 in base b" can be corresponded to the "> 1" in "the integers > 1") (for the reason why b itself is excluded (when b is prime, if b is composite, then there is no difference to include the b itself or not), see the sections below and https://www.mersenneforum.org/showpost.php?p=531632&postcount=7, the main reason is that b is the only prime ending with 0), thus the main problem in this project is very important and beautiful.

Also, the main problem in this project is hard and interesting, since the distribution of the primes are mysterious and almost completely random (https://en.wikipedia.org/wiki/Random_number, https://mathworld.wolfram.com/RandomNumber.html), and there is still no known formula of primes (https://en.wikipedia.org/wiki/Formula_for_primes, https://t5k.org/glossary/xpage/FormulasForPrimes.html, https://mathworld.wolfram.com/PrimeFormulas.html, https://mathworld.wolfram.com/Prime-GeneratingPolynomial.html, https://t5k.org/notes/faq/p_n.html, https://cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell78.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_210.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL20/Toth2/toth32.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_211.pdf), https://arxiv.org/ftp/arxiv/papers/1901/1901.01849.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_212.pdf)) which is efficiently computable (https://en.wikipedia.org/wiki/Algorithmic_efficiency), but if the Riemann hypothesis (https://en.wikipedia.org/wiki/Riemann_hypothesis, https://t5k.org/glossary/xpage/RiemannHypothesis.html, https://mathworld.wolfram.com/RiemannHypothesis.html, http://www.numericana.com/answer/open.htm#rh) is true, then there is Mills' formula (https://en.wikipedia.org/wiki/Formula_for_primes#Mills'_formula, https://en.wikipedia.org/wiki/Mills%27_constant, https://t5k.org/glossary/xpage/MillsTheorem.html, https://t5k.org/glossary/xpage/MillsPrime.html, https://t5k.org/glossary/xpage/MillsConstant.html, https://mathworld.wolfram.com/MillsTheorem.html, https://mathworld.wolfram.com/MillsPrime.html, https://mathworld.wolfram.com/MillsConstant.html, https://t5k.org/notes/proofs/A3n.html, https://t5k.org/notes/MillsConstant.html, https://www.ams.org/journals/bull/1947-53-06/S0002-9904-1947-08849-2/S0002-9904-1947-08849-2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_312.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell78.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_210.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL20/Toth2/toth32.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_211.pdf), https://oeis.org/A051254, https://oeis.org/A108739, https://oeis.org/A051021) floor(A3n), which only gives prime numbers, also there is Wright's formula (https://en.wikipedia.org/wiki/Formula_for_primes#Wright's_formula, https://arxiv.org/pdf/1705.09741v3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_218.pdf), https://oeis.org/A016104, https://oeis.org/A086238) floor(222...2α), which only gives prime numbers with no needing to assume Riemann hypothesis to be true, however, neither Mills' formula nor Wright's formula can be used to find primes, since both of these two formulas has no practical value (and neither the value of the A in Mills' formula nor the value of the α in Wright's formula is currently known), and there is no known way of calculating the constants in both of these two formulas without finding primes in the first place, another example of a formula which only gives prime numbers is a polynomial with 26 variables (https://en.wikipedia.org/wiki/Variable_(mathematics), https://mathworld.wolfram.com/Variable.html) a, b, c, ..., z (exactly the 26 Latin letters (https://en.wikipedia.org/wiki/Latin_alphabet, https://en.wikipedia.org/wiki/ISO_basic_Latin_alphabet)) and degree (https://en.wikipedia.org/wiki/Degree_of_a_polynomial, https://mathworld.wolfram.com/PolynomialDegree.html) 25, which is based on a system of Diophantine equations (https://en.wikipedia.org/wiki/Diophantine_equation, https://t5k.org/glossary/xpage/Diophantus.html, https://mathworld.wolfram.com/DiophantineEquation.html), this polynomial is (see https://en.wikipedia.org/wiki/Formula_for_primes#Formula_based_on_a_system_of_Diophantine_equations and https://t5k.org/glossary/xpage/MatijasevicPoly.html and https://web.archive.org/web/20120612174638/http://mathdl.maa.org/images/upload_library/22/Ford/JonesSatoWadaWiens.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_217.pdf)) (the variables a, b, c, ..., z must be nonnegative integers) (in fact, this polynomial can give negative nonprime numbers, such as −76, whose absolute value (https://en.wikipedia.org/wiki/Absolute_value, https://www.rieselprime.de/ziki/Absolute_value, https://mathworld.wolfram.com/AbsoluteValue.html) is not a prime, but all positive values given by this polynomial are primes):

(k + 2) × (1 − (w×z+h+jq)2 − ((g×k+2×g+k+1)×(h+j)+hz)2 − (2×n+p+q+ze)2 − (16×(k+1)3×(k+2)×(n+1)2+1−f2)2 − (e3×(e+2)×(a+1)2+1−o2)2 − ((a2−1)×y2+1−x2)2 − (16×r2×y4×(a2−1)+1−u2)2 − (((a+u2×(u2a))2−1)×(n+4×d×y)2+1−(x+c×u)2)2 - (n+l+vy)2 − ((a2−1)×l2+1−m2)2 − (a×i+k+1−li)2 − (p+l×(an−1)+b×(2×a×n+2×an2−2×n−2)−m)2 − (q+y×(ap−1)+s×(2×a×p+2×ap2−2×p−2)−x)2 − (z+p×l×(ap)+t×(2×a×pp2−1)−p×m)2)

Besides, the record for the lowest degree of such a polynomial is 4 (with 58 variables), and the record for the fewest variables of such a polynomial is 10 (with degree about 1.6×1045)

subsequence orderingdivisibility ordering
the "prime numbers > b" digit strings" in base bthe integers > 1
the set of the minimal elements of the "prime numbers > b" digit strings" in base b under the subsequence ordering (which is exactly the target of this project)the set of the minimal elements of the integers > 1 under the divisibility ordering (which is exactly the set of the prime numbers)
no common subsequence with length > 1coprime (no common divisor > 1) (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime)
proper subsequence with length > 1proper factor (https://en.wikipedia.org/wiki/Proper_factor, https://mathworld.wolfram.com/ProperFactor.html, https://mathworld.wolfram.com/ProperDivisor.html) > 1
longest common subsequence (https://en.wikipedia.org/wiki/Longest_common_subsequence)greatest common divisor (https://en.wikipedia.org/wiki/Greatest_common_divisor, https://mathworld.wolfram.com/GreatestCommonDivisor.html, http://www.numericana.com/answer/numbers.htm#gcd)
shortest common supersequence (https://en.wikipedia.org/wiki/Shortest_common_supersequence)least common multiple (https://en.wikipedia.org/wiki/Least_common_multiple, https://mathworld.wolfram.com/LeastCommonMultiple.html, http://www.numericana.com/answer/numbers.htm#lcm)
pairwise incomparable strings (no string is a subsequence of another string)pairwise incomparable numbers (no number divides another number)

This problem is an extension of the original minimal prime problem (https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_17.pdf), https://cs.uwaterloo.ca/~shallit/Papers/br10.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_18.pdf), https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_19.pdf), https://doi.org/10.1080/10586458.2015.1064048 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_20.pdf), https://scholar.colorado.edu/downloads/hh63sw661 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_16.pdf) (warning: the datas for bases 8 and 10 have errors, the data for base 8 misses the prime 6101 and the data for base 10 misses the primes 9001 and 9049 and instead wrongly includes the primes 90001, 90469, and 9000049, and the correct values of Sm for bases 8 and 10 are 15 and 26 (instead of 14 and 27), respectively, also, the "number of minimal primes base b" and the "length of the largest minimal prime base b" are not the same sizes of b but the same sizes of eγ×(b−1)×eulerphi(b), this article has this error is because it only search bases 2 ≤ b ≤ 10, and for the data of 2 ≤ b ≤ 10 for the original minimal problem, you may think that they are the same sizes of b (however, if you extend the data to b = 11, 13, 16, then you will know that they are not the same sizes of b), since bases b = 7 and b = 9 have very large differences of the "number of minimal primes base b" between the original minimal problem and this new minimal prime problem (b = 7: 9 v.s. 71, b = 9: 12 v.s. 151), and bases b = 5 and b = 8 and b = 9 have very large differences of the "length of the largest minimal prime base b" between the original minimal problem and this new minimal prime problem (b = 5: 5 v.s. 96, b = 8: 9 v.s. 221, b = 9: 4 v.s. 1161)), https://github.com/curtisbright/mepn-data, https://github.com/curtisbright/mepn, https://github.com/RaymondDevillers/primes) to cover Conjectures 'R Us Sierpinski/Riesel conjectures base b (http://www.noprimeleftbehind.net/crus/, http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-stats.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-top20.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-proven.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, https://t5k.org/bios/page.php?id=1372, https://www.rieselprime.de/ziki/Conjectures_%27R_Us, https://srbase.my-firewall.org/sr5/, https://srbase.my-firewall.org/sr5/stats.php, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), http://www.bitman.name/math/article/2005 (in Italian)) with k-values < b (thus, the main problem in this project also covers the original minimal prime problem, the only numbers in the set of the original minimal prime problem and not in the set of the main problem in this project are exactly the primes ≤ b, and there are primepi(b) such primes (where primepi is the prime-counting function (https://en.wikipedia.org/wiki/Prime-counting_function, https://t5k.org/glossary/xpage/PrimeCountingFunction.html, https://mathworld.wolfram.com/PrimeCountingFunction.html, https://oeis.org/A000720, https://oeis.org/A006880, https://oeis.org/A007053, https://faculty.lynchburg.edu/~nicely/index.html, https://faculty.lynchburg.edu/~nicely/pi/pix_0000.htm, https://faculty.lynchburg.edu/~nicely/pi/pix_0001.htm, https://faculty.lynchburg.edu/~nicely/constell.zip, http://sweet.ua.pt/tos/primes.html, https://pzktupel.de/counting/PI_01.php, https://sweet.ua.pt/tos/bib/5.4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_452.pdf), https://arxiv.org/pdf/1503.01839.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_453.pdf))), and of course all of these primes are very easily to find), i.e. finding the smallest prime of the form k×bn+1 and k×bn−1 (or proving that such prime does not exist) for all k < b (also to cover dual (http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_1.pdf), https://www.rechenkraft.net/wiki/Five_or_Bust, https://oeis.org/A076336/a076336c.html, http://www.mit.edu/~kenta/three/prime/dual-sierpinski/ezgxggdm/dualsierp-excerpt.txt, http://mit.edu/kenta/www/three/prime/dual-sierpinski/ezgxggdm/dualsierp.txt.gz, https://www.primegrid.com/download/5ob_all.html, http://www.bitman.name/math/article/1126, http://www.bitman.name/math/article/1125, https://www.mersenneforum.org/showpost.php?p=144991&postcount=1, https://www.mersenneforum.org/showthread.php?t=10761, https://www.mersenneforum.org/showthread.php?t=6545) Sierpinski/Riesel conjectures base b with k-values < b, i.e. finding the smallest prime of the form bn+k and bnk (which are the dual forms of k×bn+1 and k×bn−1, respectively) (or proving that such prime does not exist) for all k < b) (also to cover finding the smallest prime of some classic forms (or proving that such prime does not exist), such as bn+2, bn−2, bn+(b−1), bn−(b−1), 2×bn+1, 2×bn−1, (b−1)×bn+1, (b−1)×bn−1, with n ≥ 1, for the same base b (of course, for some bases b the original minimal prime base b problem already covers finding the smallest prime of these forms, e.g. the original minimal prime base b problem covers finding the smallest prime of the form (b−1)×bn+1 if and only if b−1 is not prime, and the original minimal prime base b problem covers finding the smallest prime of the form (b−1)×bn−1 if and only if neither b−1 nor b−2 is prime, but I want the problem covers finding the smallest prime of these forms for all bases b)). The original minimal prime base b problem does not cover Conjectures 'R Us Sierpinski/Riesel conjectures base b with conjectured k (http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177) < b, since in Riesel side, the prime is not minimal prime in original definition if either k−1 or b−1 (or both) is prime, and in Sierpinski side, the prime is not minimal prime in original definition if k is prime (e.g. 25×3034205−1 is not minimal prime in base 30 in original definition, since it is OT34205 in base 30, and T (= 29 in decimal) is prime, but it is minimal prime in base 30 if only primes > base are counted), but this extended version of minimal prime base b problem does.

(warning: the data in the Table 5 in https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf) has many errors, e.g. for base b = 53, k = 4 is remaining at n = 2500000 and k = 8 has first prime at n = 227183 (which should not be found when the article was written), but the table only lists {1816, 1838, 1862, 1892} for the "ks not yet eliminated" data for base b = 53, and for base b = 48, the correct "ks not yet eliminated" data (at n = 100000) is {29, 36, 62, 153, 561, 622, 701, 937, 1077, 1086, 1114, 1121, 1168}, but the table wrongly lists {29, 36, 62, 153, 422, 1174}, missing many k and wrongly includes k = 422 and k = 1174 (which are trivial k since gcd(k+1,48−1) for these two k are not 1), and for base b = 55, the correct "ks not yet eliminated" data (at n = 100000) is {36, 778, 2274}, but the table lists {1980, 2274}, this article allow k with partial or full algebraic factors to become the conjectures, thus the conjecture k for b = 55 in this article is 2500 instead of 4416 (which is the conjecture k for b = 55 in Conjectures 'R Us), and thus k = 3940 becomes a k > conjectured k and thus not considered, but the table still misses k = 36 and k = 778, and only includes k = 1980 = 36×55 (unlike Conjectures 'R Us, this article does not exclude the ks which are multiples of base (b) and where k+1 is composite))

(in fact, for any k (not only the k < b), there is always an r such that "the minimal prime in base br" covers "finding the smallest prime of the form k×bn+1 and k×bn−1 and bn+k and bnk (or proving that such prime does not exist)" (also, no matter what is the lower bound (https://en.wikipedia.org/wiki/Lower_bound, https://mathworld.wolfram.com/LowerBound.html) of allowed n, the lower bound of allowed n need not to be 1 or 2), while this is not true for the original minimal prime problem (of course, there are bases b > 36 (which are not in this project) mentioned))

However, including the base (b) itself results in automatic elimination of all possible extension numbers with "0 after 1" from the set (when the base is prime, if the base is composite, then there is no difference to include the base (b) itself or not), which is quite restrictive (since when the base is prime, then the base (b) itself is the only prime ending with 0, i.e. having trailing zero (https://en.wikipedia.org/wiki/Trailing_zero), since in any base, all numbers ending with 0 (i.e. having trailing zero) are divisible by the base (b), thus cannot be prime unless it is equal the base (b), i.e. "10" in base b, note that the numbers cannot have leading zero (https://en.wikipedia.org/wiki/Leading_zero), since typically this is not the way we write numbers (in any base), thus for all primes in our sets (i.e. all primes > base (b)), all zero digits must be "between" other digits). (for the reference of this, see https://www.mersenneforum.org/showpost.php?p=531632&postcount=7)

Besides, this problem is better than the original minimal prime problem since this problem is regardless whether 1 is considered as prime or not, i.e. no matter 1 is considered as prime or not prime (https://t5k.org/notes/faq/one.html, https://primefan.tripod.com/Prime1ProCon.html, https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_24.pdf), http://www.numericana.com/answer/numbers.htm#one), the sets in this problem are the same, while the sets in the original minimal prime problem are different, e.g. in base 10, if 1 is considered as prime, then the set in the original minimal prime problem is {1, 2, 3, 5, 7, 89, 409, 449, 499, 6469, 6949, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049}, while if 1 is not considered as prime, then the set in the original minimal prime problem is {2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049}, however, in base 10, the set in this problem is always {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, no matter 1 is considered as prime or not prime.

The third reason for excluding the single-digit primes is that they are trivial (https://en.wikipedia.org/wiki/Triviality_(mathematics), https://mathworld.wolfram.com/Trivial.html), like that Conjectures 'R Us Sierpinski/Riesel conjectures base b requires exponent n ≥ 1 for these primes (see https://www.mersenneforum.org/showpost.php?p=447679&postcount=27), n = 0 is not acceptable to avoid the trivial primes (e.g. 2×bn+1, 4×bn+1, 6×bn+1, 10×bn+1, 12×bn+1, 16×bn+1, 3×bn−1, 4×bn−1, 6×bn−1, 8×bn−1, 12×bn−1, 14×bn−1, ... cannot be quickly eliminated with n = 0, or the conjectures become much easier and uninteresting), for the same reason, this minimal prime puzzle requires ≥ b (i.e. ≥ 2 digits) for these primes, single-digit primes are not acceptable to avoid the trivial primes (e.g. families containing digit 2, 3, 5, 7, B, D, H, J, N, T, V, ... cannot be quickly eliminated with the single-digit prime, or the conjectures become much easier and uninteresting).

The fourth reason for excluding the primes ≤ b is that starting with b+1 makes the formula of the number of possible (first digit,last digit) combo of a minimal prime in base b more simple and smooth number (https://en.wikipedia.org/wiki/Smooth_number, https://mathworld.wolfram.com/SmoothNumber.html, https://oeis.org/A003586, https://oeis.org/A051037, https://oeis.org/A002473, https://oeis.org/A051038, https://oeis.org/A080197, https://oeis.org/A080681, https://oeis.org/A080682, https://oeis.org/A080683) (i.e. the greatest prime factor (http://mathworld.wolfram.com/GreatestPrimeFactor.html, https://oeis.org/A006530) is small), it is (b−1)×eulerphi(b) (https://oeis.org/A062955), where eulerphi is Euler's totient function (https://en.wikipedia.org/wiki/Euler%27s_totient_function, https://t5k.org/glossary/xpage/EulersPhi.html, https://mathworld.wolfram.com/TotientFunction.html, http://www.numericana.com/answer/modular.htm#phi, http://www.javascripter.net/math/calculators/eulertotientfunction.htm, https://oeis.org/A000010), since b−1 is the number of possible first digit (except 0, all digits can be first digit), and eulerphi(b) is the number of possible last digit (only digits coprime to b can be last digit), by rule of product (https://en.wikipedia.org/wiki/Rule_of_product), there are (b−1)×eulerphi(b) possible (first digit,last digit) combo, and if start with b, then when b is prime, there is an additional possible (first digit,last digit) combo: (1,0), and hence the formula will be (b−1)×eulerphi(b)+1 if b is prime, or (b−1)×eulerphi(b) if b is composite (the fully formula will be (b−1)×eulerphi(b)+isprime(b) or (b−1)×eulerphi(b)+floor((beulerphi(b)) / (b−1))), which is more complex, and if start with 1 (i.e. the original minimal prime problem), the formula is much more complex, since the prime digits (i.e. the single-digit primes) should be excluded, and (for such prime > b) the first digit has b−1−primepi(b) choices, and the last digit has eulerphi(b)−primepi(b)+omega(b) (https://oeis.org/A048864) choices, by the rule of product (https://en.wikipedia.org/wiki/Rule_of_product), there are (b−1−primepi(b))×(eulerphi(b)−primepi(b)+omega(b)) choices of the (first digit,last digit) combo (if for such prime ≥ b instead of > b, then the formula will be (b−1−primepi(b))×(eulerphi(b)−primepi(b)+omega(b))+1 if b is prime, or (b−1−primepi(b))×(eulerphi(b)−primepi(b)+omega(b)) if b is composite), which is much more complex, (also, the possible (first digit,last digit) combo for a prime > b in base b are exactly the (first digit,last digit) combos which there are infinitely many primes have, while this is not true when the requiring is prime ≥ b or prime ≥ 2 instead of prime > b, since this will contain the prime factors of b, which are not coprime to b and hence there is only this prime (and not infinitely many primes) have this (first digit,last digit) combo) thus the main problem in this project (i.e. the minimal prime (start with b+1) problem) is much better than the original minimal prime problem.

(in the section above, isprime(n) is the characteristic function (https://en.wikipedia.org/wiki/Indicator_function, https://mathworld.wolfram.com/CharacteristicFunction.html) of primes (i.e. 1 if n is prime, else 0) (https://oeis.org/A010051), floor is the floor function (https://en.wikipedia.org/wiki/Floor_function, https://t5k.org/glossary/xpage/FloorFunction.html, https://www.rieselprime.de/ziki/Floor_function, https://mathworld.wolfram.com/FloorFunction.html), primepi is the prime-counting function (https://en.wikipedia.org/wiki/Prime-counting_function, https://t5k.org/glossary/xpage/PrimeCountingFunction.html, https://mathworld.wolfram.com/PrimeCountingFunction.html, https://oeis.org/A000720, https://oeis.org/A006880, https://oeis.org/A007053, https://faculty.lynchburg.edu/~nicely/index.html, https://faculty.lynchburg.edu/~nicely/pi/pix_0000.htm, https://faculty.lynchburg.edu/~nicely/pi/pix_0001.htm, https://faculty.lynchburg.edu/~nicely/constell.zip, http://sweet.ua.pt/tos/primes.html, https://pzktupel.de/counting/PI_01.php, https://sweet.ua.pt/tos/bib/5.4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_452.pdf), https://arxiv.org/pdf/1503.01839.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_453.pdf)), omega is the little prime omega function (https://en.wikipedia.org/wiki/Prime_omega_function, https://oeis.org/A001221))

The fifth reason (and the main reason) for excluding the primes ≤ b is that the possible last digits of a minimal prime in base b are exactly the last digits in base b such that there are infinitely many primes, by the Dirichlet's theorem (https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions, https://t5k.org/glossary/xpage/DirichletsTheorem.html, https://mathworld.wolfram.com/DirichletsTheorem.html, http://www.numericana.com/answer/primes.htm#dirichlet), thus in this problem, we completely need not to consider the last digits which are not coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to b, while in the original minimal prime problem, this is false for the primes dividing b.

The sixth reason is that (heuristically) the more one-digit primes are contained in the set, the less primes have to be considered (since all numbers that contain one of these digits cannot be contained in the minimal set), thus one-digit primes will make this problem much easier and more uninteresting (and when single-digit primes are excluded, all base b digits may appear in large minimal primes in base b, e.g. when base b = 19 searched to length 100000, all base 19 digits except 2 and 8 still appear in the list of the 23 unsolved families), the reason is the same as why the article https://nntdm.net/papers/nntdm-25/NNTDM-25-1-036-047.pdf deals only with the minimal sets for eulerphi(n) + k (where eulerphi is the Euler's totient function (https://en.wikipedia.org/wiki/Euler%27s_totient_function, https://t5k.org/glossary/xpage/EulersPhi.html, https://mathworld.wolfram.com/TotientFunction.html, http://www.numericana.com/answer/modular.htm#phi, http://www.javascripter.net/math/calculators/eulertotientfunction.htm, https://oeis.org/A000010)) with k ≤ 5, since for k = 6, eulerphi(n) + 6 contains only two one-digit numbers (the author of that article wishes that the problem become easier, while I wish that this problem become much harder, thus I exclude the single-digit primes).

The seventh reason is that in this problem, for the linear families x{y}z (where x and z are strings (may be empty) of digits in base b, y is a digit in base b) in base b which we search, y can be any base b digit except 1 (y can be 1 if and only if base b has no repunit primes), while in the original minimal prime problem, y cannot be any prime digit (and y can be 1 if and only if base b has no repunit primes).

The eighth reason is that for the numbers in the sets in this problem, the greatest common divisor (https://en.wikipedia.org/wiki/Greatest_common_divisor, https://mathworld.wolfram.com/GreatestCommonDivisor.html, http://www.numericana.com/answer/numbers.htm#gcd) of the digits of these numbers must be 1, and the last digit must be coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to the base (b), since a prime p (when written in base b) have both "greatest common divisor (https://en.wikipedia.org/wiki/Greatest_common_divisor, https://mathworld.wolfram.com/GreatestCommonDivisor.html, http://www.numericana.com/answer/numbers.htm#gcd) of the digits of these numbers is 1" and "last digit is coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to the base (b)" if and only if p > b, thus it is not true for the numbers in the sets in the original minimal prime problem.

The minimal elements (https://en.wikipedia.org/wiki/Minimal_element) of the set (https://en.wikipedia.org/wiki/Set_(mathematics), https://mathworld.wolfram.com/Set.html) of the prime numbers (https://en.wikipedia.org/wiki/Prime_number, https://t5k.org/glossary/xpage/Prime.html, https://www.rieselprime.de/ziki/Prime, https://mathworld.wolfram.com/PrimeNumber.html, https://www.numbersaplenty.com/set/prime_number/, http://www.numericana.com/answer/primes.htm#definition, http://irvinemclean.com/maths/pfaq2.htm, https://oeis.org/A000040, https://t5k.org/lists/small/1000.txt, https://t5k.org/lists/small/10000.txt, https://t5k.org/lists/small/100000.txt, https://t5k.org/lists/small/millions/) > b in base (https://en.wikipedia.org/wiki/Radix, https://t5k.org/glossary/xpage/Radix.html, https://www.rieselprime.de/ziki/Base, https://mathworld.wolfram.com/Radix.html) b under the subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html) ordering (https://en.wikipedia.org/wiki/Partially_ordered_set, https://mathworld.wolfram.com/PartialOrder.html, https://mathworld.wolfram.com/PartiallyOrderedSet.html) is interesting because all prime numbers > b in base b except the repunits (https://en.wikipedia.org/wiki/Repunit, https://en.wikipedia.org/wiki/List_of_repunit_primes, https://t5k.org/glossary/xpage/Repunit.html, https://t5k.org/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://www.rieselprime.de/ziki/Generalized_Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://pzktupel.de/Primetables/TableRepunit.php, https://pzktupel.de/Primetables/TableRepunitGen.php, https://pzktupel.de/Primetables/TableRepunitGen.txt, https://stdkmd.net/nrr/prime/prime_rp.htm, https://stdkmd.net/nrr/prime/prime_rp.txt, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html, https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/, https://web.archive.org/web/20120227163614/http://phi.redgolpe.com/5.asp, https://web.archive.org/web/20120227163508/http://phi.redgolpe.com/4.asp, https://web.archive.org/web/20120227163610/http://phi.redgolpe.com/3.asp, https://web.archive.org/web/20120227163512/http://phi.redgolpe.com/2.asp, https://web.archive.org/web/20120227163521/http://phi.redgolpe.com/1.asp, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, https://jpbenney.blogspot.com/2022/04/another-sequence-of-note.html, http://perplexus.info/show.php?pid=8661&cid=51696, https://benvitalenum3ers.wordpress.com/2013/07/24/repunit-11111111111111-in-other-bases/, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, http://www.bitman.name/math/article/380/231/, http://www.bitman.name/math/table/379, https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf), https://t5k.org/top20/page.php?id=57, https://t5k.org/top20/page.php?id=16, https://t5k.org/primes/search.php?Comment=^Repunit&OnList=all&Number=1000000&Style=HTML, https://t5k.org/primes/search.php?Comment=Generalized%20repunit&OnList=all&Number=1000000&Style=HTML, https://oeis.org/A002275, https://oeis.org/A004022, https://oeis.org/A053696, https://oeis.org/A085104, https://oeis.org/A179625) in base b contain at least two different characters (https://en.wikipedia.org/wiki/Character_(computing)) (or digits (https://en.wikipedia.org/wiki/Numerical_digit, https://www.rieselprime.de/ziki/Digit, https://mathworld.wolfram.com/Digit.html)), since if a repdigit (https://en.wikipedia.org/wiki/Repdigit, https://mathworld.wolfram.com/Repdigit.html, https://oeis.org/A010785) in base b is a prime > b, then is must be a repunit (i.e. the repeating digit is 1) in base b, since, for example, the repdigit 77777 is divisible by 7, in any base b > 7, also, since a repunit prime in base b is a minimal prime in base b if and only if it is the smallest repunit prime in base b, thus if there exists a repunit prime in base b, then there is exactly one repunit prime in base b which is also a minimal prime in base b, thus if there exists a repunit prime in base b, then all but one minimal primes in base b contain at least two different digits, and if there does not exist a repunit prime in base b, then all minimal primes in base b contain at least two different digits.

In fact, I create this problem because I think that the single-digit primes are trivial (https://en.wikipedia.org/wiki/Triviality_(mathematics), https://mathworld.wolfram.com/Trivial.html) (like strictly non-palindromic number (https://en.wikipedia.org/wiki/Strictly_non-palindromic_number, http://www.mathpages.com/home/kmath359.htm, https://oeis.org/A016038), single-digit numbers are trivially palindromic (https://en.wikipedia.org/wiki/Palindromic_number, https://en.wikipedia.org/wiki/Palindromic_prime, https://t5k.org/glossary/xpage/PalindromicPrime.html, https://mathworld.wolfram.com/PalindromicNumber.html, https://mathworld.wolfram.com/PalindromicPrime.html, https://www.numbersaplenty.com/set/palindromic_number/, https://t5k.org/top20/page.php?id=53, https://t5k.org/primes/search.php?Comment=Palindrome&OnList=all&Number=1000000&Style=HTML, https://oeis.org/A002113, https://oeis.org/A002385), thus to test whether a number n is strictly non-palindromic, we do not consider the bases b > n, since in these bases, n is a single-digit number, thus trivially palindromic, note that all strictly non-palindromic numbers > 6 are primes), thus I do not count them. (There is someone else who also exclude the single-digit primes, but his research is about substring (https://en.wikipedia.org/wiki/Substring) instead of subsequence, see https://www.mersenneforum.org/showpost.php?p=235383&postcount=42, subsequences can contain consecutive elements which were not consecutive in the original sequence, a subsequence which consists of a consecutive run of elements from the original sequence, such as 234 from 123456, is a substring, substring is a refinement of the subsequence, subsequence is a generalization of substring, substring must be subsequence, but subsequence may not be substring, 514 is a subsequence of 352148, but not a substring of 352148, see the list below of the comparation of "subsequence" and "substring")

subsequencesubstring
https://oeis.org/A071062https://oeis.org/A033274
https://oeis.org/A130448https://oeis.org/A238334
https://oeis.org/A039995https://oeis.org/A039997
https://oeis.org/A039994https://oeis.org/A039996
https://oeis.org/A094535https://oeis.org/A093301
https://oeis.org/A350508https://oeis.org/A038103
https://oeis.org/A354113https://oeis.org/A354114
https://t5k.org/glossary/xpage/MinimalPrime.htmlhttps://www.mersenneforum.org/showpost.php?p=235098&postcount=5
longest common subsequence (https://en.wikipedia.org/wiki/Longest_common_subsequence)longest common substring (https://en.wikipedia.org/wiki/Longest_common_substring)

The longest common subsequence problem (https://en.wikipedia.org/wiki/Longest_common_subsequence) and the longest common substring problem (https://en.wikipedia.org/wiki/Longest_common_substring) are two hard problems on strings (https://en.wikipedia.org/wiki/String_(computer_science), https://mathworld.wolfram.com/String.html), the former is NP-complete (https://en.wikipedia.org/wiki/NP-complete, https://mathworld.wolfram.com/NP-CompleteProblem.html) and NP-hard (https://en.wikipedia.org/wiki/NP-hard, https://mathworld.wolfram.com/NP-HardProblem.html), while the latter is not.

(we do not research substring in this project, because for substring ordering, "the set of the minimal elements of the base 10 representations of the prime numbers > 10" are very likely to be infinite (and thus almost unsolvable), since all primes of the form 1{0}3 (i.e. https://oeis.org/A159352, for the exponents see https://oeis.org/A049054, also see https://stdkmd.net/nrr/1/10003.htm) or 3{0}1 (i.e. https://oeis.org/A259866, for the exponents see https://oeis.org/A056807, also see https://stdkmd.net/nrr/3/30001.htm) are minimal elements of the base 10 representations of the prime numbers > 10 under the substring ordering, and there is likely infinitely many primes of the form 1{0}3 and infinitely many primes of the form 3{0}1, see https://web.archive.org/web/20100628035147/http://www.math.niu.edu/~rusin/known-math/98/exp_primes and https://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Mathematics/2023_September_25#Are_there_infinitely_many_primes_of_the_form_1000%E2%80%A60007,_333%E2%80%A63331,_7111%E2%80%A6111,_or_3444%E2%80%A64447_in_base_10? and https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf) and https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf) and https://www.mersenneforum.org/showpost.php?p=564786&postcount=3 and https://www.mersenneforum.org/showpost.php?p=461665&postcount=7 and https://www.mersenneforum.org/showpost.php?p=354505&postcount=5 and https://www.mersenneforum.org/showpost.php?p=344985&postcount=293 and https://www.mersenneforum.org/showpost.php?p=625978&postcount=1027, https://www.primegrid.com/forum_thread.php?id=5093&nowrap=true#66471 and https://www.primegrid.com/forum_thread.php?id=4935&nowrap=true#63813)

It is found that both "number of minimal primes base b" and "length of the largest minimal prime base b" are roughly (https://en.wikipedia.org/wiki/Asymptotic_analysis, https://t5k.org/glossary/xpage/AsymptoticallyEqual.html, https://mathworld.wolfram.com/Asymptotic.html) eγ×(b−1)×eulerphi(b), where e = 2.718281828459... is the base of the natural logarithm (https://en.wikipedia.org/wiki/E_(mathematical_constant), https://mathworld.wolfram.com/e.html, https://oeis.org/A001113), γ = 0.577215664901 is the Euler–Mascheroni constant (https://en.wikipedia.org/wiki/Euler%27s_constant, https://t5k.org/glossary/xpage/Gamma.html, https://mathworld.wolfram.com/Euler-MascheroniConstant.html, https://oeis.org/A001620), eulerphi is Euler's totient function (https://en.wikipedia.org/wiki/Euler%27s_totient_function, https://t5k.org/glossary/xpage/EulersPhi.html, https://mathworld.wolfram.com/TotientFunction.html, http://www.numericana.com/answer/modular.htm#phi, http://www.javascripter.net/math/calculators/eulertotientfunction.htm, https://oeis.org/A000010), you can see the condensed table for bases 2 ≤ b ≤ 36 in the bottom of this article, eγ×(b−1)×eulerphi(b) is an exponential sequence (https://en.wikipedia.org/wiki/Exponential_growth, https://mathworld.wolfram.com/ExponentialGrowth.html) for (b−1)×eulerphi(b) (https://oeis.org/A062955), and since (b−1)×eulerphi(b) has polynomial growth (https://en.wikipedia.org/wiki/Polynomial, https://mathworld.wolfram.com/Polynomial.html) for b (since it is always between b−1 and b2), thus eγ×(b−1)×eulerphi(b) has exponential growth for b, and "largest minimal prime base b" is roughly beγ×(b−1)×eulerphi(b), which has double exponential growth (https://en.wikipedia.org/wiki/Double_exponential_function) for b, however, for bases b = 7 and b = 15, the estimation of "length of the largest minimal prime base b" is much higher than the real value, since these two bases are very-high Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=50442&postcount=1, https://www.mersenneforum.org/showpost.php?p=50444&postcount=1, https://www.mersenneforum.org/showpost.php?p=201642&postcount=1, https://www.mersenneforum.org/attachment.php?attachmentid=13663&d=1451910741, https://github.com/happy5214/nash, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, https://web.archive.org/web/20230928115952/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, https://web.archive.org/web/20230928120009/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, https://web.archive.org/web/20230928120025/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, https://web.archive.org/web/20230928120047/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)) bases, these two bases are "primeful" as the Conjectures 'R Us Sierpinski/Riesel conjectures (http://www.noprimeleftbehind.net/crus/, http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-stats.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-top20.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-proven.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, https://t5k.org/bios/page.php?id=1372, https://www.rieselprime.de/ziki/Conjectures_%27R_Us, https://srbase.my-firewall.org/sr5/, https://srbase.my-firewall.org/sr5/stats.php, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), http://www.bitman.name/math/article/2005 (in Italian)) in bases b = 7 and b = 15, since they are of the form 2r−1, while for bases b = 5 and b = 11 and b = 14, the estimation of "length of the largest minimal prime base b" is lower than the real value, since they are low Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=50442&postcount=1, https://www.mersenneforum.org/showpost.php?p=50444&postcount=1, https://www.mersenneforum.org/showpost.php?p=201642&postcount=1, https://www.mersenneforum.org/attachment.php?attachmentid=13663&d=1451910741, https://github.com/happy5214/nash, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, https://web.archive.org/web/20230928115952/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, https://web.archive.org/web/20230928120009/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, https://web.archive.org/web/20230928120025/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, https://web.archive.org/web/20230928120047/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)) bases (as in Conjectures 'R Us Sierpinski/Riesel conjectures (http://www.noprimeleftbehind.net/crus/, http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-stats.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-top20.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-proven.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, https://t5k.org/bios/page.php?id=1372, https://www.rieselprime.de/ziki/Conjectures_%27R_Us, https://srbase.my-firewall.org/sr5/, https://srbase.my-firewall.org/sr5/stats.php, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), http://www.bitman.name/math/article/2005 (in Italian)), bases b == 2 mod 3 are low Nash weight bases), but the estimation of "number of minimal primes base b" is always near to the real value. (there are also asymptotic analysis for other sets of primes in various bases b, such as the left-truncatable primes and the right-truncatable primes (https://en.wikipedia.org/wiki/Truncatable_prime, https://t5k.org/glossary/xpage/LeftTruncatablePrime.html, https://t5k.org/glossary/xpage/RightTruncatablePrime.html, https://mathworld.wolfram.com/TruncatablePrime.html, https://www.numbersaplenty.com/set/truncatable_prime/) in various bases b, see http://chesswanks.com/num/LTPs/ for the left-truncatable primes in bases b ≤ 120 and http://fatphil.org/maths/rtp/rtp.html for the right-truncatable primes in bases b ≤ 90, also see OEIS sequences https://oeis.org/A103443 (largest left-truncatable prime in base b) and https://oeis.org/A023107 (largest right-truncatable prime in base b) and https://oeis.org/A103463 (length of the largest left-truncatable prime in base b) and https://oeis.org/A103483 (length of the largest right-truncatable prime in base b) and https://oeis.org/A076623 (number of left-truncatable primes in base b) and https://oeis.org/A076586 (number of right-truncatable primes in base b))

(of course, you can also try to extend the main problem in this project to bases 2 ≤ b ≤ 50 or 2 ≤ b ≤ 100 (I cannot imagine the effort needed for bases b around 500 or 1000, even if strong probable primes are allowed) (in fact, the GMP (https://gmplib.org/, https://en.wikipedia.org/wiki/GNU_Multiple_Precision_Arithmetic_Library) program supports bases 2 ≤ b ≤ 50, but I only ran for bases 2 ≤ b ≤ 36) (suggestion to use the character ":" to saparate the digits for bases b > 36 (and just use decimal to write the digits), just like https://baseconvert.com/ and https://baseconvert.com/high-precision), but warning: these problems will be extremely hard (especially the bases b such that (b−1)×eulerphi(b) (https://oeis.org/A062955) is larger)!!! The difficulty of base b is roughly (https://en.wikipedia.org/wiki/Asymptotic_analysis, https://t5k.org/glossary/xpage/AsymptoticallyEqual.html, https://mathworld.wolfram.com/Asymptotic.html) eγ×(b−1)×eulerphi(b), and eγ×(b−1)×eulerphi(b) is an exponential sequence (https://en.wikipedia.org/wiki/Exponential_growth, https://mathworld.wolfram.com/ExponentialGrowth.html) for (b−1)×eulerphi(b) (https://oeis.org/A062955))

The value (b−1)×eulerphi(b) (https://oeis.org/A062955) is the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos (ordered pair (https://en.wikipedia.org/wiki/Ordered_pair, https://mathworld.wolfram.com/OrderedPair.html)) of a minimal prime in base b (these (first digit,last digit) combos are also all possible (first digit,last digit) combos (ordered pair (https://en.wikipedia.org/wiki/Ordered_pair, https://mathworld.wolfram.com/OrderedPair.html)) of a prime > b in base b) (these (first digit,last digit) combos for decimal (base b = 10) are listed in OEIS sequence https://oeis.org/A085820, except the single-digit numbers (i.e. 1, 3, 7, 9) (i.e. first digit is 0, and hence the number has leading zeros (https://en.wikipedia.org/wiki/Leading_zero)) in this sequence, the smallest primes with these (first digit,last digit) combos listed in https://oeis.org/A085820 (except the single-digit numbers (i.e. 1, 3, 7, 9) in this sequence) are (italic for primes which are not minimal primes): 11, 13, 17, 19, 211, 23, 227, 29, 31, 313, 37, 349, 41, 43, 47, 409, 521, 53, 547, 59, 61, 613, 67, 619, 71, 73, 727, 79, 811, 83, 827, 89, 911, 953, 97, 919, and the smallest minimal primes with these (first digit,last digit) combos listed in https://oeis.org/A085820 (except the single-digit numbers (i.e. 1, 3, 7, 9) in this sequence) are (0 if no such minimal prime exists): 11, 13, 17, 19, 251, 23, 227, 29, 31, 0, 37, 349, 41, 43, 47, 409, 521, 53, 557, 59, 61, 0, 67, 6469, 71, 73, 727, 79, 821, 83, 827, 89, 991, 0, 97, 9049) (they are only all "possible" (first digit,last digit) combos (ordered pair (https://en.wikipedia.org/wiki/Ordered_pair, https://mathworld.wolfram.com/OrderedPair.html)) of a minimal prime in base b, this does not mean that they must be realized, e.g. there are no minimal primes with (first digit,last digit) = (2,2) in base b = 3, and there are no minimal primes with (first digit,last digit) = (3,3), (6,3), or (9,3) in base b = 10, but it is conjectured that there are only finitely many such examples (i.e. for every sufficiently large (https://en.wikipedia.org/wiki/Sufficiently_large, https://mathworld.wolfram.com/SufficientlyLarge.html) base b, for any given such (first digit,last digit) combo, there is a minimal prime with this (first digit,last digit) combo), also, it is conjectured that all such examples have gcd(first digit, last digit, b−1) > 1 (i.e. there is a prime number which divides first digit, last digit, and b−1 simultaneously), since the first digit has b−1 choices (all digits except 0 can be the first digit), and the last digit has eulerphi(b) choices (only digits coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to b (i.e. the digits in the reduced residue system (https://en.wikipedia.org/wiki/Reduced_residue_system, https://mathworld.wolfram.com/ReducedResidueSystem.html) mod b) can be the last digit), by the rule of product (https://en.wikipedia.org/wiki/Rule_of_product), there are (b−1)×eulerphi(b) choices of the (first digit,last digit) combo.

bnumber of possible first digits of a prime > b in base b (equal b−1, since all digits except 0 can be the first digit)number of possible last digits of a prime > b in base b (equal eulerphi(b), since only digits coprime to b (i.e. the digits in the reduced residue system mod b) can be the last digit)number of possible (first digit,last digit) combos of a prime > b in base b (equal (b−1)×eulerphi(b), by the rule of product), also the relative hardness for the "minimal prime problem" in base b
2111
3224
4326
54416
65210
76636
87428
98648
109436
111010100
1211444
131212144
1413678
15148112
16158120
171616256
18176102
191818324
20198152
212012240
222110210
232222484
24238184
252420480
262512300
272618468
282712324
292828784
30298232
313030900
323116496
333220640
343316528
353424816
363512420

(Note: Not all (first digit,last digit) combos must be realized for a minimal prime in base b, e.g. there are no minimal primes with (first digit,last digit) = (2,2) in base 3, and there are no minimal primes with (first digit,last digit) = (3,3), (6,3), or (9,3) in base 10)

The probability (https://en.wikipedia.org/wiki/Probability, https://mathworld.wolfram.com/Probability.html) for a random (https://en.wikipedia.org/wiki/Random_number, https://mathworld.wolfram.com/RandomNumber.html) prime to have a given (first digit,last digit) combo (ordered pair (https://en.wikipedia.org/wiki/Ordered_pair, https://mathworld.wolfram.com/OrderedPair.html)) which is a possible (first digit,last digit) combo (ordered pair (https://en.wikipedia.org/wiki/Ordered_pair, https://mathworld.wolfram.com/OrderedPair.html)) of a prime > b in base b (i.e. "first digit" is not 0, and "last digit" is coprime to b) are all the same (for the example of decimal (base b = 10), there are OEIS sequences https://oeis.org/A077648 (first digit), https://oeis.org/A007652 (last digit), https://oeis.org/A138840 ((first digit,last digit) combo (ordered pair (https://en.wikipedia.org/wiki/Ordered_pair, https://mathworld.wolfram.com/OrderedPair.html))), https://oeis.org/A137589 (results after deletion of all digits of primes, except the first digit and the last digit, this is the same as https://oeis.org/A138840 except the single-digit primes, and this is indeed another reason for why we exclude the single-digit primes from our minimal prime problem)), i.e. they are all 1/((b−1)×eulerphi(b)) no matter which (first digit,last digit) combo (ordered pair (https://en.wikipedia.org/wiki/Ordered_pair, https://mathworld.wolfram.com/OrderedPair.html)) is given, the only condition is that "first digit" is not 0, and "last digit" is coprime to b (however, there is a hard problem: for any given base b and given (first digit,last digit) combo (ordered pair (https://en.wikipedia.org/wiki/Ordered_pair, https://mathworld.wolfram.com/OrderedPair.html)) satisfying this condition (i.e. "first digit" is not 0, and "last digit" is coprime to b), is there always an integer N such that for the set of the primes > base (b) and ≤ N, the number of primes with this (first digit,last digit) combo is more than the number of primes with any other given (first digit,last digit) combo? (i.e. the number of primes p with https://oeis.org/A138840 = https://oeis.org/A137589 (their analogs in other bases b) = any given n such that b < n < b2 and n is coprime to b, is more than the number of primes p with https://oeis.org/A138840 = https://oeis.org/A137589 (their analogs in other bases b) = any other given m (mn) such that b < m < b2 and m is coprime to b?)), for the first digit, by the Bertrand's postulate (https://en.wikipedia.org/wiki/Bertrand%27s_postulate, https://mathworld.wolfram.com/BertrandsPostulate.html), for every base b, there are infinitely many primes with 1 as the first digit, since there is at least one prime between bn and 2×bn which must necessarily start with the digit 1 in base b, and in fact, for every base b and every digit d coprime to b, there are infinitely many primes with d as the first digit in base b, since by the better results of the Bertrand's postulate (see https://en.wikipedia.org/wiki/Bertrand%27s_postulate#Better_results), which is followed by the prime number theorem (https://en.wikipedia.org/wiki/Prime_number_theorem, https://t5k.org/glossary/xpage/PrimeNumberThm.html, https://mathworld.wolfram.com/PrimeNumberTheorem.html, https://t5k.org/howmany.html, http://www.numericana.com/answer/primes.htm#pnt, https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Tables_of_special_primes), for any ε > 0 there is an N such that for all n > N there is a prime p such that n < p < n × (1+ε), for sufficiently large (https://en.wikipedia.org/wiki/Sufficiently_large, https://mathworld.wolfram.com/SufficientlyLarge.html) n, there is always a prime between d×bn and (d×bn) × (1+1/d) = (d+1)×bn (let the number ε in the formula be 1/d), which must necessarily start with the digit d in base b, also see https://t5k.org/notes/faq/BenfordsLaw.html, the primes do not follow the Benford's law (https://en.wikipedia.org/wiki/Benford%27s_law, https://t5k.org/glossary/xpage/BenfordsLaw.html, https://mathworld.wolfram.com/BenfordsLaw.html, https://www.mathpages.com/home/kmath302/kmath302.htm, https://t5k.org/notes/faq/BenfordsLaw.html, https://www.ams.org/publications/journals/notices/201702/rnoti-p132.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_261.pdf)) (only the prime factors of the numbers with exponential growth (https://en.wikipedia.org/wiki/Exponential_growth, https://mathworld.wolfram.com/ExponentialGrowth.html) (such as the repunits (https://en.wikipedia.org/wiki/Repunit, https://en.wikipedia.org/wiki/List_of_repunit_primes, https://t5k.org/glossary/xpage/Repunit.html, https://t5k.org/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://www.rieselprime.de/ziki/Generalized_Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://pzktupel.de/Primetables/TableRepunit.php, https://pzktupel.de/Primetables/TableRepunitGen.php, https://pzktupel.de/Primetables/TableRepunitGen.txt, https://stdkmd.net/nrr/prime/prime_rp.htm, https://stdkmd.net/nrr/prime/prime_rp.txt, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html, https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/, https://web.archive.org/web/20120227163614/http://phi.redgolpe.com/5.asp, https://web.archive.org/web/20120227163508/http://phi.redgolpe.com/4.asp, https://web.archive.org/web/20120227163610/http://phi.redgolpe.com/3.asp, https://web.archive.org/web/20120227163512/http://phi.redgolpe.com/2.asp, https://web.archive.org/web/20120227163521/http://phi.redgolpe.com/1.asp, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, https://jpbenney.blogspot.com/2022/04/another-sequence-of-note.html, http://perplexus.info/show.php?pid=8661&cid=51696, https://benvitalenum3ers.wordpress.com/2013/07/24/repunit-11111111111111-in-other-bases/, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, http://www.bitman.name/math/article/380/231/, http://www.bitman.name/math/table/379, https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf), https://t5k.org/top20/page.php?id=57, https://t5k.org/top20/page.php?id=16, https://t5k.org/primes/search.php?Comment=^Repunit&OnList=all&Number=1000000&Style=HTML, https://t5k.org/primes/search.php?Comment=Generalized%20repunit&OnList=all&Number=1000000&Style=HTML, https://oeis.org/A002275, https://oeis.org/A004022, https://oeis.org/A053696, https://oeis.org/A085104, https://oeis.org/A179625) and the Fibonacci numbers (https://en.wikipedia.org/wiki/Fibonacci_number, https://t5k.org/glossary/xpage/FibonacciNumber.html, https://mathworld.wolfram.com/FibonacciNumber.html, https://www.numbersaplenty.com/set/Fibonacci_number/, https://t5k.org/top20/page.php?id=39, https://t5k.org/primes/search.php?Comment=^Fibonacci%20number&OnList=all&Number=1000000&Style=HTML, https://pzktupel.de/Primetables/TableFibonacci.php, https://oeis.org/A000045, https://oeis.org/A005478, https://oeis.org/A001605)) follow, also the primes p such that (bn−1)/(b−1) is prime for non-perfectpower b (e.g. https://oeis.org/A004023 for b = 10, and https://oeis.org/A000043 for b = 2) follow), instead, all nonzero digits have the same probability (i.e. probability 1/(b−1)) for a random prime in base b, just like a positive integer in base b, for the last digit, by the Dirichlet's theorem (https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions, https://t5k.org/glossary/xpage/DirichletsTheorem.html, https://mathworld.wolfram.com/DirichletsTheorem.html, http://www.numericana.com/answer/primes.htm#dirichlet), for every base b and every digit d coprime to b, there are infinitely many primes with d as the last digit in base b (since there are infinitely many primes == d mod b (https://en.wikipedia.org/wiki/Modular_arithmetic, https://en.wikipedia.org/wiki/Congruence_relation, https://en.wikipedia.org/wiki/Modulo, https://t5k.org/glossary/xpage/Congruence.html, https://t5k.org/glossary/xpage/CongruenceClass.html, https://t5k.org/glossary/xpage/Residue.html, https://mathworld.wolfram.com/Congruence.html, https://mathworld.wolfram.com/Congruent.html, https://mathworld.wolfram.com/Residue.html, https://mathworld.wolfram.com/MinimalResidue.html, https://mathworld.wolfram.com/Mod.html), which must necessarily end with the digit d in base b), and by the prime number theorem (https://en.wikipedia.org/wiki/Prime_number_theorem, https://t5k.org/glossary/xpage/PrimeNumberThm.html, https://mathworld.wolfram.com/PrimeNumberTheorem.html, https://t5k.org/howmany.html, http://www.numericana.com/answer/primes.htm#pnt, https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Tables_of_special_primes) (extended to arithmetic progression (https://en.wikipedia.org/wiki/Primes_in_arithmetic_progression, https://t5k.org/glossary/xpage/ArithmeticSequence.html, https://mathworld.wolfram.com/PrimeArithmeticProgression.html, https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem, https://mathworld.wolfram.com/Green-TaoTheorem.html, https://t5k.org/top20/page.php?id=14, https://t5k.org/primes/search.php?Comment=Arithmetic%20progression&OnList=all&Number=1000000&Style=HTML, https://www.primegrid.com/forum_thread.php?id=7022, https://www.primegrid.com/stats_ap26.php, https://www.pzktupel.de/JensKruseAndersen/aprecords.php, http://www.primerecords.dk/aprecords.htm, https://oeis.org/A133277, https://oeis.org/A113827, https://oeis.org/A005115, https://oeis.org/A093364, https://oeis.org/A133276, https://oeis.org/A033189, https://oeis.org/A113872, https://oeis.org/A033188, https://oeis.org/A231406, https://oeis.org/A113834, https://oeis.org/A088430)), all digits coprime to b have the same probability (i.e. probability 1/eulerphi(b)) for a random prime in base b, and for the smallest prime with last digit d in base b for the digits d coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to b, the largest of these primes are https://oeis.org/A038026 (the d which make these records are https://oeis.org/A038025), or https://oeis.org/A085420 (the numbers deleted the last digit (d) are https://oeis.org/A194943) if prime > b is required (like the main problem in this project), and the smallest prime > b which has last digit d in base b are https://oeis.org/A060940, this is related to Linnik's Theorem (https://en.wikipedia.org/wiki/Linnik%27s_theorem, https://mathworld.wolfram.com/LinniksTheorem.html, https://t5k.org/glossary/xpage/LinniksConstant.html, https://mathworld.wolfram.com/LinniksConstant.html), however, according to Chebyshev's bias (https://en.wikipedia.org/wiki/Chebyshev%27s_bias, https://mathworld.wolfram.com/ChebyshevBias.html, http://www.math.uiuc.edu/~ford/wwwpapers/lehman.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_59.pdf), https://dms.umontreal.ca/~andrew/PDF/PrimeRace.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_60.pdf), http://math101.guru/wp-content/uploads/2018/09/01-A3-Presentation-v7.3EN-no.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_67.pdf), https://arxiv.org/pdf/1910.08983.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_262.pdf), https://www.ams.org/journals/mcom/2004-73-247/S0025-5718-04-01649-7/S0025-5718-04-01649-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_263.pdf), https://arxiv.org/pdf/math/0010086.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_264.pdf), https://projecteuclid.org/euclid.em/1048515870 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_265.pdf), https://www.ams.org/journals/mcom/2000-69-230/S0025-5718-99-01105-9/S0025-5718-99-01105-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_266.pdf), https://www.ams.org/journals/mcom/1978-32-142/S0025-5718-1978-0476616-X/S0025-5718-1978-0476616-X.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_267.pdf), https://oeis.org/A007350, https://oeis.org/A007352, https://oeis.org/A199547, https://oeis.org/A306891, https://oeis.org/A321856, https://oeis.org/A066520, https://oeis.org/A321857, https://oeis.org/A321859, https://oeis.org/A071838, https://oeis.org/A320857, https://oeis.org/A321860, https://oeis.org/A321858, https://oeis.org/A112632, https://oeis.org/A038698, https://oeis.org/A321862, https://oeis.org/A321864, https://oeis.org/A321861, https://oeis.org/A320858, https://oeis.org/A321865, https://oeis.org/A321863, https://oeis.org/A275939, https://oeis.org/A306499, https://oeis.org/A306500, https://oeis.org/A329224, https://oeis.org/A306502, https://oeis.org/A306503, https://oeis.org/A329225), if d1 is a quadratic residue (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://mathworld.wolfram.com/QuadraticResidue.html, https://oeis.org/A096008, https://oeis.org/A046071, https://oeis.org/A096103, https://oeis.org/A000224, https://oeis.org/A105612, https://oeis.org/A046073) mod b, d2 is a quadratic nonresidue mod b (i.e. d1 can be the last digit of a square number (https://en.wikipedia.org/wiki/Square_number, https://www.rieselprime.de/ziki/Square_number, https://mathworld.wolfram.com/SquareNumber.html, https://www.numbersaplenty.com/set/square_number/, https://oeis.org/A000290) in base b, while d2 cannot be), then for the primes ≤ N for a random positive integer N, the probability for the number of primes ending with d2 in base b is more than the number of primes ending with d1 in base b is larger than 50%, e.g. the smallest N such that the number of primes end with 1 in base b = 4 is more than the number of primes end with 3 in base b = 4 is 12203231 (26861 in decimal), and the smallest N such that the number of primes end with 1 in base b = 3 is more than the number of primes end with 2 in base b = 3 is 2011012212222201102200001 (608981813029 in decimal), however, proving that there are infinitely many primes both starting and ending with given digits (i.e. primes with given (first digit,last digit) combo (ordered pair (https://en.wikipedia.org/wiki/Ordered_pair, https://mathworld.wolfram.com/OrderedPair.html))) in base b (of course, the ending digit must be coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to b) would probably require some extensive work, possibly combining the two theorems (the Bertrand's postulate and the Dirichlet's theorem), see https://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Mathematics/2023_September_23#Are_there_infinitely_many_primes_whose_first_digit_and_last_digit_are_both_7?. (edit: now it is know that there are infinitely many primes both starting and ending with given digits (i.e. primes with given (first digit,last digit) combo (ordered pair (https://en.wikipedia.org/wiki/Ordered_pair, https://mathworld.wolfram.com/OrderedPair.html))) in any base b if the ending digit is coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to b (thus, the sequences including https://oeis.org/A062332, https://oeis.org/A062333, https://oeis.org/A062334, https://oeis.org/A062335, etc. are infinite), also, the sum of the reciprocals of these primes diverges (https://en.wikipedia.org/wiki/Divergent_series, https://mathworld.wolfram.com/DivergentSeries.html), i.e. the set of them is a large set (https://en.wikipedia.org/wiki/Large_set_(combinatorics)), see https://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Mathematics/2023_October_1#Does_the_sum_of_the_reciprocals_of_all_primes_starting_with_7_and_ending_with_7_in_base_10_diverge? for the proof)

The lengths of the minimal primes in base b (e.g. in decimal (base b = 10) the lengths of the 77 minimal primes are {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 7, 8, 8, 8, 8, 8, 12, 31}, respectively, and in base b = 16 the lengths of the 2347 minimal (probable) primes are {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23, 23, 23, 23, 23, 24, 24, 24, 25, 25, 25, 25, 25, 26, 26, 26, 27, 27, 27, 27, 29, 29, 29, 30, 30, 31, 32, 32, 32, 32, 32, 33, 33, 33, 33, 33, 34, 34, 35, 35, 36, 37, 37, 38, 38, 38, 38, 40, 40, 40, 41, 41, 42, 42, 44, 45, 45, 46, 47, 47, 49, 49, 49, 50, 50, 51, 54, 54, 56, 58, 60, 61, 62, 66, 67, 68, 73, 74, 79, 89, 101, 105, 105, 125, 130, 130, 132, 137, 146, 179, 186, 205, 210, 214, 220, 243, 249, 265, 265, 294, 307, 426, 547, 547, 796, 1053, 1066, 1519, 1717, 1965, 3545, 3703, 17806, 32235, 72787, 116139}, respectively) appear to follow the Zipf's law (https://en.wikipedia.org/wiki/Zipf%27s_law, https://mathworld.wolfram.com/ZipfsLaw.html) (which is similar to the Benford's law (https://en.wikipedia.org/wiki/Benford%27s_law, https://t5k.org/glossary/xpage/BenfordsLaw.html, https://mathworld.wolfram.com/BenfordsLaw.html, https://www.mathpages.com/home/kmath302/kmath302.htm, https://t5k.org/notes/faq/BenfordsLaw.html, https://www.ams.org/publications/journals/notices/201702/rnoti-p132.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_261.pdf)), for n ≥ 2, the number of n-digit minimal primes in base b is inversely proportional (https://en.wikipedia.org/wiki/Inversely_proportional, https://mathworld.wolfram.com/InverselyProportional.html) to n−1 (i.e. the expected value (https://en.wikipedia.org/wiki/Expected_value, https://mathworld.wolfram.com/ExpectationValue.html) is c/(n−1), where c is a fixed constant) (i.e. the graph of the points (x,y = the number of x-digit minimal primes in base b) in the xy-plane is near to the graph of y = c/(x−1) in the xy-plane for a fixed real number c), for any fixed base b, also, for n ≥ 1, the length of the nth largest minimal prime in base b is inversely proportional (https://en.wikipedia.org/wiki/Inversely_proportional, https://mathworld.wolfram.com/InverselyProportional.html) to n (i.e. the expected value (https://en.wikipedia.org/wiki/Expected_value, https://mathworld.wolfram.com/ExpectationValue.html) is c/n, where c is a fixed constant) (i.e. the graph of the points (x,y = the length of the xth largest minimal prime in base b) in the xy-plane is near to the graph of y = c/x in the xy-plane for a fixed real number c), for any fixed base b. (for more information of the Zipf's law and the Benford's law, see https://oeis.org/A008952 and https://oeis.org/A008963 and https://oeis.org/A060956 and https://oeis.org/A241299 and https://oeis.org/A244059 and https://oeis.org/A363746 and http://www.cut-the-knot.org/do_you_know/zipfLaw.shtml and https://www.fq.math.ca/Scanned/13-4/webb.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_281.pdf) and https://www.fq.math.ca/Scanned/9-1/wlodarski2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_282.pdf) and https://arxiv.org/pdf/cond-mat/0412004.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_283.pdf))

We can use the sense of https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf) to say: (note that some of the left families may cover another left family, e.g. the base 19 left family 5{H}5 covers another base 19 left family 5{H}05, and if the smallest prime in family 5{H}5 in base 19 has length n, and the family 5{H}05 in base 19 has no prime with length ≤ n, then family 5{H}05 in base 19 can be removed from the unsolved families for base 19, however, if the smallest prime in family 5{H}5 in base 19 has length n, but the family 5{H}05 in base 19 is not tested to length n or more, then family 5{H}05 in base 19 should not be removed from the unsolved families for base 19, since a number in family 5{H}05 covers the prime in family 5{H}5 with length n if and only if the length of this number is ≥ n+1; besides, the base 19 left family FH0{H} covers another base 19 left family FHHH0{H}, and if the smallest prime in family FH0{H} in base 19 has length n, and the family FHHH0{H} in base 19 has no prime with length ≤ n+1, then family FHHH0{H} in base 19 can be removed from the unsolved families for base 19, however, if the smallest prime in family FH0{H} in base 19 has length n, but the family FHHH0{H} in base 19 is not tested to length n+1 or more, then family FHHH0{H} in base 19 should not be removed from the unsolved families for base 19, since a number in family FHHH0{H} covers the prime in family FH0{H} with length n if and only if the length of this number is ≥ n+2; besides, the base 21 left family {9}D covers another base 21 left family F{9}D, and if the smallest prime in family {9}D in base 21 has length n, and the family F{9}D in base 21 has no prime with length ≤ n, then family F{9}D in base 21 can be removed from the unsolved families for base 21, however, if the smallest prime in family {9}D in base 21 has length n, but the family F{9}D in base 21 is not tested to length n or more, then family F{9}D in base 21 should not be removed from the unsolved families for base 21, since a number in family F{9}D covers the prime in family {9}D with length n if and only if the length of this number is ≥ n+1 (if a family has no primes, then we say "the smallest prime in this family has length ∞ (https://en.wikipedia.org/wiki/Infinity, https://t5k.org/glossary/xpage/Infinite.html, https://mathworld.wolfram.com/Infinity.html) (instead of 0 or −1)", see http://gladhoboexpress.blogspot.com/2019/05/prime-sandwiches-made-with-one-derbread.html and http://chesswanks.com/seq/a306861.txt (for the OEIS sequence https://oeis.org/A306861) and http://chesswanks.com/seq/a269254.txt (for the OEIS sequence https://oeis.org/A269254) (since this is more convenient, e.g. the n of the smallest prime in the base 21 family 9nD, this family has been searched to n = 100000 with no prime or probable prime found, we can use ">100000" for the n of the smallest prime in the base 21 family 9nD (while for the n of the smallest prime in the base 13 family A3nA, it is 592199), ">100000" includes infinity (since infinity is > 100000) but does not includes 0 or −1, it is still possible that there is no prime in the base 21 family 9nD, although by the heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, https://web.archive.org/web/20230911032453/https://www.utm.edu/staff/caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf), https://arxiv.org/pdf/2103.04483.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_113.pdf)) above, this is very impossible, also "the smallest n ≥ 1 such that (a×bn+c)/gcd(a+c,b−1) is prime" should be the infimum (https://en.wikipedia.org/wiki/Infimum, https://mathworld.wolfram.com/Infimum.html) of the set S of the numbers n ≥ 1 such that (a×bn+c)/gcd(a+c,b−1) is prime, and if there is no n ≥ 1 such that (a×bn+c)/gcd(a+c,b−1) is prime, then this set S is the empty set (https://en.wikipedia.org/wiki/Empty_set, https://mathworld.wolfram.com/EmptySet.html), and by the definition of "inf", the infimum of the empty set is ∞), ∞ is > any finite number, e.g. "the smallest n ≥ 1 such that k×2n+1 is prime" is ∞ for k = 78557, 157114, 271129, 271577, 314228, 322523, 327739, 482719, ..., while it is 31172165 for k = 10223 and 13018586 for k = 19249, another example is "the smallest n such that (bn−1)/(b−1) is prime" is ∞ for b = 9, 25, 32, 49, 64, 81, 121, 125, 144, 169, 216, 225, 243, 289, 324, 343, 361, 441, 484, 529, 625, 729, 784, 841, 900, 961, 1000, ..., while it is 62903 for b = 691 and 41189 for b = 693))

e.g. for bases b = 23 and b = 25:

  • We have a 50% chance of solving the "minimal prime problem" at length 1025.
  • We have a 5% chance of solving the "minimal prime problem" at length 1016.
  • We have a 95% chance of solving the "minimal prime problem" at length 1048.
  • The chances at lengths 106, 107, 108 are respectively 10−86, 10−52, and 10−33.

(The chance that an unproven probable prime in the sets is in fact composite is less than 10−2000, see https://t5k.org/notes/prp_prob.html and https://www.ams.org/journals/mcom/1989-53-188/S0025-5718-1989-0982368-4/S0025-5718-1989-0982368-4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_22.pdf))

It is extremely likely that the largest minimal prime in base b is larger than the currently largest known prime (the world record prime) (i.e. 282589933−1, with 24862048 decimal digits) (https://en.wikipedia.org/wiki/Largest_known_prime_number, https://mathworld.wolfram.com/LargePrime.html, https://t5k.org/largest.html, https://t5k.org/top20/page.php?id=3, http://www.numericana.com/answer/primes.htm#history, https://t5k.org/primes/page.php?id=125874, https://www.rieselprime.de/ziki/M51, http://factordb.com/index.php?id=1100000001257221107&open=prime, https://oeis.org/A344984), for bases b = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, i.e. they will broke the world record (https://en.wikipedia.org/wiki/World_record) like Guinness World Records (https://guinnessworldrecords.com/, https://en.wikipedia.org/wiki/Guinness_World_Records), similar example is the project "Do You Feel Lucky?" (https://www.primegrid.com/forum_thread.php?id=8422) in PrimeGrid (https://www.primegrid.com/, https://en.wikipedia.org/wiki/PrimeGrid, https://www.rieselprime.de/ziki/PrimeGrid, https://t5k.org/bios/page.php?id=950), which searches primes of the form b222+1 for bases b ≥ 846398, such primes will be larger than the currently largest known prime (the world record prime) (i.e. 282589933−1, with 24862048 decimal digits) (https://en.wikipedia.org/wiki/Largest_known_prime_number, https://mathworld.wolfram.com/LargePrime.html, https://t5k.org/largest.html, https://t5k.org/top20/page.php?id=3, http://www.numericana.com/answer/primes.htm#history, https://t5k.org/primes/page.php?id=125874, https://www.rieselprime.de/ziki/M51, http://factordb.com/index.php?id=1100000001257221107&open=prime, https://oeis.org/A344984).

We can imagine an alien force, vastly more powerful than us, landing on Earth and demanding the set of all minimal primes in base b = 17 (or 21, 26, 36) (including primality proving of all primes in this set) or they will destroy our planet. In that case, I claim, we should marshal all our computers and all our mathematicians and attempt to find the set and to prove the primality of all numbers in this set. But suppose, instead, that they ask for the set of all minimal primes in base b = 19 (or 23, 25, 27, 29, 31, 32, 33, 34, 35). In that case, I believe, we should attempt to destroy the aliens. (Maybe only the God knows the set of all minimal primes in base b = 19 (or 23, 25, 27, 29, 31, 32, 33, 34, 35)!) (just like Paul Erdős for the Ramsey numbers (https://en.wikipedia.org/wiki/Ramsey_number, https://mathworld.wolfram.com/RamseyNumber.html), I do not think that finding the set of all minimal primes in bases b = 17, 19, 21, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 36 is easier than finding the Ramsey numbers R(m,n) for m > 4, n > 4)

This problem covers finding the smallest prime in these families in the same base b (or proving that such prime does not exist), since the smallest prime in these families (if exists) must be a minimal prime in base b (since these families are of the form {x}, x{y}, {x}y, x{0}y (where x and y are any digits in base b) in base b, and the repeating digit is not 1 for all bases b > 2 (or all bases b > 3 for the family (bn+1)/2 for odd b) except the family (bn−1)/(b−1), which is the form {1}) (thus the main problem in this project covers finding the smallest prime in these families (or proving that such prime does not exist) in bases 2 ≤ b ≤ 36, since all bases b < "smallest allowed b" (i.e. the bases b < k+1 for the families of the form k×bn±1 with fixed 2 ≤ k ≤ 12 and the families of the form bn±k with fixed 2 ≤ k ≤ 4) either have a prime < 2×1017 (only count the numbers > b) or can be proven to have no primes (only count the numbers > b), the largest of the smallest prime in these families is 11×818−1 = 198158383604301823, which is the smallest prime of the form 11×8n−1 with n ≥ 1): (while the original minimal prime problem does not cover some of these forms for some bases (or all bases) b)

(of course, there are bases b > 36 (which are not in this project) mentioned in the "smaller bases b such that this family can be ruled out as only containing composites (only count the numbers > b) (by covering congruence, algebraic factorization, or combine of them)
bases b: why this family contain no primes > b" column and the "smaller bases b with the smallest (probable) prime in this family has length > 100: b (length)" column, but I will not run the bases b > 36 in the main problem in this project, since base 2 ≤ b ≤ 36 are the bases which the main problem in this project decide to go, and you can also try to extend the main problem in this project to bases 2 ≤ b ≤ 50 or 2 ≤ b ≤ 100 (I cannot imagine the effort needed for bases b around 500 or 1000, even if strong probable primes are allowed) (in fact, the GMP (https://gmplib.org/, https://en.wikipedia.org/wiki/GNU_Multiple_Precision_Arithmetic_Library) program supports bases 2 ≤ b ≤ 50, but I only ran for bases 2 ≤ b ≤ 36) (suggestion to use the character ":" to saparate the digits for bases b > 36 (and just use decimal to write the digits), just like https://baseconvert.com/ and https://baseconvert.com/high-precision), but warning: these problems will be extremely hard (especially the bases b such that (b−1)×eulerphi(b) (https://oeis.org/A062955) is larger)!!! The difficulty of base b is roughly (https://en.wikipedia.org/wiki/Asymptotic_analysis, https://t5k.org/glossary/xpage/AsymptoticallyEqual.html, https://mathworld.wolfram.com/Asymptotic.html) eγ×(b−1)×eulerphi(b), and eγ×(b−1)×eulerphi(b) is an exponential sequence (https://en.wikipedia.org/wiki/Exponential_growth, https://mathworld.wolfram.com/ExponentialGrowth.html) for (b−1)×eulerphi(b) (https://oeis.org/A062955))

(in fact, I know exactly which bases 2 ≤ b ≤ 1024 have the families listed in the table below as unsolved families, all these families in all bases 2 ≤ b ≤ 1024 have been searched to length ≥ 10000)

familycorresponding form
({x} or x{y} or {x}y or x{0}y)
the value of xthe value of ysmallest allowed bsmallest allowed nOEIS sequences for the smallest n such that this form is prime for fixed base b (such n always exist unless these families can be ruled out as only containing composites (only count the numbers > b) (by covering congruence, algebraic factorization, or combine of them) if my conjecture is true)OEIS sequences for the smallest base b such that this form is prime for fixed n (such base b always exist unless these families can be ruled out as only containing composites (by single prime factor or algebraic factorization) if the Bunyakovsky conjecture (https://en.wikipedia.org/wiki/Bunyakovsky_conjecture, https://mathworld.wolfram.com/BouniakowskyConjecture.html) is true, in fact, if the Bunyakovsky conjecture (https://en.wikipedia.org/wiki/Bunyakovsky_conjecture, https://mathworld.wolfram.com/BouniakowskyConjecture.html) is true, then all numbers not in the OEIS sequence https://oeis.org/A121719 are primes in infinitely many bases b, since if the Bunyakovsky conjecture (https://en.wikipedia.org/wiki/Bunyakovsky_conjecture, https://mathworld.wolfram.com/BouniakowskyConjecture.html) is true, then all irreducible polynomials (https://en.wikipedia.org/wiki/Irreducible_polynomial, https://mathworld.wolfram.com/IrreduciblePolynomial.html) anxn+an−1xn−1+an−2xn−2+...+a2x2+a1x+a0 which have no fixed prime factors (in fact, such prime factors must be ≤ n, i.e. ≤ the degree (https://en.wikipedia.org/wiki/Degree_of_a_polynomial, https://mathworld.wolfram.com/PolynomialDegree.html) of the polynomial) for all integers x contain infinitely many primes, see https://oeis.org/A354718 and https://oeis.org/A337164)
(although these primes need not to be minimal primes in base b, I include this only because these OEIS sequences are usable references of the primes in these families)
referencescurrent smallest base b such that this family is an unsolved family (i.e. have no known prime (or strong probable prime) members > b, nor can be ruled out as only containing composites (only count the numbers > b) (by covering congruence, algebraic factorization, or combine of them))search limit of the length of this family in this base bbases b such that this family can be ruled out as only containing composites (only count the numbers > b) (by covering congruence, algebraic factorization, or combine of them)
bases b: why this family contains no primes > b
(only list reasons such that there are bases 2 ≤ b ≤ 2048 which the reason is realized)
smaller bases b with the smallest (probable) prime in this family has length > 100: b (length)
(bn−1)/(b−1){x}122https://oeis.org/A084740
https://oeis.org/A084738 (corresponding primes)
https://oeis.org/A246005 (odd b)
https://oeis.org/A065854 (prime b)
https://oeis.org/A279068 (prime b, corresponding primes)
https://oeis.org/A360738 (n replaced by n−1)
https://oeis.org/A279069 (prime b, n replaced by n−1)
https://oeis.org/A065813 (prime b, n replaced by (n−1)/2)
https://oeis.org/A128164 (n = 2 not allowed)
https://oeis.org/A285642 (n = 2 not allowed, corresponding primes)
https://oeis.org/A066180
https://oeis.org/A084732 (corresponding primes)
(if this form is prime, then n must be a prime, see https://t5k.org/notes/proofs/Theorem2.html for the proof, this proof can be generalized to any base b, see https://en.wikipedia.org/wiki/Repunit#Properties)
http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German)
https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html
http://www.primenumbers.net/Henri/us/MersFermus.htm
http://www.bitman.name/math/table/379 (in Italian)
https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf)
https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf)
185294001b = m2: difference-of-two-squares factorization
b = m3: difference-of-two-cubes factorization
b = m5: difference-of-two-5th-powers factorization
b = m7: difference-of-two-7th-powers factorization
(note: although bases b = 4, 8, 16, 27, 36, 100, 128 have algebraic factorization (if the numbers are factored as F × G / d), F (or G) is equal to d and G (or F) is prime > b, to make the factorizations be trivial, thus these bases b have only one very small prime > b instead of "can be ruled out as only containing composites (only count the numbers > b)", thus the only smaller bases b such that this family can be ruled out as only containing composites (only count the numbers > b) are 9, 25, 32, 49, 64, 81, 121, 125, 144, 169)
35 (313)
39 (349)
47 (127)
51 (4229)
91 (4421)
92 (439)
124 (599)
135 (1171)
139 (163)
142 (1231)
152 (270217)
171 (181)
174 (3251)
182 (167)
183 (223)
184 (16703)
bn+1x{0}y1121https://oeis.org/A079706
https://oeis.org/A084712 (corresponding primes)
https://oeis.org/A228101 (n replaced by log2n)
https://oeis.org/A123669 (n = 1 not allowed, corresponding primes)
https://oeis.org/A056993
https://oeis.org/A123599 (corresponding primes)
(if this form is prime, then n must be a power of 2, see https://web.archive.org/web/20231001191526/http://yves.gallot.pagesperso-orange.fr/primes/math.html for the proof, this proof can be generalized to any base b, see https://www.mersenneforum.org/showpost.php?p=95745&postcount=3 and https://www.mersenneforum.org/showpost.php?p=96001&postcount=95)
http://jeppesn.dk/generalized-fermat.html
http://www.noprimeleftbehind.net/crus/GFN-primes.htm
https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html
https://web.archive.org/web/20231003030159/http://yves.gallot.pagesperso-orange.fr/primes/results.html
https://web.archive.org/web/20231001191355/http://yves.gallot.pagesperso-orange.fr/primes/stat.html
https://genefer.great-site.net/
3833554432b == 1 mod 2: always divisible by 2
b = m3: sum-of-two-cubes factorization
b = m5: sum-of-two-5th-powers factorization
(none)
(bn+1)/2{x}y(b−1)/2(b+1)/23
(only odd b)
2https://oeis.org/A275530
(if this form is prime, then n must be a power of 2, see https://web.archive.org/web/20231001191526/http://yves.gallot.pagesperso-orange.fr/primes/math.html for the proof, this proof can be generalized to any base b, see https://www.mersenneforum.org/showpost.php?p=95745&postcount=3 and https://www.mersenneforum.org/showpost.php?p=96001&postcount=95)
http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt (in German)3116777215b = m3: sum-of-two-cubes factorization(none)
bn+1x{0}y2131https://oeis.org/A119624
https://oeis.org/A253178 (only bases b which have possible primes)
https://oeis.org/A098872 (b divisible by 6)
https://www.mersenneforum.org/showthread.php?t=6918
https://www.mersenneforum.org/showthread.php?t=19725 (b == 11 mod 12)
https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354
365500000b == 1 mod 3: always divisible by 338 (2730)
47 (176)
101 (192276)
104 (1234)
117 (287)
122 (756)
137 (328)
147 (155)
167 (6548)
203 (106)
206 (46206)
218 (333926)
236 (161230)
248 (322)
257 (12184)
263 (958)
287 (5468)
305 (16808)
347 (124)
353 (2314)
bn−1x{y}1b−131https://oeis.org/A119591
https://oeis.org/A098873 (b divisible by 6)
https://oeis.org/A157922https://www.mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217
https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354
581600000(none)29 (137)
67 (769)
74 (133)
107 (21911)
152 (797)
161 (229)
170 (166429)
191 (971)
215 (1073)
224 (109)
233 (8621)
235 (181)
254 (2867)
260 (121)
276 (2485)
278 (43909)
284 (417)
298 (4203)
303 (40175)
308 (991)
347 (523)
380 (3787)
382 (2325)
383 (20957)
393 (108)
395 (397)
401 (113)
418 (472)
422 (541)
431 (529)
434 (1167)
449 (175)
457 (103)
473 (661)
480 (145)
503 (861)
513 (299)
515 (58467)
522 (62289)
524 (165)
536 (841)
550 (1381)
551 (2719)
572 (3805)
578 (129469)
bn+2x{0}y1231https://oeis.org/A138066
https://oeis.org/A084713 (corresponding primes)
https://oeis.org/A138067 (n = 1 not allowed)
https://oeis.org/A087576
https://oeis.org/A095302 (corresponding primes)
167100000b == 0 mod 2: always divisible by 2
b == 1 mod 3: always divisible by 3
47 (114)
89 (256)
159 (137)
bn−2{x}yb−1b−232https://oeis.org/A250200
https://oeis.org/A255707 (n = 1 allowed)
https://oeis.org/A084714 (n = 1 allowed, corresponding primes)
https://oeis.org/A292201 (prime b, n = 1 allowed)
https://oeis.org/A095303
https://oeis.org/A095304 (corresponding primes)
https://www.primepuzzles.net/puzzles/puzz_887.htm (n = 1 allowed)30530000b == 0 mod 2: always divisible by 281 (130)
97 (747)
197 (164)
209 (126)
215 (134)
221 (552)
287 (3410)
bn+1x{0}y3141https://oeis.org/A098877 (b divisible by 6)https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354
718500000b == 1 mod 2: always divisible by 2108 (271)
314 (281)
358 (9561)
386 (184)
424 (1106)
458 (108)
492 (157)
636 (142)
646 (159)
648 (647)
652 (621)
654 (217)
690 (358)
bn−1x{y}2b−141https://oeis.org/A098876 (b divisible by 6)https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354
588500000b == 1 mod 2: always divisible by 242 (2524)
202 (263)
212 (283)
238 (105)
422 (191)
432 (16003)
446 (4851)
452 (335)
464 (219)
522 (347)
532 (136)
572 (377)
582 (445)
bn+3x{0}y1341https://oeis.org/A08757771810000b == 1 mod 2: always divisible by 2
b == 0 mod 3: always divisible by 3
382 (256)
388 (109)
412 (137)
530 (1399)
548 (118)
646 (9314)
bn−3{x}yb−1b−34211926000b == 1 mod 2: always divisible by 2
b == 0 mod 3: always divisible by 3
52 (105)
94 (204)
152 (346)
154 (396)
290 (111)
302 (1061)
478 (1410)
512 (1600)
542 (1944)
676 (141)
698 (306)
754 (120)
760 (120)
1000 (330)
1006 (124)
1010 (226)
1022 (102)
1094 (1508)
1096 (135)
bn+1x{0}y4151(such base b does not exist if n is divisible by 4 because of the Aurifeuillean factorization of x4+4×y4)https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354
321717986918b == 1 mod 5: always divisible by 5
b == 14 mod 15: always divisible by some element of {3,5}
b = m4: Aurifeuillean factorization of x4+4×y4
23 (343)
bn−1x{y}3b−151(such base b does not exist if n is even because of the difference-of-two-squares factorization)https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354
2751000000b == 1 mod 3: always divisible by 3
b == 14 mod 15: always divisible by some element of {3,5}
b = m2: difference-of-two-squares factorization
b == 4 mod 5: combine of factor 5 and difference-of-two-squares factorization
47 (1556)
72 (1119850)
107 (252)
167 (1866)
212 (34414)
218 (23050)
236 (940)
240 (1402)
251 (272)
261 (820)
270 (89662)
bn+4x{0}y1451(such base b does not exist if n is divisible by 4 because of the Aurifeuillean factorization of x4+4×y4)13918000b == 0 mod 2: always divisible by 2
b == 1 mod 5: always divisible by 5
b == 14 mod 15: always divisible by some element of {3,5}
b = m4: Aurifeuillean factorization of x4+4×y4
53 (13403)
113 (10647)
bn−4{x}yb−1b−452(such base b does not exist if n is even because of the difference-of-two-squares factorization)20712000b == 0 mod 2: always divisible by 2
b == 1 mod 3: always divisible by 3
b == 14 mod 15: always divisible by some element of {3,5}
b = m2: difference-of-two-squares factorization
b == 4 mod 5: combine of factor 5 and difference-of-two-squares factorization
65 (175)
93 (105)
123 (299)
135 (165)
137 (147)
141 (395)
173 (135)
183 (113)
191 (319)
203 (107)
bn+1x{0}y5161https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354
5121000000b == 1 mod 2: always divisible by 2
b == 1 mod 3: always divisible by 3
122 (136)
170 (176)
200 (768)
248 (262)
266 (510)
308 (309756)
318 (127)
326 (400786)
332 (106)
350 (20392)
356 (596)
368 (208)
392 (152)
410 (108)
440 (826)
bn−1x{y}4b−161https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354
338300000b == 1 mod 2: always divisible by 214 (19699)
68 (13575)
112 (133)
116 (157)
196 (9850)
206 (109)
254 (15451)
320 (233)
bn+1x{0}y6171https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354
2121600000b == 1 mod 7: always divisible by 7
b == 34 mod 35: always divisible by some element of {5,7}
53 (144)
67 (4533)
93 (521)
108 (16318)
129 (16797)
144 (783)
163 (1304)
185 (171)
193 (149)
bn−1x{y}5b−171https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354
2341000000b == 1 mod 5: always divisible by 5
b == 34 mod 35: always divisible by some element of {5,7}
b = 6×m2 with m == 2, 3 mod 5: combine of factor 5 and difference-of-two-squares factorization
48 (295)
118 (211)
119 (666)
154 (1990)
178 (119)
188 (951)
bn+1x{0}y7181https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354
113610000b == 1 mod 2: always divisible by 250 (517)
62 (309)
170 (179)
194 (281)
224 (689)
236 (347)
308 (107)
338 (793)
380 (475)
382 (519)
386 (121)
398 (17473)
434 (321)
466 (181)
500 (1997)
520 (198)
522 (235)
524 (127)
598 (423)
632 (8447)
638 (265)
644 (3379)
652 (185)
674 (181)
682 (796)
724 (388)
734 (189)
764 (189)
836 (5701)
868 (274)
892 (157)
920 (491)
926 (523)
930 (218)
958 (169)
960 (128)
974 (1589)
982 (313)
1004 (54849)
1082 (2113)
1102 (820)
bn−1x{y}6b−181https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354
308300000b == 1 mod 2: always divisible by 2
b == 1 mod 3: always divisible by 3
68 (25396)
182 (210)
198 (117)
248 (3180)
260 (826)
bn+1x{0}y8191(such base b does not exist if n is divisible by 3 because of the sum-of-two-cubes factorization)https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354
861000000b == 1 mod 3: always divisible by 3
b == 20 mod 21: always divisible by some element of {3,7}
b == 47, 83 mod 195: always divisible by some element of {3,5,13}
b == 467, 4343, 9887, 25448, 35978, 41522, 42647, 57083 mod 73815: always divisible by some element of {3,5,7,19,37}
b == 722, 83813, 206672, 239432, 322523, 1283843, 1519577, 1522553 mod 1551615: always divisible by some element of {3,5,13,73,109}
b = m3: sum-of-two-cubes factorization
b = 2r such that the equation 2x == 3 mod r has no solution: no possible prime, since 8×(2r)n+1 = 2n×r+3+1, and if 2n×r+3+1 is prime, then n×r+3 must be a power of 2 (otherwise, if n×r+3 has an odd prime factor p, then 2n×r+3+1 has a sum-of-two-pth-powers factorization), and this power of 2 must be == 3 mod r, for such r which are primes see https://oeis.org/A123988, unfortunately there is no OEIS sequence for all such r or all such odd r (they are in fact combine of sum-of-two-pth-powers factorization for infinitely many odd primes p, for such r which are primes, it is combine of sum-of-two-pth-powers factorization for the odd primes p which are not qsth power residue (we only need consider the prime powers (https://oeis.org/A246655, https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html) qs dividing r−1, for qs = 2 this is quadratic residue (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html), for qs = 3 this is cubic residue (https://en.wikipedia.org/wiki/Cubic_residue, https://mathworld.wolfram.com/CubicResidue.html), for qs = 4 this is quartic residue (https://en.wikipedia.org/wiki/Quartic_reciprocity, https://mathworld.wolfram.com/BiquadraticResidue.html), for qs = 8 this is octic residue (https://en.wikipedia.org/wiki/Octic_reciprocity), for other qs see power residue symbol (https://en.wikipedia.org/wiki/Power_residue_symbol) and Dirichlet character (https://en.wikipedia.org/wiki/Dirichlet_character, https://mathworld.wolfram.com/NumberTheoreticCharacter.html, https://www.lmfdb.org/Character/Dirichlet/) and Eisenstein reciprocity (https://en.wikipedia.org/wiki/Eisenstein_reciprocity) and Artin reciprocity (https://en.wikipedia.org/wiki/Artin_reciprocity, https://mathworld.wolfram.com/ArtinsReciprocityTheorem.html)) mod r for all prime powers (https://oeis.org/A246655, https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html) qs dividing https://oeis.org/A001917 at the entry of the prime r but not dividing https://oeis.org/A094593 at the entry of the prime r, e.g. the case of b = 128 (i.e. r = 7) is combine of sum-of-two-pth-powers factorization for the odd primes p which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 (i.e. the odd primes p == 3, 5, 6 mod 7) (i.e. the odd primes p in https://oeis.org/A003625), and the case of b = 131072 (i.e. r = 17) is combine of sum-of-two-pth-powers factorization for the odd primes p which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 17 (i.e. the odd primes p == 3, 5, 6, 7, 10, 11, 12, 14 mod 17) (i.e. the odd primes p in https://oeis.org/A038890), and the case of b = 2147483648 (i.e. r = 31) is combine of sum-of-two-pth-powers factorization for the odd primes p which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 31 (i.e. the odd primes p == 3, 6, 11, 12, 13, 15, 17, 21, 22, 23, 24, 26, 27, 29, 30 mod 31) (i.e. the odd primes p in https://oeis.org/A191067), also combine of sum-of-two-pth-powers factorization for the odd primes p which are not cubic residues (https://en.wikipedia.org/wiki/Cubic_residue, https://mathworld.wolfram.com/CubicResidue.html) mod 31 (i.e. the odd primes p == 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28 mod 31), etc. and by the Dirichlet's theorem (https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions, https://t5k.org/glossary/xpage/DirichletsTheorem.html, https://mathworld.wolfram.com/DirichletsTheorem.html, http://www.numericana.com/answer/primes.htm#dirichlet), all of these sequences contain infinitely many odd primes)
23 (119216)
53 (227184)
68 (320)
bn−1x{y}7b−191(such base b does not exist if n is divisible by 3 because of the difference-of-two-cubes factorization)https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354
321800000b == 1 mod 7: always divisible by 7
b == 20 mod 21: always divisible by some element of {3,7}
b == 83, 307 mod 455: always divisible by some element of {5,7,13}
b = m3: difference-of-two-cubes factorization
b == 1266, 13593, 27292, 46353 mod 63973: combine of factors {7,13,19,37} and difference-of-two-cubes factorization
97 (192336)
101 (113)
112 (269)
131 (197)
145 (6369)
170 (15423)
194 (38361)
202 (155772)
217 (179)
237 (528)
245 (501)
252 (6288)
270 (108)
277 (1229)
282 (21413)
283 (164769)
284 (5267)
bn+1x{0}y91101https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354
724600000b == 1 mod 2: always divisible by 2
b == 1 mod 5: always divisible by 5
94 (264)
134 (184)
182 (264)
244 (1836)
248 (39511)
332 (311)
334 (340)
344 (306)
364 (166)
400 (265)
402 (127)
422 (106)
448 (372)
454 (136)
490 (469)
534 (106)
544 (4706)
592 (96870)
622 (127)
634 (190)
664 (290)
bn−1x{y}8b−1101(such base b does not exist if n is even because of the difference-of-two-squares factorization)https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354
378300000b == 1 mod 2: always divisible by 2
b = m2: difference-of-two-squares factorization
b == 4 mod 5: combine of factor 5 and difference-of-two-squares factorization
88 (172)
112 (5718)
116 (250)
130 (468)
138 (35686)
188 (3888)
198 (304)
218 (178)
258 (106)
286 (164)
292 (2928)
328 (606)
332 (946)
346 (130)
360 (316)
366 (238)
10×bn+1x{0}y101111https://oeis.org/A088782
https://oeis.org/A088622 (corresponding primes)
https://oeis.org/A089319
https://oeis.org/A089318 (corresponding primes)
https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354
1851000000b == 1 mod 11: always divisible by 11
b == 32 mod 33: always divisible by some element of {3,11}
17 (1357)
61 (166)
74 (139)
101 (1507)
137 (103)
142 (408)
173 (264235)
176 (147)
179 (337)
10×bn−1x{y}9b−1111https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354
2331000000b == 1 mod 3: always divisible by 3
b == 32 mod 33: always divisible by some element of {3,11}
17 (118)
80 (423716)
89 (250)
185 (6784)
194 (3150)
215 (144)
11×bn+1x{0}y111121https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354
560100000b == 1 mod 2: always divisible by 2
b == 1 mod 3: always divisible by 3
b == 14 mod 15: always divisible by some element of {3,5}
68 (3948)
108 (190)
110 (162)
152 (838)
222 (101)
236 (154)
294 (365)
320 (1264)
384 (491)
392 (412)
432 (226)
440 (146)
462 (762)
506 (270)
528 (249)
534 (689)
542 (4910)
11×bn−1x{y}10b−1121https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354
2141000000b == 1 mod 2: always divisible by 2
b == 1 mod 5: always divisible by 5
b == 14 mod 15: always divisible by some element of {3,5}
b = 11×m2 with m == 2, 3 mod 5: combine of factor 5 and difference-of-two-squares factorization
38 (767)
68 (199)
72 (2446)
80 (209)
102 (2071)
140 (109)
170 (109)
178 (178)
188 (183)
12×bn+1x{0}y121131https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354
163500000b == 1 mod 13: always divisible by 13
b == 142 mod 143: always divisible by some element of {11,13}
b == 562, 828, 900, 1166 mod 1729: always divisible by some element of {7,13,19}
b == 597, 1143 mod 1885: always divisible by some element of {5,13,29}
b == 296, 901, 1759, 3090, 4553, 5521, 5807, 6016, 6984, 7094, 7270, 7380, 7479, 8447, 8557, 8733, 8843, 9910, 10020, 10196, 10306, 11483, 11769, 12737, 14200, 15531, 16994, 18457 mod 19019: always divisible by some element of {7,11,13,19}
b == 563, 1433, 13212, 15097, 19848, 20718, 32497, 34382, 39133, 51782, 53667, 58418, 58452, 60337, 60883, 71067, 72952, 77737, 79622, 80168, 94267, 97022, 98583, 98907, 113552, 116307, 117868, 118192, 131967, 132513, 132837, 134398, 151252, 151798, 152122, 153683, 170537, 171083, 172968, 177753, 179638, 189822, 190368, 192253, 192287, 197038, 198923, 211572, 213568, 216323, 218208, 229987, 232853, 235608, 237493, 249272 mod 250705: always divisible by some element of {5,7,13,19,29}
30 (1024)
65 (685)
67 (136)
68 (656922)
82 (108)
87 (1215)
102 (2740)
106 (139)
159 (122)
12×bn−1x{y}11b−1131https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://www.mersenneforum.org/showthread.php?t=10354
263314000b == 1 mod 11: always divisible by 11
b == 142 mod 143: always divisible by some element of {11,13}
b == 307, 1143 mod 1595: always divisible by some element of {5,11,29}
b == 901, 6016, 7479, 18457 mod 19019: always divisible by some element of {7,11,13,19}
43 (204)
65 (1194)
98 (3600)
129 (229)
147 (113)
153 (21660)
186 (112718)
193 (117)
230 (188)
(b−1)×bn+1x{0}yb−1121https://oeis.org/A305531
https://oeis.org/A087139 (prime b, n replaced by n+1)
(such base b does not exist if n == 1 mod 6 except n = 1 because such numbers are divisible by b2b+1)https://www.rieselprime.de/ziki/Williams_prime_MP_least
https://www.rieselprime.de/ziki/Williams_prime_MP_table
https://pzktupel.de/Primetables/TableWilliams2.php
https://pzktupel.de/Primetables/Williams_2.txt
https://web.archive.org/web/20240126201446/https://pzktupel.de/Primetables/Williams2DB.txt
https://sites.google.com/view/williams-primes
http://www.bitman.name/math/table/477 (in Italian)
342300000(none)53 (961)
65 (947)
77 (829)
88 (3023)
122 (6217)
123 (865891)
127 (166)
136 (280)
158 (1621)
180 (2485)
182 (397)
185 (209)
197 (521)
202 (46774)
214 (119)
248 (605)
249 (1852)
251 (102979)
257 (1345)
269 (1437)
272 (16681)
275 (981)
282 (277)
297 (14314)
298 (60671)
307 (204)
317 (129)
319 (565)
326 (64757)
328 (1627)
329 (481)
332 (113)
338 (273)
340 (325)
(b−1)×bn−1x{y}b−2b−121https://oeis.org/A122396 (prime b, n replaced by n+1)(such base b does not exist if n == 4 mod 6 because such numbers are divisible by b2b+1)https://harvey563.tripod.com/wills.txt
https://www.rieselprime.de/ziki/Williams_prime_MM_least
https://www.rieselprime.de/ziki/Williams_prime_MM_table
https://pzktupel.de/Primetables/TableWilliams1.php
https://pzktupel.de/Primetables/Williams_1.txt
https://web.archive.org/web/20240126201427/https://pzktupel.de/Primetables/Williams1DB.txt
https://sites.google.com/view/williams-primes
http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_9.pdf)
https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_10.pdf)
http://www.bitman.name/math/table/484 (in Italian)
1282450000(none)26 (134)
38 (136212)
62 (900)
83 (21496)
91 (520)
93 (477)
98 (4984)
108 (411)
113 (286644)
125 (8740)
bn+(b−1)x{0}y1b−121https://oeis.org/A076845
https://oeis.org/A076846 (corresponding primes)
https://oeis.org/A078178 (n = 1 not allowed)
https://oeis.org/A078179 (n = 1 not allowed, corresponding primes)
https://oeis.org/A248079
(such base b does not exist if n == 5 mod 6 because such numbers are divisible by b2b+1)
https://pzktupel.de/Primetables/TableWilliams6.php
https://pzktupel.de/Primetables/W6DB.txt
https://web.archive.org/web/20231015225001/https://pzktupel.de/Primetables/Williams6DB.txt
https://sites.google.com/view/williams-primes
http://www.bitman.name/math/table/795 (in Italian)
25717000(none)32 (109)
80 (195)
107 (1401)
113 (20089)
123 (64371)
128 (505)
161 (105)
173 (11429)
179 (3357)
197 (977)
212 (109)
224 (259)
227 (157)
237 (110)
238 (117)
bn−(b−1){x}yb−1122https://oeis.org/A113516
https://oeis.org/A343589 (corresponding primes)
https://oeis.org/A113517
(such base b does not exist if n == 2 mod 6 except n = 2 because such numbers are divisible by b2b+1)
https://pzktupel.de/Primetables/TableWilliams5.php
https://pzktupel.de/Primetables/W5DB.txt
https://web.archive.org/web/20231015225036/https://pzktupel.de/Primetables/Williams5DB.txt
https://sites.google.com/view/williams-primes
http://www.bitman.name/math/table/792 (in Italian)
https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html (prime b)
http://www.bitman.name/math/table/435 (in Italian) (prime b)
9360000(none)71 (3019)
82 (169)
83 (965)
88 (2848)

(below (as well as the "left b" files), family "x{y}z" (where x and z are strings (may be empty) of digits in base b, y is a digit in base b) means sequence {xz, xyz, xyyz, xyyyz, xyyyyz, xyyyyyz, ...} (i.e. "xy+z" in regular expression (https://en.wikipedia.org/wiki/Regular_expression)), where the members are expressed as base b strings (https://en.wikipedia.org/wiki/String_(computer_science), https://mathworld.wolfram.com/String.html), like the numbers in https://stdkmd.net/nrr/aaaab.htm, https://stdkmd.net/nrr/abbbb.htm, https://stdkmd.net/nrr/aaaba.htm, https://stdkmd.net/nrr/abaaa.htm, https://stdkmd.net/nrr/abbba.htm, https://stdkmd.net/nrr/abbbc.htm, https://stdkmd.net/nrr/prime/prime_all.htm, https://stdkmd.net/nrr/prime/prime_all.txt, https://stdkmd.net/nrr/prime/prime_sequences.htm, https://stdkmd.net/nrr/prime/prime_sequences.txt, https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305200806/https://stdkmd.net/nrr/prime/primesize.txt, https://web.archive.org/web/20240305201054/https://stdkmd.net/nrr/prime/primesize.zip, https://web.archive.org/web/20240305200957/https://stdkmd.net/nrr/prime/primecount.htm, https://web.archive.org/web/20240305200920/https://stdkmd.net/nrr/prime/primecount.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt, e.g. 1{3} (in decimal) is the numbers in https://stdkmd.net/nrr/1/13333.htm#about_first, and {1}3 (in decimal) is the numbers in https://stdkmd.net/nrr/1/11113.htm#about_first, and 31{3} (in decimal) is the numbers in https://stdkmd.net/nrr/3/31333.htm#about_first, and {1}31 (in decimal) is the numbers in https://stdkmd.net/nrr/1/11131.htm#about_first, and 1{2}3 (in decimal) is the numbers in https://stdkmd.net/nrr/1/12223.htm#about_first, also, superscripts always means exponents (https://en.wikipedia.org/wiki/Exponentiation, https://www.rieselprime.de/ziki/Exponent, https://mathworld.wolfram.com/Exponent.html, https://mathworld.wolfram.com/Power.html, https://mathworld.wolfram.com/Exponentiation.html), subscripts are always used to indicate repetitions of digits, e.g. 1234567 = 123333567, all subscripts are written in decimal)

In fact, this problem covers finding the smallest prime of these form in the same base b: (where x, y, z are any digits in base b)

The primes in forms x{y}, {x}y, xy{x}, {x}yx in base b are near-repdigit primes (https://t5k.org/glossary/xpage/NearRepdigitPrime.html, https://t5k.org/top20/page.php?id=15, https://t5k.org/primes/search.php?Comment=Near-repdigit&OnList=all&Number=1000000&Style=HTML, https://pzktupel.de/Primetables/TableNRD.php, https://oeis.org/A164937, https://stdkmd.net/nrr/#factortables_nr, https://stdkmd.net/nrr/#factortables_np, https://stdkmd.net/nrr/aaaab.htm, https://stdkmd.net/nrr/abbbb.htm, https://stdkmd.net/nrr/aaaba.htm, https://stdkmd.net/nrr/abaaa.htm, https://stdkmd.net/nrr/aabaa.htm, https://stdkmd.net/nrr/prime/prime_nr.htm, https://stdkmd.net/nrr/prime/prime_nr.txt, https://stdkmd.net/nrr/prime/prime_nrpl.htm, https://stdkmd.net/nrr/prime/prime_nrpl.txt, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#nrprime, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#nrprp, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#nrpprime, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#nrpprp, http://factordb.com/tables.php?open=1, http://factordb.com/tables.php?open=3) in base b.

Proving that "the set of the minimal elements of the base b representations of the prime numbers > b under the subsequence ordering" = the set S is equivalent to (https://en.wikipedia.org/wiki/Logical_equivalence):

("the set of the minimal elements of the base b representations of the prime numbers > b under the subsequence ordering" = S is proved if and only if all these three problems are proved, i.e. "the set of the minimal elements of the base b representations of the prime numbers > b under the subsequence ordering" = S is a theorem if and only if all these three "conjectures" are theorems)

e.g. proving that "the set of the minimal elements of the base 10 representations of the prime numbers > 10 under the subsequence ordering" = {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to:

  • Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10.
  • Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite.
  • Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite, since they are contraposition (https://en.wikipedia.org/wiki/Contraposition), PQ and ¬Q ⟶ ¬P are logically equivalent (https://en.wikipedia.org/wiki/Logical_equivalence)).

(since for base b = 10, all these three problems are proved, i.e. all they are theorems, thus, "the set of the minimal elements of the base 10 representations of the prime numbers > 10 under the subsequence ordering" = {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} is also proved, i.e. "the set of the minimal elements of the base 10 representations of the prime numbers > 10 under the subsequence ordering" = {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} is also a theorem)

Theorem (https://en.wikipedia.org/wiki/Theorem, https://mathworld.wolfram.com/Theorem.html, https://t5k.org/notes/proofs/): The set of the minimal elements of the base 10 representations of the prime numbers > 10 under the subsequence ordering is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}

Proof (https://en.wikipedia.org/wiki/Mathematical_proof, https://mathworld.wolfram.com/Proof.html, https://t5k.org/notes/proofs/): (this proof uses the notation in http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_11.pdf), i.e. "XY" means "X is a subsequence of Y") (below, 𝜆 is the empty string (https://en.wikipedia.org/wiki/Empty_string)) (bold for minimal primes)

Assume p is a prime > 10, and the last digit of p must lie in {1,3,7,9}

Case 1: p ends with 1.

In this case we can write p = x1. If x contains 1, 3, 4, 6, or 7, then (respectively) 11p, 31p, 41p, 61p, or 71p. Hence we may assume all digits of x are 0, 2, 5, 8, or 9.

Case 1.1: p begins with 2.

In this case we can write p = 2y1. If 5 ◁ y, then 251p. If 8 ◁ y, then 281p. If 9 ◁ y, then 29 ◁ p. Hence we may assume all digits of y are 0 or 2.

If 22 ◁ y, then 2221p. Hence we may assume y contains zero or one 2's.

If y contains no 2's, then p ∈ 2{0}1. But then, since the sum of the digits of p is 3, p is divisible by 3, so p cannot be prime.

If y contains exactly one 2, then we can write p = 2z2w1, where z,w ∈ {0}. If 0 ◁ z and 0 ◁ w, then 20201p. Hence we may assume either z or w is empty.

If z is empty, then p ∈ 22{0}1, and the smallest prime p ∈ 22{0}1 is 22000001.

If w is empty, then p ∈ 2{0}21, and the smallest prime p ∈ 2{0}21 is 20021.

Case 1.2: p begins with 5.

In this case we can write p = 5y1. If 2 ◁ y, then 521p. If 9 ◁ y, then 59 ◁ p. Hence we may assume all digits of y are 0, 5, or 8.

If 05 ◁ y, then 5051p. If 08 ◁ y, then 5081p. If 50 ◁ y, then 5501p. If 58 ◁ y, then 5581p. If 80 ◁ y, then 5801p. If 85 ◁ y, then 5851p. Hence we may assume y ∈ {0} ∪ {5} ∪ {8}.

If y ∈ {0}, then p ∈ 5{0}1. But then, since the sum of the digits of p is 6, p is divisible by 3, so p cannot be prime.

If y ∈ {5}, then p ∈ 5{5}1 = {5}1, and the smallest prime p ∈ 5{5}1 = {5}1 is 555555555551.

If y ∈ {8}, since if 88 ◁ y, then 881 ◁ p, hence we may assume y ∈ {𝜆, 8}, and thus p ∈ {51, 581}, but 51 and 581 are both composite.

Case 1.3: p begins with 8.

In this case we can write p = 8y1. If 2 ◁ y, then 821p. If 8 ◁ y, then 881p. If 9 ◁ y, then 89 ◁ p. Hence we may assume all digits of y are 0 or 5.

If 50 ◁ y, then 8501p. Hence we may assume y ∈ {0}{5}.

If 005 ◁ y, then 80051 ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}.

If y ∈ {0}, then p ∈ 8{0}1. But then, since the sum of the digits of p is 9, p is divisible by 3, so p cannot be prime.

If y ∈ {5}, since if 55555555555 ◁ y, then 555555555551 ◁ p, hence we may assume y ∈ {𝜆, 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus p ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite.

If y ∈ 0{5}, since if 55555555555 ◁ y, then 555555555551 ◁ p, hence we may assume y ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus p ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only 80555551 is minimal prime.

Case 1.4: p begins with 9.

In this case we can write p = 9y1. If 9 ◁ y, then 991p. Hence we may assume all digits of y are 0, 2, 5, or 8.

If 00 ◁ y, then 9001p. If 22 ◁ y, then 9221p. If 55 ◁ y, then 9551p. If 88 ◁ y, then 881 ◁ p. Hence we may assume y contains at most one 0, at most one 2, at most one 5, and at most one 8.

If y only contains at most one 0 and does not contain any of {2, 5, 8}, then y ∈ {𝜆, 0}, and thus p ∈ {91, 901}, but 91 and 901 are both composite. If y only contains at most one 0 and only one of {2, 5, 8}, then the sum of the digits of p is divisible by 3, p is divisible by 3, so p cannot be prime. Hence we may assume y contains at least two of {2, 5, 8}.

If 25 ◁ y, then 251 ◁ p. If 28 ◁ y, then 281 ◁ p. If 52 ◁ y, then 521 ◁ p. If 82 ◁ y, then 821 ◁ p. Hence we may assume y contains no 2's (since if y contains 2, then y cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ y, then 9851p. Hence we may assume y ∈ {58, 580, 508, 058}, and thus p ∈ {9581, 95801, 95081, 90581}, and of these numbers only 95801 is prime, but 95801 is not minimal prime since 5801 ◁ 95801.

Case 2: p ends with 3.

In this case we can write p = x3. If x contains 1, 2, 4, 5, 7, or 8, then (respectively) 13p, 23p, 43p, 53p, 73p, or 83p. Hence we may assume all digits of x are 0, 3, 6, or 9, and thus all digits of p are 0, 3, 6, or 9. But then, since the digits of p all have a common factor 3, p is divisible by 3, so p cannot be prime.

Case 3: p ends with 7.

In this case we can write p = x7. If x contains 1, 3, 4, 6, or 9, then (respectively) 17p, 37p, 47p, 67p, or 97p. Hence we may assume all digits of x are 0, 2, 5, 7, or 8.

Case 3.1: p begins with 2.

In this case we can write p = 2y7. If 2 ◁ y, then 227p. If 5 ◁ y, then 257p. If 7 ◁ y, then 277p. Hence we may assume all digits of y are 0 or 8.

If 08 ◁ y, then 2087p. If 88 ◁ y, then 887 ◁ p. Hence we may assume y ∈ {0} ∪ 8{0}.

If y ∈ {0}, then p ∈ 2{0}7. But then, since the sum of the digits of p is 9, p is divisible by 3, so p cannot be prime.

If y ∈ 8{0}, then p ∈ 28{0}7. But then p is divisible by 7, since for n ≥ 0 we have 7 × 40n1 = 280n7, so p cannot be prime.

Case 3.2: p begins with 5.

In this case we can write p = 5y7. If 5 ◁ y, then 557p. If 7 ◁ y, then 577p. If 8 ◁ y, then 587p. Hence we may assume all digits of y are 0 or 2.

If 22 ◁ y, then 227 ◁ p. Hence we may assume y contains zero or one 2's.

If y contains no 2's, then p ∈ 5{0}7. But then, since the sum of the digits of p is 12, p is divisible by 3, so p cannot be prime.

If y contains exactly one 2, then we can write p = 5z2w7, where z,w ∈ {0}. If 0 ◁ z and 0 ◁ w, then 50207p. Hence we may assume either z or w is empty.

If z is empty, then p ∈ 52{0}7, and the smallest prime p ∈ 52{0}7 is 5200007.

If w is empty, then p ∈ 5{0}27, and the smallest prime p ∈ 5{0}27 is 5000000000000000000000000000027.

Case 3.3: p begins with 7.

In this case we can write p = 7y7. If 2 ◁ y, then 727p. If 5 ◁ y, then 757p. If 8 ◁ y, then 787p. Hence we may assume all digits of y are 0 or 7, and thus all digits of p are 0 or 7. But then, since the digits of p all have a common factor 7, p is divisible by 7, so p cannot be prime.

Case 3.4: p begins with 8.

In this case we can write p = 8y7. If 2 ◁ y, then 827p. If 5 ◁ y, then 857p. If 7 ◁ y, then 877p. If 8 ◁ y, then 887p. Hence we may assume y ∈ {0}, and thus p ∈ 8{0}7. But then, since the sum of the digits of p is 15, p is divisible by 3, so p cannot be prime.

Case 4: p ends with 9.

In this case we can write p = x9. If x contains 1, 2, 5, 7, or 8, then (respectively) 19p, 29p, 59p, 79p, or 89p. Hence we may assume all digits of x are 0, 3, 4, 6, or 9.

If 44 ◁ x, then 449p. Hence we may assume x contains zero or one 4's.

If x contains no 4's, then all digits of x are 0, 3, 6, or 9, and thus all digits of p are 0, 3, 6, or 9. But then, since the digits of p all have a common factor 3, p is divisible by 3, so p cannot be prime. Hence we may assume that x contains exactly one 4.

Case 4.1: p begins with 3.

In this case we can write p = 3y4z9, where all digits of y, z are 0, 3, 6, or 9. We must have 349p.

Case 4.2: p begins with 4.

In this case we can write p = 4y9, where all digits of y are 0, 3, 6, or 9. If 0 ◁ y, then 409p. If 3 ◁ y, then 43 ◁ p. If 9 ◁ y, then 499p. Hence we may assume y ∈ {6}, and thus p ∈ 4{6}9. But then p is divisible by 7, since for n ≥ 0 we have 7 × 6n7 = 46n9, so p cannot be prime.

Case 4.3: p begins with 6.

In this case we can write p = 6y4z9, where all digits of y, z are 0, 3, 6, or 9. If 0 ◁ z, then 409 ◁ p. If 3 ◁ z, then 43 ◁ p. If 6 ◁ z, then 6469p. If 9 ◁ z, then 499 ◁ p. Hence we may assume z is empty.

If 3 ◁ y, then 349 ◁ p. If 9 ◁ y, then 6949p. Hence we may assume all digits of y are 0 or 6.

If 06 ◁ y, then 60649p. Hence we may assume y ∈ {6}{0}.

If 666 ◁ y, then 666649p. If 00000 ◁ y, then 60000049p. Hence we may assume y ∈ {𝜆, 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus p ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only 66000049 and 66600049 are primes.

Case 4.4: p begins with 9.

In this case we can write p = 9y4z9, where all digits of y, z are 0, 3, 6, or 9. If 0 ◁ y, then 9049p. If 3 ◁ y, then 349 ◁ p. If 6 ◁ y, then 9649p. If 9 ◁ y, then 9949p. Hence we may assume y is empty.

If 0 ◁ z, then 409 ◁ p. If 3 ◁ z, then 43 ◁ p. If 9 ◁ z, then 499 ◁ p. Hence we may assume z ∈ {6}, and thus p ∈ 94{6}9, and the smallest prime p ∈ 94{6}9 is 946669.

I left as an exercise to the readers to write the proof for bases b = 2, 3, 4, 5, 6, 7, 8, 9, 12, 14, 18 (bases b = 14 and b = 18 have 650 and 549 minimal primes, respectively, they are "a little" many, thus they are "a little" difficult, but you may try them!) like http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_11.pdf) for base b = 10 and https://scholar.colorado.edu/downloads/hh63sw661 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_16.pdf) for bases 2 ≤ b ≤ 10 (warning: the datas for bases 8 and 10 have errors, the data for base 8 misses the prime 6101 and the data for base 10 misses the primes 9001 and 9049 and instead wrongly includes the primes 90001, 90469, and 9000049, and the correct values of Sm for bases 8 and 10 are 15 and 26 (instead of 14 and 27), respectively, also, the "number of minimal primes base b" and the "length of the largest minimal prime base b" are not the same sizes of b but the same sizes of eγ×(b−1)×eulerphi(b), this article has this error is because it only search bases 2 ≤ b ≤ 10, and for the data of 2 ≤ b ≤ 10 for the original minimal problem, you may think that they are the same sizes of b (however, if you extend the data to b = 11, 13, 16, then you will know that they are not the same sizes of b), since bases b = 7 and b = 9 have very large differences of the "number of minimal primes base b" between the original minimal problem and this new minimal prime problem (b = 7: 9 v.s. 71, b = 9: 12 v.s. 151), and bases b = 5 and b = 8 and b = 9 have very large differences of the "length of the largest minimal prime base b" between the original minimal problem and this new minimal prime problem (b = 5: 5 v.s. 96, b = 8: 9 v.s. 221, b = 9: 4 v.s. 1161)) for the original minimal prime problems, of course, the proof for base b = 2 is trivial, since all primes p > 2 must start and end with 1 in base 2, thus we must have 11 ◁ p, however, for some bases b like 24 (the currently "proven" base b (including the primality proving for the primes in the set) with the most number of minimal primes) and 22 (the currently "proven" base b (if unproven probable primes are allowed) with the most number of minimal primes), it is almost impossible to write the proof by hand, since base b = 24 and b = 22 have too many (3409 and 8003, respectively) minimal primes to write the proof (not to mention the "extremely hard" unproven bases b, i.e. bases b = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, you will write the proof (up to the unsolved families) until the end of time!), thus the C++ program code (for computer to compute (https://en.wikipedia.org/wiki/Computing) the proof) is made.

(in fact, the fully proof should also include the primality proving (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/prove3.html, https://t5k.org/prove/prove4.html) for all primes in the set (like https://web.archive.org/web/20020809212051/http://www.users.globalnet.co.uk/~aads/C0034177.html (for the generalized repunit prime in base b = 3 with length 4177) and https://web.archive.org/web/20020701171455/http://www.users.globalnet.co.uk/~aads/C0066883.html (for the generalized repunit prime in base b = 6 with length 6883) and https://web.archive.org/web/20020809122706/http://www.users.globalnet.co.uk/~aads/C0071699.html (for the generalized repunit prime in base b = 7 with length 1699) and https://web.archive.org/web/20020809122635/http://www.users.globalnet.co.uk/~aads/C0101031.html (for the generalized repunit prime in base b = 10 with length 1031) and https://web.archive.org/web/20020809122237/http://www.users.globalnet.co.uk/~aads/C0114801.html (for the generalized repunit prime in base b = 11 with length 4801) and https://web.archive.org/web/20020809122947/http://www.users.globalnet.co.uk/~aads/C0130991.html (for the generalized repunit prime in base b = 13 with length 991) and https://web.archive.org/web/20020809124216/http://www.users.globalnet.co.uk/~aads/C0131021.html (for the generalized repunit prime in base b = 13 with length 1021) and https://web.archive.org/web/20020809125049/http://www.users.globalnet.co.uk/~aads/C0131193.html (for the generalized repunit prime in base b = 13 with length 1193) and https://web.archive.org/web/20020809124458/http://www.users.globalnet.co.uk/~aads/C0152579.html (for the generalized repunit prime in base b = 15 with length 2579) and https://web.archive.org/web/20020809124537/http://www.users.globalnet.co.uk/~aads/C0220857.html (for the generalized repunit prime in base b = 22 with length 857) and https://web.archive.org/web/20020809152611/http://www.users.globalnet.co.uk/~aads/C0315581.html (for the generalized repunit prime in base b = 31 with length 5581) and https://web.archive.org/web/20020809124929/http://www.users.globalnet.co.uk/~aads/C0351297.html (for the generalized repunit prime in base b = 35 with length 1297) and https://stdkmd.net/nrr/pock/ (for the near-repdigit primes, although the primes 2×101755−1 and 2×103020−1 can be quickly proven primes using the N+1 primality proving (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), the primes 2×101755−1 and 2×103020−1 were proven primes by the Pocklington N−1 primality test in the page https://stdkmd.net/nrr/pock/ is because when the page https://stdkmd.net/nrr/pock/ was created (in Aug. 17, 2003), the Morrison N+1 primality test had not been discovered, only the Pocklington N−1 primality test had been discovered) and http://xenon.stanford.edu/~tjw/pp/index.html (for the generalized repunit primes) and https://t5k.org/lists/single_primes/50005cert.txt (for the prime https://t5k.org/primes/page.php?id=12806, https://t5k.org/lists/single_primes/50005bit.html) and https://www.alfredreichlg.de/10w7/cert/primo-10w7_27669.out (for the large prime factor of 1027669+7) and https://www.alfredreichlg.de/10w7/cert/primo-10w7_15093.out (for the prime 1015093+7) and https://www.alfredreichlg.de/10w7/cert/primo-10w7_10393.out (for the large prime factor of 1010393+7) and https://homes.cerias.purdue.edu/~ssw/cun/third/proofs (for the larger prime factors of bn±1 with 2 ≤ b ≤ 12) and https://oeis.hddkillers.com/A057468/3613.out (for the number 33613−23613) and https://oeis.hddkillers.com/A057468/3853.out (for the number 33853−23853) and https://oeis.hddkillers.com/A057468/3929.out (for the number 33929−23929) and https://oeis.hddkillers.com/A057468/5297.out (for the number 35297−25297) and https://oeis.hddkillers.com/A057468/7417.out (for the number 37417−27417) and https://web.archive.org/web/20150911225651/https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0104&L=nmbrthry&P=R1807&D=0 (for the prime https://t5k.org/primes/page.php?id=11084) and http://csic.som.emory.edu/~lzhou/blogs/?p=717 (for the prime https://t5k.org/primes/page.php?id=134345) and http://csic.som.emory.edu/~lzhou/blogs/?p=263 (for the primes corresponding to https://oeis.org/A181980) and https://web.archive.org/web/20131020160719/http://www.primes.viner-steward.org/andy/E/33281741.html (for the prime https://t5k.org/primes/page.php?id=82858), or using an elliptic curve primality proving (https://t5k.org/prove/prove4_2.html, https://en.wikipedia.org/wiki/Elliptic_curve_primality, https://t5k.org/glossary/xpage/ECPP.html, https://mathworld.wolfram.com/EllipticCurvePrimalityProving.html, http://irvinemclean.com/maths/pfaq7.htm, https://t5k.org/top20/page.php?id=27, https://t5k.org/primes/search.php?Comment=ECPP&OnList=all&Number=1000000&Style=HTML, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/~morain/Prgms/ecpp.english.html, https://www.multiprecision.org/cm/ecpp.html, https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1199989-X/S0025-5718-1993-1199989-X.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_256.pdf), https://arxiv.org/pdf/2404.05506.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_428.pdf)) implementation such as PRIMO (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309) or CM (https://www.multiprecision.org/cm/index.html, https://t5k.org/bios/page.php?id=5485, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/cm) to compute primality certificates (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://alfredreichlg.de/cert/certificates.tpm.html, https://alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php, http://5.199.134.130/certificates.tar.xz, http://5.199.134.130/certificates.tar.xz.SHA256SUM, http://5.199.134.130/certificates.tar.xz.par2, http://5.199.134.130/certificates.tar.xz.vol00+10.par2, http://5.199.134.130/certificates/)) and the compositeness proving for all proper subsequence of all primes in the set (usually by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) (usually to 109, this will covered by sieving (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.primegrid.com/stats_psp_sieve.php, https://www.primegrid.com/stats_pps_sieve.php, https://www.primegrid.com/stats_sr5_sieve.php, https://www.primegrid.com/stats_cw_sieve.php, https://www.primegrid.com/sieving/rsp/) for the numbers > 101000) or Fermat primality test (https://t5k.org/prove/prove2_2.html, https://en.wikipedia.org/wiki/Fermat_primality_test, https://en.wikipedia.org/wiki/Fermat_pseudoprime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Fermat_pseudoprimality_test, https://mathworld.wolfram.com/FermatPseudoprime.html, https://www.numbersaplenty.com/set/Poulet_number/, http://www.numericana.com/answer/pseudo.htm#pseudoprime, http://ntheory.org/data/psps.txt, https://www.cecm.sfu.ca/Pseudoprimes/psps-below-2-to-64.txt.bz2, https://www.cecm.sfu.ca/Pseudoprimes/factored-psps-below-2-to-64.txt.bz2, https://www.cecm.sfu.ca/Pseudoprimes/annotated-psps-below-2-to-64.txt.bz2, https://sites.google.com/view/fermat-pseudoprime, https://sites.google.com/view/bases-fermat-pseudoprime, https://oeis.org/A001567, https://oeis.org/A005935, https://oeis.org/A020136, https://oeis.org/A005936, https://oeis.org/A005937, https://oeis.org/A005938, https://oeis.org/A020137, https://oeis.org/A020138, https://oeis.org/A005939, https://oeis.org/A020139, https://oeis.org/A020140, https://oeis.org/A020141, https://oeis.org/A020142, https://oeis.org/A020143, https://oeis.org/A020144, https://oeis.org/A020145, https://oeis.org/A020146, https://oeis.org/A020147, https://oeis.org/A020148, https://oeis.org/A020149, https://oeis.org/A020150, https://oeis.org/A020151, https://oeis.org/A020152, https://oeis.org/A020153, https://oeis.org/A020154, https://oeis.org/A020155, https://oeis.org/A020156, https://oeis.org/A020157, https://oeis.org/A020158, https://oeis.org/A020159, https://oeis.org/A020160, https://oeis.org/A020161, https://oeis.org/A020162, https://oeis.org/A020163, https://oeis.org/A020164, https://oeis.org/A000864, https://oeis.org/A052155, https://oeis.org/A083737, https://oeis.org/A083739, https://oeis.org/A083876, https://oeis.org/A271221, https://oeis.org/A348258, https://oeis.org/A181780, https://oeis.org/A211455, https://oeis.org/A211456, https://oeis.org/A211457, https://oeis.org/A211458, https://oeis.org/A063994, https://oeis.org/A105222, https://oeis.org/A194946, https://oeis.org/A195327, https://oeis.org/A002997, https://oeis.org/A191311, https://oeis.org/A090086, https://oeis.org/A007535, https://oeis.org/A090087, https://oeis.org/A090085, https://oeis.org/A090088, https://oeis.org/A090089, https://oeis.org/A253233, https://oeis.org/A271801) (usually base 2 and base 3)), but in the proof above we assume that we know whether a number is prime or not)

Problems about the digits of prime numbers have a long history, and many of them are still unsolved (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://t5k.org/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, https://t5k.org/notes/conjectures/). For example, are there infinitely many primes, all of whose base-10 digits are 1? Currently, there are only six such "repunits" (https://en.wikipedia.org/wiki/Repunit, https://en.wikipedia.org/wiki/List_of_repunit_primes, https://t5k.org/glossary/xpage/Repunit.html, https://t5k.org/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://www.rieselprime.de/ziki/Generalized_Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://pzktupel.de/Primetables/TableRepunit.php, https://pzktupel.de/Primetables/TableRepunitGen.php, https://pzktupel.de/Primetables/TableRepunitGen.txt, https://stdkmd.net/nrr/prime/prime_rp.htm, https://stdkmd.net/nrr/prime/prime_rp.txt, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html, https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/, https://web.archive.org/web/20120227163614/http://phi.redgolpe.com/5.asp, https://web.archive.org/web/20120227163508/http://phi.redgolpe.com/4.asp, https://web.archive.org/web/20120227163610/http://phi.redgolpe.com/3.asp, https://web.archive.org/web/20120227163512/http://phi.redgolpe.com/2.asp, https://web.archive.org/web/20120227163521/http://phi.redgolpe.com/1.asp, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, https://jpbenney.blogspot.com/2022/04/another-sequence-of-note.html, http://perplexus.info/show.php?pid=8661&cid=51696, https://benvitalenum3ers.wordpress.com/2013/07/24/repunit-11111111111111-in-other-bases/, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, http://www.bitman.name/math/article/380/231/, http://www.bitman.name/math/table/379, https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf), https://t5k.org/top20/page.php?id=57, https://t5k.org/top20/page.php?id=16, https://t5k.org/primes/search.php?Comment=^Repunit&OnList=all&Number=1000000&Style=HTML, https://t5k.org/primes/search.php?Comment=Generalized%20repunit&OnList=all&Number=1000000&Style=HTML, https://oeis.org/A002275, https://oeis.org/A004022, https://oeis.org/A053696, https://oeis.org/A085104, https://oeis.org/A179625) known, corresponding to (10n−1)/9 for n ∈ {2, 19, 23, 317, 1031, 49081, 86453} (references for recently proven prime with n = 49081 and n = 86453: https://www.mersenneforum.org/showpost.php?p=602219&postcount=35, https://www.mersenneforum.org/showpost.php?p=630711&postcount=236, https://t5k.org/top20/page.php?id=57, https://t5k.org/top20/page.php?id=27, https://t5k.org/primes/page.php?id=133761, https://t5k.org/primes/page.php?id=136044, https://stdkmd.net/nrr/prime/prime_rp.htm, https://stdkmd.net/nrr/prime/prime_rp.txt, https://kurtbeschorner.de/db-status-3-1M.htm, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://factordb.com/cert.php?id=1100000000013937242, http://factordb.com/cert.php?id=1100000000046752372, https://stdkmd.net/nrr/cert/Phi/Phi_49081_10.zip, https://stdkmd.net/nrr/cert/Phi/Phi_86453_10.zip, http://www.ellipsa.eu/public/primo/files/ecpp49081-f4.7z.001, http://www.ellipsa.eu/public/primo/files/ecpp49081-f4.7z.002, http://www.ellipsa.eu/public/primo/files/ecpp49081-f4.7z.003, http://www.ellipsa.eu/public/primo/files/ecpp49081-f4.7z.004, http://www.ellipsa.eu/public/primo/files/ecpp49081-f4.7z.005, http://www.ellipsa.eu/public/primo/files/ecpp49081-f4.7z.006). It seems likely that four more are given by n ∈ {109297, 270343, 5794777, 8177207}, but this has not yet been rigorously proven (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://alfredreichlg.de/cert/certificates.tpm.html, https://alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php, http://5.199.134.130/certificates.tar.xz, http://5.199.134.130/certificates.tar.xz.SHA256SUM, http://5.199.134.130/certificates.tar.xz.par2, http://5.199.134.130/certificates.tar.xz.vol00+10.par2, http://5.199.134.130/certificates/). This problem also exists for other bases, e.g. for base 12, there are only nine proven such numbers, corresponding to (12n−1)/11 for n ∈ {2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951}. It seems likely that three more are given by n ∈ {37573, 46889, 769543}, but this has not yet been rigorously proven (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://alfredreichlg.de/cert/certificates.tpm.html, https://alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php, http://5.199.134.130/certificates.tar.xz, http://5.199.134.130/certificates.tar.xz.SHA256SUM, http://5.199.134.130/certificates.tar.xz.par2, http://5.199.134.130/certificates.tar.xz.vol00+10.par2, http://5.199.134.130/certificates/).

Any repunit in any base b having a composite number of digits is necessarily composite. Only repunits (in any base b) having a prime number of digits might be prime. This is a necessary but not sufficient condition, e.g. 11111111111111111111111111111111111 (the repunit with 35 (= 5 × 7, which is composite) digits) = 11111 × 1000010000100001000010000100001 = 1111111 × 10000001000000100000010000001, since 35 = 5 × 7 = 7 × 5, and this repunit factorization does not depend on the base b in which the repunit is expressed. (note that the value of the repunit (in any base b) having 1 digit is 1, and 1 is not prime (https://t5k.org/notes/faq/one.html, https://primefan.tripod.com/Prime1ProCon.html, https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_24.pdf), http://www.numericana.com/answer/numbers.htm#one)).

A repunit (in any base b) with length n can be prime only if n is prime, since otherwise bk×m−1 is a binomial number (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) which can be factored algebraically (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/53335.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/83336.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=algebraic&maxrows=10000, https://sites.google.com/view/algebraic-factors-of-xn-kyn, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://www.mersenneforum.org/showpost.php?p=96560&postcount=99, https://www.mersenneforum.org/showpost.php?p=96651&postcount=101, https://www.mersenneforum.org/showthread.php?t=21916, https://www.mersenneforum.org/showpost.php?p=196598&postcount=492, https://www.mersenneforum.org/showpost.php?p=203083&postcount=149, https://www.mersenneforum.org/showpost.php?p=206065&postcount=192, https://www.mersenneforum.org/showpost.php?p=208044&postcount=260, https://www.mersenneforum.org/showpost.php?p=210533&postcount=336, https://www.mersenneforum.org/showpost.php?p=452132&postcount=66, https://www.mersenneforum.org/showpost.php?p=451337&postcount=32, https://www.mersenneforum.org/showpost.php?p=208852&postcount=227, https://www.mersenneforum.org/showpost.php?p=232904&postcount=604, https://www.mersenneforum.org/showpost.php?p=383690&postcount=1, https://www.mersenneforum.org/showpost.php?p=207886&postcount=253, https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php, https://www.alpertron.com.ar/POLFACT.HTM, https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/). In fact, if n = 2×m is even, then bm−1 = (bm−1) × (bm+1).

This is the list of the known generalized repunit (probable) primes in bases 2 ≤ b ≤ 36 (italic for unproven probable primes (https://en.wikipedia.org/wiki/Probable_prime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Probable_prime, https://mathworld.wolfram.com/ProbablePrime.html, http://www.primenumbers.net/prptop/prptop.php, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#probableprimenumbers, https://stdkmd.net/nrr/repunit/prpfactors.htm, https://www.mersenne.ca/prp.php?show=1, https://www.alfredreichlg.de/10w7/prp.html, http://factordb.com/listtype.php?t=1)): (references: http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/table/379 (in Italian), https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf))

blengths of the generalized repunit primes in base b (written in base 10)search limitOEIS sequence
22, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, ..., 74207281, ..., 77232917, ..., 82589933, ... (the Mersenne primes (https://en.wikipedia.org/wiki/Mersenne_prime, https://en.wikipedia.org/wiki/List_of_Mersenne_primes_and_perfect_numbers, https://t5k.org/glossary/xpage/MersenneNumber.html, https://t5k.org/glossary/xpage/Mersennes.html, https://www.rieselprime.de/ziki/Mersenne_number, https://www.rieselprime.de/ziki/Mersenne_prime, https://www.rieselprime.de/ziki/List_of_known_Mersenne_primes, https://mathworld.wolfram.com/MersenneNumber.html, https://mathworld.wolfram.com/MersennePrime.html, https://pzktupel.de/Primetables/TableMersenne.php, https://t5k.org/top20/page.php?id=4, https://t5k.org/primes/search.php?Comment=Mersenne%20[[:digit:]]&OnList=all&Number=1000000&Style=HTML, https://www.mersenne.org/, https://www.mersenne.ca/, https://www.mersenne.org/primes/, https://www.mersenne.ca/prime.php, https://t5k.org/mersenne/), all are definitely primes, i.e. not merely probable primes)69105331https://oeis.org/A000043
33, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, 2215303, 2704981, 3598867, 7973131, 8530117, ...8530117https://oeis.org/A028491
42 (this is all, since (4n−1)/3 = (2n−1) × (2n+1) / 3, and both 2n−1 and 2n+1 are > 3 for n > 2, thus this factorization is nontrivial for n > 2)
53, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413, 1888279, 3300593, 4939471, 5154509, ...5154509https://oeis.org/A004061
62, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, 79987, 608099, 1365019, 3360347, ...3360347https://oeis.org/A004062
75, 13, 131, 149, 1699, 14221, 35201, 126037, 371669, 1264699, ...1264699https://oeis.org/A004063
83 (this is all, since (8n−1)/7 = (2n−1) × (4n+2n+1) / 7, and both 2n−1 and 4n+2n+1 are > 7 for n > 3, thus this factorization is nontrivial for n > 3, it only remains to check the cases n = 2 and n = 3, but (82−1)/7 = 9 = 32 is not prime)
9(not exist, since (9n−1)/8 = (3n−1) × (3n+1) / 8, and both 3n−1 and 3n+1 are > 8 for n > 2, thus this factorization is nontrivial for n > 2, it only remains to check the case n = 2, but (92−1)/8 = 10 = 2 × 5 is not prime)
102, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, 5794777, 8177207, ...10800000https://oeis.org/A004023
1117, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, 20161, 293831, 1868983, ...1868983https://oeis.org/A005808
122, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951, 37573, 46889, 769543, ...1000000https://oeis.org/A004064
135, 7, 137, 283, 883, 991, 1021, 1193, 3671, 18743, 31751, 101089, 1503503, ...1503503https://oeis.org/A016054
143, 7, 19, 31, 41, 2687, 19697, 59693, 67421, 441697, ...1000000https://oeis.org/A006032
153, 43, 73, 487, 2579, 8741, 37441, 89009, 505117, 639833, ...1000000https://oeis.org/A006033
162 (this is all, since (16n−1)/15 = (4n−1) × (4n+1) / 15, and both 4n−1 and 4n+1 are > 15 for n > 2, thus this factorization is nontrivial for n > 2)
173, 5, 7, 11, 47, 71, 419, 4799, 35149, 54919, 74509, 1990523, ...1990523https://oeis.org/A006034
182, 25667, 28807, 142031, 157051, 180181, 414269, 1270141, ...1270141https://oeis.org/A133857
1919, 31, 47, 59, 61, 107, 337, 1061, 9511, 22051, 209359, ...1000000https://oeis.org/A006035
203, 11, 17, 1487, 31013, 48859, 61403, 472709, 984349, ...1000000https://oeis.org/A127995
213, 11, 17, 43, 271, 156217, 328129, ...1000000https://oeis.org/A127996
222, 5, 79, 101, 359, 857, 4463, 9029, 27823, ...1000000https://oeis.org/A127997
235, 3181, 61441, 91943, 121949, 221411, ...1000000https://oeis.org/A204940
243, 5, 19, 53, 71, 653, 661, 10343, 49307, 115597, 152783, ...1000000https://oeis.org/A127998
25(not exist, since (25n−1)/24 = (5n−1) × (5n+1) / 24, and both 5n−1 and 5n+1 are > 24 for n > 2, thus this factorization is nontrivial for n > 2, it only remains to check the case n = 2, but (252−1)/24 = 26 = 2 × 13 is not prime)
267, 43, 347, 12421, 12473, 26717, ...1000000https://oeis.org/A127999
273 (this is all, since (27n−1)/26 = (3n−1) × (9n+3n+1) / 26, and both 3n−1 and 9n+3n+1 are > 26 for n > 3, thus this factorization is nontrivial for n > 3, it only remains to check the cases n = 2 and n = 3, but (272−1)/26 = 28 = 22×7 is not prime)
282, 5, 17, 457, 1423, 115877, ...1000000https://oeis.org/A128000
295, 151, 3719, 49211, 77237, ...1000000https://oeis.org/A181979
302, 5, 11, 163, 569, 1789, 8447, 72871, 78857, 82883, ...1000000https://oeis.org/A098438
317, 17, 31, 5581, 9973, 54493, 101111, 535571, ...1000000https://oeis.org/A128002
32(not exist, since (32n−1)/31 = (2n−1) × (16n+8n+4n+2n+1) / 31, and both 2n−1 and 16n+8n+4n+2n+1 are > 31 for n > 5, thus this factorization is nontrivial for n > 5, it only remains to check the cases n = 2 and n = 3 and n = 4 and n = 5, but (322−1)/31 = 33 = 3 × 11 and (323−1)/31 = 1057 = 7 × 151 and (324−1)/31 = 33825 = 3 × 52 × 11 × 41 and (325−1)/31 = 1082401 = 601 × 1801 are not primes)
333, 197, 3581, 6871, 183661, ...1000000https://oeis.org/A209120
3413, 1493, 5851, 6379, 125101, ...1000000https://oeis.org/A185073
35313, 1297, 568453, ...1000000https://oeis.org/A348170
362 (this is all, since (36n−1)/35 = (6n−1) × (6n+1) / 35, and both 6n−1 and 6n+1 are > 35 for n > 2, thus this factorization is nontrivial for n > 2)

Determining the set of the minimal elements of a arbitrary set of strings under the subsequence ordering is in general unsolvable, and can be difficult even when this set is relatively simple (such as the base b representations of the prime numbers > b), also, determining the set of the minimal elements of a arbitrary set of strings under the subsequence ordering may be an open problem (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://t5k.org/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, http://www.numericana.com/answer/open.htm, https://t5k.org/notes/conjectures/) or NP-complete (https://en.wikipedia.org/wiki/NP-complete, https://mathworld.wolfram.com/NP-CompleteProblem.html) or an undecidable problem (https://en.wikipedia.org/wiki/Undecidable_problem, https://mathworld.wolfram.com/Undecidable.html), or an example of Gödel's incompleteness theorems (https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems, https://mathworld.wolfram.com/GoedelsFirstIncompletenessTheorem.html) (like the continuum hypothesis (https://en.wikipedia.org/wiki/Continuum_hypothesis, https://mathworld.wolfram.com/ContinuumHypothesis.html) and the halting problem (https://en.wikipedia.org/wiki/Halting_problem, https://mathworld.wolfram.com/HaltingProblem.html)), or as hard as the unsolved problems in mathematics, such as the Riemann hypothesis (https://en.wikipedia.org/wiki/Riemann_hypothesis, https://t5k.org/glossary/xpage/RiemannHypothesis.html, https://mathworld.wolfram.com/RiemannHypothesis.html, http://www.numericana.com/answer/open.htm#rh) and the abc conjecture (https://en.wikipedia.org/wiki/Abc_conjecture, https://mathworld.wolfram.com/abcConjecture.html, http://www.numericana.com/answer/open.htm#abc) and the Schinzel's hypothesis H (https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H, https://mathworld.wolfram.com/SchinzelsHypothesis.html, http://www.numericana.com/answer/open.htm#h) (which are the three famous hard problems in number theory (https://en.wikipedia.org/wiki/Number_theory, https://www.rieselprime.de/ziki/Number_theory, https://mathworld.wolfram.com/NumberTheory.html)) or their generalized conjectures including the generalized Riemann hypothesis (https://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis, https://mathworld.wolfram.com/GeneralizedRiemannHypothesis.html) and the grand Riemann hypothesis (https://en.wikipedia.org/wiki/Grand_Riemann_hypothesis) (both of them are generalized conjectures of the Riemann hypothesis) and the n conjecture (https://en.wikipedia.org/wiki/N_conjecture) (which is a generalized conjecture of the abc conjecture) and the Bateman–Horn conjecture (https://en.wikipedia.org/wiki/Bateman%E2%80%93Horn_conjecture) (which is a generalized conjecture of the Schinzel's hypothesis H).

The following is a "semi-algorithm" (https://en.wikipedia.org/wiki/Semi-algorithm) that is guaranteed to produce the minimal elements of a arbitrary set of strings under the subsequence ordering, but it is not so easy to implement:

  1. M :=
  2. while (L) do
  3. choose x, a shortest string in L
  4. M := M ∪ {x}
  5. L := Lsup({x})

In practice, for arbitrary L, we cannot feasibly carry out step 5. Instead, we work with L', some regular overapproximation to L, until we can show L' = (which implies L = ). In practice, L' is usually chosen to be a finite union of sets of the form L1{L2}L3, where each of L1, L2, L3 is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime > b or not.

To solve this problem (i.e. to compute (https://en.wikipedia.org/wiki/Computing) the set of the minimal elements of the base b representations of the prime numbers > b under the subsequence ordering), we need to determine whether a given family contains a prime. In practice, if family x{Y}z (where x and z are strings (may be empty) of digits in base b, Y is a set of digits in base b) could not be ruled out as only containing composites and Y contains two or more digits, then a relatively small prime > b could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2n−2 strings of length n in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case Y contains only one digit, this family is of the form x{y}z, and there is only a single string of each length > (the length of x + the length of z), and it is not known if the following decision problem (https://en.wikipedia.org/wiki/Decision_problem, https://mathworld.wolfram.com/DecisionProblem.html) is recursively solvable:

Problem: Given strings x, z (may be empty), a digit y, and a base b (x does not start with the digit 0, z ends with a digit which coprime to b, y is not 0 if x is empty, y is coprime to b if z is empty), does there exist a prime number whose base-b expansion is of the form xynz for some n ≥ 0? (Note: In this problem, the base b may be > 36, although the bases b > 36 are not in this project) (If we say "yes", then we should find a such prime (the smallest such prime may be very large, e.g. > 1025000, and if so, then we should use (probable) primality testing (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/index.html) programs (https://www.rieselprime.de/ziki/Primality_testing_program) such as PFGW (https://sourceforge.net/projects/openpfgw/, https://t5k.org/bios/page.php?id=175, https://www.rieselprime.de/ziki/PFGW, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/pfgw_win_4.0.3) or LLR (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64) to find it, and before using these programs, we should use sieving (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.primegrid.com/stats_psp_sieve.php, https://www.primegrid.com/stats_pps_sieve.php, https://www.primegrid.com/stats_sr5_sieve.php, https://www.primegrid.com/stats_cw_sieve.php, https://www.primegrid.com/sieving/rsp/) programs (https://www.rieselprime.de/ziki/Sieving_program) such as srsieve (or sr1/2/5sieve) (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, https://www.mersenneforum.org/showpost.php?p=631129&postcount=1, http://www.rieselprime.de/dl/CRUS_pack.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve_1.8.2, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve-other-programs, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2cl.exe) to remove the numbers either having small prime factors or having algebraic factors) and prove its primality (by N−1 primality test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1) or N+1 primality test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) or elliptic curve primality proving (https://t5k.org/prove/prove4_2.html, https://en.wikipedia.org/wiki/Elliptic_curve_primality, https://t5k.org/glossary/xpage/ECPP.html, https://mathworld.wolfram.com/EllipticCurvePrimalityProving.html, http://irvinemclean.com/maths/pfaq7.htm, https://t5k.org/top20/page.php?id=27, https://t5k.org/primes/search.php?Comment=ECPP&OnList=all&Number=1000000&Style=HTML, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/~morain/Prgms/ecpp.english.html, https://www.multiprecision.org/cm/ecpp.html, https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1199989-X/S0025-5718-1993-1199989-X.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_256.pdf), https://arxiv.org/pdf/2404.05506.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_428.pdf)) implementation such as PRIMO (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309) or CM (https://www.multiprecision.org/cm/index.html, https://t5k.org/bios/page.php?id=5485, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/cm) to compute primality certificates (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://alfredreichlg.de/cert/certificates.tpm.html, https://alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php, http://5.199.134.130/certificates.tar.xz, http://5.199.134.130/certificates.tar.xz.SHA256SUM, http://5.199.134.130/certificates.tar.xz.par2, http://5.199.134.130/certificates.tar.xz.vol00+10.par2, http://5.199.134.130/certificates/)) (and if we want to solve the main problem in this project, we should check whether this prime is the smallest such prime or not, i.e. prove all smaller numbers of the form xynz with n ≥ 0 are composite, usually by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) or Fermat primality test (https://t5k.org/prove/prove2_2.html, https://en.wikipedia.org/wiki/Fermat_primality_test, https://en.wikipedia.org/wiki/Fermat_pseudoprime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Fermat_pseudoprimality_test, https://mathworld.wolfram.com/FermatPseudoprime.html, https://www.numbersaplenty.com/set/Poulet_number/, http://www.numericana.com/answer/pseudo.htm#pseudoprime, http://ntheory.org/data/psps.txt, https://www.cecm.sfu.ca/Pseudoprimes/psps-below-2-to-64.txt.bz2, https://www.cecm.sfu.ca/Pseudoprimes/factored-psps-below-2-to-64.txt.bz2, https://www.cecm.sfu.ca/Pseudoprimes/annotated-psps-below-2-to-64.txt.bz2, https://sites.google.com/view/fermat-pseudoprime, https://sites.google.com/view/bases-fermat-pseudoprime, https://oeis.org/A001567, https://oeis.org/A005935, https://oeis.org/A020136, https://oeis.org/A005936, https://oeis.org/A005937, https://oeis.org/A005938, https://oeis.org/A020137, https://oeis.org/A020138, https://oeis.org/A005939, https://oeis.org/A020139, https://oeis.org/A020140, https://oeis.org/A020141, https://oeis.org/A020142, https://oeis.org/A020143, https://oeis.org/A020144, https://oeis.org/A020145, https://oeis.org/A020146, https://oeis.org/A020147, https://oeis.org/A020148, https://oeis.org/A020149, https://oeis.org/A020150, https://oeis.org/A020151, https://oeis.org/A020152, https://oeis.org/A020153, https://oeis.org/A020154, https://oeis.org/A020155, https://oeis.org/A020156, https://oeis.org/A020157, https://oeis.org/A020158, https://oeis.org/A020159, https://oeis.org/A020160, https://oeis.org/A020161, https://oeis.org/A020162, https://oeis.org/A020163, https://oeis.org/A020164, https://oeis.org/A000864, https://oeis.org/A052155, https://oeis.org/A083737, https://oeis.org/A083739, https://oeis.org/A083876, https://oeis.org/A271221, https://oeis.org/A348258, https://oeis.org/A181780, https://oeis.org/A211455, https://oeis.org/A211456, https://oeis.org/A211457, https://oeis.org/A211458, https://oeis.org/A063994, https://oeis.org/A105222, https://oeis.org/A194946, https://oeis.org/A195327, https://oeis.org/A002997, https://oeis.org/A191311, https://oeis.org/A090086, https://oeis.org/A007535, https://oeis.org/A090087, https://oeis.org/A090085, https://oeis.org/A090088, https://oeis.org/A090089, https://oeis.org/A253233, https://oeis.org/A271801)), and if we say "no", then we should prove that such prime does not exist, may by covering congruence (http://irvinemclean.com/maths/siercvr.htm, http://irvinemclean.com/maths/siernums.htm, http://irvinemclean.com/maths/brier2.htm, https://web.archive.org/web/20221230035324/https://sites.google.com/site/robertgerbicz/coveringsets, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/coveringsets, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/showpost.php?p=638165&postcount=1, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/covset, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/covset-dynam, http://www.numericana.com/answer/primes.htm#sierpinski, http://irvinemclean.com/maths/sierpin.htm, http://irvinemclean.com/maths/sierpin2.htm, http://irvinemclean.com/maths/sierpin3.htm, http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://www.ams.org/journals/mcom/1975-29-129/S0025-5718-1975-0376583-0/S0025-5718-1975-0376583-0.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_27.pdf), https://www.ams.org/journals/mcom/1983-40-161/S0025-5718-1983-0679453-8/S0025-5718-1983-0679453-8.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_40.pdf), https://web.archive.org/web/20231002155518/http://yves.gallot.pagesperso-orange.fr/papers/smallbrier.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_48.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://arxiv.org/pdf/2209.10646.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_52.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL14/Jones/jones12.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_80.pdf), https://web.archive.org/web/20081119135435/http://math.crg4.com/a094076.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_102.pdf), http://www.renyi.hu/~p_erdos/1950-07.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_103.pdf), http://www.fq.math.ca/Scanned/40-3/paulsen.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_331.pdf), http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_1.pdf), https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), http://www.primepuzzles.net/puzzles/puzz_614.htm, http://www.primepuzzles.net/problems/prob_029.htm, http://www.primepuzzles.net/problems/prob_030.htm, http://www.primepuzzles.net/problems/prob_036.htm, http://www.primepuzzles.net/problems/prob_049.htm, https://www.rieselprime.de/Related/LiskovetsGallot.htm, https://www.rieselprime.de/Related/RieselTwinSG.htm, https://stdkmd.net/nrr/coveringset.htm (warning: the case "381w" in this page is in fact combine of covering congruence and algebraic factorization, since it is a combine of two prime factors {3,37} and a difference-of-two-cubes factorization), https://stdkmd.net/nrr/9/91113.htm#prime_period, https://stdkmd.net/nrr/9/94449.htm#prime_period, https://stdkmd.net/nrr/9/95559.htm#prime_period, https://web.archive.org/web/20070220134129/http://www.users.globalnet.co.uk/~perry/maths/wildeprimes/wildeprimes.htm, https://www.rose-hulman.edu/~rickert/Compositeseq/, https://oeis.org/A137985/a137985.txt, http://web.archive.org/web/20081104104617/http://www.csm.astate.edu/~wpaulsen/primemaze/mazeisol.html, https://math.stackexchange.com/questions/1151875/prove-that-f-n-37111111-111-is-never-prime, https://math.stackexchange.com/questions/1153333/prove-that-the-number-19-cdot8n17-is-not-prime-n-in-mathbbz, https://www.quora.com/How-can-I-prove-that-14-n-+-11-for-all-natural-n-is-never-a-prime-number, https://www.quora.com/If-p-is-a-prime-number-does-there-exist-a-natural-number-n-such-that-2-n-p-is-also-a-prime-number-If-so-are-there-infinitely-many-possible-values-of-n, https://oeis.org/A244561, https://oeis.org/A244562, https://oeis.org/A244563, https://oeis.org/A244564, https://oeis.org/A244565, https://oeis.org/A244566, https://oeis.org/A270271, https://oeis.org/A244070, https://oeis.org/A244071, https://oeis.org/A244072, https://oeis.org/A244073, https://oeis.org/A244074, https://oeis.org/A244076, https://oeis.org/A251057, https://oeis.org/A251757, https://oeis.org/A244545, https://oeis.org/A244549, https://oeis.org/A244211, https://oeis.org/A244351, https://oeis.org/A243969, https://oeis.org/A243974, https://oeis.org/A146563, https://oeis.org/A257647, https://oeis.org/A258154, https://oeis.org/A289110, https://oeis.org/A257861, https://oeis.org/A306151, https://oeis.org/A305473, https://oeis.org/A069568 (the Ray Chandler comment for a(37), also the Toshitaka Suzuki comment for the first 6 "a(n) = −1"), https://oeis.org/A069568/a069568.txt, http://list.seqfan.eu/pipermail/seqfan/2023-December/074965.html, https://en.wikipedia.org/wiki/Covering_set, https://www.rieselprime.de/ziki/Covering_set, https://mathworld.wolfram.com/SierpinskisCompositeNumberTheorem.html) (i.e. finding a finite set (https://en.wikipedia.org/wiki/Finite_set, https://mathworld.wolfram.com/FiniteSet.html) S of primes p such that all numbers in a given family are divisible (https://en.wikipedia.org/wiki/Divides, https://t5k.org/glossary/xpage/Divides.html, https://t5k.org/glossary/xpage/Divisor.html, https://www.rieselprime.de/ziki/Factor, https://mathworld.wolfram.com/Divides.html, https://mathworld.wolfram.com/Divisor.html, https://mathworld.wolfram.com/Divisible.html, http://www.numericana.com/answer/primes.htm#divisor) by some element of S (this is equivalent to finding a positive integer N such that all numbers in a given family are not coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to N, this N is usually a factor of a small generalized repunit number (https://en.wikipedia.org/wiki/Repunit, https://en.wikipedia.org/wiki/List_of_repunit_primes, https://t5k.org/glossary/xpage/Repunit.html, https://t5k.org/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://www.rieselprime.de/ziki/Generalized_Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://pzktupel.de/Primetables/TableRepunit.php, https://pzktupel.de/Primetables/TableRepunitGen.php, https://pzktupel.de/Primetables/TableRepunitGen.txt, https://stdkmd.net/nrr/prime/prime_rp.htm, https://stdkmd.net/nrr/prime/prime_rp.txt, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html, https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/, https://web.archive.org/web/20120227163614/http://phi.redgolpe.com/5.asp, https://web.archive.org/web/20120227163508/http://phi.redgolpe.com/4.asp, https://web.archive.org/web/20120227163610/http://phi.redgolpe.com/3.asp, https://web.archive.org/web/20120227163512/http://phi.redgolpe.com/2.asp, https://web.archive.org/web/20120227163521/http://phi.redgolpe.com/1.asp, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.fermatquotient.com/PrimSerien/GenRepuP.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, https://jpbenney.blogspot.com/2022/04/another-sequence-of-note.html, http://perplexus.info/show.php?pid=8661&cid=51696, https://benvitalenum3ers.wordpress.com/2013/07/24/repunit-11111111111111-in-other-bases/, http://www.bitman.name/math/article/380/231/, http://www.bitman.name/math/table/379, http://www.bitman.name/math/table/488, https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_5.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf), https://t5k.org/top20/page.php?id=57, https://t5k.org/top20/page.php?id=16, https://t5k.org/primes/search.php?Comment=^Repunit&OnList=all&Number=1000000&Style=HTML, https://t5k.org/primes/search.php?Comment=Generalized%20repunit&OnList=all&Number=1000000&Style=HTML, https://oeis.org/A002275, https://oeis.org/A004022, https://oeis.org/A053696, https://oeis.org/A085104, https://oeis.org/A179625) in base b, e.g. all numbers in the family 2{5} in base 11 are not coprime to 6, gcd((5×11n−1)/2, 6) can only be 2 or 3, and cannot be 1, also equivalent to finding a prime p such that the least prime factor (http://mathworld.wolfram.com/LeastPrimeFactor.html, https://oeis.org/A020639) of all numbers in a given family is ≤ p, also equivalent to finding a prime p such that all numbers in a given family are not p-rough numbers (https://en.wikipedia.org/wiki/Rough_number, https://mathworld.wolfram.com/RoughNumber.html, https://oeis.org/A007310, https://oeis.org/A007775, https://oeis.org/A008364, https://oeis.org/A008365, https://oeis.org/A008366, https://oeis.org/A166061, https://oeis.org/A166063)), by modular arithmetic (https://en.wikipedia.org/wiki/Modular_arithmetic, https://en.wikipedia.org/wiki/Congruence_relation, https://en.wikipedia.org/wiki/Modulo, https://t5k.org/glossary/xpage/Congruence.html, https://t5k.org/glossary/xpage/CongruenceClass.html, https://t5k.org/glossary/xpage/Residue.html, https://mathworld.wolfram.com/Congruence.html, https://mathworld.wolfram.com/Congruent.html, https://mathworld.wolfram.com/Residue.html, https://mathworld.wolfram.com/MinimalResidue.html, https://mathworld.wolfram.com/Mod.html)), algebraic factorization (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/53335.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/83336.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=algebraic&maxrows=10000, https://sites.google.com/view/algebraic-factors-of-xn-kyn, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://www.mersenneforum.org/showpost.php?p=96560&postcount=99, https://www.mersenneforum.org/showpost.php?p=96651&postcount=101, https://www.mersenneforum.org/showthread.php?t=21916, https://www.mersenneforum.org/showpost.php?p=196598&postcount=492, https://www.mersenneforum.org/showpost.php?p=203083&postcount=149, https://www.mersenneforum.org/showpost.php?p=206065&postcount=192, https://www.mersenneforum.org/showpost.php?p=208044&postcount=260, https://www.mersenneforum.org/showpost.php?p=210533&postcount=336, https://www.mersenneforum.org/showpost.php?p=452132&postcount=66, https://www.mersenneforum.org/showpost.php?p=451337&postcount=32, https://www.mersenneforum.org/showpost.php?p=208852&postcount=227, https://www.mersenneforum.org/showpost.php?p=232904&postcount=604, https://www.mersenneforum.org/showpost.php?p=383690&postcount=1, https://www.mersenneforum.org/showpost.php?p=207886&postcount=253, https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php, https://www.alpertron.com.ar/POLFACT.HTM, https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) (which includes difference-of-two-squares factorization (https://en.wikipedia.org/wiki/Difference_of_two_squares) and sum/difference-of-two-cubes factorization (https://en.wikipedia.org/wiki/Sum_of_two_cubes) and Sophie Germain's identity (https://en.wikipedia.org/wiki/Sophie_Germain%27s_identity, https://www.theoremoftheday.org/Binomial/GermainId/TotDGermainIdentity.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_478.pdf)) and difference-of-two-nth-powers factorization with n > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and sum/difference-of-two-nth-powers factorization with odd n > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and Aurifeuillean factorization (https://en.wikipedia.org/wiki/Aurifeuillean_factorization, https://www.rieselprime.de/ziki/Aurifeuillian_factor, https://mathworld.wolfram.com/AurifeuilleanFactorization.html, http://www.numericana.com/answer/numbers.htm#aurifeuille, https://web.archive.org/web/20231002141924/http://colin.barker.pagesperso-orange.fr/lpa/cycl_fac.htm, http://list.seqfan.eu/oldermail/seqfan/2017-March/017363.html, http://myfactorcollection.mooo.com:8090/source/cyclo.cpp, http://myfactorcollection.mooo.com:8090/LCD_2_199, http://myfactorcollection.mooo.com:8090/LCD_2_998, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/source/cyclo.cpp, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/LucasCD/LCD_2_199, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/LucasCD/LCD_2_998, https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean, https://www.unshlump.com/hcn/aurif.html, https://www.mersenneforum.org/showthread.php?t=10439, https://www.mersenneforum.org/showpost.php?p=515828&postcount=8, https://maths-people.anu.edu.au/~brent/pd/rpb135.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_97.pdf), https://www.ams.org/journals/mcom/2006-75-253/S0025-5718-05-01766-7/S0025-5718-05-01766-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_138.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb127.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_164.pdf), https://www.jams.jp/scm/contents/Vol-2-3/2-3-16.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_167.pdf), https://web.archive.org/web/20130702000532/http://xyyxf.at.tut.by/aurifeuillean.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_443.pdf))), or combine of them (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm (bases b = 55 (k = 2500), b = 63 (k = 3511808 and 27000000), b = 200 (k = 16), b = 225 (k = 114244)), http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm (simple cases (i.e. combine of single prime factor and difference-of-two-squares factorization) such as bases b = 12, 19, 24, 28, 33, 39, 40, 51, 52, 54, 60, complex cases (i.e. other situation) such as bases b = 30 (k = 1369), b = 95 (k = 324), b = 270 (k = 3600), b = 498 (k = 93025), b = 540 (k = 61009), b = 936 (k = 64 and 13689 and 59904), b = 940 (k = 19044), b = 957 (k = 64), b = 1005 (k = 17424 and 85264 and 179776 and 202500), see http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base540-algebraic.htm and http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base936-algebraic.htm and http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base1005-algebraic.htm), https://web.archive.org/web/20070220134129/http://www.users.globalnet.co.uk/~perry/maths/wildeprimes/wildeprimes.htm (case k = 38), https://oeis.org/A069568 (the Ray Chandler comment for a(38), also the Toshitaka Suzuki comment for a(38) and "the general form"), https://www.mersenneforum.org/showthread.php?t=11143, https://www.mersenneforum.org/showthread.php?t=10279, https://www.mersenneforum.org/showthread.php?t=10204, https://www.mersenneforum.org/showpost.php?p=123774&postcount=15, https://www.mersenneforum.org/showpost.php?p=151864&postcount=104, https://www.mersenneforum.org/showpost.php?p=202043&postcount=148, https://www.mersenneforum.org/showpost.php?p=202153&postcount=152, https://www.mersenneforum.org/showpost.php?p=208082&postcount=212, https://www.mersenneforum.org/showpost.php?p=208859&postcount=282, https://www.mersenneforum.org/showpost.php?p=209779&postcount=316, https://www.mersenneforum.org/showpost.php?p=210142&postcount=275, https://www.mersenneforum.org/showpost.php?p=120932&postcount=11, https://math.stackexchange.com/questions/1683082/does-every-sierpinski-number-have-a-finite-congruence-covering, https://math.stackexchange.com/questions/3766036/what-are-some-small-riesel-numbers-without-a-covering-set, https://math.stackexchange.com/questions/760966/is-324455n-ever-prime, https://math.stackexchange.com/questions/625049/a-prime-of-the-form-38111111-ldots, https://www.fq.math.ca/Scanned/33-3/izotov.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_46.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf) (bases b = 63 (k = 3511808), b = 2070 (k = 324)), https://oeis.org/A213353, https://oeis.org/A233469))

An algorithm to solve this problem, for example, would allow us to decide if there are any additional Fermat primes (https://en.wikipedia.org/wiki/Fermat_number, https://t5k.org/glossary/xpage/FermatNumber.html, https://www.rieselprime.de/ziki/Fermat_number, https://mathworld.wolfram.com/FermatNumber.html, https://mathworld.wolfram.com/FermatPrime.html, https://pzktupel.de/Primetables/TableFermat.php, http://www.prothsearch.com/fermat.html, https://t5k.org/top20/page.php?id=8, https://t5k.org/primes/search.php?Comment=Divides&OnList=all&Number=1000000&Style=HTML, http://www.fermatsearch.org/, https://64ordle.au/fermat/, https://64ordle.au/fermat/small/, https://www.primegrid.com/forum_thread.php?id=8778, https://www.primegrid.com/stats_div_llr.php, https://www.primegrid.com/primes/primes.php?project=DIV&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.rieselprime.de/ziki/PrimeGrid_Fermat_Divisor_Search, http://www.fermatsearch.org/factors/faclist.php, http://www.fermatsearch.org/factors/composite.php) (of the form 22n+1) other than the known ones (corresponding to n = 0, 1, 2, 3, 4). To see this, take b = 2, x = 1, y = 0, and z = 0161. Since if 2n+1 is prime then n must be a power of two (https://web.archive.org/web/20231001191526/http://yves.gallot.pagesperso-orange.fr/primes/math.html), a prime of the form xynz in base b must be a new Fermat prime. Besides, it would allow us to decide if there are infinitely many Mersenne primes (https://en.wikipedia.org/wiki/Mersenne_prime, https://en.wikipedia.org/wiki/List_of_Mersenne_primes_and_perfect_numbers, https://t5k.org/glossary/xpage/MersenneNumber.html, https://t5k.org/glossary/xpage/Mersennes.html, https://www.rieselprime.de/ziki/Mersenne_number, https://www.rieselprime.de/ziki/Mersenne_prime, https://www.rieselprime.de/ziki/List_of_known_Mersenne_primes, https://mathworld.wolfram.com/MersenneNumber.html, https://mathworld.wolfram.com/MersennePrime.html, https://pzktupel.de/Primetables/TableMersenne.php, https://t5k.org/top20/page.php?id=4, https://t5k.org/primes/search.php?Comment=Mersenne%20[[:digit:]]&OnList=all&Number=1000000&Style=HTML, https://www.mersenne.org/, https://www.mersenne.ca/, https://www.mersenne.org/primes/, https://www.mersenne.ca/prime.php, https://t5k.org/mersenne/) (of the form 2p−1 with prime p). To see this, take b = 2, x = 𝜆 (the empty string (https://en.wikipedia.org/wiki/Empty_string)), y = 1, and z = 1n+1, where n is the exponent of the Mersenne prime which we want to know whether it is the largest Mersenne prime or not. Since if 2n−1 is prime then n must be a prime (https://t5k.org/notes/proofs/Theorem2.html, https://www.mersenne.org/various/math.php), a prime of the form xynz in base b must be a new Mersenne prime. Also, it would allow us to decide whether 78557 is the smallest Sierpinski number (i.e. odd numbers k such that k×2n+1 is composite for all n ≥ 1) (http://www.prothsearch.com/sierp.html, https://www.primegrid.com/forum_thread.php?id=1647, https://www.primegrid.com/forum_thread.php?id=972, https://www.primegrid.com/forum_thread.php?id=1750, https://www.primegrid.com/forum_thread.php?id=5758, https://www.primegrid.com/stats_sob_llr.php, https://www.primegrid.com/stats_psp_llr.php, https://www.primegrid.com/stats_esp_llr.php, https://www.primegrid.com/primes/primes.php?project=SOB&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=PSP&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=ESP&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://web.archive.org/web/20160405211049/http://www.seventeenorbust.com/, https://en.wikipedia.org/wiki/Sierpi%C5%84ski_number, https://t5k.org/glossary/xpage/SierpinskiNumber.html, https://www.rieselprime.de/ziki/Sierpi%C5%84ski_number, https://www.rieselprime.de/ziki/Sierpi%C5%84ski_problem, https://www.rieselprime.de/ziki/Proth_2_Sierpinski, https://www.rieselprime.de/ziki/Proth_2_Count-0, https://mathworld.wolfram.com/SierpinskiNumberoftheSecondKind.html, https://en.wikipedia.org/wiki/Seventeen_or_Bust, https://www.rieselprime.de/ziki/Seventeen_or_Bust, https://www.rieselprime.de/ziki/PrimeGrid_Seventeen_or_Bust, https://www.rieselprime.de/ziki/PrimeGrid_Prime_Sierpi%C5%84ski_Problem, https://web.archive.org/web/20190929190947/https://primes.utm.edu/glossary/xpage/ColbertNumber.html, https://mathworld.wolfram.com/ColbertNumber.html, http://www.numericana.com/answer/primes.htm#sierpinski, http://www.bitman.name/math/article/204 (in Italian), http://jpenne.free.fr/Sierpeven/, https://www.ams.org/journals/mcom/1983-40-161/S0025-5718-1983-0679453-8/S0025-5718-1983-0679453-8.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_40.pdf), https://www.fq.math.ca/Scanned/33-3/izotov.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_46.pdf), http://www.digizeitschriften.de/download/PPN378850199_0015/PPN378850199_0015___log24.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_213.pdf), https://www.ams.org/journals/mcom/1981-37-155/S0025-5718-1981-0616376-2/S0025-5718-1981-0616376-2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_214.pdf), https://www.ams.org/journals/mcom/1983-41-164/S0025-5718-1983-0717710-7/S0025-5718-1983-0717710-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_215.pdf), http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://oeis.org/A076336) and whether 509203 is the smallest Riesel number (i.e. odd numbers k such that k×2n−1 is composite for all n ≥ 1) (http://www.prothsearch.com/rieselprob.html, https://web.archive.org/web/20081120153544/http://www.15k.org/RieselProblem.htm, https://web.archive.org/web/20081120154015/http://www.15k.org/EvenRieselProblem.htm, https://www.primegrid.com/forum_thread.php?id=1731, https://www.primegrid.com/stats_trp_llr.php, https://www.primegrid.com/primes/primes.php?project=TRP&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://web.archive.org/web/20061021145019/http://rieselsieve.com/, https://web.archive.org/web/20061021153313/http://stats.rieselsieve.com//queue.php, https://en.wikipedia.org/wiki/Riesel_number, https://t5k.org/glossary/xpage/RieselNumber.html, https://www.rieselprime.de/ziki/Riesel_number, https://www.rieselprime.de/ziki/Riesel_problem_1, https://www.rieselprime.de/ziki/Riesel_problem_2, https://www.rieselprime.de/ziki/Riesel_problem_3, https://www.rieselprime.de/ziki/Riesel_problem_4, https://www.rieselprime.de/ziki/Riesel_2_Riesel, https://www.rieselprime.de/ziki/Riesel_2_Count-0, https://mathworld.wolfram.com/RieselNumber.html, https://en.wikipedia.org/wiki/Riesel_Sieve, https://www.rieselprime.de/ziki/Riesel_Sieve, https://www.rieselprime.de/ziki/PrimeGrid_The_Riesel_Problem, http://www.bitman.name/math/article/203 (in Italian), http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://oeis.org/A076337, https://oeis.org/A101036), etc.

(Currently, whether 65537 is the largest Fermat prime, whether there are infinitely many Mersenne primes, whether 78557 is the smallest Sierpinski number, whether 509203 is the smallest Riesel number, are all unsolved problems (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://t5k.org/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, http://www.numericana.com/answer/open.htm, https://t5k.org/notes/conjectures/))

Also, there are some examples in decimal (i.e. base b = 10): (references: https://stdkmd.net/nrr/prime/prime_sequences.htm, https://stdkmd.net/nrr/prime/prime_sequences.txt, https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305200957/https://stdkmd.net/nrr/prime/primecount.htm, https://web.archive.org/web/20240305200920/https://stdkmd.net/nrr/prime/primecount.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt) (see https://sites.google.com/view/smallest-quasi-repdigit-primes for more examples)

xyzanswerfactorization of the first 200 numbers in this family (n is the number of digits in the "{}", start with the smallest allowed n in the table (usually n = 0, unless other n are mentioned))
502801Yes! But the smallest such prime is very large, it is 50280839811, its algebraic form is 5028×1083982+1, its factordb entry is http://factordb.com/index.php?id=1100000000765961536, it can be proven prime by the N−1 primality proving (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1)http://factordb.com/index.php?query=5028*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
70189𝜆 (the empty string (https://en.wikipedia.org/wiki/Empty_string))Yes! But the smallest such prime is very large, it is 70189881309, its algebraic form is 7019×10881309−1, its factordb entry is http://factordb.com/index.php?id=1100000000628445542, it can be proven prime by the N+1 primality proving (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2)http://factordb.com/index.php?query=7019*10%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
917501No! All numbers of this form is divisible by some element of {7,11,13,37}, since the algebraic form of this family is 9175×10n+1+1, it is divisible by 7 if n == 3 mod 6, divisible by 11 if n == 0 mod 2, divisible by 13 if n == 1 mod 6, divisible by 37 if n == 2 mod 3http://factordb.com/index.php?query=9175*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
101759𝜆 (the empty string (https://en.wikipedia.org/wiki/Empty_string))No! All numbers of this form is divisible by some element of {7,11,13,37}, since the algebraic form of this family is 10176×10n−1, it is divisible by 7 if n == 1 mod 6, divisible by 11 if n == 0 mod 2, divisible by 13 if n == 5 mod 6, divisible by 37 if n == 0 mod 3http://factordb.com/index.php?query=10176*10%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
766601Nobody knows! We cannot find a prime in this family, nor can prove that this family only contain composites (by covering congruence, algebraic factorization, or combine of them), the algebraic form of this family is 7666×10n+1+1, and if such prime exists, then it must have n > 5000000http://factordb.com/index.php?query=7666*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
44209𝜆 (the empty string (https://en.wikipedia.org/wiki/Empty_string))Nobody knows! We cannot find a prime in this family, nor can prove that this family only contain composites (by covering congruence, algebraic factorization, or combine of them), the algebraic form of this family is 4421×10n−1, and if such prime exists, then it must have n > 4850000http://factordb.com/index.php?query=4421*10%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
717Yes! But the smallest such prime is very large, it is 71109057, its algebraic form is (64×1010906+53)/9, its factordb entry is http://factordb.com/index.php?id=1000000000008860930, its primality certificate is http://factordb.com/cert.php?id=1000000000008860930 and https://stdkmd.net/nrr/cert/7/71117_10906.ziphttp://factordb.com/index.php?query=%2864*10%5E%28n%2B1%29%2B53%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
949No! All numbers of this form is divisible by some element of {3,7,11,13}, since the algebraic form of this family is (85×10n+1+41)/9, it is divisible by 3 if n == 0 mod 3, divisible by 7 if n == 5 mod 6, divisible by 11 if n == 0 mod 2, divisible by 13 if n == 1 mod 6http://factordb.com/index.php?query=%2885*10%5E%28n%2B1%29%2B41%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
10 (at least two 0)1Nobody knows! We cannot find a prime in this family, nor can prove that this family only contain composites (by covering congruence, algebraic factorization, or combine of them), the algebraic form of this family is 10n+1+1, and if such prime exists, then it must have n ≥ 2147483648http://factordb.com/index.php?query=10%5E%28n%2B1%29%2B1&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
561𝜆 (the empty string (https://en.wikipedia.org/wiki/Empty_string))Yes! But the smallest such prime is very large, it is 56118470, its algebraic form is (505×1018470−1)/9, its factordb entry is http://factordb.com/index.php?id=1100000000301454592, its primality certificate is http://factordb.com/cert.php?id=1100000000301454592 and https://stdkmd.net/nrr/cert/5/56111_18470.ziphttp://factordb.com/index.php?query=%28505*10%5En-1%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
381𝜆 (the empty string (https://en.wikipedia.org/wiki/Empty_string))No! All numbers of this form is either divisible by some element of {3,37} or has a difference-of-two-cubes factorization, since the algebraic form of this family is (343×10n−1)/9, it is divisible by 3 if n == 1 mod 3, divisible by 37 if n == 2 mod 3, and can be factored to (343×10n−1)/9 = (7×10n/3−1) × (49×10n/3+7×10n/3+1) / 9 if n == 0 mod 3http://factordb.com/index.php?query=%28343*10%5En-1%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1761𝜆 (the empty string (https://en.wikipedia.org/wiki/Empty_string))No! All numbers of this form is divisible by some element of {3,7,11,13}, since the algebraic form of this family is (1585×10n−1)/9, it is divisible by 3 if n == 1 mod 3, divisible by 7 if n == 5 mod 6, divisible by 11 if n == 0 mod 2, divisible by 13 if n == 3 mod 6http://factordb.com/index.php?query=%281585*10%5En-1%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
6031𝜆 (the empty string (https://en.wikipedia.org/wiki/Empty_string))Nobody knows! We cannot find a prime in this family, nor can prove that this family only contain composites (by covering congruence, algebraic factorization, or combine of them), the algebraic form of this family is (5428×10n−1)/9, and if such prime exists, then it must have n > 300000http://factordb.com/index.php?query=%285428*10%5En-1%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
𝜆 (the empty string (https://en.wikipedia.org/wiki/Empty_string))299Yes! But the smallest such prime is very large, it is 21915199, its algebraic form is (2×1019153+691)/9, its factordb entry is http://factordb.com/index.php?id=1100000000301493137, its primality certificate is http://factordb.com/cert.php?id=1100000000301493137 and https://stdkmd.net/nrr/cert/2/2w99_19153.ziphttp://factordb.com/index.php?query=%282*10%5E%28n%2B2%29%2B691%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
𝜆 (the empty string (https://en.wikipedia.org/wiki/Empty_string))5 (at least one 5)37No! All numbers of this form is divisible by some element of {3,7,13,37}, since the algebraic form of this family is (5×10n+2−167)/9, it is divisible by 3 if n == 1 mod 3, divisible by 7 if n == 2 mod 6, divisible by 13 if n == 5 mod 6, divisible by 37 if n == 0 mod 3http://factordb.com/index.php?query=%285*10%5E%28n%2B2%29-167%29%2F9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
449 (at least two 9)𝜆 (the empty string (https://en.wikipedia.org/wiki/Empty_string))Yes! But the smallest such prime is very large, it is 44911959, its algebraic form is 45×1011959−1, its factordb entry is http://factordb.com/index.php?id=1100000000291927010, it can be proven prime by the N+1 primality proving (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2)http://factordb.com/index.php?query=45*10%5En-1&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
42103Almost surely Yes! But the smallest such (probable) prime is very large, it is 4210160193, its algebraic form is 421×1016020+3, its factordb entry is http://factordb.com/index.php?id=1100000002392921307, but since this number is only a probable prime and not definitely prime, we cannot definitely say "Yes!" (this family has no known definitely prime)http://factordb.com/index.php?query=421*10%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
40403No! All numbers of this form is divisible by some element of {7,11,13,37}, since the algebraic form of this family is 404×10n+1+3, it is divisible by 7 if n == 5 mod 6, divisible by 11 if n == 0 mod 2, divisible by 13 if n == 1 mod 6, divisible by 37 if n == 0 mod 3http://factordb.com/index.php?query=404*10%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
99499Almost surely Yes! But the smallest such (probable) prime is very large, it is 9943401999, its algebraic form is (895×1034021+491)/9, its factordb entry is http://factordb.com/index.php?id=1100000002454717990, but since this number is only a probable prime and not definitely prime, we cannot definitely say "Yes!" (this family has no known definitely prime)http://factordb.com/index.php?query=%28895*10%5E%28n%2B2%29%2B491%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2567Nobody knows! We cannot find a prime in this family, nor can prove that this family only contain composites (by covering congruence, algebraic factorization, or combine of them), the algebraic form of this family is (23×10n+2+103)/9, and if such prime exists, then it must have n > 30000http://factordb.com/index.php?query=%2823*10%5E%28n%2B2%29%2B103%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
7183Almost surely Yes! But the smallest such (probable) prime is very large, it is 711846683, its algebraic form is (64×1018468+647)/9, its factordb entry is http://factordb.com/index.php?id=1100000000301454024, but since this number is only a probable prime and not definitely prime, we cannot definitely say "Yes!" (this family has no known definitely prime)http://factordb.com/index.php?query=%2864*10%5E%28n%2B2%29%2B647%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3201No! All numbers of this form is divisible by some element of {3,7,11,13}, since the algebraic form of this family is (29×10n+2−191)/9, it is divisible by 3 if n == 1 mod 3, divisible by 7 if n == 0 mod 6, divisible by 11 if n == 1 mod 2, divisible by 13 if n == 2 mod 6http://factordb.com/index.php?query=%2829*10%5E%28n%2B2%29-191%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show

(however, it can be shown that there is no base b ≥ 2 and family xynz (with fixed strings x, z (may be empty), fixed digit y, and variable n) in base b (x does not start with the digit 0, z ends with a digit which coprime to b, y is not 0 if x is empty, y is coprime to b if z is empty) which only contains primes, moreover, it can be shown that there is no base b ≥ 2 and family xynz (with fixed strings x, z (may be empty), fixed digit y, and variable n) in base b (x does not start with the digit 0, z ends with a digit which coprime to b, y is not 0 if x is empty, y is coprime to b if z is empty) which is prime for almost all (https://en.wikipedia.org/wiki/Almost_all, https://mathworld.wolfram.com/AlmostAll.html) n, since the number xynz with the smallest n making xynz > b (if n = 0 already makes xynz > b, then n = 0) (i.e. n = 2 if both x and z are empty, n = 1 if one of x and z is empty, the other has length 1, n = 0 otherwise) is > b and coprime to b, thus xynz must have a prime factor p which does not divide b, and if p divides b−1, then p divides xyk×p+nz for all natural number k, otherwise, p divides xyk×ordp(b)+nz for all natural number k (where ordp(b) is the multiplicative order (https://en.wikipedia.org/wiki/Multiplicative_order, https://t5k.org/glossary/xpage/Order.html, https://mathworld.wolfram.com/MultiplicativeOrder.html, http://www.numbertheory.org/php/order.html, https://oeis.org/A250211, https://oeis.org/A139366, https://oeis.org/A323376, https://oeis.org/A057593, https://oeis.org/A086145) of b mod p), thus there must be infinitely many composites in the family xynz, this theorem also proves that there is no (first kind or second kind) Cunningham chain (https://en.wikipedia.org/wiki/Cunningham_chain, https://t5k.org/glossary/xpage/CunninghamChain.html, https://mathworld.wolfram.com/CunninghamChain.html, https://t5k.org/top20/page.php?id=19, https://t5k.org/top20/page.php?id=20, https://t5k.org/primes/search.php?Comment=Cunningham%20chain&OnList=all&Number=1000000&Style=HTML, https://www.pzktupel.de/JensKruseAndersen/CC.php, http://www.primerecords.dk/Cunningham_Chain_records.htm, http://www.primenumbers.net/Henri/us/NouvTh1us.htm, https://web.archive.org/web/20050406075848/http://ksc9.th.com/warut/cunningham.html, https://oeis.org/A005602, https://oeis.org/A005603, https://oeis.org/A057331, https://oeis.org/A057330) with infinite length, since the first kind Cunningham chain is b = 2, y = 1, z = 𝜆 (the empty string (https://en.wikipedia.org/wiki/Empty_string)), with any given x, and the second kind Cunningham chain is b = 2, y = 0, z = 1, with any given x, also, for a proof for the special case of b = 2, y = 0, either x or z (or both) is 1, of this theorem, see https://oeis.org/A076336/a076336.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_377.pdf))

My conjecture: If family xynz (with fixed strings x, z (may be empty), fixed digit y, and variable n) in base b (with fixed b ≥ 2) (x does not start with the digit 0, z ends with a digit which coprime to b, y is not 0 if x is empty, y is coprime to b if z is empty) cannot be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them), then family xynz in base b contains infinitely many primes (this is equivalent to: If form (a×bn+c)/gcd(a+c,b−1) (with fixed integers a ≥ 1, b ≥ 2, c ≠ 0 (with gcd(a,c) = 1 and gcd(b,c) = 1), and variable n) cannot be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them), then form (a×bn+c)/gcd(a+c,b−1) contains infinitely many primes)

(of course, I hope that this conjecture is true, but this conjecture appears to be harder than the Schinzel's hypothesis H (https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H, https://mathworld.wolfram.com/SchinzelsHypothesis.html, http://www.numericana.com/answer/open.htm#h) and even the Bateman–Horn conjecture (https://en.wikipedia.org/wiki/Bateman%E2%80%93Horn_conjecture), and in fact, no single case is proven to contain infinitely many primes, like the Bunyakovsky conjecture (https://en.wikipedia.org/wiki/Bunyakovsky_conjecture, https://mathworld.wolfram.com/BouniakowskyConjecture.html) and the Dickson's conjecture (https://en.wikipedia.org/wiki/Dickson%27s_conjecture, https://t5k.org/glossary/xpage/DicksonsConjecture.html), no single case is proven to contain infinitely many primes, except the cases in the Dirichlet's theorem (https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions, https://t5k.org/glossary/xpage/DirichletsTheorem.html, https://mathworld.wolfram.com/DirichletsTheorem.html, http://www.numericana.com/answer/primes.htm#dirichlet) (i.e. the polynomials with degree (https://en.wikipedia.org/wiki/Degree_of_a_polynomial, https://mathworld.wolfram.com/PolynomialDegree.html) 1 in the Bunyakovsky conjecture, or the set with 1 polynomial in the Dickson's conjecture))

(this conjecture (https://en.wikipedia.org/wiki/Conjecture, https://t5k.org/glossary/xpage/Conjecture.html, https://mathworld.wolfram.com/Conjecture.html) is very important for the main problem in this project, since if this conjecture is in fact false, then there will may be some unsolved families which in fact contain no primes, thus the main problem in this project in corresponding bases b will may be unsolvable (at least in current technology, unless someone finds a new theorem (i.e. other than covering congruence, algebraic factorization, or combine of them) to prove that some families contain no primes, but I do not think that this is possible), however, this conjecture is currently to far to prove, much far than the Bunyakovsky conjecture (https://en.wikipedia.org/wiki/Bunyakovsky_conjecture, https://mathworld.wolfram.com/BouniakowskyConjecture.html) and even the Schinzel's hypothesis H (https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H, https://mathworld.wolfram.com/SchinzelsHypothesis.html, http://www.numericana.com/answer/open.htm#h), besides, this conjecture is reasonable, since there is a heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, https://web.archive.org/web/20230911032453/https://www.utm.edu/staff/caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf), https://arxiv.org/pdf/2103.04483.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_113.pdf)) that all families which cannot be ruled out as only containing composites or only containing finitely many primes (by covering congruence, algebraic factorization, or combine of them) contain infinitely many primes (references: https://t5k.org/mersenne/heuristic.html, https://t5k.org/notes/faq/NextMersenne.html, https://t5k.org/glossary/xpage/Repunit.html, https://web.archive.org/web/20100628035147/http://www.math.niu.edu/~rusin/known-math/98/exp_primes, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, https://mathoverflow.net/questions/268918/density-of-primes-in-sequences-of-the-form-anb, https://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Mathematics/2023_September_25#Are_there_infinitely_many_primes_of_the_form_1000%E2%80%A60007,_333%E2%80%A63331,_7111%E2%80%A6111,_or_3444%E2%80%A64447_in_base_10?, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf), https://www.mersenneforum.org/showthread.php?t=12327, https://oeis.org/A234285 (the comment by Farideh Firoozbakht, although this comment is not true, there is no prime for s = 509203 and s = −78557, s = 509203 has a covering set of {3, 5, 7, 13, 17, 241}, and s = −78557 has a covering set of {3, 5, 7, 13, 19, 37, 73}), https://www.mersenneforum.org/showpost.php?p=564786&postcount=3, https://www.mersenneforum.org/showpost.php?p=461665&postcount=7, https://www.mersenneforum.org/showpost.php?p=354505&postcount=5, https://www.mersenneforum.org/showpost.php?p=344985&postcount=293, https://www.mersenneforum.org/showpost.php?p=625978&postcount=1027, https://www.primegrid.com/forum_thread.php?id=5093&nowrap=true#66471, https://www.primegrid.com/forum_thread.php?id=4935&nowrap=true#63813, also the graphs https://t5k.org/gifs/lg_lg_Mn.gif (for the family {1} in base b = 2) and https://t5k.org/gifs/repunits.gif (for the family {1} in base b = 10) and https://www.mersenneforum.org/attachment.php?attachmentid=4010&d=1642088235 (for the family 2{0}1 in base b = 3)), since by the prime number theorem (https://en.wikipedia.org/wiki/Prime_number_theorem, https://t5k.org/glossary/xpage/PrimeNumberThm.html, https://mathworld.wolfram.com/PrimeNumberTheorem.html, https://t5k.org/howmany.html, http://www.numericana.com/answer/primes.htm#pnt, https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Tables_of_special_primes) the chance (https://en.wikipedia.org/wiki/Probability, https://mathworld.wolfram.com/Probability.html) that a random (https://en.wikipedia.org/wiki/Random_number, https://mathworld.wolfram.com/RandomNumber.html) n-digit base b number is prime is approximately (https://en.wikipedia.org/wiki/Asymptotic_analysis, https://t5k.org/glossary/xpage/AsymptoticallyEqual.html, https://mathworld.wolfram.com/Asymptotic.html) 1/n (more accurately, the chance is approximately 1/(n×ln(b)), where ln is the natural logarithm (https://en.wikipedia.org/wiki/Natural_logarithm, https://t5k.org/glossary/xpage/Log.html, https://mathworld.wolfram.com/NaturalLogarithm.html), i.e. the logarithm with base e = 2.718281828459... (https://en.wikipedia.org/wiki/E_(mathematical_constant), https://mathworld.wolfram.com/e.html, https://oeis.org/A001113)). If one conjectures the numbers x{y}z behave similarly (i.e. the numbers x{y}z is a pseudorandom sequence (https://en.wikipedia.org/wiki/Pseudorandomness, https://mathworld.wolfram.com/PseudorandomNumber.html, https://people.seas.harvard.edu/~salil/pseudorandomness/pseudorandomness-Aug12.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_197.pdf))) you would expect 1/1 + 1/2 + 1/3 + 1/4 + ... = ∞ (https://en.wikipedia.org/wiki/Harmonic_series_(mathematics), https://mathworld.wolfram.com/HarmonicSeries.html) primes of the form x{y}z (of course, this does not always happen, since some x{y}z families can be ruled out as only containing composites (only count the numbers > b) (by covering congruence, algebraic factorization, or combine of them), and every family has its own Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=50442&postcount=1, https://www.mersenneforum.org/showpost.php?p=50444&postcount=1, https://www.mersenneforum.org/showpost.php?p=201642&postcount=1, https://www.mersenneforum.org/attachment.php?attachmentid=13663&d=1451910741, https://github.com/happy5214/nash, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, https://web.archive.org/web/20230928115952/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, https://web.archive.org/web/20230928120009/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, https://web.archive.org/web/20230928120025/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, https://web.archive.org/web/20230928120047/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)), see https://www.mersenneforum.org/showpost.php?p=564786&postcount=3 and https://www.primegrid.com/forum_thread.php?id=4935&nowrap=true#63813, families which can be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them) have Nash weight (or difficulty) 0, and families which cannot be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them) have positive Nash weight (or difficulty), just like that for the polynomials in the Bunyakovsky conjecture (https://en.wikipedia.org/wiki/Bunyakovsky_conjecture, https://mathworld.wolfram.com/BouniakowskyConjecture.html), every polynomial with degree (https://en.wikipedia.org/wiki/Degree_of_a_polynomial, https://mathworld.wolfram.com/PolynomialDegree.html) ≥ 2 has its own Hardy-Littlewood constant (https://mathworld.wolfram.com/Hardy-LittlewoodConstants.html, https://oeis.org/A221712, https://oeis.org/A221713, https://oeis.org/A331941, https://oeis.org/A199401, https://oeis.org/A337607, https://oeis.org/A331940, https://oeis.org/A331949, https://oeis.org/A003521, https://oeis.org/A003420, https://oeis.org/A003419, https://oeis.org/A003421, https://oeis.org/A331945, https://oeis.org/A331946, https://oeis.org/A331947, https://oeis.org/A331948, https://oeis.org/A332708, https://oeis.org/A332707, https://oeis.org/A342547, https://oeis.org/A342569, https://oeis.org/A342549, https://oeis.org/A342566, https://oeis.org/A331950, https://oeis.org/A342548, https://oeis.org/A221712/a221712.gp.txt, https://oeis.org/A331950/a331950.gp.txt, https://kconrad.math.uconn.edu/articles/hlconst.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_412.pdf), https://oeis.org/A221712/a221712.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_413.pdf)), polynomials which either are reducible (https://en.wikipedia.org/wiki/Irreducible_polynomial, https://mathworld.wolfram.com/IrreduciblePolynomial.html) or have a fixed prime factor (in fact, such prime factors must be ≤ the degree (https://en.wikipedia.org/wiki/Degree_of_a_polynomial, https://mathworld.wolfram.com/PolynomialDegree.html) of the polynomials) (i.e. polynomials which do not satisfy the conditions of the Bunyakovsky conjecture) have Hardy-Littlewood constant 0, and polynomials which are irreducible (https://en.wikipedia.org/wiki/Irreducible_polynomial, https://mathworld.wolfram.com/IrreduciblePolynomial.html) and have no fixed prime factors (in fact, such prime factors must be ≤ the degree (https://en.wikipedia.org/wiki/Degree_of_a_polynomial, https://mathworld.wolfram.com/PolynomialDegree.html) of the polynomials) (i.e. polynomials which satisfy the conditions of the Bunyakovsky conjecture) have positive Hardy-Littlewood constant (see https://oeis.org/A354718 and https://oeis.org/A337164), and for the examples of polynomials with degree (https://en.wikipedia.org/wiki/Degree_of_a_polynomial, https://mathworld.wolfram.com/PolynomialDegree.html) 1, see https://oeis.org/A034693 (smallest k such that n×k+1 is prime) and https://oeis.org/A034694 (smallest prime of the form n×k+1) and https://oeis.org/A053989 (smallest k such that n×k−1 is prime) and https://oeis.org/A038700 (smallest prime of the form n×k−1) and Chebyshev's bias (https://en.wikipedia.org/wiki/Chebyshev%27s_bias, https://mathworld.wolfram.com/ChebyshevBias.html, http://www.math.uiuc.edu/~ford/wwwpapers/lehman.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_59.pdf), https://dms.umontreal.ca/~andrew/PDF/PrimeRace.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_60.pdf), http://math101.guru/wp-content/uploads/2018/09/01-A3-Presentation-v7.3EN-no.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_67.pdf), https://arxiv.org/pdf/1910.08983.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_262.pdf), https://www.ams.org/journals/mcom/2004-73-247/S0025-5718-04-01649-7/S0025-5718-04-01649-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_263.pdf), https://arxiv.org/pdf/math/0010086.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_264.pdf), https://projecteuclid.org/euclid.em/1048515870 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_265.pdf), https://www.ams.org/journals/mcom/2000-69-230/S0025-5718-99-01105-9/S0025-5718-99-01105-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_266.pdf), https://www.ams.org/journals/mcom/1978-32-142/S0025-5718-1978-0476616-X/S0025-5718-1978-0476616-X.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_267.pdf), https://oeis.org/A007350, https://oeis.org/A007352, https://oeis.org/A199547, https://oeis.org/A306891, https://oeis.org/A321856, https://oeis.org/A066520, https://oeis.org/A321857, https://oeis.org/A321859, https://oeis.org/A071838, https://oeis.org/A320857, https://oeis.org/A321860, https://oeis.org/A321858, https://oeis.org/A112632, https://oeis.org/A038698, https://oeis.org/A321862, https://oeis.org/A321864, https://oeis.org/A321861, https://oeis.org/A320858, https://oeis.org/A321865, https://oeis.org/A321863, https://oeis.org/A275939, https://oeis.org/A306499, https://oeis.org/A306500, https://oeis.org/A329224, https://oeis.org/A306502, https://oeis.org/A306503, https://oeis.org/A329225) (if r1 is a quadratic residue (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://mathworld.wolfram.com/QuadraticResidue.html, https://oeis.org/A096008, https://oeis.org/A046071, https://oeis.org/A096103, https://oeis.org/A000224, https://oeis.org/A105612, https://oeis.org/A046073) mod n, r2 is a quadratic nonresidue mod n, then for the primes ≤ N for a random positive integer N, the probability for the number of primes == r2 mod n is more than the number of primes == r1 mod n is larger than 50%), but it is at least a reasonable conjecture in the absence of evidence to the contrary), there are approximately (https://en.wikipedia.org/wiki/Asymptotic_analysis, https://t5k.org/glossary/xpage/AsymptoticallyEqual.html, https://mathworld.wolfram.com/Asymptotic.html) (eγ×W×N−1/1−1/2−1/3−...−1/(length(x)+length(z)−1))/ln(b) primes in the family x{y}z in base b with length ≤ N (where e = 2.718281828459... is the base of the natural logarithm (https://en.wikipedia.org/wiki/E_(mathematical_constant), https://mathworld.wolfram.com/e.html, https://oeis.org/A001113), γ = 0.577215664901 is the Euler–Mascheroni constant (https://en.wikipedia.org/wiki/Euler%27s_constant, https://t5k.org/glossary/xpage/Gamma.html, https://mathworld.wolfram.com/Euler-MascheroniConstant.html, https://oeis.org/A001620), W is the Nash weight (or difficulty) of the family x{y}z in base b (W = 0 if and only if the family x{y}z in base b can be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them)), ln is the natural logarithm (https://en.wikipedia.org/wiki/Natural_logarithm, https://t5k.org/glossary/xpage/Log.html, https://mathworld.wolfram.com/NaturalLogarithm.html) (i.e. the logarithm with base e = 2.718281828459... (https://en.wikipedia.org/wiki/E_(mathematical_constant), https://mathworld.wolfram.com/e.html, https://oeis.org/A001113))).

(Note: Families with higher Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=50442&postcount=1, https://www.mersenneforum.org/showpost.php?p=50444&postcount=1, https://www.mersenneforum.org/showpost.php?p=201642&postcount=1, https://www.mersenneforum.org/attachment.php?attachmentid=13663&d=1451910741, https://github.com/happy5214/nash, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, https://web.archive.org/web/20230928115952/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, https://web.archive.org/web/20230928120009/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, https://web.archive.org/web/20230928120025/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, https://web.archive.org/web/20230928120047/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)) usually have smaller first prime (since the expected number of primes in a given range of n is larger, thus the expected number of the smallest n such that this family gives prime is smaller), but not always, e.g. the base 16 family {3}AF has Nash weight (or difficulty) higher than the base 16 family {4}DD, but the base 16 family {3}AF has the first (probable) prime at length 116139, while the base 16 family {4}DD has the first (probable) prime at length 72787)

(Note: Although most mathematician think that there are only finitely many Fermat primes, and possibly there are only the five known ones (i.e. 3, 5, 17, 257, 65537), and more generally, there are only finitely many generalized Fermat primes to any even base b, and there are only finitely many generalized half Fermat primes to any odd base b (if there are only finitely many generalized Fermat primes to any even base b, and there are only finitely many generalized half Fermat primes to any odd base b, then the unsolved family {F}G in base b = 31 and then the unsolved family 4{0}1 in base b = 32 and then the unsolved family G{0}1 in base b = 32 may have no primes), see https://www.mersenneforum.org/showpost.php?p=116415&postcount=1 and https://www.mersenneforum.org/showpost.php?p=447711&postcount=5 and https://www.mersenneforum.org/showpost.php?p=412191&postcount=3 and https://www.mersenneforum.org/showpost.php?p=447998&postcount=38 and https://www.mersenneforum.org/showpost.php?p=586121&postcount=16 and http://jeppesn.dk/generalized-fermat.html and https://www.primepuzzles.net/conjectures/conj_004.htm and https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf) and https://arxiv.org/pdf/1605.01371.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_29.pdf) and https://oeis.org/A171381 and https://oeis.org/A182331 and https://oeis.org/A078680 and https://oeis.org/history?seq=A257495 (the discussion by Thomas Ordowski at Wed Jun 05 01:18), but I do not think that, since I do not believe that in the sea of infinity, there are no single n > 4 such that 22n+1 is prime, also, all of "Is there an n > 0 such that 38n+1 is prime?", "Is there an n > 0 such that 50n+1 is prime?", "Is there an n > 0 such that 21181×2n+1 is prime?", "Is there an n > 0 such that 1597×6n−1 is prime?", "Is there an n > 0 such that 4×32n+1 is prime?", "Is there an n > 0 such that 4×53n+1 is prime?", "Is there an n > 0 such that 23n+458 is prime?", "Is there an n > 0 such that 38n+31 is prime?" are open problems (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://t5k.org/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, http://www.numericana.com/answer/open.htm, https://t5k.org/notes/conjectures/), since currently there is no known n > 0 such that these formulas give primes, and currently it cannot be proven that there is no n > 0 such that these formulas give primes, their situation are completely the same (for all these forms, nobody knows whether there is such an n or not!), while "Is there an n > 0 such that 78557×2n+1 is prime?", "Is there an n > 0 such that 84687×6n−1 is prime?", "Is there an n > 0 such that 8n+1 is prime?", "Is there an n > 0 such that 4×9n−1 is prime?", "Is there an n > 0 such that 25×12n−1 is prime?", "Is there an n > 0 such that (9n−1)/8 is prime?", "Is there an n > 0 such that 10223×2n+1 is prime?", "Is there an n > 0 such that 36772×6n−1 is prime?", "Is there an n > 0 such that 8×23n+1 is prime?", "Is there an n > 0 such that 25×30n−1 is prime?", "Is there an n > 0 such that 17n+32 is prime?", "Is there an n > 0 such that 29n+10 is prime?" are not open problems (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://t5k.org/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, http://www.numericana.com/answer/open.htm, https://t5k.org/notes/conjectures/), since the answer of them are currently known ("No" for the first four, "Yes" for the last four), and there are known proofs (https://en.wikipedia.org/wiki/Mathematical_proof, https://mathworld.wolfram.com/Proof.html, https://t5k.org/notes/proofs/) of them, thus they are solved problems, maybe the smallest n > 4 such that 22n+1 is prime is around googol (https://en.wikipedia.org/wiki/Googol, https://mathworld.wolfram.com/Googol.html), or around googolplex (https://en.wikipedia.org/wiki/Googolplex, https://mathworld.wolfram.com/Googolplex.html), or around Graham's number (https://en.wikipedia.org/wiki/Graham%27s_number, https://mathworld.wolfram.com/GrahamsNumber.html), or around TREE(3) (https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem, https://mathworld.wolfram.com/KruskalsTreeTheorem.html)? I think that every family xynz (with fixed strings x, z (may be empty), fixed digit y, and variable n) in every base b (with fixed b ≥ 2) contains infinitely many primes unless this family can be proven to only contain composites or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them (of course, there are also generalized Fermat forms and generalized half Fermat forms which can be proven to only contain composites (whose situation are different to other generalized Fermat forms and generalized half Fermat forms, and the same to other forms which can be proven to only contain composites or only contain finitely many primes, e.g. the situation of 78557×2n+1 and 84687×6n−1 and 8n+1 and 4×9n−1 are the same (since all of them can be proven to only contain composites, although the reason of they can be proven to only contain composites are not completely the same, the reason is "covering congruence" for the first two and "algebraic factorization" for the last two), while the situation of 8n+1 and 38n+1 are different (since 8n+1 can be proven to only contain composites, while 38n+1 cannot), the situation of 38n+1 and 21181×2n+1 and 1597×6n−1 and 4×32n+1 are the same (since whether they contain a prime are open problems, currently there is no known n such that they give primes, and currently it cannot be proven that there is no n such that they give primes)), note the difference of them, see https://www.mersenneforum.org/showpost.php?p=210408&postcount=45 and https://www.mersenneforum.org/showpost.php?p=199373&postcount=67), see https://web.archive.org/web/20160418084212/http://staff.spd.dcu.ie/johnbcos/download/Fermat%20material/Fermat%206/Fermat_6.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_465.pdf) and https://oeis.org/A234285 (the comment by Farideh Firoozbakht, although this comment is not true, there is no prime for s = 509203 and s = −78557, s = 509203 has a covering set of {3, 5, 7, 13, 17, 241}, and s = −78557 has a covering set of {3, 5, 7, 13, 19, 37, 73}))

(this conjecture is for exponential sequences (https://en.wikipedia.org/wiki/Exponential_growth, https://mathworld.wolfram.com/ExponentialGrowth.html) (a×bn+c)/gcd(a+c,b−1) (with fixed integers a ≥ 1, b ≥ 2, c ≠ 0, gcd(a, c) = 1, gcd(b, c) = 1, and variable n), there is also a similar conjecture for polynomial sequences (https://en.wikipedia.org/wiki/Polynomial, https://mathworld.wolfram.com/Polynomial.html) a0+a1x+a2x2+a3x3+...+an−1xn−1+anxn (with fixed n, a0, a1, a2, ..., an and variable x): the Bunyakovsky conjecture (https://en.wikipedia.org/wiki/Bunyakovsky_conjecture, https://mathworld.wolfram.com/BouniakowskyConjecture.html), the condition is similar to this conjecture (divisible by small primes and algebraic factorizations), the main difference is that polynomial sequence cannot have a covering congruence with > 1 primes, nor have a combine of covering congruence and algebraic factorization)

This conjecture will imply:

We call families of the form x{y}z (where x and z are strings (may be empty) of digits in base b, y is a digit in base b) "linear" families, and we reduce these families by removing all trailing digits y from x, and removing all leading digits y from z, to make the families be easier, e.g. family 12333{3}33345 in base b is reduced to family 12{3}45 in base b, since they are in fact the same family. Our algorithm then proceeds as follows:

    1. M := {minimal primes in base b of length 2 or 3}, L := union of all x{Y}z (where x and z are strings (may be empty) of digits in base b) such that x ≠ 0 and gcd(z, b) = 1 and Y is the set of digits y in base b such that xyz has no subsequence in M.
    1. While L contains nonlinear families (families which are not linear families): Explore each family of L, and update L. Examine each family of L by:
  • 2.1. Let w be the shortest string in the family. If w has a subsequence in M, then remove the family from L. If w represents a prime, then add w to M and remove the family from L.
  • 2.2. If possible, simplify the family.
  • 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers > b), and if so then remove the family from L.
    1. Update L, after each split examine the new families as in step 2.

e.g. in decimal (base b = 10):

M := {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991}

L := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9}

and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1

and since the family 2{0}1 can be proven to contain no primes > base (since all numbers in this family are divisible by 3), it can be removed

and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1

221 and 2021 are composites, but 20021 is prime, thus add 20021 to L

none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to L

and since the family 3{0,3,6,9}3 can be proven to contain no primes > base (since all numbers in this family are divisible by 3), it can be removed

etc.

The process of exploring/examining/splitting a nonlinear family can be concisely expressed in a tree (https://en.wikipedia.org/wiki/Tree_(graph_theory), https://mathworld.wolfram.com/Tree.html) of decompositions, for an example of a tree of decompositions for a nonlinear family (for the family 1{0,1,6}1 in base b = 9) see figure 1 of https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_17.pdf) and figure 1 of https://cs.uwaterloo.ca/~shallit/Papers/br10.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_18.pdf) and figure 1 of https://doi.org/10.1080/10586458.2015.1064048 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_20.pdf). (in fact, for the main problem in this project, we should consider the family 1{0,1,6,7}1 (in base b = 9) instead of 1{0,1,6}1 (in base b = 9), but these three articles are for the original minimal prime problem (i.e. prime > b is not required))

Shrinking the family x{Y}z (where x and z are strings (may be empty) of digits in base b, Y is a set of digits in base b)

  • If yY and the string xyyz represents a prime > b in base b (in this case, add this prime to the list) or has a subsequence which represents a prime > b in base b, then x{Y}z can be replaced with x{Y \ y}zx{Y \ y}y{Y \ y}z.
  • If y1Y and y2Y and y1y2 and the string xy1y2z represents a prime > b in base b (in this case, add this prime to the list) or has a subsequence which represents a prime > b in base b, then x{Y}z can be replaced with x{Y \ y1}{Y \ y2}z.
  • If y1Y and y2Y and y1y2 and both the strings xy1y2z and xy2y1z represent a prime > b in base b (in this case, add this prime to the list) or have a subsequence which represents a prime > b in base b, then x{Y}z can be replaced with x{Y \ y1}zx{Y \ y2}z.

e.g. in decimal (base b = 10):

  • 2221 is a prime > 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1.
  • 227 is a prime > 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7.
  • 449 is a prime > 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9.
  • Both 5051 and 5501 are primes > 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1.
  • 8501 is a prime > 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1.
  • 887 is a prime > 10, and it is a subsequence of 2887, also 2087 is a prime > 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7.
  • 349 and 449 are primes > 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes > 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9.
  • 251, 281, 521, 821, 881 are primes > 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes > 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}.

If the methods we have discussed cannot be used to rule out or shrink x{Y}z where Y = {y1, y2, ..., yn}, then we can replace x{Y}z by xy1{Y}zxy2{Y}z ∪ ... ∪ xyn{Y}z and re-run the methods on this new language.

If all remain families are linear families (i.e. of the form x{y}z), then we search the smallest (probable) primes in these families and add these primes to the list.

e.g. in decimal (base b = 10):

  • The smallest prime in the family 5{0}27 is 5000000000000000000000000000027.
  • The smallest prime in the family {5}1 is 555555555551.
  • The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not minimal prime since 555555555551 is a subsequence of 8555555555555555555551.

There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the minimal primes in base b for 2 ≤ b ≤ 16 and b = 18, 20, 22, 24, 30. The bases b = 17, 19, 21, 23, 25 ≤ b ≤ 29, 31 ≤ b ≤ 36 are solved with the exception of 793+? families of the form x{y}z (where x and z are strings (may be empty) of digits in base b, y is a digit in base b).

(for the examples of nonlinear families, see https://web.archive.org/web/20240305201337/https://stdkmd.net/nrr/prime/primecount3.htm and https://web.archive.org/web/20240305201316/https://stdkmd.net/nrr/prime/primecount3.txt (only base 10 families), nonlinear families usually have small primes if they cannot be ruled out as only containing composites by covering congruence, see the section above)

We should first make data up to linear families (i.e. only linear families left) (see https://github.com/curtisbright/mepn-data/commit/7acfa0656d3c6b759f95a031f475a30f7664a122 for the original minimal prime problem in bases 2 ≤ b ≤ 26), then searching each left linear family to certain limit of length (say length 1000) (just like the new base script for Conjectures 'R Us Sierpinski/Riesel conjectures base b (http://www.noprimeleftbehind.net/crus/, http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-stats.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-top20.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-proven.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, https://t5k.org/bios/page.php?id=1372, https://www.rieselprime.de/ziki/Conjectures_%27R_Us, https://srbase.my-firewall.org/sr5/, https://srbase.my-firewall.org/sr5/stats.php, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), http://www.bitman.name/math/article/2005 (in Italian)), see http://www.noprimeleftbehind.net/crus/new-bases-5.1.txt and https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/CRUS_pack/scripts/new-bases-5.1.txt, also see https://github.com/curtisbright/mepn-data/commit/e6b2b806f341e9dc5b96662edba2caf3220c98b7 for the original minimal prime problem in bases 2 ≤ b ≤ 28), then find the smallest prime in each left linear family (use SRSIEVE (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, https://www.mersenneforum.org/showpost.php?p=631129&postcount=1, http://www.rieselprime.de/dl/CRUS_pack.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve_1.8.2, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve-other-programs, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2cl.exe) to sieve, then use LLR (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64) or PFGW (https://sourceforge.net/projects/openpfgw/, https://t5k.org/bios/page.php?id=175, https://www.rieselprime.de/ziki/PFGW, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/pfgw_win_4.0.3) to test the probable-primality of the remain numbers, then use PRIMO (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309) to prove the primality of the probable primes < 1025000).

Some x{y}z (where x and z are strings (may be empty) of digits in base b, y is a digit in base b) families can be proven to contain no primes > b, by covering congruence (http://irvinemclean.com/maths/siercvr.htm, http://irvinemclean.com/maths/siernums.htm, http://irvinemclean.com/maths/brier2.htm, https://web.archive.org/web/20221230035324/https://sites.google.com/site/robertgerbicz/coveringsets, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/coveringsets, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/showpost.php?p=638165&postcount=1, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/covset, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/covset-dynam, http://www.numericana.com/answer/primes.htm#sierpinski, http://irvinemclean.com/maths/sierpin.htm, http://irvinemclean.com/maths/sierpin2.htm, http://irvinemclean.com/maths/sierpin3.htm, http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://www.ams.org/journals/mcom/1975-29-129/S0025-5718-1975-0376583-0/S0025-5718-1975-0376583-0.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_27.pdf), https://www.ams.org/journals/mcom/1983-40-161/S0025-5718-1983-0679453-8/S0025-5718-1983-0679453-8.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_40.pdf), https://web.archive.org/web/20231002155518/http://yves.gallot.pagesperso-orange.fr/papers/smallbrier.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_48.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://arxiv.org/pdf/2209.10646.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_52.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL14/Jones/jones12.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_80.pdf), https://web.archive.org/web/20081119135435/http://math.crg4.com/a094076.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_102.pdf), http://www.renyi.hu/~p_erdos/1950-07.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_103.pdf), http://www.fq.math.ca/Scanned/40-3/paulsen.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_331.pdf), http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_1.pdf), https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), http://www.primepuzzles.net/puzzles/puzz_614.htm, http://www.primepuzzles.net/problems/prob_029.htm, http://www.primepuzzles.net/problems/prob_030.htm, http://www.primepuzzles.net/problems/prob_036.htm, http://www.primepuzzles.net/problems/prob_049.htm, https://www.rieselprime.de/Related/LiskovetsGallot.htm, https://www.rieselprime.de/Related/RieselTwinSG.htm, https://stdkmd.net/nrr/coveringset.htm (warning: the case "381w" in this page is in fact combine of covering congruence and algebraic factorization, since it is a combine of two prime factors {3,37} and a difference-of-two-cubes factorization), https://stdkmd.net/nrr/9/91113.htm#prime_period, https://stdkmd.net/nrr/9/94449.htm#prime_period, https://stdkmd.net/nrr/9/95559.htm#prime_period, https://web.archive.org/web/20070220134129/http://www.users.globalnet.co.uk/~perry/maths/wildeprimes/wildeprimes.htm, https://www.rose-hulman.edu/~rickert/Compositeseq/, https://oeis.org/A137985/a137985.txt, http://web.archive.org/web/20081104104617/http://www.csm.astate.edu/~wpaulsen/primemaze/mazeisol.html, https://math.stackexchange.com/questions/1151875/prove-that-f-n-37111111-111-is-never-prime, https://math.stackexchange.com/questions/1153333/prove-that-the-number-19-cdot8n17-is-not-prime-n-in-mathbbz, https://www.quora.com/How-can-I-prove-that-14-n-+-11-for-all-natural-n-is-never-a-prime-number, https://www.quora.com/If-p-is-a-prime-number-does-there-exist-a-natural-number-n-such-that-2-n-p-is-also-a-prime-number-If-so-are-there-infinitely-many-possible-values-of-n, https://oeis.org/A244561, https://oeis.org/A244562, https://oeis.org/A244563, https://oeis.org/A244564, https://oeis.org/A244565, https://oeis.org/A244566, https://oeis.org/A270271, https://oeis.org/A244070, https://oeis.org/A244071, https://oeis.org/A244072, https://oeis.org/A244073, https://oeis.org/A244074, https://oeis.org/A244076, https://oeis.org/A251057, https://oeis.org/A251757, https://oeis.org/A244545, https://oeis.org/A244549, https://oeis.org/A244211, https://oeis.org/A244351, https://oeis.org/A243969, https://oeis.org/A243974, https://oeis.org/A146563, https://oeis.org/A257647, https://oeis.org/A258154, https://oeis.org/A289110, https://oeis.org/A257861, https://oeis.org/A306151, https://oeis.org/A305473, https://oeis.org/A069568 (the Ray Chandler comment for a(37), also the Toshitaka Suzuki comment for the first 6 "a(n) = −1"), https://oeis.org/A069568/a069568.txt, http://list.seqfan.eu/pipermail/seqfan/2023-December/074965.html, https://en.wikipedia.org/wiki/Covering_set, https://www.rieselprime.de/ziki/Covering_set, https://mathworld.wolfram.com/SierpinskisCompositeNumberTheorem.html) (i.e. finding a finite set (https://en.wikipedia.org/wiki/Finite_set, https://mathworld.wolfram.com/FiniteSet.html) S of primes p such that all numbers in a given family are divisible (https://en.wikipedia.org/wiki/Divides, https://t5k.org/glossary/xpage/Divides.html, https://t5k.org/glossary/xpage/Divisor.html, https://www.rieselprime.de/ziki/Factor, https://mathworld.wolfram.com/Divides.html, https://mathworld.wolfram.com/Divisor.html, https://mathworld.wolfram.com/Divisible.html, http://www.numericana.com/answer/primes.htm#divisor) by some element of S (this is equivalent to finding a positive integer N such that all numbers in a given family are not coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to N, this N is usually a factor of a small generalized repunit number (https://en.wikipedia.org/wiki/Repunit, https://en.wikipedia.org/wiki/List_of_repunit_primes, https://t5k.org/glossary/xpage/Repunit.html, https://t5k.org/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://www.rieselprime.de/ziki/Generalized_Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://pzktupel.de/Primetables/TableRepunit.php, https://pzktupel.de/Primetables/TableRepunitGen.php, https://pzktupel.de/Primetables/TableRepunitGen.txt, https://stdkmd.net/nrr/prime/prime_rp.htm, https://stdkmd.net/nrr/prime/prime_rp.txt, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html, https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/, https://web.archive.org/web/20120227163614/http://phi.redgolpe.com/5.asp, https://web.archive.org/web/20120227163508/http://phi.redgolpe.com/4.asp, https://web.archive.org/web/20120227163610/http://phi.redgolpe.com/3.asp, https://web.archive.org/web/20120227163512/http://phi.redgolpe.com/2.asp, https://web.archive.org/web/20120227163521/http://phi.redgolpe.com/1.asp, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.fermatquotient.com/PrimSerien/GenRepuP.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, https://jpbenney.blogspot.com/2022/04/another-sequence-of-note.html, http://perplexus.info/show.php?pid=8661&cid=51696, https://benvitalenum3ers.wordpress.com/2013/07/24/repunit-11111111111111-in-other-bases/, http://www.bitman.name/math/article/380/231/, http://www.bitman.name/math/table/379, http://www.bitman.name/math/table/488, https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_5.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf), https://t5k.org/top20/page.php?id=57, https://t5k.org/top20/page.php?id=16, https://t5k.org/primes/search.php?Comment=^Repunit&OnList=all&Number=1000000&Style=HTML, https://t5k.org/primes/search.php?Comment=Generalized%20repunit&OnList=all&Number=1000000&Style=HTML, https://oeis.org/A002275, https://oeis.org/A004022, https://oeis.org/A053696, https://oeis.org/A085104, https://oeis.org/A179625) in base b, e.g. all numbers in the family 2{5} in base 11 are not coprime to 6, gcd((5×11n−1)/2, 6) can only be 2 or 3, and cannot be 1, also equivalent to finding a prime p such that the least prime factor (http://mathworld.wolfram.com/LeastPrimeFactor.html, https://oeis.org/A020639) of all numbers in a given family is ≤ p, also equivalent to finding a prime p such that all numbers in a given family are not p-rough numbers (https://en.wikipedia.org/wiki/Rough_number, https://mathworld.wolfram.com/RoughNumber.html, https://oeis.org/A007310, https://oeis.org/A007775, https://oeis.org/A008364, https://oeis.org/A008365, https://oeis.org/A008366, https://oeis.org/A166061, https://oeis.org/A166063)), by modular arithmetic (https://en.wikipedia.org/wiki/Modular_arithmetic, https://en.wikipedia.org/wiki/Congruence_relation, https://en.wikipedia.org/wiki/Modulo, https://t5k.org/glossary/xpage/Congruence.html, https://t5k.org/glossary/xpage/CongruenceClass.html, https://t5k.org/glossary/xpage/Residue.html, https://mathworld.wolfram.com/Congruence.html, https://mathworld.wolfram.com/Congruent.html, https://mathworld.wolfram.com/Residue.html, https://mathworld.wolfram.com/MinimalResidue.html, https://mathworld.wolfram.com/Mod.html)), algebraic factorization (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/53335.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/83336.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=algebraic&maxrows=10000, https://sites.google.com/view/algebraic-factors-of-xn-kyn, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://www.mersenneforum.org/showpost.php?p=96560&postcount=99, https://www.mersenneforum.org/showpost.php?p=96651&postcount=101, https://www.mersenneforum.org/showthread.php?t=21916, https://www.mersenneforum.org/showpost.php?p=196598&postcount=492, https://www.mersenneforum.org/showpost.php?p=203083&postcount=149, https://www.mersenneforum.org/showpost.php?p=206065&postcount=192, https://www.mersenneforum.org/showpost.php?p=208044&postcount=260, https://www.mersenneforum.org/showpost.php?p=210533&postcount=336, https://www.mersenneforum.org/showpost.php?p=452132&postcount=66, https://www.mersenneforum.org/showpost.php?p=451337&postcount=32, https://www.mersenneforum.org/showpost.php?p=208852&postcount=227, https://www.mersenneforum.org/showpost.php?p=232904&postcount=604, https://www.mersenneforum.org/showpost.php?p=383690&postcount=1, https://www.mersenneforum.org/showpost.php?p=207886&postcount=253, https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php, https://www.alpertron.com.ar/POLFACT.HTM, https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) (which includes difference-of-two-squares factorization (https://en.wikipedia.org/wiki/Difference_of_two_squares) and sum/difference-of-two-cubes factorization (https://en.wikipedia.org/wiki/Sum_of_two_cubes) and Sophie Germain's identity (https://en.wikipedia.org/wiki/Sophie_Germain%27s_identity, https://www.theoremoftheday.org/Binomial/GermainId/TotDGermainIdentity.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_478.pdf)) and difference-of-two-nth-powers factorization with n > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and sum/difference-of-two-nth-powers factorization with odd n > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and Aurifeuillean factorization (https://en.wikipedia.org/wiki/Aurifeuillean_factorization, https://www.rieselprime.de/ziki/Aurifeuillian_factor, https://mathworld.wolfram.com/AurifeuilleanFactorization.html, http://www.numericana.com/answer/numbers.htm#aurifeuille, https://web.archive.org/web/20231002141924/http://colin.barker.pagesperso-orange.fr/lpa/cycl_fac.htm, http://list.seqfan.eu/oldermail/seqfan/2017-March/017363.html, http://myfactorcollection.mooo.com:8090/source/cyclo.cpp, http://myfactorcollection.mooo.com:8090/LCD_2_199, http://myfactorcollection.mooo.com:8090/LCD_2_998, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/source/cyclo.cpp, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/LucasCD/LCD_2_199, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/LucasCD/LCD_2_998, https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean, https://www.unshlump.com/hcn/aurif.html, https://www.mersenneforum.org/showthread.php?t=10439, https://www.mersenneforum.org/showpost.php?p=515828&postcount=8, https://maths-people.anu.edu.au/~brent/pd/rpb135.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_97.pdf), https://www.ams.org/journals/mcom/2006-75-253/S0025-5718-05-01766-7/S0025-5718-05-01766-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_138.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb127.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_164.pdf), https://www.jams.jp/scm/contents/Vol-2-3/2-3-16.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_167.pdf), https://web.archive.org/web/20130702000532/http://xyyxf.at.tut.by/aurifeuillean.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_443.pdf))), or combine of them (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm (bases b = 55 (k = 2500), b = 63 (k = 3511808 and 27000000), b = 200 (k = 16), b = 225 (k = 114244)), http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm (simple cases (i.e. combine of single prime factor and difference-of-two-squares factorization) such as bases b = 12, 19, 24, 28, 33, 39, 40, 51, 52, 54, 60, complex cases (i.e. other situation) such as bases b = 30 (k = 1369), b = 95 (k = 324), b = 270 (k = 3600), b = 498 (k = 93025), b = 540 (k = 61009), b = 936 (k = 64 and 13689 and 59904), b = 940 (k = 19044), b = 957 (k = 64), b = 1005 (k = 17424 and 85264 and 179776 and 202500), see http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base540-algebraic.htm and http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base936-algebraic.htm and http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base1005-algebraic.htm), https://web.archive.org/web/20070220134129/http://www.users.globalnet.co.uk/~perry/maths/wildeprimes/wildeprimes.htm (case k = 38), https://oeis.org/A069568 (the Ray Chandler comment for a(38), also the Toshitaka Suzuki comment for a(38) and "the general form"), https://www.mersenneforum.org/showthread.php?t=11143, https://www.mersenneforum.org/showthread.php?t=10279, https://www.mersenneforum.org/showthread.php?t=10204, https://www.mersenneforum.org/showpost.php?p=123774&postcount=15, https://www.mersenneforum.org/showpost.php?p=151864&postcount=104, https://www.mersenneforum.org/showpost.php?p=202043&postcount=148, https://www.mersenneforum.org/showpost.php?p=202153&postcount=152, https://www.mersenneforum.org/showpost.php?p=208082&postcount=212, https://www.mersenneforum.org/showpost.php?p=208859&postcount=282, https://www.mersenneforum.org/showpost.php?p=209779&postcount=316, https://www.mersenneforum.org/showpost.php?p=210142&postcount=275, https://www.mersenneforum.org/showpost.php?p=120932&postcount=11, https://math.stackexchange.com/questions/1683082/does-every-sierpinski-number-have-a-finite-congruence-covering, https://math.stackexchange.com/questions/3766036/what-are-some-small-riesel-numbers-without-a-covering-set, https://math.stackexchange.com/questions/760966/is-324455n-ever-prime, https://math.stackexchange.com/questions/625049/a-prime-of-the-form-38111111-ldots, https://www.fq.math.ca/Scanned/33-3/izotov.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_46.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf) (bases b = 63 (k = 3511808), b = 2070 (k = 324)), https://oeis.org/A213353, https://oeis.org/A233469), for me, there is nothing worse then searching a family for a long time that later is proven to contain no primes (e.g. we searched the base 16 families {C}D and {C}DD to length 10000 and searched the base 21 family B0{H}6H to length 20000 and searched the base 30 family A{0}9J to length 10000 in the past, and later we proved that the base 16 families {C}D and {C}DD both have Aurifeuillean factorization of x4+4×y4 and the base 21 family B0{H}6H has a single trivial factor 4637 and the base 30 family A{0}9J has a covering set of {7, 13, 19, 31} with period 6, this is because the original program cannot remove the families which can be ruled out as only containing composites by these conditions: "Aurifeuillean factorization of x4+4×y4" and "single trivial prime factor > b2" and "covering congruence with period > 4", now the program can remove the families which can be ruled out as only containing composites by these conditions: "single trivial prime factor < b3" and "covering congruence with period ≤ 24" and "difference of r-th powers with r ≤ 5" and "Aurifeuillean factorization of x4+4×y4").

The multiplicative order (https://en.wikipedia.org/wiki/Multiplicative_order, https://t5k.org/glossary/xpage/Order.html, https://mathworld.wolfram.com/MultiplicativeOrder.html, http://www.numbertheory.org/php/order.html, https://oeis.org/A250211, https://oeis.org/A139366, https://oeis.org/A323376, https://oeis.org/A057593, https://oeis.org/A086145) is very important in this problem, since if a prime p divides the number with n digits in family x{y}z in base b, then p also divides the number with k×ordp(b)+n digits (where ordp(b) is the multiplicative order (https://en.wikipedia.org/wiki/Multiplicative_order, https://t5k.org/glossary/xpage/Order.html, https://mathworld.wolfram.com/MultiplicativeOrder.html, http://www.numbertheory.org/php/order.html, https://oeis.org/A250211, https://oeis.org/A139366, https://oeis.org/A323376, https://oeis.org/A057593, https://oeis.org/A086145) of b mod p) in family x{y}z in base b for all nonnegative integer k (unless ordp(b) = 1, i.e. p divides b−1, in this case p also divides the number with k×p+n digits in family x{y}z in base b for all nonnegative integer k), the period of "divisible by p" for a prime p in family x{y}z in base b (if some but not all numbers in family x{y}z in base b are divisible by p, of course, if all numbers in family x{y}z in base b are divisible by p, then the period of "divisible by p" for a prime p in family x{y}z in base b is 1) is ordp(b) (by Fermat's little theorem (https://en.wikipedia.org/wiki/Fermat%27s_little_theorem, https://t5k.org/glossary/xpage/FermatsLittleTheorem.html, https://mathworld.wolfram.com/FermatsLittleTheorem.html, https://t5k.org/notes/proofs/FermatsLittleTheorem.html, http://www.numericana.com/answer/modular.htm#fermat), ordp(b) must divide p−1, if and only if ordp(b) is exactly p−1, then b is a primitive root (https://en.wikipedia.org/wiki/Primitive_root_modulo_n, https://mathworld.wolfram.com/PrimitiveRoot.html, https://oeis.org/A060749, https://oeis.org/A001918, https://oeis.org/A071894, https://oeis.org/A002199, https://oeis.org/A008330, https://oeis.org/A219028, https://oeis.org/A046147, https://oeis.org/A046145, https://oeis.org/A174621, https://oeis.org/A248207, https://oeis.org/A046146, https://oeis.org/A046144, https://oeis.org/A219027, https://oeis.org/A033948, https://oeis.org/A033949, https://oeis.org/A138305, https://oeis.org/A122028, https://oeis.org/A002233, https://oeis.org/A223942, https://oeis.org/A219429, https://oeis.org/A138304, https://oeis.org/A002230, https://oeis.org/A002229, https://oeis.org/A114885, https://oeis.org/A023048, https://oeis.org/A133432, https://oeis.org/A133433, https://oeis.org/A318239, https://oeis.org/A066529, https://oeis.org/A056619, https://oeis.org/A023049, https://oeis.org/A280015, https://oeis.org/A029932, https://oeis.org/A002231, https://oeis.org/A079061, https://oeis.org/A079060, https://oeis.org/A214158, https://oeis.org/A259484, https://oeis.org/A103309, https://oeis.org/A103310, https://oeis.org/A084739, https://oeis.org/A133434, https://oeis.org/A263984, https://oeis.org/A001122, https://oeis.org/A019334, https://oeis.org/A019335, https://oeis.org/A019336, https://oeis.org/A019337, https://oeis.org/A019338, https://oeis.org/A001913, https://oeis.org/A019339, https://oeis.org/A019340, https://oeis.org/A019341, https://oeis.org/A019342, https://oeis.org/A019343, https://oeis.org/A019344, https://oeis.org/A019345, https://oeis.org/A019346, https://oeis.org/A019347, https://oeis.org/A019348, https://oeis.org/A019349, https://oeis.org/A019350, https://oeis.org/A019351, https://oeis.org/A019352, https://oeis.org/A019353, https://oeis.org/A019354, https://oeis.org/A019355, https://oeis.org/A019356, https://oeis.org/A019357, https://oeis.org/A019358, https://oeis.org/A019359, https://oeis.org/A019360, https://oeis.org/A019361, https://oeis.org/A167791, https://oeis.org/A167792, https://oeis.org/A167793, https://oeis.org/A167794, https://oeis.org/A167795, https://oeis.org/A167796, https://oeis.org/A167797, https://oeis.org/A240028, https://oeis.org/A240030, https://oeis.org/A240032, https://oeis.org/A240094, https://oeis.org/A240096, https://oeis.org/A240101, https://oeis.org/A240103, https://oeis.org/A240106, https://oeis.org/A240108, https://oeis.org/A001123, https://oeis.org/A001124, https://oeis.org/A001125, https://oeis.org/A001126, https://oeis.org/A061323, https://oeis.org/A061324, https://oeis.org/A061325, https://oeis.org/A061326, https://oeis.org/A061327, https://oeis.org/A061328, https://oeis.org/A061329, https://oeis.org/A061330, https://oeis.org/A061331, https://oeis.org/A061332, https://oeis.org/A061333, https://oeis.org/A061334, https://oeis.org/A061335, https://oeis.org/A061730, https://oeis.org/A061731, https://oeis.org/A061732, https://oeis.org/A061733, https://oeis.org/A061734, https://oeis.org/A061735, https://oeis.org/A061736, https://oeis.org/A061737, https://oeis.org/A061738, https://oeis.org/A055578, https://oeis.org/A101710, http://www.bluetulip.org/2014/programs/primitive.html, http://www.numbertheory.org/php/lprimroot.html, http://sweet.ua.pt/tos/p_roots.html, https://www.ams.org/journals/bull/1942-48-10/S0002-9904-1942-07767-6/S0002-9904-1942-07767-6.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_105.pdf), https://oeis.org/A002223/a002223.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_495.pdf), https://arxiv.org/pdf/2206.14193.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_496.pdf), https://arxiv.org/pdf/1503.04519.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_497.pdf), https://oeis.org/A005180/a005180.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_498.pdf), https://arxiv.org/pdf/math/9807104.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_500.pdf), https://arxiv.org/pdf/1908.11497.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_499.pdf), https://www.ams.org/journals/mcom/2009-78-266/S0025-5718-08-02090-5/S0025-5718-08-02090-5.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_187.pdf)) mod p, and this is studying in Artin's conjecture on primitive roots (https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots, https://mathworld.wolfram.com/ArtinsConjecture.html, http://www.numericana.com/answer/constants.htm#artin), which is an unsolved problem in mathematics) unless p divides b−1, in this case the period of "divisible by p" for such prime p in family x{y}z in base b is simply p, of course, there are cases that a prime p (only consider the primes p not dividing b) divides no numbers in family x{y}z in base b (i.e. the equation (https://en.wikipedia.org/wiki/Equation, https://mathworld.wolfram.com/Equation.html) (a×bn+c)/gcd(a+c,b−1) == 0 mod p has no solution, i.e. the discrete logarithm (https://en.wikipedia.org/wiki/Discrete_logarithm, https://mathworld.wolfram.com/DiscreteLogarithm.html, https://www.alpertron.com.ar/DILOG.HTM, http://www.numbertheory.org/php/discrete_log.html) logb(−c/a) mod p (where c/a is the division in the finite field Zp, which does not exist when a is divisible by p (since in this case a is a zero divisor (https://en.wikipedia.org/wiki/Zero_divisor, https://mathworld.wolfram.com/ZeroDivisor.html) in the finite field Zp), and the equation a×bn+c = 0 in the finite field Zp has no solution if a is divisible by p) has no solution) (this occurs for (p2p×ordp(b)−1)/p2 of the families if p does not divide b−1, or (p−1)/p2 of the families if p divides b−1), and there are cases that a prime p (only consider the primes p not dividing b) divides all numbers in family x{y}z in base b (i.e. all n are solutions of the equation (https://en.wikipedia.org/wiki/Equation, https://mathworld.wolfram.com/Equation.html) (a×bn+c)/gcd(a+c,b−1) == 0 mod p, i.e. all n are solutions of the discrete logarithm (https://en.wikipedia.org/wiki/Discrete_logarithm, https://mathworld.wolfram.com/DiscreteLogarithm.html, https://www.alpertron.com.ar/DILOG.HTM, http://www.numbertheory.org/php/discrete_log.html) logb(−c/a) mod p (where c/a is the division in the finite field Zp, which does not exist when a is divisible by p (since in this case a is a zero divisor (https://en.wikipedia.org/wiki/Zero_divisor, https://mathworld.wolfram.com/ZeroDivisor.html) in the finite field Zp), and the equation a×bn+c = 0 in the finite field Zp has no solution if a is divisible by p)) (this occurs for 1/p2 of the families), e.g. the primes p = 7 and p = 11 and p = 13 and p = 37 and p = 41 divides no numbers in families 1{0}7 or 7{0}1 or {3}1 or 2{3} or 73{1} or 4{6}3 in decimal (base b = 10), see https://oeis.org/A262083 for families 1{0}z in base b = 10, also see these sequences for the primes p (only consider the primes p not dividing b) which divide some numbers in family x{y}z in base b: https://oeis.org/A014662 (1{0}1 in base b = 2), https://oeis.org/A256396 (11{0}1 and 1{0}11 in base b = 2, they are dual families), https://oeis.org/A001915 (10{1} and {1}01 in base b = 2, they are dual families), https://oeis.org/A001916 (100{1} and {1}011 in base b = 2, they are dual families), https://oeis.org/A028416 (1{0}1 in base b = 10) (in fact, all primes p (only consider the primes p not dividing b) divide some numbers in family x{y}z in base b if and only if both x and z are empty, i.e. the family is the repdigit (https://en.wikipedia.org/wiki/Repdigit, https://mathworld.wolfram.com/Repdigit.html, https://oeis.org/A010785) family {y} in base b), in all other cases (i.e. some but not all numbers in family x{y}z in base b are divisible by p, only consider the primes p not dividing b, this occurs for ordp(b)/p of the families if p does not divide b−1, or (p−1)/p of the families if p divides b−1), and p divides the number with n digits in family x{y}z in base b, then p divides the number with m digits in family x{y}z in base b if and only if m == n mod ordp(b), unless ordp(b) = 1, i.e. p divides b−1, in this case p divides the number with m digits in family x{y}z in base b if and only if m == n mod p, in general, if there is a prime p which divides the number with m digits in family x{y}z in base b if and only if m has a congruence mod 2, then the family x{y}z in base b has low Nash weight (or difficulty), also, if there are two prime p and q such that either p or q divides the number with m digits in family x{y}z in base b if and only if m has two congruences mod 3 or three congruences mod 4, then the family x{y}z in base b has extremely low Nash weight (or difficulty), the primes p such that ordp(b) = n are exactly the prime factors of the Zsigmondy number (https://en.wikipedia.org/wiki/Zsigmondy%27s_theorem, https://mathworld.wolfram.com/ZsigmondyTheorem.html) Zs(n, b, 1), Zs(n, b, 1) = Φn(b)/gcd(Φn(b), n) (where Φ is the cyclotomic polynomial (https://en.wikipedia.org/wiki/Cyclotomic_polynomial, https://mathworld.wolfram.com/CyclotomicPolynomial.html, http://www.numericana.com/answer/polynomial.htm#cyclotomic, https://stdkmd.net/nrr/repunit/repunitnote.htm#cyclotomic, https://oeis.org/A013595, https://oeis.org/A013596, https://oeis.org/A253240) if n ≠ 2, Zs(n, 2, 1) = odd part (http://mathworld.wolfram.com/OddPart.html, https://oeis.org/A000265) of n+1, the numbers Zs(n, b, 1) are https://oeis.org/A323748, the prime factors of Zs(n, b, 1) for odd n are exactly the primitive prime factors of bn−1, the prime factors of Zs(n, b, 1) for even n are exactly the primitive prime factors of bn/2+1, and if Zs(n, b, 1) is a prime or prime power (https://oeis.org/A246655, https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html) pr with r ≥ 1, then p is the only one prime such that the multiplicative order of b mod p is n, and p is a unique prime (https://web.archive.org/web/20220602040014/https://en.wikipedia.org/wiki/Unique_prime_number, https://t5k.org/glossary/xpage/UniquePrime.html, https://mathworld.wolfram.com/UniquePrime.html, https://t5k.org/top20/page.php?id=62, https://t5k.org/top20/page.php?id=44, https://t5k.org/primes/search.php?Comment=^Unique&OnList=all&Number=1000000&Style=HTML, https://t5k.org/primes/search.php?Comment=Generalized%20Unique&OnList=all&Number=1000000&Style=HTML, https://stdkmd.net/nrr/cert/Phi/, https://stdkmd.net/nrr/repunit/prpfactors.htm, https://web.archive.org/web/20231015225027/https://www.utm.edu/staff/caldwell/preprints/unique.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_66.pdf), https://oeis.org/A040017, https://oeis.org/A051627, https://oeis.org/A007615, https://oeis.org/A007498) in base b, references: https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=1039706119 (list of the ordp(b) for 2 ≤ b ≤ 128 and primes p ≤ 4096), https://en.wikipedia.org/w/index.php?title=Wikipedia:Sandbox&oldid=1040004339 (list of primes p such that ordp(b) = n for 2 ≤ b ≤ 64 and 1 ≤ n ≤ 64), also factorization of bn±1: https://homes.cerias.purdue.edu/~ssw/cun/index.html (2 ≤ b ≤ 12), https://homes.cerias.purdue.edu/~ssw/cun/pmain524.txt (2 ≤ b ≤ 12), https://doi.org/10.1090/conm/022 (2 ≤ b ≤ 12), https://www.mersenneforum.org/attachment.php?attachmentid=7727&d=1330555980 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_7.pdf) (2 ≤ b ≤ 12), https://web.archive.org/web/20190315214330/http://cage.ugent.be/~jdemeyer/cunningham/ (2 ≤ b ≤ 12), http://myfactorcollection.mooo.com:8090/cgi-bin/showCustomRep?CustomList=B&EN=&LM= (2 ≤ b ≤ 12), http://myfactorcollection.mooo.com:8090/cgi-bin/showREGComps?REGCompList=F&REGSortList=A&LabelList=E&REGHeader=&REGExp= (2 ≤ b ≤ 12), https://maths-people.anu.edu.au/~brent/factors.html (13 ≤ b ≤ 99), https://arxiv.org/pdf/1004.3169.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_206.pdf) (13 ≤ b ≤ 99), https://maths-people.anu.edu.au/~brent/pd/rpb134t.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_208.pdf) (13 ≤ b ≤ 99), http://myfactorcollection.mooo.com:8090/cgi-bin/showANCustomRep?LevelList=A&BaseRangeList=A&EN=&LM= (13 ≤ b ≤ 99), http://myfactorcollection.mooo.com:8090/cgi-bin/showANCustomRep?LevelList=B&BaseRangeList=A&EN=&LM= (13 ≤ b ≤ 99), https://web.archive.org/web/20220513215832/http://myfactorcollection.mooo.com:8090/cgi-bin/showCustomRep?CustomList=A&EN=&LM= (13 ≤ b ≤ 99), http://myfactorcollection.mooo.com:8090/cgi-bin/showANComps?LevelList=A&BaseRangeList=A&REGSortList=A&LabelList=E&REGHeader=&REGExp= (13 ≤ b ≤ 99), http://myfactorcollection.mooo.com:8090/cgi-bin/showANComps?LevelList=B&BaseRangeList=A&REGSortList=A&LabelList=E&REGHeader=&REGExp= (13 ≤ b ≤ 99), http://maths-people.anu.edu.au/~brent/ftp/rpb200t.txt.gz (13 ≤ b ≤ 99), http://maths-people.anu.edu.au/~brent/ftp/factors/comps.gz (13 ≤ b ≤ 99), https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html (2 ≤ b ≤ 999), https://mers.sourceforge.io/factoredM.txt (b = 2, only primitive factors), https://oeis.org/A250197/a250197_2.txt (b = 2, only primitive factors), https://web.archive.org/web/20130531074320/http://www.euronet.nl/users/bota/medium-p-odd.txt (b = 2, +1 side, "1201 ≤ n ≤ 1999, odd n" or "2003 ≤ n ≤ 9973, prime n"), https://web.archive.org/web/20120314040812/http://www.euronet.nl/users/bota/medium-p-even4k.txt (b = 2, +1 side, 1200 ≤ n ≤ 2400, n is divisible by 4), https://web.archive.org/web/20120313035810/http://www.euronet.nl/users/bota/medium-m-odd.txt (b = 2, −1 side, 1201 ≤ n ≤ 2499, odd n), https://www.mersenne.org/report_exponent/ (b = 2, −1 side, prime n), https://www.mersenne.org/report_factors/ (b = 2, −1 side, prime n), https://www.mersenne.org/report_PRP/ (b = 2, −1 side, prime n), https://www.mersenne.org/report_exponent/?exp_lo=2&exp_hi=1000&full=1&ancient=1&expired=1&ecmhist=1&swversion=1 (b = 2, −1 side, prime n), https://www.mersenne.org/report_exponent/?exp_lo=1001&exp_hi=2000&full=1&ancient=1&expired=1&ecmhist=1&swversion=1 (b = 2, −1 side, prime n), https://www.mersenne.org/report_factors/?dispdate=1&exp_hi=999999937 (b = 2, −1 side, prime n), https://www.mersenne.org/report_PRP/?swv=1 (b = 2, −1 side, prime n), https://www.mersenne.ca/prp.php?show=2 (b = 2, −1 side, prime n), https://www.mersenne.ca/exponent/browse/1/9999 (b = 2, −1 side, prime n), https://2721.hddkillers.com/billion/controls/?candnum=1&exponent=1&currentdepth=1&activedepth=1&digits=1&factors=1&searcher=1&candtype=2&rowlimit=0&highstrength=0&highsearcher= (b = 2, −1 side, n ≥ 3321928097, prime n), https://web.archive.org/web/20190222204546/http://home.earthlink.net/~elevensmooth/Billion.html (b = 2, −1 side, n ≥ 3321928097, prime n), https://web.archive.org/web/20211128174912/http://mprime.s3-website.us-west-1.amazonaws.com/mersenne/MERSENNE_FF_with_factors.txt (b = 2, −1 side, prime n), https://web.archive.org/web/20210726214248/http://mprime.s3-website.us-west-1.amazonaws.com/wagstaff/WAGSTAFF_FF_with_factors.txt (b = 2, +1 side, prime n), https://web.archive.org/web/20231126215849/http://bearnol.is-a-geek.com/Mersenneplustwo/Mersenneplustwo.html (b = 2, +1 side, prime n such that 2n−1 is also prime), https://www-users.york.ac.uk/~ss44/cyc/m/mersenne.htm (b = 2, −1 side, prime n, n ≤ 263), https://planetmath.org/tableoffactorsofsmallmersennenumbers (b = 2, −1 side, prime n, n ≤ 199), https://web.archive.org/web/20180525070353/http://home.earthlink.net/~elevensmooth/ElevenFactors.html (b = 2, n divides 1663200), https://web.archive.org/web/20180209113238/http://home.earthlink.net/~elevensmooth/Progress.html (b = 2, n divides 1663200), https://stdkmd.net/nrr/repunit/ (b = 10), https://stdkmd.net/nrr/repunit/10001.htm (b = 10, +1 side), https://stdkmd.net/nrr/repunit/phin10.htm (b = 10, only primitive factors), https://stdkmd.net/nrr/repunit/Phin10.txt (b = 10, only primitive factors), https://stdkmd.net/nrr/repunit/Phin10.txt.lz (b = 10, only primitive factors), https://stdkmd.net/nrr/repunit/Phin10.txt.gz (b = 10, only primitive factors), https://stdkmd.net/nrr/repunit/Phin10ex.txt (b = 10, only primitive factors), https://stdkmd.net/nrr/repunit/Phin10ex.txt.lz (b = 10, only primitive factors), https://stdkmd.net/nrr/repunit/Phin10ex.txt.gz (b = 10, only primitive factors), https://kurtbeschorner.de/ (b = 10, only primitive factors), https://kurtbeschorner.de/fact-2500.htm (b = 10, only primitive factors), https://repunit-koide.jimdofree.com/ (b = 10), https://repunit-koide.jimdofree.com/app/download/10317119350/Repunit100-20240726.pdf?t=1725017883 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_506.pdf) (b = 10), https://gmplib.org/~tege/repunit.html (b = 10, only primitive factors), https://gmplib.org/~tege/fac10m.txt (b = 10, −1 side, only primitive factors), https://gmplib.org/~tege/fac10p.txt (b = 10, +1 side, only primitive factors), https://web.archive.org/web/20070708171301/http://www.ludwigsgymnasium.de/unterr/mathe/prim/zehnp.htm (b = 10, +1 side, only primitive factors), http://chesswanks.com/pxp/repfactors.html (b = 10), https://web.archive.org/web/20191127073837/http://oddperfect.org/composites.html (prime b, −1 side, prime n), https://web.archive.org/web/20170701232547/http://oddperfect.org/cleared.html (prime b, −1 side, prime n), https://www.lirmm.fr/~ochem/opn/checkfacts.txt (prime b, −1 side, prime n), https://www.lirmm.fr/~ochem/opn/checkfacts.txt.gz (prime b, −1 side, prime n), http://myfactorcollection.mooo.com:8090/oddperfect/Sep8_2024/opfactors.gz (prime b, −1 side, prime n, bn < 10850), https://web.archive.org/web/20081006071311/http://www-staff.maths.uts.edu.au/~rons/fact/fact.htm (2 ≤ b ≤ 9973, prime b), http://myfactorcollection.mooo.com:8090/cgi-bin/showCROPComps?OPCompList=A&OPSortList=A&LabelList=E&OPHeader=&OPExp= (2 ≤ b ≤ 9973, prime b, −1 side, prime n), http://myfactorcollection.mooo.com:8090/cgi-bin/showCROPComps?OPCompList=B&OPSortList=A&LabelList=E&OPHeader=&OPExp= (2 ≤ b ≤ 9973, prime b, −1 side, prime n), https://homes.cerias.purdue.edu/~ssw/bell/r1 (3 ≤ b ≤ 179, prime b, n = b, −1 side), https://homes.cerias.purdue.edu/~ssw/bell/r2 (3 ≤ b ≤ 179, prime b, n = b, +1 side), https://homes.cerias.purdue.edu/~ssw/bell/bell4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_86.pdf) (11 ≤ b ≤ 173, prime b, n = b, −1 side), https://homes.cerias.purdue.edu/~ssw/bell/bell.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_87.pdf) (3 ≤ b ≤ 179, prime b, n = b), http://www.prothsearch.com/fermat.html (b = 2, +1 side, power-of-2 n), http://www.prothsearch.com/GFN03.html (b = 3, +1 side, power-of-2 n), http://www.prothsearch.com/GFN05.html (b = 5, +1 side, power-of-2 n), http://www.prothsearch.com/GFN06.html (b = 6, +1 side, power-of-2 n), http://www.prothsearch.com/GFN07.html (b = 7, +1 side, power-of-2 n), http://www.prothsearch.com/GFN10.html (b = 10, +1 side, power-of-2 n), http://www.prothsearch.com/GFN11.html (b = 11, +1 side, power-of-2 n), http://www.prothsearch.com/GFN12.html (b = 12, +1 side, power-of-2 n), http://www.prothsearch.com/GFNfacs.html ("3 ≤ b ≤ 12, +1 side, power-of-2 n" and "b = 2, +1 side, 3-times-power-of-2 n"), http://www.prothsearch.com/GFNsmall.html ("3 ≤ b ≤ 12, +1 side, power-of-2 n" and "b = 2, +1 side, 3-times-power-of-2 n"), http://www.prothsearch.com/OriginalGFNs.html ("3 ≤ b ≤ 12, +1 side, power-of-2 n" and "b = 2, +1 side, 3-times-power-of-2 n"), https://web.archive.org/web/20070910080730/http://members.cox.net/jfoug/GFNFacts_Riesel.html ("3 ≤ b ≤ 12, +1 side, power-of-2 n" and "b = 2, +1 side, 3-times-power-of-2 n"), https://web.archive.org/web/20070914091821/http://members.cox.net/jfoug/GFNFacts_SearchLimits.html ("3 ≤ b ≤ 12, +1 side, power-of-2 n" and "b = 2, +1 side, 3-times-power-of-2 n"), https://web.archive.org/web/20070914092135/http://members.cox.net/jfoug/GFNFacts_ECMComposites.html ("3 ≤ b ≤ 12, +1 side, power-of-2 n" and "b = 2, +1 side, 3-times-power-of-2 n"), https://mers.sourceforge.io/MMPstats.txt (b = 2, −1 side, prime n such that n+1 is power-of-2), http://www.doublemersennes.org/factors.php (b = 2, −1 side, prime n such that n+1 is power-of-2), http://www.doublemersennes.org/history.php (b = 2, −1 side, prime n such that n+1 is power-of-2), http://www.doublemersennes.org/sieving/validi.php (b = 2, −1 side, prime n such that n+1 is power-of-2), http://www.fermatsearch.org/factors/faclist.php (b = 2, +1 side, power-of-2 n), http://www.fermatsearch.org/factors/composite.php (b = 2, +1 side, power-of-2 n), https://www.alpertron.com.ar/MODFERM.HTM (b = 2, power-of-3 n), http://myfactors.mooo.com/ (2 ≤ b ≤ 1100000), http://myfactorcollection.mooo.com:8090/dbio.html (2 ≤ b ≤ 1100000), http://myfactorcollection.mooo.com:8090/interactive.html (2 ≤ b ≤ 1100000) (the lattices saparated to two lattices means the number has Aurifeuillean factorization, and for such lattices, the left lattice is for the Aurifeuillean L part, and the right lattice is for the Aurifeuillean M part), http://myfactorcollection.mooo.com:8090/brentdata/Aug31_2024/factors.gz (2 ≤ b ≤ 1100000), http://maths-people.anu.edu.au/~brent/ftp/factors/factors.gz (2 ≤ b ≤ 9999, only prime factors > 109), http://www.asahi-net.or.jp/~KC2H-MSM/cn/old/index.htm (2 ≤ b ≤ 1000, only primitive factors), http://www.asahi-net.or.jp/~KC2H-MSM/cn/index.htm (2 ≤ b ≤ 1000, only primitive factors), https://web.archive.org/web/20050922233702/http://user.ecc.u-tokyo.ac.jp/~g440622/cn/index.html (2 ≤ b ≤ 1000, only primitive factors), https://web.archive.org/web/20070629012309/http://subsite.icu.ac.jp/people/mitsuo/enbunsu/table.html (2 ≤ b ≤ 1000, only primitive factors), also for the factors of bn±1 with 2 ≤ b ≤ 400 and 1 ≤ n ≤ 400 and for the first holes of bn±1 with 2 ≤ b ≤ 400 see the links in the list below, also, the period of "difference-of-two-squares factorization" in any base b is 2 if b is not square, 1 if b is square; the period of "sum/difference-of-two-pth-powers factorization with odd prime p" is p if b is not p-th power, 1 if b is p-th power; the period of "Aurifeuillean factorization of x4+4×y4" is 4 if b is not square, 2 if b is square but not 4th power, 1 if b is 4th power, (for more information, see https://stdkmd.net/nrr/1/10003.htm#prime_period, https://stdkmd.net/nrr/3/30001.htm#prime_period, https://stdkmd.net/nrr/1/11113.htm#prime_period, https://stdkmd.net/nrr/3/31111.htm#prime_period, https://stdkmd.net/nrr/1/13333.htm#prime_period, https://stdkmd.net/nrr/3/33331.htm#prime_period, https://stdkmd.net/nrr/1/11131.htm#prime_period, https://stdkmd.net/nrr/1/13111.htm#prime_period, https://stdkmd.net/nrr/3/31333.htm#prime_period, https://stdkmd.net/nrr/3/33313.htm#prime_period, https://stdkmd.net/nrr/1/13331.htm#prime_period, https://stdkmd.net/nrr/3/31113.htm#prime_period, https://oeis.org/A014664, https://oeis.org/A062117, https://oeis.org/A002371, https://oeis.org/A108974, https://oeis.org/A129733, https://oeis.org/A046107, https://oeis.org/A204845, http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm) the family x{y}z in base b can be proven to contain no primes > b (or only contain finitely many primes > b) if and only if these residue classes with these periods gives a complete residue system (https://en.wikipedia.org/wiki/Covering_system, https://mathworld.wolfram.com/CompleteResidueSystem.html).

range of bases bthe factors of bn±1 with 1 ≤ n ≤ 100the factors of bn±1 with 101 ≤ n ≤ 200the factors of bn±1 with 201 ≤ n ≤ 300the factors of bn±1 with 301 ≤ n ≤ 400the first holes of bn±1 and their known prime factors
2 ≤ b ≤ 100http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=100&FExp=1&TExp=100&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=2&TBase=100&FExp=1&TExp=100&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=2&TBase=100&FExp=1&TExp=100&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=100&FExp=101&TExp=200&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=2&TBase=100&FExp=101&TExp=200&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=2&TBase=100&FExp=101&TExp=200&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=100&FExp=201&TExp=300&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=2&TBase=100&FExp=201&TExp=300&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=2&TBase=100&FExp=201&TExp=300&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=100&FExp=301&TExp=400&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=2&TBase=100&FExp=301&TExp=400&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=2&TBase=100&FExp=301&TExp=400&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)http://myfactorcollection.mooo.com:8090/cgi-bin/showFH?FBase=2&TBase=100&c0=&Expanded=, http://myfactorcollection.mooo.com:8090/cgi-bin/showCRHoles?BaseRangeList=A (bases 13 ≤ b ≤ 100 instead of 2 ≤ b ≤ 100, also list known prime factors only for bn < 10255), http://maths-people.anu.edu.au/~brent/ftp/factors/holes.txt (bases 13 ≤ b ≤ 100 instead of 2 ≤ b ≤ 100, also list known prime factors only for bn < 10255)
101 ≤ b ≤ 200http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=101&TBase=200&FExp=1&TExp=100&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=101&TBase=200&FExp=1&TExp=100&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=101&TBase=200&FExp=1&TExp=100&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=101&TBase=200&FExp=101&TExp=200&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=101&TBase=200&FExp=101&TExp=200&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=101&TBase=200&FExp=101&TExp=200&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=101&TBase=200&FExp=201&TExp=300&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=101&TBase=200&FExp=201&TExp=300&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=101&TBase=200&FExp=201&TExp=300&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=101&TBase=200&FExp=301&TExp=400&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=101&TBase=200&FExp=301&TExp=400&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=101&TBase=200&FExp=301&TExp=400&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)http://myfactorcollection.mooo.com:8090/cgi-bin/showFH?FBase=101&TBase=200&c0=&Expanded=, http://myfactorcollection.mooo.com:8090/cgi-bin/showCRHoles?BaseRangeList=B (bases 101 ≤ b ≤ 199 instead of 101 ≤ b ≤ 200, also list known prime factors only for bn < 10255)
201 ≤ b ≤ 300http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=201&TBase=300&FExp=1&TExp=100&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=201&TBase=300&FExp=1&TExp=100&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=201&TBase=300&FExp=1&TExp=100&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=201&TBase=300&FExp=101&TExp=200&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=201&TBase=300&FExp=101&TExp=200&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=201&TBase=300&FExp=101&TExp=200&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=201&TBase=300&FExp=201&TExp=300&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=201&TBase=300&FExp=201&TExp=300&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=201&TBase=300&FExp=201&TExp=300&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=201&TBase=300&FExp=301&TExp=400&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=201&TBase=300&FExp=301&TExp=400&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=201&TBase=300&FExp=301&TExp=400&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)http://myfactorcollection.mooo.com:8090/cgi-bin/showFH?FBase=201&TBase=300&c0=&Expanded=, http://myfactorcollection.mooo.com:8090/cgi-bin/showCRHoles?BaseRangeList=C (bases 200 ≤ b ≤ 299 instead of 201 ≤ b ≤ 300, also list known prime factors only for bn < 10255)
301 ≤ b ≤ 400http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=301&TBase=400&FExp=1&TExp=100&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=301&TBase=400&FExp=1&TExp=100&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=301&TBase=400&FExp=1&TExp=100&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=301&TBase=400&FExp=101&TExp=200&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=301&TBase=400&FExp=101&TExp=200&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=301&TBase=400&FExp=101&TExp=200&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=301&TBase=400&FExp=201&TExp=300&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=301&TBase=400&FExp=201&TExp=300&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=301&TBase=400&FExp=201&TExp=300&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=301&TBase=400&FExp=301&TExp=400&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=301&TBase=400&FExp=301&TExp=400&c0=&LM= (only primitive factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showComps?FBase=301&TBase=400&FExp=301&TExp=400&FDigits=&TDigits=&FTL=&TTL=&c0=&MaxCount=&LabelList=E&SortList=A&CompExp= (remaining composites)http://myfactorcollection.mooo.com:8090/cgi-bin/showFH?FBase=301&TBase=400&c0=&Expanded=, http://myfactorcollection.mooo.com:8090/cgi-bin/showCRHoles?BaseRangeList=D (bases 300 ≤ b ≤ 400 instead of 301 ≤ b ≤ 400, also list known prime factors only for bn < 10255)

The above section only includes the multiplicative order (https://en.wikipedia.org/wiki/Multiplicative_order, https://t5k.org/glossary/xpage/Order.html, https://mathworld.wolfram.com/MultiplicativeOrder.html, http://www.numbertheory.org/php/order.html, https://oeis.org/A250211, https://oeis.org/A139366, https://oeis.org/A323376, https://oeis.org/A057593, https://oeis.org/A086145) of the base (b) mod the primes (i.e. ordp(b) with prime p), if you want to calculate the multiplicative order of the base (b) mod a composite number c coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to b, factor (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) c to product of distinct prime powers (https://oeis.org/A246655, https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html), and calculate the multiplicative order of b mod pe (i.e. ordpe(b)) for all these prime powers pe, and ordpe(b) = pmax(er(b,p),0)×ordp(b), where r(b,p) is the largest integer s such that ps divides bp−1−1, the primes p such that r(b,p) > 1 are called generalized Wieferich prime (https://en.wikipedia.org/wiki/Wieferich_prime, https://t5k.org/glossary/xpage/WieferichPrime.html, https://mathworld.wolfram.com/WieferichPrime.html, https://www.primegrid.com/forum_thread.php?id=9436, https://www.primegrid.com/stats_ww.php, https://oeis.org/A001220) base b.

This is a list for all known generalized Wieferich primes in bases 2 ≤ b ≤ 36 (r(b,p) = 2 for the generalized Wieferich primes p in base b with no orders listed (for these generalized Wieferich primes p, the orders are 1), and r(b,p) = (the order listed) + 1 for the generalized Wieferich primes p in base b with orders listed): (references: http://www.fermatquotient.com/FermatQuotienten/FermQ_Sort.txt, http://www.fermatquotient.com/FermatQuotienten/FermQ_Sorg.txt, http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math11/fer_quo.htm, http://download2.polytechnic.edu.na/pub4/sourceforge/w/wi/wieferich/results/table.txt (although this page is not available in the web and unfortunately has no archive page in the wayback machine), https://web.archive.org/web/20140809030451/http://www1.uni-hamburg.de/RRZ/W.Keller/FermatQuotient.html (prime bases b), https://web.archive.org/web/20140810051836/http://www.cecm.sfu.ca/~mjm/WieferichBarker/WPBS1.html (prime bases b), http://www.bitman.name/math/table/489 (in Italian), http://www.urticator.net/essay/6/624.html, http://go.helms-net.de/math/expdioph/fermatquot_ge2_table1.htm, http://wayback.cecm.sfu.ca/~mjm/WieferichBarker/, https://web.archive.org/web/20160417130531/http://home.earthlink.net/~oddperfect/FermatQuotients.html (prime bases b), https://web.archive.org/web/20060925172546/http://www.lrz-muenchen.de/~hr/tmp/A039951.txt, http://www.primepuzzles.net/puzzles/puzz_762.htm, http://www.sci.kobe-u.ac.jp/old/seminar/pdf/2008_yamazaki.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_63.pdf), https://kconrad.math.uconn.edu/blurbs/gradnumthy/integersradical.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_183.pdf), https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1182246-5/S0025-5718-1993-1182246-5.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_185.pdf), https://www.ams.org/journals/mcom/2005-74-250/S0025-5718-04-01666-7/S0025-5718-04-01666-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_246.pdf), https://www.maa.org/sites/default/files/321929430448.pdf.bannered.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_248.pdf), https://www.maa.org/sites/default/files/321929430448.pdf.bannered.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_249.pdf), http://go.helms-net.de/math/expdioph/fermatquotients.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_64.pdf))

(Also OEIS sequences for the smallest generalized Wieferich primes p in base b: https://oeis.org/A039951, https://oeis.org/A174422 (prime b), https://oeis.org/A268352 (2nd smallest prime), https://oeis.org/A178871 (2nd smallest prime, prime b), https://oeis.org/A096082 (p = 2 not allowed), https://oeis.org/A255838 (p > b required, prime b), https://oeis.org/A247072 (p > sqrt(b) required))

bgeneralized Wieferich primes in base b (written in base 10)search limitOEIS sequence
21093, 3511, ...6×1017https://oeis.org/A001220
311, 1006003, ...1.2×1015https://oeis.org/A014127
41093, 3511, ...6×1017the same as https://oeis.org/A001220
52, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801, ...1.2×1015https://oeis.org/A123692
666161, 534851, 3152573, ...1.885×1014https://oeis.org/A212583
75, 491531, ...1.2×1015https://oeis.org/A123693
83, 1093, 3511, ...6×1017the same as https://oeis.org/A001220 plus the prime 3
92 (order 2), 11, 1006003, ...1.2×1015the same as https://oeis.org/A014127 plus the prime 2
103, 487, 56598313, ...1.885×1014https://oeis.org/A045616
1171, ...1.885×1014
122693, 123653, ...1.885×1014https://oeis.org/A111027
132, 863, 1747591, ...1.885×1014https://oeis.org/A128667
1429, 353, 7596952219, ...1.885×1014https://oeis.org/A234810
1529131, 119327070011, ...1.885×1014https://oeis.org/A242741
161093, 3511, ...6×1017the same as https://oeis.org/A001220
172 (order 3), 3, 46021, 48947, 478225523351, ...1.885×1014https://oeis.org/A128668
185, 7 (order 2), 37, 331, 33923, 1284043, ...1.885×1014https://oeis.org/A244260
193, 7 (order 2), 13, 43, 137, 63061489, ...1.885×1014https://oeis.org/A090968
20281, 46457, 9377747, 122959073, ...1.885×1014https://oeis.org/A242982
212, ...1.885×1014
2213, 673, 1595813, 492366587, 9809862296159, ...1.885×1014https://oeis.org/A298951
2313, 2481757, 13703077, 15546404183, 2549536629329, ...1.885×1014https://oeis.org/A128669
245, 25633, ...1.885×1014
252 (order 2), 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801, ...1.2×1015the same as https://oeis.org/A123692
263 (order 2), 5, 71, 486999673, 6695256707, ...1.885×1014https://oeis.org/A306255
2711, 1006003, ...1.2×1015the same as https://oeis.org/A014127
283 (order 2), 19, 23, ...1.885×1014
292, ...1.885×1014
307, 160541, 94727075783, ...1.885×1014https://oeis.org/A306256
317, 79, 6451, 2806861, ...1.885×1014https://oeis.org/A331424
325, 1093, 3511, ...6×1017the same as https://oeis.org/A001220 plus the prime 5
332 (order 4), 233, 47441, 9639595369, ...1.885×1014
3446145917691, ...1.885×1014
353, 1613, 3571, ...1.885×1014
3666161, 534851, 3152573, ...1.885×1014the same as https://oeis.org/A212583

(a×bn+c)/gcd(a+c,b−1) (with a ≥ 1, b ≥ 2, c ≠ 0, gcd(a,c) = 1, gcd(b,c) = 1) has algebraic factorization if and only if either "there is an integer r > 1 such that a×bn and −c are both rth powers" (in this case, (a×bn+c)/gcd(a+c,b−1) has sum-of-two-rth-powers factorization if c > 0, or difference-of-two-rth-powers factorization if c < 0, although there is no "sum-of-two-rth-powers factorization" for even r, but no such situation (i.e. c > 0 and r is even) exists, since if c > 0 then −c < 0, but negative numbers cannot be squares, however, if r is even then all rth powers are squares (since if s divides r, then all rth powers are sth powers), thus, −c cannot be an rth power if c > 0 and r is even) or "one of a×bn and c is a 4th power, and the other is of the form 4×m4" (in this case, (a×bn+c)/gcd(a+c,b−1) has Aurifeuillean factorization of x4+4×y4), thus, if |c| (the absolute value (https://en.wikipedia.org/wiki/Absolute_value, https://www.rieselprime.de/ziki/Absolute_value, https://mathworld.wolfram.com/AbsoluteValue.html) of c) is not a perfect power (https://oeis.org/A001597, https://en.wikipedia.org/wiki/Perfect_power, https://mathworld.wolfram.com/PerfectPower.html, https://www.numbersaplenty.com/set/perfect_power/), then there is no n such that (a×bn+c)/gcd(a+c,b−1) has algebraic factorization, also, if there is no n such that a×bn is a perfect power (https://oeis.org/A001597, https://en.wikipedia.org/wiki/Perfect_power, https://mathworld.wolfram.com/PerfectPower.html, https://www.numbersaplenty.com/set/perfect_power/), then there is no n such that (a×bn+c)/gcd(a+c,b−1) has algebraic factorization.

We can show that: For the case of covering congruence, the numbers in the family are not equal to any element in S, if n makes the numbers > b, thus these factorizations are nontrivial (https://en.wikipedia.org/wiki/Triviality_(mathematics), https://mathworld.wolfram.com/Trivial.html); for the case of algebraic factorization (if the numbers are factored as F × G / d), both F and G are > d, if n makes the numbers > b, thus these factorizations are nontrivial (the exceptions are the base 9 family {1} and the base 25 family {1} and the base 27 family 9{G} and the base 32 family {1}. For the base 9 family {1}, the algebraic form is (9n−1)/8 with n ≥ 2, and can be factored to (3n−1) × (3n+1) / 8, if n ≥ 3, then both 3n−1 and 3n+1 are > 8, thus these factorizations are nontrivial, it only remains to check the case n = 2, but the number with n = 2 is 10 = 2 × 5 is not prime; for the base 25 family {1}, the algebraic form is (25n−1)/24 with n ≥ 2, and can be factored to (5n−1) × (5n+1) / 24, if n ≥ 3, then both 5n−1 and 5n+1 are > 24, thus these factorizations are nontrivial, it only remains to check the case n = 2, but the number with n = 2 is 26 = 2 × 13 is not prime; for the base 27 family 9{G}, the algebraic form is (125×27n−8)/13 with n ≥ 1, and can be factored to (5×3n−2) × (25×9n+10×3n+4) / 13, if n ≥ 2, then both 5×3n−2 and 25×9n+10×3n+4 are > 13, thus these factorizations are nontrivial, it only remains to check the case n = 1, but the number with n = 1 is 259 = 7 × 37 is not prime; for the base 32 family {1}, the algebraic form is (32n−1)/31 with n ≥ 2, and can be factored to (2n−1) × (16n+8n+4n+2n+1) / 31, if n ≥ 6, then both 2n−1 and 16n+8n+4n+2n+1 are > 31, thus these factorizations are nontrivial, it only remains to check the cases n = 2, 3, 4, 5, but the numbers with n = 2, 3, 4, 5 are 33 = 3 × 11, 1057 = 7 × 151, 33825 = 3 × 52 × 11 × 41, 1082401 = 601 × 1801 are not primes); for the case of combine of covering congruence and algebraic factorization (if the numbers are factored as F × G / d), the numbers in the family are not equal to any element in S and both F and G are > d, if n makes the numbers > b, thus these factorizations are nontrivial.

type for a family can be ruled out as only containing composites (only count the numbers > b)possible bases bsuch bases 2 ≤ b ≤ 36
covering congruence with 1 primeany base b
(however, all such families in base b = 2 end with 0 and thus have trailing zeros (https://en.wikipedia.org/wiki/Trailing_zero) and thus not counted)
(2), 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36
covering congruence with 2 primesb such that b+1 is not a prime power (https://oeis.org/A246655, https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html)
(however, the case b = 5 has only four such families: {1}3, {1}4, 3{1}, 4{1}, all of them are covered by the prime 111 (31 in decimal), thus the smallest base b with families which have covering congruence with 2 primes is b = 9)
(5), 9, 11, 13, 14, 17, 19, 20, 21, 23, 25, 27, 29, 32, 33, 34, 35
covering congruence with 3 primes and period 3b such that omega(b2+b+1) ≥ 3 (where omega is the omega function (https://en.wikipedia.org/wiki/Prime_omega_function, https://oeis.org/A001221), the number of distinct primes dividing n)16, 25
covering congruence with 3 primes and period 4b such that b+1 is not a power of 2 (https://oeis.org/A000079, https://en.wikipedia.org/wiki/Power_of_two) and b2+1 is not a prime power (https://oeis.org/A246655, https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html)
(however, bases b = 5, 8, 9, 11, 12, 18 has no such families, base 8 family 6{4}7 is covered by the prime 42207)
(5), (8), (9), (11), (12), 13, 17, (18), 19, 21, 22, 23, 25, 27, 28, 29, 30, 32, 33, 34, 35
algebraic factorization with difference of two squaresb such that b is square (https://oeis.org/A000290, https://en.wikipedia.org/wiki/Square_number, https://www.rieselprime.de/ziki/Square_number, https://mathworld.wolfram.com/SquareNumber.html, https://www.numbersaplenty.com/set/square_number/)
(however, base b = 4 has no such families, the family {1} has the prime 11 (5 in decimal))
(4), 9, 16, 25, 36
algebraic factorization with sum/difference of two cubesb such that b is cube (https://oeis.org/A000578, https://en.wikipedia.org/wiki/Cube_(algebra), https://mathworld.wolfram.com/CubicNumber.html, https://www.numbersaplenty.com/set/cubic_number/)
(however, base b = 8 has no such families which have algebraic factorization with difference of two cubes, the family {1} has the prime 111 (73 in decimal), base b = 8 only have families which have algebraic factorization with sum of two cubes)
8, 27
algebraic factorization with sum/difference of two 5th powersb such that b is 5th power (https://oeis.org/A000584, https://en.wikipedia.org/wiki/Fifth_power_(algebra))32
algebraic factorization with x4+4×y4b such that b is 4th power (https://oeis.org/A000583, https://en.wikipedia.org/wiki/Fourth_power, https://mathworld.wolfram.com/BiquadraticNumber.html)16
combine of covering congruence with 1 prime and algebraic factorization with difference of two squaresb such that b is not square (https://oeis.org/A000290, https://en.wikipedia.org/wiki/Square_number, https://www.rieselprime.de/ziki/Square_number, https://mathworld.wolfram.com/SquareNumber.html, https://www.numbersaplenty.com/set/square_number/) and b+1 has a prime factor p == 1 mod 412, 14, 19, 24, 28, 29, 33, 34

(You can see the factorization (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) of the numbers in these families in factordb (http://factordb.com/, https://www.rieselprime.de/ziki/Factoring_Database), you have to convert them to algebraic ((a×bn+c)/gcd(a+c,b−1)) form, and you will find that all numbers in these families have status (http://factordb.com/status.html, http://factordb.com/distribution.php) either "FF" or "CF", and no numbers in these families have status (http://factordb.com/status.html, http://factordb.com/distribution.php) "C" (i.e. in http://factordb.com/listtype.php?t=3) (also see https://stdkmd.net/nrr/wanted.htm#nofactorknown) (i.e. no numbers in these families are "genuine composites", i.e. compositeness is proved but no proper (prime or composite) factor is yet known) (of course, also no numbers in these families have status (http://factordb.com/status.html, http://factordb.com/distribution.php) "U", "P", "PRP" (i.e. in http://factordb.com/listtype.php?t=2, http://factordb.com/listtype.php?t=4, http://factordb.com/listtype.php?t=1), since all numbers in these families are known to be composite), and all numbers in these families which are not too large to be primality tested can be decided whether they are semiprimes (https://en.wikipedia.org/wiki/Semiprime, https://t5k.org/glossary/xpage/Semiprime.html, https://mathworld.wolfram.com/Semiprime.html, https://www.numbersaplenty.com/set/semiprime/, https://oeis.org/A001358) or not (for the examples, see https://oeis.org/A085724 and https://oeis.org/A092559 and https://oeis.org/A080892 and https://oeis.org/A081715), and the sieve file for these families will be empty after sieving (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.primegrid.com/stats_psp_sieve.php, https://www.primegrid.com/stats_pps_sieve.php, https://www.primegrid.com/stats_sr5_sieve.php, https://www.primegrid.com/stats_cw_sieve.php, https://www.primegrid.com/sieving/rsp/) with SRSIEVE (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, https://www.mersenneforum.org/showpost.php?p=631129&postcount=1, http://www.rieselprime.de/dl/CRUS_pack.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve_1.8.2, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve-other-programs, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2cl.exe) if the sieving program (i.e. SRSIEVE) was updated so that it also removes the n such that a×bn+c has algebraic factors, and their Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=50442&postcount=1, https://www.mersenneforum.org/showpost.php?p=50444&postcount=1, https://www.mersenneforum.org/showpost.php?p=201642&postcount=1, https://www.mersenneforum.org/attachment.php?attachmentid=13663&d=1451910741, https://github.com/happy5214/nash, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, https://web.archive.org/web/20230928115952/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, https://web.archive.org/web/20230928120009/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, https://web.archive.org/web/20230928120025/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, https://web.archive.org/web/20230928120047/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)) is zero, e.g. for the family 3{0}95 in base 13, its algebraic ((a×bn+c)/gcd(a+c,b−1)) form is 3×13n+2+122, and in factordb you will find that all numbers in this family are divisible by some element of {5,7,17}, see http://factordb.com/index.php?query=3*13%5E%28n%2B2%29%2B122&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show; for the family {7}D in base 21, its algebraic ((a×bn+c)/gcd(a+c,b−1)) form is (7×21n+1+113)/20, and in factordb you will find that all numbers in this family are divisible by some element of {2,13,17}, see http://factordb.com/index.php?query=%287*21%5E%28n%2B1%29%2B113%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (note: for this family n = 0 is not allowed, since we only consider the numbers > base); for the family 30{F}A0F in base 16, its algebraic ((a×bn+c)/gcd(a+c,b−1)) form is 49×16n+3−1521, and in factordb you will find that no numbers in this family have a prime factor with decimal length > ((the decimal length of the number + 1)/2), and all numbers in this family have two nearly equal (prime or composite) factors, see http://factordb.com/index.php?query=49*16%5E%28n%2B3%29-1521&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show; for the family 5{1} in base 25, its algebraic ((a×bn+c)/gcd(a+c,b−1)) form is (121×25n−1)/24, and in factordb you will find that no numbers in this family have a prime factor with decimal length > ((the decimal length of the number + 1)/2), and all numbers in this family have two nearly equal (prime or composite) factors, see http://factordb.com/index.php?query=%28121*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (note: for this family n = 0 is not allowed, since we only consider the numbers > base); for the family {D}5 in base 14, its algebraic ((a×bn+c)/gcd(a+c,b−1)) form is 14n+1−9, and in factordb you will find that all numbers with even n in this family are divisible by 5, and you will find that no numbers with odd n in this family have a prime factor with decimal length > ((the decimal length of the number + 1)/2), and all numbers with odd n in this family have two nearly equal (prime or composite) factors, see http://factordb.com/index.php?query=14%5E%28n%2B1%29-9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (note: for this family n = 0 is not allowed, since we only consider the numbers > base); for the family 7{9} in base 17, its algebraic ((a×bn+c)/gcd(a+c,b−1)) form is (121×17n−9)/16, and in factordb you will find that all numbers with odd n in this family are divisible by 2, and you will find that no numbers with even n in this family have a prime factor with decimal length > ((the decimal length of the number + 1)/2), and all numbers with even n in this family have two nearly equal (prime or composite) factors, see http://factordb.com/index.php?query=%28121*17%5En-9%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (note: for this family n = 0 is not allowed, since we only consider the numbers > base). In contrast, you can see the factorization of the numbers in unsolved families in base b (which are listed in the "left b" file) in factordb, you will find some numbers in these families which have neither small prime factors (say < 1016) nor two nearly equal (prime or composite) factors, also you will find some numbers in these families which have no known proper factor (https://en.wikipedia.org/wiki/Proper_factor, https://mathworld.wolfram.com/ProperFactor.html, https://mathworld.wolfram.com/ProperDivisor.html) > 1 (i.e. you will find some numbers in these families with status (http://factordb.com/status.html, http://factordb.com/distribution.php) "C" (instead of "CF" or "FF") (i.e. in http://factordb.com/listtype.php?t=3) (also see https://stdkmd.net/nrr/wanted.htm#nofactorknown) (and maybe some large numbers in these families with status (http://factordb.com/status.html, http://factordb.com/distribution.php) "U", i.e. in http://factordb.com/listtype.php?t=2) in factordb (http://factordb.com/), i.e. some numbers in these families are "genuine composites" (i.e. compositeness of these numbers are proved but no proper (prime or composite) factors of them are yet known)), and the sieve file for these families will be empty after sieving (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.primegrid.com/stats_psp_sieve.php, https://www.primegrid.com/stats_pps_sieve.php, https://www.primegrid.com/stats_sr5_sieve.php, https://www.primegrid.com/stats_cw_sieve.php, https://www.primegrid.com/sieving/rsp/) with SRSIEVE (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, https://www.mersenneforum.org/showpost.php?p=631129&postcount=1, http://www.rieselprime.de/dl/CRUS_pack.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve_1.8.2, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve-other-programs, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2cl.exe) even if the sieving program (i.e. SRSIEVE) was updated so that it also removes the n such that a×bn+c has algebraic factors, and they have positive Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=50442&postcount=1, https://www.mersenneforum.org/showpost.php?p=50444&postcount=1, https://www.mersenneforum.org/showpost.php?p=201642&postcount=1, https://www.mersenneforum.org/attachment.php?attachmentid=13663&d=1451910741, https://github.com/happy5214/nash, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, https://web.archive.org/web/20230928115952/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, https://web.archive.org/web/20230928120009/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, https://web.archive.org/web/20230928120025/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, https://web.archive.org/web/20230928120047/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)), and they have prime candidates (the Nash weight (or difficulty) tells you how many candidates remain after sieving a certain number of terms to a certain depth (say 109), if the sieving program (i.e. SRSIEVE) was updated so that it also removes the n such that a×bn+c has algebraic factors), we can use the sense of http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://stdkmd.net/nrr/1/10003.htm#prime_period, https://stdkmd.net/nrr/3/30001.htm#prime_period, https://stdkmd.net/nrr/1/11113.htm#prime_period, https://stdkmd.net/nrr/3/31111.htm#prime_period, https://stdkmd.net/nrr/1/13333.htm#prime_period, https://stdkmd.net/nrr/3/33331.htm#prime_period, https://stdkmd.net/nrr/1/11131.htm#prime_period, https://stdkmd.net/nrr/1/13111.htm#prime_period, https://stdkmd.net/nrr/3/31333.htm#prime_period, https://stdkmd.net/nrr/3/33313.htm#prime_period, https://stdkmd.net/nrr/1/13331.htm#prime_period, https://stdkmd.net/nrr/3/31113.htm#prime_period, https://www.mersenneforum.org/showpost.php?p=138737&postcount=24, https://www.mersenneforum.org/showpost.php?p=153508&postcount=147, to show this, we use discrete logarithm (https://en.wikipedia.org/wiki/Discrete_logarithm, https://mathworld.wolfram.com/DiscreteLogarithm.html, https://www.alpertron.com.ar/DILOG.HTM, http://www.numbertheory.org/php/discrete_log.html) to solve the equation (https://en.wikipedia.org/wiki/Equation, https://mathworld.wolfram.com/Equation.html) (a×bn+c)/gcd(a+c,b−1) == 0 mod p (i.e. solve the equation (https://en.wikipedia.org/wiki/Equation, https://mathworld.wolfram.com/Equation.html) a×bn+c = 0 in the finite field (https://en.wikipedia.org/wiki/Finite_field, https://mathworld.wolfram.com/FiniteField.html) Zp (i.e. solve the discrete logarithm (https://en.wikipedia.org/wiki/Discrete_logarithm, https://mathworld.wolfram.com/DiscreteLogarithm.html, https://www.alpertron.com.ar/DILOG.HTM, http://www.numbertheory.org/php/discrete_log.html) logb(−c/a) mod p (where c/a is the division in the finite field Zp, which does not exist when a is divisible by p (since in this case a is a zero divisor (https://en.wikipedia.org/wiki/Zero_divisor, https://mathworld.wolfram.com/ZeroDivisor.html) in the finite field Zp), and the equation a×bn+c = 0 in the finite field Zp has no solution if a is divisible by p)) in the case that p does not divide b−1, or solve the equation (https://en.wikipedia.org/wiki/Equation, https://mathworld.wolfram.com/Equation.html) a×bn+c = 0 in the finite ring (https://en.wikipedia.org/wiki/Finite_ring) Zpr+1 (i.e. solve the discrete logarithm (https://en.wikipedia.org/wiki/Discrete_logarithm, https://mathworld.wolfram.com/DiscreteLogarithm.html, https://www.alpertron.com.ar/DILOG.HTM, http://www.numbertheory.org/php/discrete_log.html) logb(−c/a) mod pr+1 (where c/a is the division in the finite ring Zpr+1, which does not exist when a is divisible by p (since in this case a is a zero divisor (https://en.wikipedia.org/wiki/Zero_divisor, https://mathworld.wolfram.com/ZeroDivisor.html) in the finite ring Zpr+1), and the equation a×bn+c = 0 in the finite ring Zpr+1 has no solution if a is divisible by p)) in the case that p divides b−1 and the number r satisfies that gcd(a+c,b−1) is divisible by pr but not by pr+1 (i.e. r is the largest number such that pr divides gcd(a+c,b−1))) to find the n (find the congruence of n mod ordp(b) if p does not divide b−1, or find the congruence of n mod p if p divides b−1) such that (a×bn+c)/gcd(a+c,b−1) is divisible by p for small primes p, e.g. for the unsolved family {9}D in base b = 21 (its algebraic form is (9×21n+1+71)/20, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=%289*21%5E%28n%2B1%29%2B71%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show): )

  • The algebraic form of 9nD (in base b = 21) is (9×21n+1+71)/20, and 71 is not a perfect power, thus the family 9nD (in base b = 21) has no algebraic factorization for any n
  • 9nD (in base b = 21) is divisible by 2 if and only if n == 1 mod 2
  • 9nD (in base b = 21) is divisible by 43 if and only if n == 2 mod 7
  • 9nD (in base b = 21) is divisible by 97 if and only if n == 2 mod 96
  • 9nD (in base b = 21) is divisible by 5 if and only if n == 3 mod 5
  • 9nD (in base b = 21) is divisible by 547 if and only if n == 3 mod 39
  • 9nD (in base b = 21) is divisible by 13 if and only if n == 0 mod 4
  • 9nD (in base b = 21) is divisible by 109 if and only if n == 4 mod 27
  • 9nD (in base b = 21) is divisible by 1297 if and only if n == 4 mod 648
  • 9nD (in base b = 21) is divisible by 23 if and only if n == 6 mod 22
  • 9nD (in base b = 21) is divisible by 37 if and only if n == 6 mod 18
  • 9nD (in base b = 21) is divisible by 138679 if and only if n == 10 mod 69339
  • 9nD (in base b = 21) is divisible by 31 if and only if n == 14 mod 30
  • 9nD (in base b = 21) is divisible by 957158401 if and only if n == 22 mod 957158400
  • 9nD (in base b = 21) is divisible by 113 if and only if n == 26 mod 112
  • 9nD (in base b = 21) is divisible by 149 if and only if n == 26 mod 148
  • 9nD (in base b = 21) is divisible by 83 if and only if n == 34 mod 41
  • 9nD (in base b = 21) is divisible by 151 if and only if n == 34 mod 75
  • 9nD (in base b = 21) is divisible by 317 if and only if n == 34 mod 316
  • 9nD (in base b = 21) is divisible by 571 if and only if n == 46 mod 285

etc.

and it does not appear to be any covering set (of either primes or algebraic factors, or both), thus this form cannot be ruled out as only containing composites (only count the numbers > b) (by covering congruence, algebraic factorization, or combine of them), and its Nash weight (or difficulty) is positive, and it has prime candidate, and hence there must be a prime at some point.

Also, you can see the prime factorizations (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) of the numbers in the family 9nD in base b = 21: (for more information, see http://factordb.com/index.php?query=%289*21%5E%28n%2B1%29%2B71%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

ncurrently known prime factorization of the number 9nD in base b = 21
12 × 101
243 × 97
325 × 5 × 547
413 × 109 × 1297
52 × 19297379
623 × 37 × 952397
722 × 19 × 3491 × 64151
85 × 13 × 373 × 14742257
10138679 × 1136616757
1431 × 20605229 × 47991331061
185 × 44600683 × 127256483 × 210081439
22957158401 × 1211357903339029702951
26113 × 149 × 293877403 × 45572469859265068620757
3483 × 151 × 317 × 116959 × 1833983 × 10007937292019458273806456107
46571 × 172443379 × 637142732633278216747911831698921177222682613261423
5464514074883 × 1952284707517 × 18839791743425715061337512664154919843840071132209
621987 × 2179 × 4491749 × 31018210441926961332833324941 × 148778702451046425986212833742378968279623
6610169 × 669662569903 × 2563132092...4328784141 (73-digit prime)
702107831759082832524772111929 × 194079650340743929919158086605141 × 8297867485221151080164383803051187

and it does not appear to be any covering congruence of primes, since if there is a covering congruence of primes for the family {9}D in base b = 21, then the period must be at least lcm(2, ord43(21), ord13(21), ord23(21), ord138679(21), ord31(21), 5, ord957158401(21), ord113(21), ord83(21), ord571(21), ord64514074883(21), ord1987(21), ord10169(21), ord2107831759082832524772111929(21)) (note: since the primes p = 2 and p = 5 divides b−1 = 21−1 = 20, and for these two primes, ordp(21) = 1, thus the smallest n such that Rn(21) (where "Rn(b)" means the repunit in base b with length n, i.e. Rn(b) = (bn−1)/(b−1) (see https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization)) is divisible by p is the primes p themselves instead of ordp(21) = 1, thus their period is the primes p themselves instead of ordp(21) = 1, their situation is like the prime p = 3 in decimal (b = 10)) (where lcm is the least common multiple (https://en.wikipedia.org/wiki/Least_common_multiple, https://mathworld.wolfram.com/LeastCommonMultiple.html, http://www.numericana.com/answer/numbers.htm#lcm), ordn(b) is the multiplicative order (https://en.wikipedia.org/wiki/Multiplicative_order, https://t5k.org/glossary/xpage/Order.html, https://mathworld.wolfram.com/MultiplicativeOrder.html, http://www.numbertheory.org/php/order.html, https://oeis.org/A250211, https://oeis.org/A139366, https://oeis.org/A323376, https://oeis.org/A057593, https://oeis.org/A086145) of b mod n) = lcm(2, 7, 4, 22, 69339, 30, 5, 957158400, 112, 41, 285, 32257037441, 662, 2542, 2107831759082832524772111928) = 7191035432210805538926222608335822284250318907919283200 > 7.19103 × 1054, which is very impossible, and since 71 is not a perfect power, thus the family 9nD (in base b = 21) = (9×21n+1+71)/20 has no algebraic factorization for any n, thus the possbility that the family {9}D in base b = 21 can be ruled out as only containing composites (only count the numbers > b) (by covering congruence, algebraic factorization, or combine of them) is very small (< 1/(7.19103 × 1054)), thus we can almost sure that there is a prime > b in the family {9}D in base b = 21, in fact, we can almost sure that there are infinitely many primes > b in the family {9}D in base b = 21.

For an example of an unsolved family in bases 2 ≤ b ≤ 36 with a very low (but still positive) Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=50442&postcount=1, https://www.mersenneforum.org/showpost.php?p=50444&postcount=1, https://www.mersenneforum.org/showpost.php?p=201642&postcount=1, https://www.mersenneforum.org/attachment.php?attachmentid=13663&d=1451910741, https://github.com/happy5214/nash, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, https://web.archive.org/web/20230928115952/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, https://web.archive.org/web/20230928120009/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, https://web.archive.org/web/20230928120025/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, https://web.archive.org/web/20230928120047/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)), the family 5{H}5 in base b = 19: (its algebraic form is (107×19n+1−233)/18, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=%28107*19%5E%28n%2B1%29-233%29%2F18&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

  • 5Hn5 (in base b = 19) is divisible by 2 if and only if n == 0 mod 2 (and to this step, only n == 1 mod 2 gives possible candidates for prime numbers!)
  • 5Hn5 (in base b = 19) is divisible by 3 if and only if n == 1 mod 3 (and to this step, only n == 3, 5 mod 6 gives possible candidates for prime numbers!)
  • 5Hn5 (in base b = 19) is divisible by 7 if and only if n == 5 mod 6 (and to this step, only n == 3 mod 6 gives possible candidates for prime numbers!)
  • 5Hn5 (in base b = 19) is divisible by 13 if and only if n == 9 mod 12 (and to this step, only n == 3 mod 12 gives possible candidates for prime numbers!)
  • 5Hn5 (in base b = 19) is divisible by 17 if and only if n == 3 mod 8 (and to this step, only n == 15 mod 24 gives possible candidates for prime numbers!)

Thus the only interesting cases to search for possible primes are when n == 15 mod 24 (and thus the Nash weight (or difficulty) of this family is less than 1/24), and the least prime factor (http://mathworld.wolfram.com/LeastPrimeFactor.html, https://oeis.org/A020639) of 5Hn5 (in base b = 19) is ≤ 17 if n is not == 15 mod 24, and 5Hn5 (in base b = 19) is not 19-rough number (https://en.wikipedia.org/wiki/Rough_number, https://mathworld.wolfram.com/RoughNumber.html, https://oeis.org/A007310, https://oeis.org/A007775, https://oeis.org/A008364, https://oeis.org/A008365, https://oeis.org/A008366, https://oeis.org/A166061, https://oeis.org/A166063) if n is not == 15 mod 24, and thus if we sieve (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.primegrid.com/stats_psp_sieve.php, https://www.primegrid.com/stats_pps_sieve.php, https://www.primegrid.com/stats_sr5_sieve.php, https://www.primegrid.com/stats_cw_sieve.php, https://www.primegrid.com/sieving/rsp/) the family 5Hn5 (in base b = 19) with primes ≤ 17, then all n not == 15 mod 24 will be removed.

And for n == 15 mod 24, we have: (all "composites without known proper factor > 1" in the list below have been trial factored (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) to 1012, also none of these composites have algebraic factors)

ncurrently known prime factorization of the number 5Hn5 in base b = 19
1511 × 29 × 47 × 71 × 439 × 15287 × 240013247
39281 × 9833 × 99133 × 281717 × 390527 × 278727989572543299217912827703
6311063830209439375789 × 3719086087...5959852157 (64-digit prime)
873889 × 21599 × 5720809 × 832699912393 × 5037448295...0434343479 (87-digit composite without known proper factor > 1, this composite has already checked with P−1 to B1 = 50000 and 3 times P+1 to B1 = 150000 and 10 times ECM to B1 = 250000 (these can be checked for composites < 10300), see http://factordb.com/sequences.php?se=1&aq=%28107*19%5E88-233%29%2F18&action=last20&fr=0&to=100, the "Check for factors" box shows "Already checked", this is the ECM effort t-level (https://oeis.org/wiki/OEIS_sequences_needing_factors#T-levels, https://stdkmd.net/nrr/wanted.htm (the "ECM" column of the three tables), https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, http://myfactorcollection.mooo.com:8090/lists.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the lists in the boxes), http://myfactorcollection.mooo.com:8090/dbio.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the DB inputs/outputs in the boxes), https://github.com/brubsby/t-level, https://www.mersenneforum.org/showthread.php?t=29615) t30 (see http://www.loria.fr/~zimmerma/records/ecm/params.html and https://www.rieselprime.de/ziki/Elliptic_curve_method#Choosing_the_best_parameters_for_ECM), i.e. the prime factors of this composite number are probably > 1030)
11189 × 1109493015...7064024897 (143-digit composite without known proper factor > 1, this composite has already checked with P−1 to B1 = 50000 and 3 times P+1 to B1 = 150000 and 10 times ECM to B1 = 250000 (these can be checked for composites < 10300), see http://factordb.com/sequences.php?se=1&aq=%28107*19%5E112-233%29%2F18&action=last20&fr=0&to=100, the "Check for factors" box shows "Already checked", this is the ECM effort t-level (https://oeis.org/wiki/OEIS_sequences_needing_factors#T-levels, https://stdkmd.net/nrr/wanted.htm (the "ECM" column of the three tables), https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, http://myfactorcollection.mooo.com:8090/lists.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the lists in the boxes), http://myfactorcollection.mooo.com:8090/dbio.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the DB inputs/outputs in the boxes), https://github.com/brubsby/t-level, https://www.mersenneforum.org/showthread.php?t=29615) t30 (see http://www.loria.fr/~zimmerma/records/ecm/params.html and https://www.rieselprime.de/ziki/Elliptic_curve_method#Choosing_the_best_parameters_for_ECM), i.e. the prime factors of this composite number are probably > 1030)
13511 × 75193 × 5848316996...0023641331 (169-digit composite without known proper factor > 1, this composite has already checked with P−1 to B1 = 50000 and 3 times P+1 to B1 = 150000 and 10 times ECM to B1 = 250000 (these can be checked for composites < 10300), see http://factordb.com/sequences.php?se=1&aq=%28107*19%5E136-233%29%2F18&action=last20&fr=0&to=100, the "Check for factors" box shows "Already checked", this is the ECM effort t-level (https://oeis.org/wiki/OEIS_sequences_needing_factors#T-levels, https://stdkmd.net/nrr/wanted.htm (the "ECM" column of the three tables), https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, http://myfactorcollection.mooo.com:8090/lists.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the lists in the boxes), http://myfactorcollection.mooo.com:8090/dbio.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the DB inputs/outputs in the boxes), https://github.com/brubsby/t-level, https://www.mersenneforum.org/showthread.php?t=29615) t30 (see http://www.loria.fr/~zimmerma/records/ecm/params.html and https://www.rieselprime.de/ziki/Elliptic_curve_method#Choosing_the_best_parameters_for_ECM), i.e. the prime factors of this composite number are probably > 1030)
15926021 × 1645306722881711 × 5534986552...1734587403 (186-digit composite without known proper factor > 1, this composite has already checked with P−1 to B1 = 50000 and 3 times P+1 to B1 = 150000 and 10 times ECM to B1 = 250000 (these can be checked for composites < 10300), see http://factordb.com/sequences.php?se=1&aq=%28107*19%5E160-233%29%2F18&action=last20&fr=0&to=100, the "Check for factors" box shows "Already checked", this is the ECM effort t-level (https://oeis.org/wiki/OEIS_sequences_needing_factors#T-levels, https://stdkmd.net/nrr/wanted.htm (the "ECM" column of the three tables), https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, http://myfactorcollection.mooo.com:8090/lists.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the lists in the boxes), http://myfactorcollection.mooo.com:8090/dbio.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the DB inputs/outputs in the boxes), https://github.com/brubsby/t-level, https://www.mersenneforum.org/showthread.php?t=29615) t30 (see http://www.loria.fr/~zimmerma/records/ecm/params.html and https://www.rieselprime.de/ziki/Elliptic_curve_method#Choosing_the_best_parameters_for_ECM), i.e. the prime factors of this composite number are probably > 1030)
18329 × 4002910224...4472103837 (235-digit composite without known proper factor > 1, this composite has already checked with P−1 to B1 = 50000 and 3 times P+1 to B1 = 150000 and 10 times ECM to B1 = 250000 (these can be checked for composites < 10300), see http://factordb.com/sequences.php?se=1&aq=%28107*19%5E184-233%29%2F18&action=last20&fr=0&to=100, the "Check for factors" box shows "Already checked", this is the ECM effort t-level (https://oeis.org/wiki/OEIS_sequences_needing_factors#T-levels, https://stdkmd.net/nrr/wanted.htm (the "ECM" column of the three tables), https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, http://myfactorcollection.mooo.com:8090/lists.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the lists in the boxes), http://myfactorcollection.mooo.com:8090/dbio.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the DB inputs/outputs in the boxes), https://github.com/brubsby/t-level, https://www.mersenneforum.org/showthread.php?t=29615) t30 (see http://www.loria.fr/~zimmerma/records/ecm/params.html and https://www.rieselprime.de/ziki/Elliptic_curve_method#Choosing_the_best_parameters_for_ECM), i.e. the prime factors of this composite number are probably > 1030)

and it does not appear to be any covering congruence of primes, besides, since 233 is not a perfect power, thus the family 5Hn5 (in base b = 19) has no algebraic factorization for any n, thus its Nash weight (or difficulty) is positive (although very low), so there must be a prime at some point (and the corresponding n must be == 15 mod 24).

And for n == 15 mod 24, we have: (all "composites without known proper factor > 1" in the list below have been trial factored (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) to 109, also none of these composites have algebraic factors)

Also the family C{H}C in base b = 19: (its algebraic form is (233×19n+1−107)/18, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=%28233*19%5E%28n%2B1%29-107%29%2F18&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) (in fact, it is the dual (http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_1.pdf), https://www.rechenkraft.net/wiki/Five_or_Bust, https://oeis.org/A076336/a076336c.html, http://www.mit.edu/~kenta/three/prime/dual-sierpinski/ezgxggdm/dualsierp-excerpt.txt, http://mit.edu/kenta/www/three/prime/dual-sierpinski/ezgxggdm/dualsierp.txt.gz, https://www.primegrid.com/download/5ob_all.html, http://www.bitman.name/math/article/1126, http://www.bitman.name/math/article/1125, https://www.mersenneforum.org/showpost.php?p=144991&postcount=1, https://www.mersenneforum.org/showthread.php?t=10761, https://www.mersenneforum.org/showthread.php?t=6545) family of the family 5{H}5 in base b = 19, thus they have the same Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=50442&postcount=1, https://www.mersenneforum.org/showpost.php?p=50444&postcount=1, https://www.mersenneforum.org/showpost.php?p=201642&postcount=1, https://www.mersenneforum.org/attachment.php?attachmentid=13663&d=1451910741, https://github.com/happy5214/nash, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, https://web.archive.org/web/20230928115952/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, https://web.archive.org/web/20230928120009/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, https://web.archive.org/web/20230928120025/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, https://web.archive.org/web/20230928120047/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)), a family can be proven to only contain composites or only contain finitely many primes by covering congruence, algebraic factorization, or combine of them if and only if its dual family can also be proven to only contain composites or only contain finitely many primes by covering congruence, algebraic factorization, or combine of them (note: this is not true for all reasons for a family can be proven to only contain composites or only contain finitely many primes, e.g. 8×128n+1, 32×128n+1, 64×128n+1 can be proven to only contain composites, while their dual families (16×128n+1, 4×128n+1, 2×128n+1, respectively) cannot be proven to only contain composites or only contain finitely many primes, in fact, the latter two of them have known primes (4×1282+1 and 2×1281+1, respectively), the reason for these three families is combine of sum-of-two-pth-powers factorization for the odd primes p which are not quadratic residues (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod 7 (i.e. the odd primes p == 3, 5, 6 mod 7) (i.e. the odd primes p in https://oeis.org/A003625), and by the Dirichlet's theorem (https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions, https://t5k.org/glossary/xpage/DirichletsTheorem.html, https://mathworld.wolfram.com/DirichletsTheorem.html, http://www.numericana.com/answer/primes.htm#dirichlet), there are infinitely many such odd primes p, thus, the reason for these three families is combine of sum-of-two-pth-powers factorization for infinitely many odd primes p (and the period is also infinite), and such infinite-cover does not apply to this), every family has a corresponding dual family (although the dual family may not corresponding to minimal primes), (assuming a ≥ 1, b ≥ 2 (b is the base), c ≠ 0, gcd(a,c) = 1, gcd(b,c) = 1) if gcd(a,b) = 1, then the dual family of a×bn+1 is bn+a, and the dual family of a×bn−1 is bna, more generally, if gcd(a,b) = 1 and c > 0, then the dual family of (a×bn+c)/gcd(a+c,b−1) is (c×bn+a)/gcd(a+c,b−1), and the dual family of (a×bnc)/gcd(a+c,b−1) is (c×bna)/gcd(a+c,b−1), however, when gcd(a,b) > 1, some dual families may not be easy to be observe, e.g. the dual family of 5×10n+27 (the family 5{0}27 in decimal (base b = 10), the family corresponding to the largest minimal prime in decimal (base b = 10), i.e. 5×1028+27, or 502827) is 54×10n+1 (the family 54{0}1 in decimal (base b = 10), which does not correspond to any minimal prime), and the dual family of (5×10n−41)/9 (the family {5}1 in decimal (base b = 10), the family corresponding to the second-largest minimal prime in decimal (base b = 10), i.e. (5×1012−41)/9, or 5111) is (82×10n−1)/9 (the family 9{1} in decimal (base b = 10), which does not correspond to any minimal prime), besides, only families of the form bn+1 (with even b), (bn+1)/2 (with odd b), (bn−1)/(b−1), (sqrt(bbn−1)/(sqrt(b)−1) (with square b), (sqrt(bbn+1)/(sqrt(b)+1) (with square b) are self-dual)

  • CHnC (in base b = 19) is divisible by 2 if and only if n == 0 mod 2 (and to this step, only n == 1 mod 2 gives possible candidates for prime numbers!)
  • CHnC (in base b = 19) is divisible by 3 if and only if n == 0 mod 3 (and to this step, only n == 1, 5 mod 6 gives possible candidates for prime numbers!)
  • CHnC (in base b = 19) is divisible by 7 if and only if n == 5 mod 6 (and to this step, only n == 1 mod 6 gives possible candidates for prime numbers!)
  • CHnC (in base b = 19) is divisible by 13 if and only if n == 1 mod 12 (and to this step, only n == 7 mod 12 gives possible candidates for prime numbers!)
  • CHnC (in base b = 19) is divisible by 17 if and only if n == 3 mod 8 (and to this step, only n == 7 mod 24 gives possible candidates for prime numbers!)

Thus the only interesting cases to search for possible primes are when n == 7 mod 24 (and thus the Nash weight (or difficulty) of this family is less than 1/24), and the least prime factor (http://mathworld.wolfram.com/LeastPrimeFactor.html, https://oeis.org/A020639) of CHnC (in base b = 19) is ≤ 17 if n is not == 7 mod 24, and 5Hn5 (in base b = 19) is not 19-rough number (https://en.wikipedia.org/wiki/Rough_number, https://mathworld.wolfram.com/RoughNumber.html, https://oeis.org/A007310, https://oeis.org/A007775, https://oeis.org/A008364, https://oeis.org/A008365, https://oeis.org/A008366, https://oeis.org/A166061, https://oeis.org/A166063) if n is not == 7 mod 24, and thus if we sieve (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.primegrid.com/stats_psp_sieve.php, https://www.primegrid.com/stats_pps_sieve.php, https://www.primegrid.com/stats_sr5_sieve.php, https://www.primegrid.com/stats_cw_sieve.php, https://www.primegrid.com/sieving/rsp/) the family 5Hn5 (in base b = 19) with primes ≤ 17, then all n not == 7 mod 24 will be removed.

And for n == 7 mod 24, we have: (all "composites without known proper factor > 1" in the list below have been trial factored (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) to 1012, also none of these composites have algebraic factors)

ncurrently known prime factorization of the number CHnC in base b = 19
72347 × 4937 × 18973
31951820073 × 682709071131761 × 1657326648944531039
555039891 × 5312207 × 44999186966208286552983113 × 4379094008236471173214119625150027
79409 × 709 × 11467 × 37699 × 10660810089661 × 1282461768425116416287 × 76403952268139600660997011 × 197365776416390077077003257867
103112 × 1049 × 6997 × 1425558955...0356672779 (126-digit prime)
12783 × 127498425829 × 5860861017...1703217721 (152-digit composite without known proper factor > 1, this composite has already checked with P−1 to B1 = 50000 and 3 times P+1 to B1 = 150000 and 10 times ECM to B1 = 250000 (these can be checked for composites < 10300), see http://factordb.com/sequences.php?se=1&aq=%28233*19%5E128-107%29%2F18&action=last20&fr=0&to=100, the "Check for factors" box shows "Already checked", this is the ECM effort t-level (https://oeis.org/wiki/OEIS_sequences_needing_factors#T-levels, https://stdkmd.net/nrr/wanted.htm (the "ECM" column of the three tables), https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, http://myfactorcollection.mooo.com:8090/lists.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the lists in the boxes), http://myfactorcollection.mooo.com:8090/dbio.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the DB inputs/outputs in the boxes), https://github.com/brubsby/t-level, https://www.mersenneforum.org/showthread.php?t=29615) t30 (see http://www.loria.fr/~zimmerma/records/ecm/params.html and https://www.rieselprime.de/ziki/Elliptic_curve_method#Choosing_the_best_parameters_for_ECM), i.e. the prime factors of this composite number are probably > 1030)
15129 × 89 × 1896473 × 5672167 × 1094326921...1399117877 (180-digit composite without known proper factor > 1, this composite has already checked with P−1 to B1 = 50000 and 3 times P+1 to B1 = 150000 and 10 times ECM to B1 = 250000 (these can be checked for composites < 10300), see http://factordb.com/sequences.php?se=1&aq=%28233*19%5E152-107%29%2F18&action=last20&fr=0&to=100, the "Check for factors" box shows "Already checked", this is the ECM effort t-level (https://oeis.org/wiki/OEIS_sequences_needing_factors#T-levels, https://stdkmd.net/nrr/wanted.htm (the "ECM" column of the three tables), https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, http://myfactorcollection.mooo.com:8090/lists.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the lists in the boxes), http://myfactorcollection.mooo.com:8090/dbio.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the DB inputs/outputs in the boxes), https://github.com/brubsby/t-level, https://www.mersenneforum.org/showthread.php?t=29615) t30 (see http://www.loria.fr/~zimmerma/records/ecm/params.html and https://www.rieselprime.de/ziki/Elliptic_curve_method#Choosing_the_best_parameters_for_ECM), i.e. the prime factors of this composite number are probably > 1030)
1751488391508...4544758887 (227-digit composite without known proper factor > 1, this composite has already checked with P−1 to B1 = 50000 and 3 times P+1 to B1 = 150000 and 10 times ECM to B1 = 250000 (these can be checked for composites < 10300), see http://factordb.com/sequences.php?se=1&aq=%28233*19%5E176-107%29%2F18&action=last20&fr=0&to=100, the "Check for factors" box shows "Already checked", this is the ECM effort t-level (https://oeis.org/wiki/OEIS_sequences_needing_factors#T-levels, https://stdkmd.net/nrr/wanted.htm (the "ECM" column of the three tables), https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, http://myfactorcollection.mooo.com:8090/lists.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the lists in the boxes), http://myfactorcollection.mooo.com:8090/dbio.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the DB inputs/outputs in the boxes), https://github.com/brubsby/t-level, https://www.mersenneforum.org/showthread.php?t=29615) t30 (see http://www.loria.fr/~zimmerma/records/ecm/params.html and https://www.rieselprime.de/ziki/Elliptic_curve_method#Choosing_the_best_parameters_for_ECM), i.e. the prime factors of this composite number are probably > 1030)
19952093221742579 × 1830029658442819 × 7648288486...8219234607 (228-digit composite without known proper factor > 1, this composite has already checked with P−1 to B1 = 50000 and 3 times P+1 to B1 = 150000 and 10 times ECM to B1 = 250000 (these can be checked for composites < 10300), see http://factordb.com/sequences.php?se=1&aq=%28233*19%5E200-107%29%2F18&action=last20&fr=0&to=100, the "Check for factors" box shows "Already checked", this is the ECM effort t-level (https://oeis.org/wiki/OEIS_sequences_needing_factors#T-levels, https://stdkmd.net/nrr/wanted.htm (the "ECM" column of the three tables), https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, http://myfactorcollection.mooo.com:8090/lists.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the lists in the boxes), http://myfactorcollection.mooo.com:8090/dbio.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the DB inputs/outputs in the boxes), https://github.com/brubsby/t-level, https://www.mersenneforum.org/showthread.php?t=29615) t30 (see http://www.loria.fr/~zimmerma/records/ecm/params.html and https://www.rieselprime.de/ziki/Elliptic_curve_method#Choosing_the_best_parameters_for_ECM), i.e. the prime factors of this composite number are probably > 1030)

and it does not appear to be any covering congruence of primes, besides, since 107 is not a perfect power, thus the family CHnC (in base b = 19) has no algebraic factorization for any n, thus its Nash weight (or difficulty) is positive (although very low), so there must be a prime at some point (and the corresponding n must be == 7 mod 24).

And for n == 7 mod 24, we have: (all "composites without known proper factor > 1" in the list below have been trial factored (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) to 109, also none of these composites have algebraic factors)

Another example of a family and its dual family are both unsolved families is the base b = 25 family FB{0}H (its algebraic form is 386×25n+1+17, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=386*25%5E%28n%2B1%29%2B17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show) and the base b = 25 family H{0}FB (its algebraic form is 17×25n+2+386, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=17*25%5E%28n%2B2%29%2B386&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show).

Also two examples of a family and its dual family both have known (probable) primes (and these two (probable) primes are both minimal primes (assuming their primality)) but both have no easy primes (i.e. lengths ≤ 1000 in base b) are (the base b = 17 families 109{0}D and D{0}109 are dual families, and the base b = 21 families 1{0}5D and 5D{0}1 are also dual families):

b
(2 ≤ b ≤ 36)
familyalgebraic formbase-b form of the smallest (probable) prime in this family (write "dn" if there are 5 or more (n) consecutive same digits d)length of the smallest (probable) prime in this familylength of the smallest (probable) prime in this family in decimalalgebraic ((a×bn+c)/gcd(a+c,b−1)) form of the smallest (probable) prime in this familyfactordb entry of the smallest (probable) prime in this familythe smallest (probable) prime in this family written in base b (use lower case letters instead of upper case letters)the smallest (probable) prime in this family written in decimalfactorization of the first 200 numbers in corresponding family (n is the number of digits in the "{}", start with the smallest n making the number > b (if n = 0 already makes the number > b, then start with n = 0))
17109{0}D298×17n+1+1310906387D63917863298×176388+13http://factordb.com/index.php?id=1100000003782940751&open=primehttp://factordb.com/index.php?showid=1100000003782940751&base=17http://factordb.com/index.php?showid=1100000003782940751http://factordb.com/index.php?query=298*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17D{0}10913×17n+3+298D011691091173144413×171172+298http://factordb.com/index.php?id=1100000003782940735&open=primehttp://factordb.com/index.php?showid=1100000003782940735&base=17http://factordb.com/index.php?showid=1100000003782940735http://factordb.com/index.php?query=13*17%5E%28n%2B3%29%2B298&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
211{0}5D21n+2+11810113955D11398150702111397+118http://factordb.com/index.php?id=1100000003815090811&open=primehttp://factordb.com/index.php?showid=1100000003815090811&base=21http://factordb.com/index.php?showid=1100000003815090811http://factordb.com/index.php?query=21%5E%28n%2B2%29%2B118&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
215D{0}1118×21n+1+15D01984811985126247118×2119849+1http://factordb.com/index.php?id=1100000000777265872&open=primehttp://factordb.com/index.php?showid=1100000000777265872&base=21http://factordb.com/index.php?showid=1100000000777265872http://factordb.com/index.php?query=118*21%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show

The same holds for all other unsolved families in bases 2 ≤ b ≤ 36, thus we can almost sure that all unsolved families in bases 2 ≤ b ≤ 36 have a prime, in fact, we can almost sure that all unsolved families in bases 2 ≤ b ≤ 36 have infinitely many primes. (if the family has algebraic factorization for some n, then we should only consider the n such that this family does not have algebraic factorization and check whether there is a covering congruence for these n, e.g. the unsolved family 2EBn is base b = 23, whose algebraic form is (121×23n−1)/2, which has a difference-of-two-squares factorization if n is even (factored to (11×23n/2−1) × (11×23n/2+1) / 2), thus we should only consider the odd n and check whether there is a covering congruence for the odd n)

Also, there are unsolved families in bases 2 ≤ b ≤ 36 which are impossible to be ruled out as only containing composites (only count the numbers > b) (by covering congruence, algebraic factorization, or combine of them), note the difference of the unsolved families in bases 2 ≤ b ≤ 36 which are very unlikely to be ruled out as only containing composites (only count the numbers > b) (by covering congruence, algebraic factorization, or combine of them), the latter still has a possibility, such as the three families above (family {9}D in base b = 21 and family 5{H}5 in base b = 19 and family C{H}C in base b = 19), the former has no possibility, such as the three examples below:

Family 1{7} in base b = 17: (its algebraic form is (23×17n−7)/16, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

Since 7 is not a perfect power, thus the family 17n (in base b = 17) has no algebraic factorization for any n, thus if this family can be ruled out as only containing composites, then it must have a covering congruence, and if the period of its covering congruence is m, consider the prime factor p dividing (23×17n−7)/16 for all n == 0 mod m, p must divide (23×170−7)/16 = −1, which is a contradiction.

Family 9{6}M in base b = 25: (its algebraic form is (37×25n+1+63)/4, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=%2837*25%5E%28n%2B1%29%2B63%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

Since 63 is not a perfect power, thus the family 96nM (in base b = 25) has no algebraic factorization for any n, thus if this family can be ruled out as only containing composites, then it must have a covering congruence, and if the period of its covering congruence is m, consider the prime factor p dividing (37×25n+1+63)/4 for all n == −1 mod m, p must divide (37×25(−1)+1+63)/4 = 25, and thus p must be 5 (since 5 is the only prime factor of 25), but no numbers in the family 9{6}M (in base b = 25) are divisible by 5 (since all numbers in this family are == 22 mod 25, thus are == 2 mod 5), which is a contradiction.

Family E{D} in base b = 25: (its algebraic form is (349×25n−13)/24, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=%28349*25%5En-13%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

Since 13 is not a perfect power, thus the family EDn (in base b = 25) has no algebraic factorization for any n, thus if this family can be ruled out as only containing composites, then it must have a covering congruence, and if the period of its covering congruence is m, consider the prime factor p dividing (349×25n−13)/24 for all n == −1 mod m, p must divide the numerator of (349×25−1−13)/24 = 1/25, i.e. p divides 1, which is a contradiction.

Family 1{F}5 in base b = 27: (its algebraic form is (41×27n+1−275)/26, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=%2841*27%5E%28n%2B1%29-275%29%2F26&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

Since 275 is not a perfect power, thus the family 1Fn5 (in base b = 27) has no algebraic factorization for any n, thus if this family can be ruled out as only containing composites, then it must have a covering congruence, and if the period of its covering congruence is m, consider the prime factor p dividing (41×27n+1−275)/26 for all n == −1 mod m, p must divide (41×27(−1)+1−275)/26 = −9, and thus p must be 3 (since 3 is the only prime factor of −9), but no numbers in the family 1{F}5 (in base b = 27) are divisible by 3 (since all numbers in this family are == 5 mod 27, thus are == 2 mod 3), which is a contradiction.

In fact, the corresponding families for the largest minimal primes in bases b = 7, 10, 14, 22, 30 are such families, since:

Family {3}1 in base b = 7: (its algebraic form is (7n+1−5)/2, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=%287%5E%28n%2B1%29-5%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

Since 5 is not a perfect power, thus the family 3n1 (in base b = 7) has no algebraic factorization for any n, thus if this family can be ruled out as only containing composites, then it must have a covering congruence, and if the period of its covering congruence is m, consider the prime factor p dividing (7n+1−5)/2 for all n == 0 mod m, p must divide (70+1−5)/2 = 1, which is a contradiction.

Family 5{0}27 in base b = 10: (its algebraic form is 5×10n+2+27, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=5*10%5E%28n%2B2%29%2B27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

Since 27 is a cube, thus 5×10n+2+27 has sum-of-two-cubes factorization if 5×10n+2 is a cube (if not, then it has no algebraic factorization), but 5×10n+2 cannot be a perfect power, since the prime factorization of 5×10n+2 is 2n+2 × 5n+3, but n+2 and n+3 must be coprime, thus 2n+2 × 5n+3 cannot be a perfect power, thus the family 50n27 (in base b = 10) has no algebraic factorization for any n, thus if this family can be ruled out as only containing composites, then it must have a covering congruence, and if the period of its covering congruence is m, consider the prime factor p dividing 5×10n+2+27 for all n == −2 mod m, p must divide 5×10(−2)+2+27 = 32, and thus p must be 2 (since 2 is the only prime factor of 32), but no numbers in the family 5{0}27 (in base b = 10) are divisible by 2 (since all numbers in this family are == 7 mod 10, thus are == 1 mod 2), which is a contradiction.

Family 4{D} in base b = 14: (its algebraic form is 5×14n−1, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=5*14%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

Since 1 is an rth power for all r, thus 5×14n−1 has difference-of-two-rth-powers factorization if 5×14n is an rth power (if not, then it has no algebraic factorization), but 5×14n cannot be a perfect power, after all, 5×14n is divisible by 5 but not 52, thus the family 4Dn (in base b = 14) has no algebraic factorization for any n, thus if this family can be ruled out as only containing composites, then it must have a covering congruence, and if the period of its covering congruence is m, consider the prime factor p dividing 5×14n−1 for all n == 0 mod m, p must divide 5×140−1 = 4, and thus p must be 2 (since 2 is the only prime factor of 4), but no numbers in the family 4{D} (in base b = 14) are divisible by 2 (since all numbers in this family are == 13 mod 14, thus are == 1 mod 2), which is a contradiction.

Family B{K}5 in base b = 22: (its algebraic form is (251×22n+1−335)/21, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=%28251*22%5E%28n%2B1%29-335%29%2F21&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

Since 335 is not a perfect power, thus the family BKn5 (in base b = 22) has no algebraic factorization for any n, thus if this family can be ruled out as only containing composites, then it must have a covering congruence, and if the period of its covering congruence is m, consider the prime factor p dividing (251×22n+1−335)/21 for all n == −1 mod m, p must divide (251×22(−1)+1−335)/21 = −4, and thus p must be 2 (since 2 is the only prime factor of −4), but no numbers in the family B{K}5 (in base b = 22) are divisible by 2 (since all numbers in this family are == 5 mod 22, thus are == 1 mod 2), which is a contradiction.

Family O{T} in base b = 30: (its algebraic form is 25×30n−1, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=25*30%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

Since 1 is an rth power for all r, thus 25×30n−1 has difference-of-two-rth-powers factorization if 25×30n is an rth power (if not, then it has no algebraic factorization), and indeed 25×30n is a square if n == 0 mod 2, but 25×30n cannot be a perfect power if n == 1 mod 2, since the prime factorization of 25×30n is 2n × 3n × 5n+2, but n and n+2 must be coprime if n == 1 mod 2, thus 2n × 3n × 5n+2 cannot be a perfect power if n == 1 mod 2, thus the family OTn (in base b = 30) has no algebraic factorization for any n == 1 mod 2, thus if this family can be ruled out as only containing composites, then it must have a covering congruence on the n == 1 mod 2, and if the period of its covering congruence is m, consider the prime factor p dividing 25×30n−1 for all n == −1 mod 2×m (such n must be == 1 mod 2), p must divide the numerator of 25×30−1−1 = −1/6, i.e. p divides 1, which is a contradiction.

e.g. (only list the families which all numbers do not contain "prime > b" subsequence) (for the case of covering congruence, we can show that the corresponding numbers are > all elements in the sets if the corresponding numbers are > b, thus these factorizations are nontrivial; and for the case of algebraic factorization, we can show that both factors are > 1 if the corresponding numbers are > b, thus these factorizations are nontrivial; for the case of combine of them, we can show that for the part of covering congruence, the corresponding numbers are > all elements in the sets if the corresponding numbers are > b, and for the part of algebraic factorization, both factors are > 1 if the corresponding numbers are > b, thus these factorizations are nontrivial)

For the factor patterns of some families: (list the first 20 numbers in these families, start with the smallest number > b in these families) (all numbers are written in base b, and these factorizations are nontrivial (https://en.wikipedia.org/wiki/Triviality_(mathematics), https://mathworld.wolfram.com/Trivial.html), i.e. both of these two factors are > 1, of course, one or two of these two factors may be composite)

Example 1, family 4{6}9 in base b = 10: (the period of the factor pattern is 1, and its algebraic form is (14×10n+1+7)/3, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=%2814*10%5E%28n%2B1%29%2B7%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

  • 49 = 7 × 7
  • 469 = 7 × 67
  • 4669 = 7 × 667
  • 46669 = 7 × 6667
  • 466669 = 7 × 66667
  • 4666669 = 7 × 666667
  • 46666669 = 7 × 6666667
  • 466666669 = 7 × 66666667
  • 4666666669 = 7 × 666666667
  • 46666666669 = 7 × 6666666667
  • 466666666669 = 7 × 66666666667
  • 4666666666669 = 7 × 666666666667
  • 46666666666669 = 7 × 6666666666667
  • 466666666666669 = 7 × 66666666666667
  • 4666666666666669 = 7 × 666666666666667
  • 46666666666666669 = 7 × 6666666666666667
  • 466666666666666669 = 7 × 66666666666666667
  • 4666666666666666669 = 7 × 666666666666666667
  • 46666666666666666669 = 7 × 6666666666666666667
  • 466666666666666666669 = 7 × 66666666666666666667

Example 2, family 28{0}7 in base b = 10: (the period of the factor pattern is 1, and its algebraic form is 28×10n+1+7, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=28*10%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

  • 287 = 7 × 41
  • 2807 = 7 × 401
  • 28007 = 7 × 4001
  • 280007 = 7 × 40001
  • 2800007 = 7 × 400001
  • 28000007 = 7 × 4000001
  • 280000007 = 7 × 40000001
  • 2800000007 = 7 × 400000001
  • 28000000007 = 7 × 4000000001
  • 2800000000007 = 7 × 4000000001
  • 28000000000007 = 7 × 40000000001
  • 280000000000007 = 7 × 400000000001
  • 2800000000000007 = 7 × 4000000000001
  • 28000000000000007 = 7 × 40000000000001
  • 280000000000000007 = 7 × 400000000000001
  • 2800000000000000007 = 7 × 4000000000000001
  • 28000000000000000007 = 7 × 40000000000000001
  • 280000000000000000007 = 7 × 400000000000000001
  • 2800000000000000000007 = 7 × 4000000000000000001
  • 28000000000000000000007 = 7 × 40000000000000000001

Example 3, family {1} in base b = 9: (the period of the factor pattern is 2, and its algebraic form is (9n−1)/8, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=%289%5En-1%29%2F8&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

  • 11 = 2 × 5
  • 111 = 7 × 14
  • 1111 = 22 × 45
  • 11111 = 67 × 144
  • 111111 = 222 × 445
  • 1111111 = 667 × 1444
  • 11111111 = 2222 × 4445
  • 111111111 = 6667 × 14444
  • 1111111111 = 22222 × 44445
  • 11111111111 = 66667 × 144444
  • 111111111111 = 222222 × 444445
  • 1111111111111 = 666667 × 1444444
  • 11111111111111 = 2222222 × 4444445
  • 111111111111111 = 6666667 × 14444444
  • 1111111111111111 = 22222222 × 44444445
  • 11111111111111111 = 66666667 × 144444444
  • 111111111111111111 = 222222222 × 444444445
  • 1111111111111111111 = 666666667 × 1444444444
  • 11111111111111111111 = 2222222222 × 4444444445
  • 111111111111111111111 = 6666666667 × 14444444444

Example 4, family 3{8} in base b = 9: (the period of the factor pattern is 2, and its algebraic form is 4×9n−1, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=4*9%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

  • 38 = 5 × 7
  • 388 = 18 × 21
  • 3888 = 58 × 61
  • 38888 = 188 × 201
  • 388888 = 588 × 601
  • 3888888 = 1888 × 2001
  • 38888888 = 5888 × 6001
  • 388888888 = 18888 × 20001
  • 3888888888 = 58888 × 60001
  • 38888888888 = 188888 × 200001
  • 388888888888 = 588888 × 600001
  • 3888888888888 = 1888888 × 2000001
  • 38888888888888 = 5888888 × 6000001
  • 388888888888888 = 18888888 × 20000001
  • 3888888888888888 = 58888888 × 60000001
  • 38888888888888888 = 188888888 × 200000001
  • 388888888888888888 = 588888888 × 600000001
  • 3888888888888888888 = 1888888888 × 2000000001
  • 38888888888888888888 = 5888888888 × 6000000001
  • 388888888888888888888 = 18888888888 × 20000000001

Example 5, family 1{0}1 in base b = 8: (the period of the factor pattern is 3, and its algebraic form is 8n+1+1, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=8%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

  • 11 = 3 × 3
  • 101 = 5 × 15
  • 1001 = 11 × 71
  • 10001 = 21 × 361
  • 100001 = 41 × 1741
  • 1000001 = 101 × 7701
  • 10000001 = 201 × 37601
  • 100000001 = 401 × 177401
  • 1000000001 = 1001 × 777001
  • 10000000001 = 2001 × 3776001
  • 100000000001 = 4001 × 17774001
  • 1000000000001 = 10001 × 77770001
  • 10000000000001 = 20001 × 377760001
  • 100000000000001 = 40001 × 1777740001
  • 1000000000000001 = 100001 × 7777700001
  • 10000000000000001 = 200001 × 37777600001
  • 100000000000000001 = 400001 × 177777400001
  • 1000000000000000001 = 1000001 × 777777000001
  • 10000000000000000001 = 2000001 × 3777776000001
  • 100000000000000000001 = 4000001 × 17777774000001

Example 6, family {8}5 in base b = 9: (the period of the factor pattern is 2, and its algebraic form is 9n+1−4, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=9%5E%28n%2B1%29-4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

  • 85 = 7 × 12
  • 885 = 27 × 32
  • 8885 = 87 × 102
  • 88885 = 287 × 302
  • 888885 = 887 × 1002
  • 8888885 = 2887 × 3002
  • 88888885 = 8887 × 10002
  • 888888885 = 28887 × 30002
  • 8888888885 = 88887 × 100002
  • 88888888885 = 288887 × 300002
  • 888888888885 = 888887 × 1000002
  • 8888888888885 = 2888887 × 3000002
  • 88888888888885 = 8888887 × 10000002
  • 888888888888885 = 28888887 × 30000002
  • 8888888888888885 = 88888887 × 100000002
  • 88888888888888885 = 288888887 × 300000002
  • 888888888888888885 = 888888887 × 1000000002
  • 8888888888888888885 = 2888888887 × 3000000002
  • 88888888888888888885 = 8888888887 × 10000000002

Example 7, family {2}5 in base b = 11: (the period of the factor pattern is 2, and its algebraic form is (5×11n−1)/2, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=%285*11%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

  • 25 = 3 × 9
  • 255 = 2 × 128
  • 2555 = 3 × 919
  • 25555 = 2 × 12828
  • 255555 = 3 × 91919
  • 2555555 = 2 × 1282828
  • 25555555 = 3 × 9191919
  • 255555555 = 2 × 128282828
  • 2555555555 = 3 × 919191919
  • 25555555555 = 2 × 12828282828
  • 255555555555 = 3 × 91919191919
  • 2555555555555 = 2 × 1282828282828
  • 25555555555555 = 3 × 9191919191919
  • 255555555555555 = 2 × 128282828282828
  • 2555555555555555 = 3 × 919191919191919
  • 25555555555555555 = 2 × 12828282828282828
  • 255555555555555555 = 3 × 91919191919191919
  • 2555555555555555555 = 2 × 1282828282828282828
  • 25555555555555555555 = 3 × 9191919191919191919
  • 255555555555555555555 = 2 × 128282828282828282828

Example 8, family {B}9B in base b = 12: (the period of the factor pattern is 2, and its algebraic form is 12n+2−25, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=12%5E%28n%2B2%29-25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

  • 9B = 7 × 15
  • B9B = 11 × AB
  • BB9B = B7 × 105
  • BBB9B = 11 × B0AB
  • BBBB9B = BB7 × 1005
  • BBBBB9B = 11 × B0B0AB
  • BBBBBB9B = BBB7 × 10005
  • BBBBBBB9B = 11 × B0B0B0AB
  • BBBBBBBB9B = BBBB7 × 100005
  • BBBBBBBBB9B = 11 × B0B0B0B0AB
  • BBBBBBBBBB9B = BBBBB7 × 1000005
  • BBBBBBBBBBB9B = 11 × B0B0B0B0B0AB
  • BBBBBBBBBBBB9B = BBBBBB7 × 10000005
  • BBBBBBBBBBBBB9B = 11 × B0B0B0B0B0B0AB
  • BBBBBBBBBBBBBB9B = BBBBBBB7 × 100000005
  • BBBBBBBBBBBBBBB9B = 11 × B0B0B0B0B0B0B0AB
  • BBBBBBBBBBBBBBBB9B = BBBBBBBB7 × 1000000005
  • BBBBBBBBBBBBBBBBB9B = 11 × B0B0B0B0B0B0B0B0AB
  • BBBBBBBBBBBBBBBBBB9B = BBBBBBBBB7 × 10000000005
  • BBBBBBBBBBBBBBBBBBB9B = 11 × B0B0B0B0B0B0B0B0B0AB

Example 9, family B{0}1 in base b = 14: (the period of the factor pattern is 2, and its algebraic form is 11×14n+1+1, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=11*14%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

  • B1 = 5 × 23
  • B01 = 3 × 395
  • B001 = 5 × 22B3
  • B0001 = 3 × 39495
  • B00001 = 5 × 22B2B3
  • B000001 = 3 × 3949495
  • B0000001 = 5 × 22B2B2B3
  • B00000001 = 3 × 394949495
  • B000000001 = 5 × 22B2B2B2B3
  • B0000000001 = 3 × 39494949495
  • B00000000001 = 5 × 22B2B2B2B2B3
  • B000000000001 = 3 × 3949494949495
  • B0000000000001 = 5 × 22B2B2B2B2B2B3
  • B00000000000001 = 3 × 394949494949495
  • B000000000000001 = 5 × 22B2B2B2B2B2B2B3
  • B0000000000000001 = 3 × 39494949494949495
  • B00000000000000001 = 5 × 22B2B2B2B2B2B2B2B3
  • B000000000000000001 = 3 × 3949494949494949495
  • B0000000000000000001 = 5 × 22B2B2B2B2B2B2B2B2B3
  • B00000000000000000001 = 3 × 394949494949494949495

Example 10, family 3{0}95 in base b = 13: (the period of the factor pattern is 4, and its algebraic form is 3×13n+2+122, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=3*13%5E%28n%2B2%29%2B122&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

  • 395 = 14 × 2B
  • 3095 = 7 × 58A
  • 30095 = 5 × 7A71
  • 300095 = 7 × 5758A
  • 3000095 = 14 × 23A92B
  • 30000095 = 7 × 575758A
  • 300000095 = 5 × 7A527A71
  • 3000000095 = 7 × 57575758A
  • 30000000095 = 14 × 23A923A92B
  • 300000000095 = 7 × 5757575758A
  • 3000000000095 = 5 × 7A527A527A71
  • 30000000000095 = 7 × 575757575758A
  • 300000000000095 = 14 × 23A923A923A92B
  • 3000000000000095 = 7 × 57575757575758A
  • 30000000000000095 = 5 × 7A527A527A527A71
  • 300000000000000095 = 7 × 5757575757575758A
  • 3000000000000000095 = 14 × 23A923A923A923A92B
  • 30000000000000000095 = 7 × 575757575757575758A
  • 300000000000000000095 = 5 × 7A527A527A527A527A71
  • 3000000000000000000095 = 7 × 57575757575757575758A

Example 11, family {4}D in base b = 16: (the period of the factor pattern is 3, and its algebraic form is (4×16n+1+131)/15, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29%2B131%29%2F15&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

  • 4D = 7 × B
  • 44D = 3 × 16F
  • 444D = D × 541
  • 4444D = 7 × 9C0B
  • 44444D = 3 × 16C16F
  • 444444D = D × 540541
  • 4444444D = 7 × 9C09C0B
  • 44444444D = 3 × 16C16C16F
  • 444444444D = D × 540540541
  • 4444444444D = 7 × 9C09C09C0B
  • 44444444444D = 3 × 16C16C16C16F
  • 444444444444D = D × 540540540541
  • 4444444444444D = 7 × 9C09C09C09C0B
  • 44444444444444D = 3 × 16C16C16C16C16F
  • 444444444444444D = D × 540540540540541
  • 4444444444444444D = 7 × 9C09C09C09C09C0B
  • 44444444444444444D = 3 × 16C16C16C16C16C16F
  • 444444444444444444D = D × 540540540540540541
  • 4444444444444444444D = 7 × 9C09C09C09C09C09C0B
  • 44444444444444444444D = 3 × 16C16C16C16C16C16C16F

Example 12, family {C}D in base b = 16: (the period of the factor pattern is 4, and its algebraic form is (4×16n+1+1)/5, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29%2B1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

  • CD = 5 × 29
  • CCD = 1D × 71
  • CCCD = 6D × 1E1
  • CCCCD = 18D × 841
  • CCCCCD = 64D × 2081
  • CCCCCCD = 19CD × 7F01
  • CCCCCCCD = 66CD × 1FE01
  • CCCCCCCCD = 198CD × 80401
  • CCCCCCCCCD = 664CD × 200801
  • CCCCCCCCCCD = 199CCD × 7FF001
  • CCCCCCCCCCCD = 666CCD × 1FFE001
  • CCCCCCCCCCCCD = 1998CCD × 8004001
  • CCCCCCCCCCCCCD = 6664CCD × 20008001
  • CCCCCCCCCCCCCCD = 1999CCCD × 7FFF0001
  • CCCCCCCCCCCCCCCD = 6666CCCD × 1FFFE0001
  • CCCCCCCCCCCCCCCCD = 19998CCCD × 800040001
  • CCCCCCCCCCCCCCCCCD = 66664CCCD × 2000080001
  • CCCCCCCCCCCCCCCCCCD = 19999CCCCD × 7FFFF00001
  • CCCCCCCCCCCCCCCCCCCD = 66666CCCCD × 1FFFFE00001
  • CCCCCCCCCCCCCCCCCCCCD = 199998CCCCD × 80000400001

Example 13, family 1{9} in base b = 17: (the period of the factor pattern is 2, and its algebraic form is (25×17n−9)/16, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=%2825*17%5En-9%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

  • 19 = 2 × D
  • 199 = B × 27
  • 1999 = 2 × D4D
  • 19999 = AB × 287
  • 199999 = 2 × D4D4D
  • 1999999 = AAB × 2887
  • 19999999 = 2 × D4D4D4D
  • 199999999 = AAAB × 28887
  • 1999999999 = 2 × D4D4D4D4D
  • 19999999999 = AAAAB × 288887
  • 199999999999 = 2 × D4D4D4D4D4D
  • 1999999999999 = AAAAAB × 2888887
  • 19999999999999 = 2 × D4D4D4D4D4D4D
  • 199999999999999 = AAAAAAB × 28888887
  • 1999999999999999 = 2 × D4D4D4D4D4D4D4D
  • 19999999999999999 = AAAAAAAB × 288888887
  • 199999999999999999 = 2 × D4D4D4D4D4D4D4D4D
  • 1999999999999999999 = AAAAAAAAB × 2888888887
  • 19999999999999999999 = 2 × D4D4D4D4D4D4D4D4D4D
  • 199999999999999999999 = AAAAAAAAAB × 28888888887

Example 14, family 1{6} in base b = 19: (the period of the factor pattern is 2, and its algebraic form is (4×19n−1)/3, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=%284*19%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

  • 16 = 5 × 5
  • 166 = D × 1I
  • 1666 = 5 × 515
  • 16666 = CD × 1II
  • 166666 = 5 × 51515
  • 1666666 = CCD × 1III
  • 16666666 = 5 × 5151515
  • 166666666 = CCCD × 1IIII
  • 1666666666 = 5 × 515151515
  • 16666666666 = CCCCD × 1IIIII
  • 166666666666 = 5 × 51515151515
  • 1666666666666 = CCCCCD × 1IIIIII
  • 16666666666666 = 5 × 5151515151515
  • 166666666666666 = CCCCCCD × 1IIIIIII
  • 1666666666666666 = 5 × 515151515151515
  • 16666666666666666 = CCCCCCCD × 1IIIIIIII
  • 166666666666666666 = 5 × 51515151515151515
  • 1666666666666666666 = CCCCCCCCD × 1IIIIIIIII
  • 16666666666666666666 = 5 × 5151515151515151515
  • 166666666666666666666 = CCCCCCCCCD × 1IIIIIIIIII

Example 15, family {D}GA in base b = 23: (the period of the factor pattern is 12, and its algebraic form is (13×23n+2+1439)/22, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=%2813*23%5E%28n%2B2%29%2B1439%29%2F22&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

  • GA = 6 × 2H
  • DGA = 5 × 2H2
  • DDGA = 6 × 262H
  • DDDGA = 7 × 1LF28
  • DDDDGA = 6 × 26262H
  • DDDDDGA = 3A × 3M03M1
  • DDDDDDGA = 6 × 2626262H
  • DDDDDDDGA = 1E × 8A74GC44
  • DDDDDDDDGA = 6 × 262626262H
  • DDDDDDDDDGA = 7 × 1LF1LF1LF28
  • DDDDDDDDDDGA = 6 × 26262626262H
  • DDDDDDDDDDDGA = 3A × 3M03M03M03M1
  • DDDDDDDDDDDDGA = 6 × 2626262626262H
  • DDDDDDDDDDDDDGA = 5 × 2GBL2GBL2GBL2H2
  • DDDDDDDDDDDDDDGA = 6 × 262626262626262H
  • DDDDDDDDDDDDDDDGA = 7 × 1LF1LF1LF1LF1LF28
  • DDDDDDDDDDDDDDDDGA = 6 × 26262626262626262H
  • DDDDDDDDDDDDDDDDDGA = 3A × 3M03M03M03M03M03M1
  • DDDDDDDDDDDDDDDDDDGA = 6 × 2626262626262626262H
  • DDDDDDDDDDDDDDDDDDDGA = 1E × 8A74GC4257J08A74GC44

Example 16, family 2{1} in base b = 25: (the period of the factor pattern is 2, and its algebraic form is (49×25n−1)/24, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=%2849*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

  • 21 = 3 × H
  • 211 = 14 × 1J
  • 2111 = 2N × HC
  • 21111 = 144 × 1IJ
  • 211111 = 2MN × HCC
  • 2111111 = 1444 × 1IIJ
  • 21111111 = 2MMN × HCCC
  • 211111111 = 14444 × 1IIIJ
  • 2111111111 = 2MMMN × HCCCC
  • 21111111111 = 144444 × 1IIIIJ
  • 211111111111 = 2MMMMN × HCCCCC
  • 2111111111111 = 1444444 × 1IIIIIJ
  • 21111111111111 = 2MMMMMN × HCCCCCC
  • 211111111111111 = 14444444 × 1IIIIIIJ
  • 2111111111111111 = 2MMMMMMN × HCCCCCCC
  • 21111111111111111 = 144444444 × 1IIIIIIIJ
  • 211111111111111111 = 2MMMMMMMN × HCCCCCCCC
  • 2111111111111111111 = 1444444444 × 1IIIIIIIIJ
  • 21111111111111111111 = 2MMMMMMMMN × HCCCCCCCCC
  • 211111111111111111111 = 14444444444 × 1IIIIIIIIIJ

Example 17, family 7{Q} in base b = 27: (the period of the factor pattern is 3, and its algebraic form is 8×27n−1, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=8*27%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

  • 7Q = 5 × 1G
  • 7QQ = H × CJ
  • 7QQQ = 1Q × 421
  • 7QQQQ = 5Q × 1961
  • 7QQQQQ = HQ × C0I1
  • 7QQQQQQ = 1QQ × 40201
  • 7QQQQQQQ = 5QQ × 190601
  • 7QQQQQQQQ = HQQ × C00I01
  • 7QQQQQQQQQ = 1QQQ × 4002001
  • 7QQQQQQQQQQ = 5QQQ × 19006001
  • 7QQQQQQQQQQQ = HQQQ × C000I001
  • 7QQQQQQQQQQQQ = 1QQQQ × 400020001
  • 7QQQQQQQQQQQQQ = 5QQQQ × 1900060001
  • 7QQQQQQQQQQQQQQ = HQQQQ × C0000I0001
  • 7QQQQQQQQQQQQQQQ = 1QQQQQ × 40000200001
  • 7QQQQQQQQQQQQQQQQ = 5QQQQQ × 190000600001
  • 7QQQQQQQQQQQQQQQQQ = HQQQQQ × C00000I00001
  • 7QQQQQQQQQQQQQQQQQQ = 1QQQQQQ × 4000002000001
  • 7QQQQQQQQQQQQQQQQQQQ = 5QQQQQQ × 19000006000001
  • 7QQQQQQQQQQQQQQQQQQQQ = HQQQQQQ × C000000I000001

Example 18, family A{0}9J in base b = 30: (the period of the factor pattern is 6, and its algebraic form is 10×30n+2+289, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=10*30%5E%28n%2B2%29%2B289&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

  • A9J = 7 × 1E7
  • A09J = 11 × 9KJ
  • A009J = J × FNL1
  • A0009J = 11 × 9K9KJ
  • A00009J = D × N296SD
  • A000009J = 11 × 9K9K9KJ
  • A0000009J = 7 × 1CPLCPLE7
  • A00000009J = 11 × 9K9K9K9KJ
  • A000000009J = J × FNKFNKFNL1
  • A0000000009J = 11 × 9K9K9K9K9KJ
  • A00000000009J = D × N296RKN296SD
  • A000000000009J = 11 × 9K9K9K9K9K9KJ
  • A0000000000009J = 7 × 1CPLCPLCPLCPLE7
  • A00000000000009J = 11 × 9K9K9K9K9K9K9KJ
  • A000000000000009J = J × FNKFNKFNKFNKFNL1
  • A0000000000000009J = 11 × 9K9K9K9K9K9K9K9KJ
  • A00000000000000009J = D × N296RKN296RKN296SD
  • A000000000000000009J = 11 × 9K9K9K9K9K9K9K9K9KJ
  • A0000000000000000009J = 7 × 1CPLCPLCPLCPLCPLCPLE7
  • A00000000000000000009J = 11 × 9K9K9K9K9K9K9K9K9K9KJ

Example 19, family {1} in base b = 32: (the period of the factor pattern is 5, and its algebraic form is (32n−1)/31, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=%2832%5En-1%29%2F31&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

  • 11 = 3 × B
  • 111 = 7 × 4N
  • 1111 = F × 26F
  • 11111 = IP × 1O9
  • 111111 = 1V × GOSV
  • 1111111 = 3V × 8AAQV
  • 11111111 = 7V × 44KMMV
  • 111111111 = FV × 2266EEV
  • 1111111111 = IP × 1O9001O9
  • 11111111111 = 1VV × GGOOSSUV
  • 111111111111 = 3VV × 88AAAQQUV
  • 1111111111111 = 7VV × 444KKMMMUV
  • 11111111111111 = FVV × 222666EEEUV
  • 111111111111111 = IP × 1O9001O9001O9
  • 1111111111111111 = 1VVV × GGGOOOSSSUV
  • 11111111111111111 = 3VVV × 888AAAAQQQUUV
  • 111111111111111111 = 7VVV × 4444KKKMMMMUUV
  • 1111111111111111111 = FVVV × 22226666EEEEUUV
  • 11111111111111111111 = IP × 1O9001O9001O9001O9

Example 20, family 3{7} in base b = 36: (the period of the factor pattern is 2, and its algebraic form is (16×36n−1)/5, and for the factorization of the numbers in the family (n is the number of digits in the "{}", start with the smallest n making the number > b) see http://factordb.com/index.php?query=%2816*36%5En-1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show)

  • 37 = 5 × N
  • 377 = T × 3Z
  • 3777 = 4T × NZ
  • 37777 = ST × 3ZZ
  • 377777 = 4ST × NZZ
  • 3777777 = SST × 3ZZZ
  • 37777777 = 4SST × NZZZ
  • 377777777 = SSST × 3ZZZZ
  • 3777777777 = 4SSST × NZZZZ
  • 37777777777 = SSSST × 3ZZZZZ
  • 377777777777 = 4SSSST × NZZZZZ
  • 3777777777777 = SSSSST × 3ZZZZZZ
  • 37777777777777 = 4SSSSST × NZZZZZZ
  • 377777777777777 = SSSSSST × 3ZZZZZZZ
  • 3777777777777777 = 4SSSSSST × NZZZZZZZ
  • 37777777777777777 = SSSSSSST × 3ZZZZZZZZ
  • 377777777777777777 = 4SSSSSSST × NZZZZZZZZ
  • 3777777777777777777 = SSSSSSSST × 3ZZZZZZZZZ
  • 37777777777777777777 = 4SSSSSSSST × NZZZZZZZZZ
  • 377777777777777777777 = SSSSSSSSST × 3ZZZZZZZZZZ

(only list the families which can be proven to be not covered by any prime > b (this would include all such families of the form x{0}y and all such families of the form x{y} (unless y = 1) and all such families of the form {x}y (unless x = 1)), the only exceptions are the base 21 family B0{H}6H (which is covered by the smallest prime in the family B{H}6H in base 21 (if such prime exists)) and the base 8 family 6{4}7 (which is covered by the prime 42207 in base 8))

(Note: There are families with more that one covering congruence, for more such examples see https://oeis.org/A263391 and https://oeis.org/A263392, also the case "444...44407" in https://sites.google.com/view/smallest-quasi-repdigit-primes, also the cases "936Rn" and "1222Rn" and "2739Rn" in https://web.archive.org/web/20070220134129/http://www.users.globalnet.co.uk/~perry/maths/wildeprimes/wildeprimes.htm)

(Note: Some of these families not only can be proven to contain no primes (only count the numbers > b), even can be proven to contain no semiprimes (https://en.wikipedia.org/wiki/Semiprime, https://t5k.org/glossary/xpage/Semiprime.html, https://mathworld.wolfram.com/Semiprime.html, https://www.numbersaplenty.com/set/semiprime/, https://oeis.org/A001358) (i.e. the product of two primes (not necessary distinct)) (only count the numbers > b), see the table below, but this is out of the main problem in this project)

(for the factorization of the numbers in these families, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=snfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#smallpolynomial, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=gnfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) may be used, they have SNFS polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt and https://alfredreichlg.de/10w7/10w7.txt and https://web.archive.org/web/20020320010222/http://proth.cjb.net/ and http://web.archive.org/web/20111104173105/http://www.immortaltheory.com/NumberTheory/2nm3_db.txt and https://www.asahi-net.or.jp/~KC2H-MSM/mathland/aba/index.htm and https://www.alpertron.com.ar/BRILLIANT.HTM and https://www.alpertron.com.ar/BRILLIANT3.HTM and https://www.alpertron.com.ar/BRILLIANT4.HTM and https://www.alpertron.com.ar/BRILLIANT2.HTM and https://oeis.org/wiki/Factors_of_33*2%5En%2B1 and https://oeis.org/wiki/Factors_of_33*2%5En-1 and https://web.archive.org/web/20111018190410/http://www.sr5.psp-project.de/s5stats.html (section "Top ten factors:") and https://web.archive.org/web/20111018190339/http://www.sr5.psp-project.de/r5stats.html (section "Top ten factors:") and https://brnikat.com/nums/cullen_woodall/cw.html and https://oeis.org/wiki/OEIS_sequences_needing_factors#Near_powers.2C_factorials.2C_and_primorials (sections "near-powers with b = 2" and "near-powers with b = 3" and "near-powers with b = 5" and "near-powers with b = 6" and "near-powers with b = 7" and "near-powers with b = 10" and "near-powers with b > 10") and https://www.mersenneforum.org/showthread.php?t=9554 and https://www.mersenneforum.org/showthread.php?t=9167 and https://www.mersenneforum.org/showpost.php?p=644144&postcount=5, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm and https://www.mersenneforum.org/showthread.php?t=12962)

(link to the factorization of the first 200 numbers in these families (start with the smallest number in these families which is > b), to show the small prime factors (and their period) (i.e. show whether these numbers are rough numbers) and the algebraic factors (and their period) and let you feel the Nash weight (or difficulty) of these families (the Nash weight (or difficulty) of these families are all zero, since they can be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them)), all small prime factors (< 1012, by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172)) and all algebraic factors (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/53335.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/83336.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=algebraic&maxrows=10000, https://sites.google.com/view/algebraic-factors-of-xn-kyn, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://www.mersenneforum.org/showpost.php?p=96560&postcount=99, https://www.mersenneforum.org/showpost.php?p=96651&postcount=101, https://www.mersenneforum.org/showthread.php?t=21916, https://www.mersenneforum.org/showpost.php?p=196598&postcount=492, https://www.mersenneforum.org/showpost.php?p=203083&postcount=149, https://www.mersenneforum.org/showpost.php?p=206065&postcount=192, https://www.mersenneforum.org/showpost.php?p=208044&postcount=260, https://www.mersenneforum.org/showpost.php?p=210533&postcount=336, https://www.mersenneforum.org/showpost.php?p=452132&postcount=66, https://www.mersenneforum.org/showpost.php?p=451337&postcount=32, https://www.mersenneforum.org/showpost.php?p=208852&postcount=227, https://www.mersenneforum.org/showpost.php?p=232904&postcount=604, https://www.mersenneforum.org/showpost.php?p=383690&postcount=1, https://www.mersenneforum.org/showpost.php?p=207886&postcount=253, https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php, https://www.alpertron.com.ar/POLFACT.HTM, https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) of the first 200 numbers (start with the smallest n making the number > b (if n = 0 already makes the number > b, then start with n = 0)) in these families were added to factordb)

(this table is sorted by: single trivial prime factor (2{0}1 in base b = 10 through D{6}R in base b = 28) → covering congruence with 2 primes ({1}5 in base b = 9 through {X}5 in base b = 34) → covering congruence with ≥ 3 primes (6{4}7 in base b = 8 through {G}L in base b = 32) → algebraic factorization ({1} in base b = 9 through 9{S}IJ in base b = 36) → combine of covering congruence and algebraic factorization (8{D} in base b = 14 through {X}P in base b = 34))

b
(2 ≤ b ≤ 36)
familyalgebraic ((a×bn+c)/d) form of this family (n is the number of digits in the "{}", also the lower bound of n to make the numbers > b)
(note: d divides gcd(a+c,b−1), but d need not be gcd(a+c,b−1), d = gcd(a+c,b−1) if and only if the numbers in the family are not divisible by some prime factor of b−1, i.e. the numbers in the family are coprime to b−1, in fact, d = gcd(a+c,b−1)/gcd(a+c,b−1,(largest trivial factor of the family)))
why this family contain no primes > bfactorization of the first 200 numbers in this family (n is the number of digits in the "{}", start with the smallest n making the number > b (if n = 0 already makes the number > b, then start with n = 0))
(only for linear families)
102{0}12×10n+1+1 (n ≥ 0)always divisible by 3http://factordb.com/index.php?query=2*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
102{0}72×10n+1+7 (n ≥ 0)always divisible by 3
(in fact, always divisible by 9)
http://factordb.com/index.php?query=2*10%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
105{0}15×10n+1+1 (n ≥ 0)always divisible by 3http://factordb.com/index.php?query=5*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
105{0}75×10n+1+7 (n ≥ 0)always divisible by 3http://factordb.com/index.php?query=5*10%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
108{0}18×10n+1+1 (n ≥ 0)always divisible by 3
(in fact, always divisible by 9)
http://factordb.com/index.php?query=8*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
108{0}78×10n+1+7 (n ≥ 0)always divisible by 3http://factordb.com/index.php?query=8*10%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1028{0}728×10n+1+7 (n ≥ 0)always divisible by 7http://factordb.com/index.php?query=28*10%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
104{6}9(14×10n+1+7)/3 (n ≥ 0)always divisible by 7http://factordb.com/index.php?query=%2814*10%5E%28n%2B1%29%2B7%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
10families ending with 0, 2, 4, 6, or 8always divisible by 2
10families ending with 0 or 5always divisible by 5
10{0,3,6,9}always divisible by 3
(nonlinear family)
10{0,7}always divisible by 7
(nonlinear family)
any base (b)families ending with digits d which are not coprime to balways divisible by gcd(d,b)
any base (b)families whose digits all have a common factor d > 1always divisible by d
31{0}13n+1+1 (n ≥ 0)always divisible by 2http://factordb.com/index.php?query=3%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
42{0}12×4n+1+1 (n ≥ 0)always divisible by 3http://factordb.com/index.php?query=2*4%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
51{0}15n+1+1 (n ≥ 0)always divisible by 2http://factordb.com/index.php?query=5%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
51{0}35n+1+3 (n ≥ 0)always divisible by 2
(in fact, always divisible by 4)
http://factordb.com/index.php?query=5%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
53{0}13×5n+1+1 (n ≥ 0)always divisible by 2
(in fact, always divisible by 4)
http://factordb.com/index.php?query=3*5%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
511{0}36×5n+1+3 (n ≥ 0)always divisible by 3http://factordb.com/index.php?query=6*5%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
53{0}113×5n+2+6 (n ≥ 0)always divisible by 3http://factordb.com/index.php?query=3*5%5E%28n%2B2%29%2B6&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
64{0}14×6n+1+1 (n ≥ 0)always divisible by 5http://factordb.com/index.php?query=4*6%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
71{0}17n+1+1 (n ≥ 0)always divisible by 2http://factordb.com/index.php?query=7%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
71{0}37n+1+3 (n ≥ 0)always divisible by 2http://factordb.com/index.php?query=7%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
71{0}57n+1+5 (n ≥ 0)always divisible by 2
(in fact, always divisible by 6)
http://factordb.com/index.php?query=7%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
73{0}13×7n+1+1 (n ≥ 0)always divisible by 2http://factordb.com/index.php?query=3*7%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
73{0}53×7n+1+5 (n ≥ 0)always divisible by 2http://factordb.com/index.php?query=3*7%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
75{0}15×7n+1+1 (n ≥ 0)always divisible by 2
(in fact, always divisible by 6)
http://factordb.com/index.php?query=5*7%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
75{0}35×7n+1+3 (n ≥ 0)always divisible by 2http://factordb.com/index.php?query=5*7%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
71{0}27n+1+2 (n ≥ 0)always divisible by 3http://factordb.com/index.php?query=7%5E%28n%2B1%29%2B2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
72{0}12×7n+1+1 (n ≥ 0)always divisible by 3http://factordb.com/index.php?query=2*7%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
74{0}54×7n+1+5 (n ≥ 0)always divisible by 3http://factordb.com/index.php?query=4*7%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
75{0}45×7n+1+4 (n ≥ 0)always divisible by 3http://factordb.com/index.php?query=5*7%5E%28n%2B1%29%2B4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
71{0}1{0}17m+n+2+7n+1+1 (m+n ≥ 0)always divisible by 3
(nonlinear family)
71{0}3{0}57m+n+2+3×7n+1+5 (m+n ≥ 0)always divisible by 3
(nonlinear family)
71{0}5{0}37m+n+2+5×7n+1+3 (m+n ≥ 0)always divisible by 3
(nonlinear family)
73{0}1{0}53×7m+n+2+7n+1+5 (m+n ≥ 0)always divisible by 3
(nonlinear family)
73{0}5{0}13×7m+n+2+5×7n+1+1 (m+n ≥ 0)always divisible by 3
(nonlinear family)
75{0}1{0}35×7m+n+2+7n+1+3 (m+n ≥ 0)always divisible by 3
(nonlinear family)
75{0}3{0}15×7m+n+2+3×7n+1+1 (m+n ≥ 0)always divisible by 3
(nonlinear family)
71{0}1{0}1{0}17r+m+n+3+7m+n+2+7n+1+1 (r+m+n ≥ 0)always divisible by 2
(nonlinear family)
71{0}1{0}27m+n+2+7n+1+2 (m+n ≥ 0)always divisible by 2
(nonlinear family)
71{0}2{0}17m+n+2+2×7n+1+1 (m+n ≥ 0)always divisible by 2
(nonlinear family)
72{0}1{0}12×7m+n+2+7n+1+1 (m+n ≥ 0)always divisible by 2
(nonlinear family)
74{0}5{0}54×7m+n+2+5×7n+1+5 (m+n ≥ 0)always divisible by 2
(nonlinear family)
75{0}4{0}55×7m+n+2+4×7n+1+5 (m+n ≥ 0)always divisible by 2
(nonlinear family)
75{0}5{0}45×7m+n+2+5×7n+1+4 (m+n ≥ 0)always divisible by 2
(nonlinear family)
82{0}52×8n+1+5 (n ≥ 0)always divisible by 7http://factordb.com/index.php?query=2*8%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
84{0}34×8n+1+3 (n ≥ 0)always divisible by 7http://factordb.com/index.php?query=4*8%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
86{0}16×8n+1+1 (n ≥ 0)always divisible by 7http://factordb.com/index.php?query=6*8%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
844{0}336×8n+1+3 (n ≥ 0)always divisible by 3http://factordb.com/index.php?query=36*8%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
86{0}116×8n+2+9 (n ≥ 0)always divisible by 3http://factordb.com/index.php?query=6*8%5E%28n%2B2%29%2B9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
86{0}416×8n+2+33 (n ≥ 0)always divisible by 3http://factordb.com/index.php?query=6*8%5E%28n%2B2%29%2B33&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
83{4}1(25×8n+1−25)/7 (n ≥ 0)always divisible by 5
(in fact, always divisible by 25)
(in fact, also difference-of-two-cubes factorization)
(25×8n+1−25)/7 = 25 × (2n−1) × (4n+2n+1) / 7
(special example, as the numbers with length ≥ 10 in this family contain "prime > b" subsequence, this prime is 481)
http://factordb.com/index.php?query=%2825*8%5E%28n%2B1%29-25%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
9{7}62(7×9n+2−119)/8 (n ≥ 0)always divisible by 7http://factordb.com/index.php?query=%287*9%5E%28n%2B2%29-119%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
112{5}3(5×11n+1−5)/2 (n ≥ 0)always divisible by 5
(in fact, always divisible by 25)
http://factordb.com/index.php?query=%285*11%5E%28n%2B1%29-5%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
113{5}2(7×11n+1−7)/2 (n ≥ 0)always divisible by 5
(in fact, always divisible by 35)
http://factordb.com/index.php?query=%287*11%5E%28n%2B1%29-7%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
113{5}9(7×11n+1+7)/2 (n ≥ 0)always divisible by 7http://factordb.com/index.php?query=%287*11%5E%28n%2B1%29%2B7%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11A{5}2(21×11n+1−7)/2 (n ≥ 0)always divisible by 7http://factordb.com/index.php?query=%2821*11%5E%28n%2B1%29-7%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11A{5}9(21×11n+1+7)/2 (n ≥ 0)always divisible by 7http://factordb.com/index.php?query=%2821*11%5E%28n%2B1%29%2B7%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
12A{0}2110×12n+2+25always divisible by 5http://factordb.com/index.php?query=10*12%5E%28n%2B2%29%2B25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
133{6}A(7×13n+1+7)/2 (n ≥ 0)always divisible by 7http://factordb.com/index.php?query=%287*13%5E%28n%2B1%29%2B7%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
135{A}C(35×13n+1+7)/6 (n ≥ 0)always divisible by 7http://factordb.com/index.php?query=%2835*13%5E%28n%2B1%29%2B7%29%2F6&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13A{6}3(21×13n+1−7)/2 (n ≥ 0)always divisible by 7http://factordb.com/index.php?query=%2821*13%5E%28n%2B1%29-7%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13C{A}5(77×13n+1−35)/6 (n ≥ 0)always divisible by 7http://factordb.com/index.php?query=%2877*13%5E%28n%2B1%29-35%29%2F6&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
148{6}9(110×14n+1+33)/13 (n ≥ 0)always divisible by 11http://factordb.com/index.php?query=%28110*14%5E%28n%2B1%29%2B33%29%2F13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1440{4}9(732×14n+1+61)/13 (n ≥ 0)always divisible by 61http://factordb.com/index.php?query=%28732*14%5E%28n%2B1%29%2B61%29%2F13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
159{6}8(66×15n+1+11)/7 (n ≥ 0)always divisible by 11http://factordb.com/index.php?query=%2866*15%5E%28n%2B1%29%2B11%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
162{C}3(14×16n+1−49)/5 (n ≥ 0)always divisible by 7http://factordb.com/index.php?query=%2814*16%5E%28n%2B1%29-49%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16A{4}1(154×16n+1−49)/15 (n ≥ 0)always divisible by 7http://factordb.com/index.php?query=%28154*16%5E%28n%2B1%29-49%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
168{C}F(44×16n+1+11)/5 (n ≥ 0)always divisible by 11
(in fact, also Aurifeuillean factorization of x4+4×y4
(44×16n+1+11)/5 = 11 × (2×4n+1−2×2n+1+1) × (2×4n+1+2×2n+1+1) / 5
http://factordb.com/index.php?query=%2844*16%5E%28n%2B1%29%2B11%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
168{A}F(26×16n+1+13)/3 (n ≥ 0)always divisible by 13http://factordb.com/index.php?query=%2826*16%5E%28n%2B1%29%2B13%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16A{6}9(52×16n+1+13)/5 (n ≥ 0)always divisible by 13
(in fact, also Aurifeuillean factorization of x4+4×y4
(52×16n+1+13)/5 = 13 × (2×4n+1−2×2n+1+1) × (2×4n+1+2×2n+1+1) / 5
http://factordb.com/index.php?query=%2852*16%5E%28n%2B1%29%2B13%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
21B0{H}6H(4637×21n+2−4637)/20 (n ≥ 0)always divisible by 4637http://factordb.com/index.php?query=%284637*21%5E%28n%2B2%29-4637%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
284{O}9(44×28n+1−143)/9 (n ≥ 0)always divisible by 11http://factordb.com/index.php?query=%2844*28%5E%28n%2B1%29-143%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28N{6}R(209×28n+1+187)/9 (n ≥ 0)always divisible by 11http://factordb.com/index.php?query=%28209*28%5E%28n%2B1%29%2B187%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28D{6}R(119×28n+1+187)/9 (n ≥ 0)always divisible by 17http://factordb.com/index.php?query=%28119*28%5E%28n%2B1%29%2B187%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
9{1}5(9n+1+31)/8 (n ≥ 1)always divisible by some element of {2,5}
divisible by 2 if n is odd, divisible by 5 if n is even
http://factordb.com/index.php?query=%289%5E%28n%2B1%29%2B31%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
9{1}61(9n+2+359)/8 (n ≥ 0)always divisible by some element of {2,5}
divisible by 2 if n is odd, divisible by 5 if n is even
http://factordb.com/index.php?query=%289%5E%28n%2B2%29%2B359%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
916{1}(121×9n−1)/8 (n ≥ 0)always divisible by some element of {2,5}
divisible by 2 if n is odd, divisible by 5 if n is even
(in fact, also difference-of-two-squares factorization)
(121×9n−1)/8 = (11×3n−1) × (11×3n+1) / 8
http://factordb.com/index.php?query=%28121*9%5En-1%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
92{7}(23×9n−7)/8 (n ≥ 1)always divisible by some element of {2,5}
divisible by 2 if n is even, divisible by 5 if n is odd
http://factordb.com/index.php?query=%2823*9%5En-7%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
9{3}5(3×9n+1+13)/8 (n ≥ 1)always divisible by some element of {2,5}
divisible by 2 if n is odd, divisible by 5 if n is even
http://factordb.com/index.php?query=%283*9%5E%28n%2B1%29%2B13%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
9{3}8(3×9n+1+37)/8 (n ≥ 1)always divisible by some element of {2,5}
divisible by 2 if n is even, divisible by 5 if n is odd
http://factordb.com/index.php?query=%283*9%5E%28n%2B1%29%2B37%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
9{3}05(3×9n+2−203)/8 (n ≥ 1)always divisible by some element of {2,5}
divisible by 2 if n is odd, divisible by 5 if n is even
http://factordb.com/index.php?query=%283*9%5E%28n%2B2%29-203%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
95{1}(41×9n−1)/8 (n ≥ 1)always divisible by some element of {2,5}
divisible by 2 if n is odd, divisible by 5 if n is even
http://factordb.com/index.php?query=%2841*9%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
95{7}(47×9n−7)/8 (n ≥ 1)always divisible by some element of {2,5}
divisible by 2 if n is odd, divisible by 5 if n is even
http://factordb.com/index.php?query=%2847*9%5En-7%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
96{1}(49×9n−1)/8 (n ≥ 1)always divisible by some element of {2,5}
divisible by 2 if n is even, divisible by 5 if n is odd
(in fact, also difference-of-two-squares factorization)
(49×9n−1)/8 = (7×3n−1) × (7×3n+1) / 8
http://factordb.com/index.php?query=%2849*9%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
9{7}2(7×9n+1−47)/8 (n ≥ 1)always divisible by some element of {2,5}
divisible by 2 if n is even, divisible by 5 if n is odd
http://factordb.com/index.php?query=%287*9%5E%28n%2B1%29-47%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
9{7}5(7×9n+1−23)/8 (n ≥ 1)always divisible by some element of {2,5}
divisible by 2 if n is odd, divisible by 5 if n is even
http://factordb.com/index.php?query=%287*9%5E%28n%2B1%29-23%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
9{7}05(3×9n+2−527)/8 (n ≥ 1)always divisible by some element of {2,5}
divisible by 2 if n is odd, divisible by 5 if n is even
http://factordb.com/index.php?query=%287*9%5E%28n%2B2%29-527%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
9{1}6{1}(9m+n+1+40×9n−1)/8 (m+n ≥ 1)always divisible by some element of {2,5}
(nonlinear family)
divisible by 2 if m+n is even, divisible by 5 if m+n is odd
9{7}2{7}(7×9m+n+1−40×9n−7)/8 (m+n ≥ 1)always divisible by some element of {2,5}
(nonlinear family)
divisible by 2 if m+n is even, divisible by 5 if m+n is odd
95{0}{1}(40×9m+n+9n−1)/8 (m+n ≥ 1)always divisible by some element of {2,5}
(nonlinear family)
divisible by 2 if n is odd, divisible by 5 if n is even
95{0}{7}(40×9m+n+7×9n−7)/8 (m+n ≥ 1)always divisible by some element of {2,5}
(nonlinear family)
divisible by 2 if n is odd, divisible by 5 if n is even
9{1}{0}5(9m+n+1−9n+1+40)/8 (m+n ≥ 1)always divisible by some element of {2,5}
(nonlinear family)
divisible by 2 if m is odd, divisible by 5 if m is even
9{3}{0}5(3×9m+n+1−3×9n+1+40)/8 (m+n ≥ 1)always divisible by some element of {2,5}
(nonlinear family)
divisible by 2 if m is odd, divisible by 5 if m is even
9{7}{0}5(7×9m+n+1−7×9n+1+40)/8 (m+n ≥ 1)always divisible by some element of {2,5}
(nonlinear family)
divisible by 2 if m is odd, divisible by 5 if m is even
112{5}(5×11n−1)/2 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is even, divisible by 3 if n is odd
http://factordb.com/index.php?query=%285*11%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
113{1}(31×11n−1)/10 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is odd, divisible by 3 if n is even
(special example, as the numbers with length ≥ 18 in this family contain "prime > b" subsequence, this prime is 117)
http://factordb.com/index.php?query=%2831*11%5En-1%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
113{5}(7×11n−1)/2 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is odd, divisible by 3 if n is even
http://factordb.com/index.php?query=%287*11%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
113{7}(37×11n−7)/10 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is odd, divisible by 3 if n is even
(in fact, always divisible by some element of {22,3})
divisible by 22 if n is odd, divisible by 3 if n is even
http://factordb.com/index.php?query=%2837*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
114{1}(41×11n−1)/10 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is even, divisible by 3 if n is odd
(in fact, always divisible by some element of {22,3})
divisible by 22 if n is even, divisible by 3 if n is odd
(special example, as the numbers with length ≥ 18 in this family contain "prime > b" subsequence, this prime is 117)
http://factordb.com/index.php?query=%2841*11%5En-1%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
114{7}(47×11n−7)/10 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is even, divisible by 3 if n is odd
(in fact, always divisible by some element of {22,3})
divisible by 22 if n is even, divisible by 3 if n is odd
http://factordb.com/index.php?query=%2847*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
118{5}(17×11n−1)/2 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is even, divisible by 3 if n is odd
(in fact, always divisible by some element of {22,3})
divisible by 22 if n is even, divisible by 3 if n is odd
http://factordb.com/index.php?query=%2817*11%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
119{1}(91×11n−1)/10 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is odd, divisible by 3 if n is even
(in fact, always divisible by some element of {22,3})
divisible by 22 if n is odd, divisible by 3 if n is even
(special example, as the numbers with length ≥ 18 in this family contain "prime > b" subsequence, this prime is 117)
http://factordb.com/index.php?query=%2891*11%5En-1%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
119{5}(19×11n−1)/2 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is odd, divisible by 3 if n is even
(in fact, always divisible by some element of {22,3})
divisible by 22 if n is odd, divisible by 3 if n is even
http://factordb.com/index.php?query=%2819*11%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
119{7}(97×11n−7)/10 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is odd, divisible by 3 if n is even
http://factordb.com/index.php?query=%2897*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11A{1}(101×11n−1)/10 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is even, divisible by 3 if n is odd
(special example, as the numbers with length ≥ 18 in this family contain "prime > b" subsequence, this prime is 117)
http://factordb.com/index.php?query=%28101*11%5En-1%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11A{7}(107×11n−7)/10 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is even, divisible by 3 if n is odd
http://factordb.com/index.php?query=%28107*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11{5}2(11n+1−7)/2 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is even, divisible by 3 if n is odd
http://factordb.com/index.php?query=%2811%5E%28n%2B1%29-7%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11{1}3(11n+1+19)/10 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is odd, divisible by 3 if n is even
(special example, as the numbers with length ≥ 18 in this family contain "prime > b" subsequence, this prime is 117)
http://factordb.com/index.php?query=%2811%5E%28n%2B1%29%2B19%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11{5}3(11n+1−5)/2 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is odd, divisible by 3 if n is even
http://factordb.com/index.php?query=%2811%5E%28n%2B1%29-5%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11{7}3(7×11n+1−47)/10 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is odd, divisible by 3 if n is even
(in fact, always divisible by some element of {22,3})
divisible by 22 if n is odd, divisible by 3 if n is even
http://factordb.com/index.php?query=%287*11%5E%28n%2B1%29-47%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11{1}4(11n+1+29)/10 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is even, divisible by 3 if n is odd
(in fact, always divisible by some element of {22,3})
divisible by 22 if n is even, divisible by 3 if n is odd
(special example, as the numbers with length ≥ 18 in this family contain "prime > b" subsequence, this prime is 117)
http://factordb.com/index.php?query=%2811%5E%28n%2B1%29%2B29%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11{7}4(7×11n+1−37)/10 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is even, divisible by 3 if n is odd
(in fact, always divisible by some element of {22,3})
divisible by 22 if n is even, divisible by 3 if n is odd
http://factordb.com/index.php?query=%287*11%5E%28n%2B1%29-37%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11{5}8(11n+1+5)/2 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is even, divisible by 3 if n is odd
(in fact, always divisible by some element of {22,3})
divisible by 22 if n is even, divisible by 3 if n is odd
http://factordb.com/index.php?query=%2811%5E%28n%2B1%29%2B5%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11{1}9(11n+1+79)/10 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is odd, divisible by 3 if n is even
(in fact, always divisible by some element of {22,3})
divisible by 22 if n is odd, divisible by 3 if n is even
(special example, as the numbers with length ≥ 18 in this family contain "prime > b" subsequence, this prime is 117)
http://factordb.com/index.php?query=%2811%5E%28n%2B1%29%2B79%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11{5}9(11n+1+7)/2 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is odd, divisible by 3 if n is even
(in fact, always divisible by some element of {22,3})
divisible by 22 if n is odd, divisible by 3 if n is even
http://factordb.com/index.php?query=%2811%5E%28n%2B1%29%2B7%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11{7}9(7×11n+1+13)/10 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is odd, divisible by 3 if n is even
http://factordb.com/index.php?query=%287*11%5E%28n%2B1%29%2B13%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11{1}A(11n+1+89)/10 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if the length is odd, divisible by 3 if the length is even
(special example, as the numbers with length ≥ 18 in this family contain "prime > b" subsequence, this prime is 117)
http://factordb.com/index.php?query=%2811%5E%28n%2B1%29%2B89%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11{7}A(7×11n+1+23)/10 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if the length is odd, divisible by 3 if the length is even
http://factordb.com/index.php?query=%287*11%5E%28n%2B1%29%2B23%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
113{0}{5}(6×11m+n+11n−1)/2always divisible by some element of {2,3}
(nonlinear family)
divisible by 2 if n is odd, divisible by 3 if n is even
11{5}{0}3(11m+n+1−11n+1+6)/2always divisible by some element of {2,3}
(nonlinear family)
divisible by 2 if m is odd, divisible by 3 if m is even
144{0}14×14n+1+1 (n ≥ 0)always divisible by some element of {3,5}
divisible by 3 if n is even, divisible by 5 if n is odd
http://factordb.com/index.php?query=4*14%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
14B{0}111×14n+1+1 (n ≥ 0)always divisible by some element of {3,5}
divisible by 3 if n is odd, divisible by 5 if n is even
http://factordb.com/index.php?query=11*14%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
143{D}4×14n−1 (n ≥ 1)always divisible by some element of {3,5}
divisible by 3 if n is even, divisible by 5 if n is odd
http://factordb.com/index.php?query=4*14%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
14A{D}11×14n−1 (n ≥ 1)always divisible by some element of {3,5}
divisible by 3 if n is odd, divisible by 5 if n is even
http://factordb.com/index.php?query=11*14%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
141{0}B14n+1+11 (n ≥ 0)always divisible by some element of {3,5}
divisible by 3 if n is odd, divisible by 5 if n is even
http://factordb.com/index.php?query=14%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
14{D}314n+1−11 (n ≥ 1)always divisible by some element of {3,5}
divisible by 3 if n is even, divisible by 5 if n is odd
http://factordb.com/index.php?query=14%5E%28n%2B1%29-11&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
14{4}9(4×14n+1+61)/13 (n ≥ 1)always divisible by some element of {3,5}
divisible by 3 if n is even, divisible by 5 if n is odd
http://factordb.com/index.php?query=%284*14%5E%28n%2B1%29%2B61%29%2F13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
14{8}5(8×14n+1−47)/13 (n ≥ 1)always divisible by some element of {3,5}
divisible by 3 if n is odd, divisible by 5 if n is even
http://factordb.com/index.php?query=%288*14%5E%28n%2B1%29-47%29%2F13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
172{B}(43×17n−11)/16 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is even, divisible by 3 if n is odd
(in fact, always divisible by some element of {2,32})
divisible by 2 if n is even, divisible by 32 if n is odd
http://factordb.com/index.php?query=%2843*17%5En-11%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
179{B}(155×17n−11)/16 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is odd, divisible by 3 if n is even
(in fact, always divisible by some element of {2,32})
divisible by 2 if n is odd, divisible by 32 if n is even
http://factordb.com/index.php?query=%28155*17%5En-11%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17{B}2(11×17n+1−155)/16 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is even, divisible by 3 if n is odd
(in fact, always divisible by some element of {2,32})
divisible by 2 if n is even, divisible by 32 if n is odd
http://factordb.com/index.php?query=%2811*17%5E%28n%2B1%29-155%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17{B}9(11×17n+1−43)/16 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is odd, divisible by 3 if n is even
(in fact, always divisible by some element of {2,32})
divisible by 2 if n is odd, divisible by 32 if n is even
http://factordb.com/index.php?query=%2811*17%5E%28n%2B1%29-43%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
194{9}(9×19n−1)/2 (n ≥ 1)always divisible by some element of {2,5}
divisible by 2 if n is even, divisible by 5 if n is odd
(in fact, always divisible by some element of {22,5})
divisible by 22 if n is even, divisible by 5 if n is odd
http://factordb.com/index.php?query=%289*19%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
195{9}(11×19n−1)/2 (n ≥ 1)always divisible by some element of {2,5}
divisible by 2 if n is odd, divisible by 5 if n is even
(in fact, always divisible by some element of {22,5})
divisible by 22 if n is odd, divisible by 5 if n is even
http://factordb.com/index.php?query=%2811*19%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
19{9}4(19n+1−11)/2 (n ≥ 1)always divisible by some element of {2,5}
divisible by 2 if n is even, divisible by 5 if n is odd
(in fact, always divisible by some element of {22,5})
divisible by 22 if n is even, divisible by 5 if n is odd
http://factordb.com/index.php?query=%2819%5E%28n%2B1%29-11%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
19{9}5(19n+1−9)/2 (n ≥ 1)always divisible by some element of {2,5}
divisible by 2 if n is odd, divisible by 5 if n is even
(in fact, always divisible by some element of {22,5})
divisible by 22 if n is odd, divisible by 5 if n is even
http://factordb.com/index.php?query=%2819%5E%28n%2B1%29-9%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
208{0}18×20n+1+1 (n ≥ 0)always divisible by some element of {3,7}
divisible by 3 if n is odd, divisible by 7 if n is even
http://factordb.com/index.php?query=8*20%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
20D{0}113×20n+1+1 (n ≥ 0)always divisible by some element of {3,7}
divisible by 3 if n is even, divisible by 7 if n is odd
http://factordb.com/index.php?query=13*20%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
207{J}8×20n−1 (n ≥ 1)always divisible by some element of {3,7}
divisible by 3 if n is odd, divisible by 7 if n is even
http://factordb.com/index.php?query=8*20%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
20C{J}13×20n−1 (n ≥ 1)always divisible by some element of {3,7}
divisible by 3 if n is even, divisible by 7 if n is odd
http://factordb.com/index.php?query=13*20%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
201{0}D20n+1+13 (n ≥ 0)always divisible by some element of {3,7}
divisible by 3 if n is even, divisible by 7 if n is odd
http://factordb.com/index.php?query=20%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
20{J}720n+1−13 (n ≥ 1)always divisible by some element of {3,7}
divisible by 3 if n is odd, divisible by 7 if n is even
http://factordb.com/index.php?query=20%5E%28n%2B1%29-13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25D{1}(313×25n−1)/24 (n ≥ 1)always divisible by some element of {2,13}
divisible by 2 if n is odd, divisible by 13 if n is even
http://factordb.com/index.php?query=%28313*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25E{1}(337×25n−1)/24 (n ≥ 1)always divisible by some element of {2,13}
divisible by 2 if n is even, divisible by 13 if n is odd
http://factordb.com/index.php?query=%28337*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25{1}D(25n+1+287)/24 (n ≥ 1)always divisible by some element of {2,13}
divisible by 2 if n is odd, divisible by 13 if n is even
http://factordb.com/index.php?query=%2825%5E%28n%2B1%29%2B287%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25{1}E(25n+1+311)/24 (n ≥ 1)always divisible by some element of {2,13}
divisible by 2 if n is even, divisible by 13 if n is odd
http://factordb.com/index.php?query=%2825%5E%28n%2B1%29%2B311%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
251E{1}(937×25n−1)/24 (n ≥ 0)always divisible by some element of {2,13}
divisible by 2 if n is odd, divisible by 13 if n is even
http://factordb.com/index.php?query=%28937*25%5En-1%29%2F24&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
251F{1}(961×25n−1)/24 (n ≥ 0)always divisible by some element of {2,13}
divisible by 2 if n is even, divisible by 13 if n is odd
(in fact, also difference-of-two-squares factorization)
(961×25n−1)/24 = (31×5n−1) × (31×5n+1) / 24
http://factordb.com/index.php?query=%28961*25%5En-1%29%2F24&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
253G{3}(729×25n−1)/8 (n ≥ 0)always divisible by some element of {2,13}
divisible by 2 if n is odd, divisible by 13 if n is even
(in fact, also difference-of-two-squares factorization)
(729×25n−1)/8 = (27×5n−1) × (27×5n+1) / 8
http://factordb.com/index.php?query=%28729*25%5En-1%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25D0{3}(2601×25n−1)/8 (n ≥ 0)always divisible by some element of {2,13}
divisible by 2 if n is odd, divisible by 13 if n is even
(in fact, also difference-of-two-squares factorization)
(2601×25n−1)/8 = (51×5n−1) × (51×5n+1) / 8
http://factordb.com/index.php?query=%282601*25%5En-1%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25G6{3}(3249×25n−1)/8 (n ≥ 0)always divisible by some element of {2,13}
divisible by 2 if n is even, divisible by 13 if n is odd
(in fact, also difference-of-two-squares factorization)
(3249×25n−1)/8 = (57×5n−1) × (57×5n+1) / 8
http://factordb.com/index.php?query=%283249*25%5En-1%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
294{0}14×29n+1+1 (n ≥ 0)always divisible by some element of {3,5}
divisible by 3 if n is even, divisible by 5 if n is odd
http://factordb.com/index.php?query=4*29%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
29Q{0}126×29n+1+1 (n ≥ 0)always divisible by some element of {3,5}
divisible by 3 if n is odd, divisible by 5 if n is even
http://factordb.com/index.php?query=26*29%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
293{S}4×29n−1 (n ≥ 1)always divisible by some element of {3,5}
divisible by 3 if n is even, divisible by 5 if n is odd
http://factordb.com/index.php?query=4*29%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
29P{S}26×29n−1 (n ≥ 1)always divisible by some element of {3,5}
divisible by 3 if n is odd, divisible by 5 if n is even
http://factordb.com/index.php?query=26*29%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
291{0}429n+1+4 (n ≥ 0)always divisible by some element of {3,5}
divisible by 3 if n is even, divisible by 5 if n is odd
http://factordb.com/index.php?query=29%5E%28n%2B1%29%2B4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
291{0}Q29n+1+26 (n ≥ 0)always divisible by some element of {3,5}
divisible by 3 if n is odd, divisible by 5 if n is even
http://factordb.com/index.php?query=29%5E%28n%2B1%29%2B26&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
29{S}P29n+1−4 (n ≥ 1)always divisible by some element of {3,5}
divisible by 3 if n is odd, divisible by 5 if n is even
http://factordb.com/index.php?query=29%5E%28n%2B1%29-4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
29{S}329n+1−26 (n ≥ 1)always divisible by some element of {3,5}
divisible by 3 if n is even, divisible by 5 if n is odd
http://factordb.com/index.php?query=29%5E%28n%2B1%29-26&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
292{H}(73×29n−17)/28 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is even, divisible by 3 if n is odd
(in fact, also always divisible by some element of {2,5})
divisible by 2 if n is even, divisible by 5 if n is odd
http://factordb.com/index.php?query=%2873*29%5En-17%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
29F{H}(437×29n−17)/28 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is even, divisible by 3 if n is odd
(in fact, also always divisible by some element of {2,5})
divisible by 2 if n is even, divisible by 5 if n is odd
http://factordb.com/index.php?query=%28437*29%5En-17%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
29{H}2(17×29n+1−437)/28 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is even, divisible by 3 if n is odd
(in fact, also always divisible by some element of {2,5})
divisible by 2 if n is even, divisible by 5 if n is odd
http://factordb.com/index.php?query=%2817*29%5E%28n%2B1%29-437%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
29{H}F(17×29n+1−73)/28 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is even, divisible by 3 if n is odd
(in fact, also always divisible by some element of {2,5})
divisible by 2 if n is even, divisible by 5 if n is odd
http://factordb.com/index.php?query=%2817*29%5E%28n%2B1%29-73%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
293{D}(97×29n−13)/28 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is even, divisible by 3 if n is odd
(in fact, also always divisible by some element of {2,5})
divisible by 2 if n is even, divisible by 5 if n is odd
http://factordb.com/index.php?query=%2897*29%5En-13%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
29A{D}(293×29n−13)/28 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is even, divisible by 3 if n is odd
(in fact, also always divisible by some element of {2,5})
divisible by 2 if n is even, divisible by 5 if n is odd
http://factordb.com/index.php?query=%28293*29%5En-13%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
29{D}3(13×29n+1−293)/28 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is even, divisible by 3 if n is odd
(in fact, also always divisible by some element of {2,5})
divisible by 2 if n is even, divisible by 5 if n is odd
http://factordb.com/index.php?query=%2813*29%5E%28n%2B1%29-293%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
29{D}A(13×29n+1−97)/28 (n ≥ 1)always divisible by some element of {2,3}
divisible by 2 if n is even, divisible by 3 if n is odd
(in fact, also always divisible by some element of {2,5})
divisible by 2 if n is even, divisible by 5 if n is odd
http://factordb.com/index.php?query=%2813*29%5E%28n%2B1%29-97%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
295{B}(151×29n−11)/28 (n ≥ 1)always divisible by some element of {2,5}
divisible by 2 if n is odd, divisible by 5 if n is even
(in fact, also always divisible by some element of {3,5})
divisible by 3 if n is odd, divisible by 5 if n is even
http://factordb.com/index.php?query=%28151*29%5En-11%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
296{B}(179×29n−11)/28 (n ≥ 1)always divisible by some element of {2,5}
divisible by 2 if n is even, divisible by 5 if n is odd
(in fact, also always divisible by some element of {3,5})
divisible by 3 if n is even, divisible by 5 if n is odd
http://factordb.com/index.php?query=%28179*29%5En-11%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
29{B}5(11×29n+1−179)/28 (n ≥ 1)always divisible by some element of {2,5}
divisible by 2 if n is odd, divisible by 5 if n is even
(in fact, also always divisible by some element of {3,5})
divisible by 3 if n is odd, divisible by 5 if n is even
http://factordb.com/index.php?query=%2811*29%5E%28n%2B1%29-179%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
29{B}6(11×29n+1−151)/28 (n ≥ 1)always divisible by some element of {2,5}
divisible by 2 if n is even, divisible by 5 if n is odd
(in fact, also always divisible by some element of {3,5})
divisible by 3 if n is even, divisible by 5 if n is odd
http://factordb.com/index.php?query=%2811*29%5E%28n%2B1%29-151%29%2F28&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
32A{0}110×32n+1+1 (n ≥ 0)always divisible by some element of {3,11}
divisible by 3 if n is even, divisible by 11 if n is odd
http://factordb.com/index.php?query=10*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
32N{0}123×32n+1+1 (n ≥ 0)always divisible by some element of {3,11}
divisible by 3 if n is odd, divisible by 11 if n is even
http://factordb.com/index.php?query=23*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
329{V}10×32n−1 (n ≥ 1)always divisible by some element of {3,11}
divisible by 3 if n is even, divisible by 11 if n is odd
http://factordb.com/index.php?query=10*32%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
32M{V}23×32n−1 (n ≥ 1)always divisible by some element of {3,11}
divisible by 3 if n is odd, divisible by 11 if n is even
http://factordb.com/index.php?query=23*32%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
321{0}N32n+1+23 (n ≥ 0)always divisible by some element of {3,11}
divisible by 3 if n is odd, divisible by 11 if n is even
http://factordb.com/index.php?query=32%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
32{V}932n+1−23 (n ≥ 1)always divisible by some element of {3,11}
divisible by 3 if n is even, divisible by 11 if n is odd
http://factordb.com/index.php?query=32%5E%28n%2B1%29-23&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
346{0}16×34n+1+1 (n ≥ 0)always divisible by some element of {5,7}
divisible by 5 if n is even, divisible by 7 if n is odd
http://factordb.com/index.php?query=6*34%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
345{X}6×34n−1 (n ≥ 1)always divisible by some element of {5,7}
divisible by 5 if n is even, divisible by 7 if n is odd
http://factordb.com/index.php?query=6*34%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
34S{X}29×34n−1 (n ≥ 1)always divisible by some element of {5,7}
divisible by 5 if n is odd, divisible by 7 if n is even
http://factordb.com/index.php?query=29*34%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
34{X}534n+1−29 (n ≥ 1)always divisible by some element of {5,7}
divisible by 5 if n is even, divisible by 7 if n is odd
http://factordb.com/index.php?query=34%5E%28n%2B1%29-29&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
86{4}7(46×8n+1+17)/7 (n ≥ 0)always divisible by some element of {3,5,13}
divisible by 3 if the length is odd, divisible by 5 if the length is == 2 mod 4, divisible by 13 if the length is == 0 mod 4
(special example, as the numbers with length ≥ 222 in this family contain "prime > b" subsequence, this prime is 42207)
http://factordb.com/index.php?query=%2846*8%5E%28n%2B1%29%2B17%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1395{0}3122×13n+1+3 (n ≥ 0)always divisible by some element of {5,7,17}
divisible by 7 if n is even, divisible by 5 if n == 3 mod 4, divisible by 17 if n == 1 mod 4
http://factordb.com/index.php?query=122*13%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
133{0}953×13n+2+122 (n ≥ 0)always divisible by some element of {5,7,17}
divisible by 7 if n is odd, divisible by 5 if n == 2 mod 4, divisible by 17 if n == 0 mod 4
http://factordb.com/index.php?query=3*13%5E%28n%2B2%29%2B122&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16{4}D(4×16n+1+131)/15 (n ≥ 1)always divisible by some element of {3,7,13}
divisible by 3 if n == 2 mod 3, divisible by 7 if n == 1 mod 3, divisible by 13 if n == 0 mod 3
http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29%2B131%29%2F15&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16{8}F(8×16n+1+97)/15 (n ≥ 1)always divisible by some element of {3,7,13}
divisible by 3 if n == 0 mod 3, divisible by 7 if n == 2 mod 3, divisible by 13 if n == 1 mod 3
http://factordb.com/index.php?query=%288*16%5E%28n%2B1%29%2B97%29%2F15&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
177F{0}D134×17n+1+13 (n ≥ 0)always divisible by some element of {3,5,29}
divisible by 3 if n is odd, divisible by 5 if n == 2 mod 4, divisible by 29 if n == 0 mod 4
http://factordb.com/index.php?query=134*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17D{0}7F13×17n+2+134 (n ≥ 0)always divisible by some element of {3,5,29}
divisible by 3 if n is even, divisible by 5 if n == 3 mod 4, divisible by 29 if n == 1 mod 4
http://factordb.com/index.php?query=13*17%5E%28n%2B2%29%2B134&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
21{7}D(7×21n+1+113)/20 (n ≥ 1)always divisible by some element of {2,13,17}
divisible by 2 if n is odd, divisible by 13 if n == 0 mod 4, divisible by 17 if n == 2 mod 4
http://factordb.com/index.php?query=%287*21%5E%28n%2B1%29%2B113%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
217{0}FJ7×21n+2+334 (n ≥ 0)always divisible by some element of {11,13,17}
divisible by 11 if n is even, divisible by 13 if n == 3 mod 4, divisible by 17 if n == 1 mod 4
http://factordb.com/index.php?query=7*21%5E%28n%2B2%29%2B334&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
237L{0}1182×23n+1+1 (n ≥ 0)always divisible by some element of {3,5,53}
divisible by 3 if n is odd, divisible by 5 if n == 2 mod 4, divisible by 53 if n == 0 mod 4
http://factordb.com/index.php?query=182*23%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
231{0}7L23n+2+182 (n ≥ 0)always divisible by some element of {3,5,53}
divisible by 3 if n is even, divisible by 5 if n == 3 mod 4, divisible by 53 if n == 1 mod 4
http://factordb.com/index.php?query=23%5E%28n%2B2%29%2B182&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
23{D}GA(13×23n+2+1439)/22 (n ≥ 0)always divisible by some element of {2,5,7,37,79}
divisible by 2 if n is even, divisible by 5 if n == 1 mod 4, divisible by 7 if n == 0 mod 3, divisible by 37 if n == 7 mod 12, divisible by 79 if n == 2 mod 3
(in fact, also always divisible by some element of {3,5,7,37,79})
divisible by 3 if n is even, divisible by 5 if n == 1 mod 4, divisible by 7 if n == 0 mod 3, divisible by 37 if n == 7 mod 12, divisible by 79 if n == 2 mod 3
http://factordb.com/index.php?query=%2813*23%5E%28n%2B2%29%2B1439%29%2F22&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
23L{5}L(467×23n+1+347)/22 (n ≥ 0)always divisible by some element of {2,5,7,13,37}
divisible by 2 if n is even, divisible by 5 if n == 1 mod 4, divisible by 7 if n == 0 mod 3, divisible by 13 if n == 1 mod 6, divisible by 37 if n == 11 mod 12
(in fact, also always divisible by some element of {3,5,7,13,37})
divisible by 3 if n is even, divisible by 5 if n == 1 mod 4, divisible by 7 if n == 0 mod 3, divisible by 13 if n == 1 mod 6, divisible by 37 if n == 11 mod 12
http://factordb.com/index.php?query=%28467*23%5E%28n%2B1%29%2B347%29%2F22&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
259{N}(239×25n−23)/24 (n ≥ 1)always divisible by some element of {3,7,31}
divisible by 3 if n == 0 mod 3, divisible by 7 if n == 2 mod 3, divisible by 31 if n == 1 mod 3
http://factordb.com/index.php?query=%28239*25%5En-23%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25{N}E(23×25n+1−239)/24 (n ≥ 1)always divisible by some element of {3,7,31}
divisible by 3 if n == 2 mod 3, divisible by 7 if n == 0 mod 3, divisible by 31 if n == 1 mod 3
http://factordb.com/index.php?query=%2823*25%5E%28n%2B1%29-239%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2527{1}(1369×25n−1)/24 (n ≥ 0)always divisible by some element of {3,7,31}
divisible by 3 if n == 0 mod 3, divisible by 7 if n == 2 mod 3, divisible by 31 if n == 1 mod 3
(in fact, also difference-of-two-squares factorization)
(1369×25n−1)/24 = (37×5n−1) × (37×5n+1) / 24
http://factordb.com/index.php?query=%281369*25%5En-1%29%2F24&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
27JP{0}1538×27n+1+1 (n ≥ 0)always divisible by some element of {5,7,73}
divisible by 7 if n is odd, divisible by 5 if n == 2 mod 4, divisible by 73 if n == 0 mod 4
http://factordb.com/index.php?query=538*27%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
271{0}JP27n+2+538 (n ≥ 0)always divisible by some element of {5,7,73}
divisible by 7 if n is even, divisible by 5 if n == 3 mod 4, divisible by 73 if n == 1 mod 4
http://factordb.com/index.php?query=27%5E%28n%2B2%29%2B538&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
27J{0}219×27n+1+2 (n ≥ 0)always divisible by some element of {5,7,73}
divisible by 7 if n is odd, divisible by 5 if n == 0 mod 4, divisible by 73 if n == 2 mod 4
http://factordb.com/index.php?query=19*27%5E%28n%2B1%29%2B2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
29{2}{5}(2×29m+n+3×29n−5)/28always divisible by some element of {2,3,5}
(nonlinear family)
divisible by 2 if n is even, divisible by 3 if m and n are both even or both odd, divisible by 5 if m is even
29{5}{2}(5×29m+n−3×29n−2)/28always divisible by some element of {2,3,5}
(nonlinear family)
divisible by 2 if m is even, divisible by 3 if m and n are both even or both odd, divisible by 5 if n is even
29{M}{P}(22×29m+n+3×29n−25)/28always divisible by some element of {2,3,5}
(nonlinear family)
divisible by 2 if n is even, divisible by 3 if m and n are both even or both odd, divisible by 5 if m is even
29{P}{M}(25×29m+n−3×29n−22)/28always divisible by some element of {2,3,5}
(nonlinear family)
divisible by 2 if m is even, divisible by 3 if m and n are both even or both odd, divisible by 5 if n is even
30A{0}9J10×30n+2+289 (n ≥ 0)always divisible by some element of {7,13,19,31}
divisible by 7 if n == 0 mod 3, divisible by 13 if n == 4 mod 6, divisible by 19 if n == 2 mod 3, divisible by 31 if n is odd
http://factordb.com/index.php?query=10*30%5E%28n%2B2%29%2B289&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
31A{N}(323×31n−23)/30 (n ≥ 1)always divisible by some element of {2,13,37}
divisible by 2 if n is even, divisible by 13 if n == 3 mod 4, divisible by 37 if n == 1 mod 4
http://factordb.com/index.php?query=%28323*31%5En-23%29%2F30&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
31{N}D(23×31n+1−323)/30 (n ≥ 1)always divisible by some element of {2,13,37}
divisible by 2 if n is odd, divisible by 13 if n == 0 mod 4, divisible by 37 if n == 2 mod 4
http://factordb.com/index.php?query=%2823*31%5E%28n%2B1%29-323%29%2F30&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
31D{R}(139×31n−9)/10 (n ≥ 1)always divisible by some element of {2,13,37}
divisible by 2 if n is odd, divisible by 13 if n == 0 mod 4, divisible by 37 if n == 2 mod 4
http://factordb.com/index.php?query=%28139*31%5En-9%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
31{R}E(9×31n+1−139)/10 (n ≥ 1)always divisible by some element of {2,13,37}
divisible by 2 if n is even, divisible by 13 if n == 3 mod 4, divisible by 37 if n == 1 mod 4
http://factordb.com/index.php?query=%289*31%5E%28n%2B1%29-139%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
31O{5}(145×31n−1)/6 (n ≥ 1)always divisible by some element of {2,3,7,19}
divisible by 2 if n is even, divisible by 3 if n == 0 mod 3, divisible by 7 if n == 1 mod 6, divisible by 19 if n == 5 mod 6
http://factordb.com/index.php?query=%28145*31%5En-1%29%2F6&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
31J{T}(599×31n−29)/30 (n ≥ 1)always divisible by some element of {2,3,7,19}
divisible by 2 if n is odd, divisible by 3 if n == 1 mod 3, divisible by 7 if n == 2 mod 6, divisible by 19 if n == 0 mod 6
http://factordb.com/index.php?query=%28599*31%5En-29%29%2F30&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
31{T}A(29×31n+1−599)/30 (n ≥ 1)always divisible by some element of {2,3,7,19}
divisible by 2 if n is even, divisible by 3 if n == 1 mod 3, divisible by 7 if n == 3 mod 6, divisible by 19 if n == 5 mod 6
http://factordb.com/index.php?query=%2829*31%5E%28n%2B1%29-599%29%2F30&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
328{0}V8×32n+1+31 (n ≥ 0)always divisible by some element of {3,5,41}
divisible by 3 if n is odd, divisible by 5 if n == 2 mod 4, divisible by 41 if n == 0 mod 4
http://factordb.com/index.php?query=8*32%5E%28n%2B1%29%2B31&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
32{G}L(16×32n+1+139)/31 (n ≥ 1)always divisible by some element of {3,5,41}
divisible by 3 if n is even, divisible by 5 if n == 3 mod 4, divisible by 41 if n == 1 mod 4
http://factordb.com/index.php?query=%2816*32%5E%28n%2B1%29%2B139%29%2F31&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
9{1}(9n−1)/8 (n ≥ 2)difference-of-two-squares factorization
(9n−1)/8 = (3n−1) × (3n+1) / 8
http://factordb.com/index.php?query=%289%5En-1%29%2F8&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
81{0}18n+1+1 (n ≥ 0)sum-of-two-cubes factorization
8n+1+1 = (2n+1+1) × (4n+1−2n+1+1)
http://factordb.com/index.php?query=8%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
93{1}(25×9n−1)/8 (n ≥ 1)difference-of-two-squares factorization
(25×9n−1)/8 = (5×3n−1) × (5×3n+1) / 8
http://factordb.com/index.php?query=%2825*9%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
93{8}4×9n−1 (n ≥ 1)difference-of-two-squares factorization
4×9n−1 = (2×3n−1) × (2×3n+1)
http://factordb.com/index.php?query=4*9%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
9{8}59n+1−4 (n ≥ 1)difference-of-two-squares factorization
9n+1−4 = (3n+1−2) × (3n+1+2)
http://factordb.com/index.php?query=9%5E%28n%2B1%29-4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
93{8}354×9n+2−49 (n ≥ 0)difference-of-two-squares factorization
4×9n+2−49 = (2×3n+2−7) × (2×3n+2+7)
http://factordb.com/index.php?query=4*9%5E%28n%2B2%29-49&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
168{F}9×16n−1 (n ≥ 1)difference-of-two-squares factorization
9×16n−1 = (3×4n−1) × (3×4n+1)
http://factordb.com/index.php?query=9*16%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16{F}716n+1−9 (n ≥ 1)difference-of-two-squares factorization
16n+1−9 = (4n+1−3) × (4n+1+3)
http://factordb.com/index.php?query=16%5E%28n%2B1%29-9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16{4}1(4×16n+1−49)/15 (n ≥ 1)difference-of-two-squares factorization
(4×16n+1−49)/15 = (2×4n+1−7) × (2×4n+1+7) / 15
http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29-49%29%2F15&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16B{4}1(169×16n+1−49)/15 (n ≥ 0)difference-of-two-squares factorization
(169×16n+1−49)/15 = (13×4n+1−7) × (13×4n+1+7) / 15
http://factordb.com/index.php?query=%28169*16%5E%28n%2B1%29-49%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
161{5}(4×16n−1)/3 (n ≥ 1)difference-of-two-squares factorization
(4×16n−1)/3 = (2×4n−1) × (2×4n+1) / 3
http://factordb.com/index.php?query=%284*16%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
168{5}(25×16n−1)/3 (n ≥ 1)difference-of-two-squares factorization
(25×16n−1)/3 = (5×4n−1) × (5×4n+1) / 3
http://factordb.com/index.php?query=%2825*16%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1610{5}(49×16n−1)/3 (n ≥ 1)difference-of-two-squares factorization
(49×16n−1)/3 = (7×4n−1) × (7×4n+1) / 3
http://factordb.com/index.php?query=%2849*16%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16A1{5}(484×16n−1)/3 (n ≥ 0)difference-of-two-squares factorization
(484×16n−1)/3 = (22×4n−1) × (22×4n+1) / 3
http://factordb.com/index.php?query=%28484*16%5En-1%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
167{3}(36×16n−1)/5 (n ≥ 1)difference-of-two-squares factorization
(36×16n−1)/5 = (6×4n−1) × (6×4n+1) / 5
http://factordb.com/index.php?query=%2836*16%5En-1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
163{F}AF4×16n+2−81 (n ≥ 0)difference-of-two-squares factorization
4×16n+2−81 = (2×4n+2−9) × (2×4n+2+9)
http://factordb.com/index.php?query=4*16%5E%28n%2B2%29-81&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1630{F}AF49×16n+2−81 (n ≥ 0)difference-of-two-squares factorization
49×16n+2−81 = (7×4n+2−9) × (7×4n+2+9)
http://factordb.com/index.php?query=49*16%5E%28n%2B2%29-81&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
163{F}A0F4×16n+3−1521 (n ≥ 0)difference-of-two-squares factorization
4×16n+3−1521 = (2×4n+3−39) × (2×4n+3+39)
http://factordb.com/index.php?query=4*16%5E%28n%2B3%29-1521&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1630{F}A0F49×16n+3−1521 (n ≥ 0)difference-of-two-squares factorization
49×16n+3−1521 = (7×4n+3−39) × (7×4n+3+39)
http://factordb.com/index.php?query=49*16%5E%28n%2B3%29-1521&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16{3}23(16n+2−81)/5 (n ≥ 0)difference-of-two-squares factorization
(16n+2−81)/5 = (4n+2−9) × (4n+2+9) / 5
(in fact, difference-of-two-4th-powers factorization)
(16n+2−81)/5 = (2n+2−3) × (2n+2+3) × (4n+2+9) / 5
http://factordb.com/index.php?query=%2816%5E%28n%2B2%29-81%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16{5}45(16n+2−49)/3 (n ≥ 0)difference-of-two-squares factorization
(16n+2−49)/3 = (4n+2−7) × (4n+2+7) / 3
http://factordb.com/index.php?query=%2816%5E%28n%2B2%29-49%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16{C}B(4×16n+1−9)/5 (n ≥ 1)difference-of-two-squares factorization
(4×16n+1−9)/5 = (2×4n+1−3) × (2×4n+1+3) / 5
http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29-9%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16{C}D(4×16n+1+1)/5 (n ≥ 1)Aurifeuillean factorization of x4+4×y4
(4×16n+1+1)/5 = (2×4n+1−2×2n+1+1) × (2×4n+1+2×2n+1+1) / 5
http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29%2B1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16{C}DD(4×16n+2+81)/5 (n ≥ 0)Aurifeuillean factorization of x4+4×y4
(4×16n+2+81)/5 = (2×4n+2−6×2n+2+9) × (2×4n+2+6×2n+2+9) / 5
http://factordb.com/index.php?query=%284*16%5E%28n%2B2%29%2B81%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25{1}(25n−1)/24 (n ≥ 2)difference-of-two-squares factorization
(25n−1)/24 = (5n−1) × (5n+1) / 24
http://factordb.com/index.php?query=%2825%5En-1%29%2F24&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
252{1}(49×25n−1)/24 (n ≥ 1)difference-of-two-squares factorization
(49×25n−1)/24 = (7×5n−1) × (7×5n+1) / 24
http://factordb.com/index.php?query=%2849*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
255{1}(121×25n−1)/24 (n ≥ 1)difference-of-two-squares factorization
(121×25n−1)/24 = (11×5n−1) × (11×5n+1) / 24
http://factordb.com/index.php?query=%28121*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
257{1}(169×25n−1)/24 (n ≥ 1)difference-of-two-squares factorization
(169×25n−1)/24 = (13×5n−1) × (13×5n+1) / 24
http://factordb.com/index.php?query=%28169*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25C{1}(289×25n−1)/24 (n ≥ 1)difference-of-two-squares factorization
(289×25n−1)/24 = (17×5n−1) × (17×5n+1) / 24
http://factordb.com/index.php?query=%28289*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25F{1}(361×25n−1)/24 (n ≥ 1)difference-of-two-squares factorization
(361×25n−1)/24 = (19×5n−1) × (19×5n+1) / 24
http://factordb.com/index.php?query=%28361*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25M{1}(529×25n−1)/24 (n ≥ 1)difference-of-two-squares factorization
(529×25n−1)/24 = (23×5n−1) × (23×5n+1) / 24
http://factordb.com/index.php?query=%28529*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
257C{1}(4489×25n−1)/24 (n ≥ 0)difference-of-two-squares factorization
(4489×25n−1)/24 = (67×5n−1) × (67×5n+1) / 24
http://factordb.com/index.php?query=%284489*25%5En-1%29%2F24&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25D5{1}(7921×25n−1)/24 (n ≥ 0)difference-of-two-squares factorization
(7921×25n−1)/24 = (89×5n−1) × (89×5n+1) / 24
http://factordb.com/index.php?query=%287921*25%5En-1%29%2F24&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
251{3}(9×25n−1)/8 (n ≥ 1)difference-of-two-squares factorization
(9×25n−1)/8 = (3×5n−1) × (3×5n+1) / 8
http://factordb.com/index.php?query=%289*25%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
251{8}(4×25n−1)/3 (n ≥ 1)difference-of-two-squares factorization
(4×25n−1)/3 = (2×5n−1) × (2×5n+1) / 3
http://factordb.com/index.php?query=%284*25%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
255{8}(16×25n−1)/3 (n ≥ 1)difference-of-two-squares factorization
(16×25n−1)/3 = (4×5n−1) × (4×5n+1) / 3
http://factordb.com/index.php?query=%2816*25%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25A{3}(81×25n−1)/8 (n ≥ 1)difference-of-two-squares factorization
(81×25n−1)/8 = (9×5n−1) × (9×5n+1) / 8
http://factordb.com/index.php?query=%2881*25%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25L{8}(64×25n−1)/3 (n ≥ 1)difference-of-two-squares factorization
(64×25n−1)/3 = (8×5n−1) × (8×5n+1) / 3
http://factordb.com/index.php?query=%2864*25%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25{3}2(25n+1−9)/8 (n ≥ 1)difference-of-two-squares factorization
(25n+1−9)/8 = (5n+1−3) × (5n+1+3) / 8
http://factordb.com/index.php?query=%2825%5E%28n%2B1%29-9%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25{8}3(25n+1−16)/3 (n ≥ 1)difference-of-two-squares factorization
(25n+1−16)/3 = (5n+1−4) × (5n+1+4) / 3
http://factordb.com/index.php?query=%2825%5E%28n%2B1%29-16%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25{8}7(25n+1−4)/3 (n ≥ 1)difference-of-two-squares factorization
(25n+1−4)/3 = (5n+1−2) × (5n+1+2) / 3
http://factordb.com/index.php?query=%2825%5E%28n%2B1%29-4%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25{1}0E(25n+2−289)/24 (n ≥ 1)difference-of-two-squares factorization
(25n+2−289)/24 = (5n+2−17) × (5n+2+17) / 24
http://factordb.com/index.php?query=%2825%5E%28n%2B2%29-289%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25{1}0J(25n+2−169)/24 (n ≥ 1)difference-of-two-squares factorization
(25n+2−169)/24 = (5n+2−13) × (5n+2+13) / 24
(special example, as the numbers with length ≥ 8690 in this family contain "prime > b" subsequence, this prime is 18688J)
http://factordb.com/index.php?query=%2825%5E%28n%2B2%29-169%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25{1}0L(25n+2−121)/24 (n ≥ 1)difference-of-two-squares factorization
(25n+2−121)/24 = (5n+2−11) × (5n+2+11) / 24
(special example, as the numbers with length ≥ 88 in this family contain "prime > b" subsequence, this prime is 186L)
http://factordb.com/index.php?query=%2825%5E%28n%2B2%29-121%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25{3}2I(25n+2−81)/8 (n ≥ 0)difference-of-two-squares factorization
(25n+2−81)/8 = (5n+2−9) × (5n+2+9) / 8
http://factordb.com/index.php?query=%2825%5E%28n%2B2%29-81%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25{8}5I(25n+2−196)/3 (n ≥ 0)difference-of-two-squares factorization
(25n+2−196)/3 = (5n+2−14) × (5n+2+14) / 3
http://factordb.com/index.php?query=%2825%5E%28n%2B2%29-196%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25{8}7C(25n+2−64)/3 (n ≥ 0)difference-of-two-squares factorization
(25n+2−64)/3 = (5n+2−8) × (5n+2+8) / 3
http://factordb.com/index.php?query=%2825%5E%28n%2B2%29-64%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
253{O}G4×25n+1−9 (n ≥ 0)difference-of-two-squares factorization
4×25n+1−9 = (2×5n+1−3) × (2×5n+1+3)
http://factordb.com/index.php?query=4*25%5E%28n%2B1%29-9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
258{O}L9×25n+1−4 (n ≥ 0)difference-of-two-squares factorization
9×25n+1−4 = (3×5n+1−2) × (3×5n+1+2)
http://factordb.com/index.php?query=9*25%5E%28n%2B1%29-4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25F{O}G16×25n+1−9 (n ≥ 0)difference-of-two-squares factorization
16×25n+1−9 = (4×5n+1−3) × (4×5n+1+3)
http://factordb.com/index.php?query=16*25%5E%28n%2B1%29-9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
258{O}99×25n+1−16 (n ≥ 0)difference-of-two-squares factorization
9×25n+1−16 = (3×5n+1−4) × (3×5n+1+4)
http://factordb.com/index.php?query=9*25%5E%28n%2B1%29-16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25G{8}7(49×25n+1−4)/3 (n ≥ 0)difference-of-two-squares factorization
(49×25n+1−4)/3 = (7×5n+1−2) × (7×5n+1+2) / 3
http://factordb.com/index.php?query=%2849*25%5E%28n%2B1%29-4%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
256{3}2(49×25n+1−9)/8 (n ≥ 0)difference-of-two-squares factorization
(49×25n+1−9)/8 = (7×5n+1−3) × (7×5n+1+3) / 8
http://factordb.com/index.php?query=%2849*25%5E%28n%2B1%29-9%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25G{8}3(49×25n+1−16)/3 (n ≥ 0)difference-of-two-squares factorization
(49×25n+1−16)/3 = (7×5n+1−4) × (7×5n+1+4) / 3
http://factordb.com/index.php?query=%2849*25%5E%28n%2B1%29-16%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
251{8}7H(4×25n+2−49)/3 (n ≥ 0)difference-of-two-squares factorization
(4×25n+2−49)/3 = (2×5n+2−7) × (2×5n+2+7) / 3
(special example, as the numbers with length ≥ 78 in this family contain "prime > b" subsequence, this prime is 875H)
http://factordb.com/index.php?query=%284*25%5E%28n%2B2%29-49%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
255{8}7H(16×25n+2−49)/3 (n ≥ 0)difference-of-two-squares factorization
(16×25n+2−49)/3 = (4×5n+2−7) × (4×5n+2+7) / 3
(special example, as the numbers with length ≥ 78 in this family contain "prime > b" subsequence, this prime is 875H)
http://factordb.com/index.php?query=%2816*25%5E%28n%2B2%29-49%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
278{0}18×27n+1+1 (n ≥ 0)sum-of-two-cubes factorization
8×27n+1+1 = (2×3n+1+1) × (4×9n+1−2×3n+1+1)
http://factordb.com/index.php?query=8*27%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
271{0}827n+1+8 (n ≥ 0)sum-of-two-cubes factorization
27n+1+8 = (3n+1+2) × (9n+1−2×3n+1+4)
http://factordb.com/index.php?query=27%5E%28n%2B1%29%2B8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
27{D}E(27n+1+1)/2 (n ≥ 1)sum-of-two-cubes factorization
(27n+1+1)/2 = (3n+1+1) × (9n+1−3n+1+1) / 2
http://factordb.com/index.php?query=%2827%5E%28n%2B1%29%2B1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
277{Q}8×27n−1 (n ≥ 1)difference-of-two-cubes factorization
8×27n−1 = (2×3n−1) × (4×9n+2×3n+1)
http://factordb.com/index.php?query=8*27%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
27{Q}J27n+1−8 (n ≥ 1)difference-of-two-cubes factorization
27n+1−8 = (3n+1−2) × (9n+1+2×3n+1+4)
http://factordb.com/index.php?query=27%5E%28n%2B1%29-8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
279{G}(125×27n−8)/13 (n ≥ 1)difference-of-two-cubes factorization
(125×27n−8)/13 = (5×3n−2) × (25×9n+10×3n+4) / 13
http://factordb.com/index.php?query=%28125*27%5En-8%29%2F13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
321{0}132n+1+1 (n ≥ 0)sum-of-two-5th-powers factorization
32n+1+1 = (2n+1+1) × (16n+1−8n+1+4n+1−2n+1+1)
http://factordb.com/index.php?query=32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
32{1}(32n−1)/31 (n ≥ 2)difference-of-two-5th-powers factorization
(32n−1)/31 = (2n−1) × (16n+8n+4n+2n+1) / 31
http://factordb.com/index.php?query=%2832%5En-1%29%2F31&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
363{7}(16×36n−1)/5 (n ≥ 1)difference-of-two-squares factorization
(16×36n−1)/5 = (4×6n−1) × (4×6n+1) / 5
http://factordb.com/index.php?query=%2816*36%5En-1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
363{Z}4×36n−1 (n ≥ 1)difference-of-two-squares factorization
4×36n−1 = (2×6n−1) × (2×6n+1)
http://factordb.com/index.php?query=4*36%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
368{Z}9×36n−1 (n ≥ 1)difference-of-two-squares factorization
9×36n−1 = (3×6n−1) × (3×6n+1)
http://factordb.com/index.php?query=9*36%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36O{Z}25×36n−1 (n ≥ 1)difference-of-two-squares factorization
25×36n−1 = (5×6n−1) × (5×6n+1)
http://factordb.com/index.php?query=25*36%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36{Z}B36n+1−25 (n ≥ 1)difference-of-two-squares factorization
36n+1−25 = (6n+1−5) × (6n+1+5)
http://factordb.com/index.php?query=36%5E%28n%2B1%29-25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
368{Z}B9×36n+1−25 (n ≥ 0)difference-of-two-squares factorization
9×36n+1−25 = (3×6n+1−5) × (3×6n+1+5)
http://factordb.com/index.php?query=9*36%5E%28n%2B1%29-25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36F{Z}B16×36n+1−25 (n ≥ 0)difference-of-two-squares factorization
16×36n+1−25 = (4×6n+1−5) × (4×6n+1+5)
http://factordb.com/index.php?query=16*36%5E%28n%2B1%29-25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36{Z}RZ36n+2−289 (n ≥ 0)difference-of-two-squares factorization
36n+2−289 = (6n+2−17) × (6n+2+17)
http://factordb.com/index.php?query=36%5E%28n%2B2%29-289&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36O{Z}RZ25×36n+2−289 (n ≥ 0)difference-of-two-squares factorization
25×36n+2−289 = (5×6n+2−17) × (5×6n+2+17)
http://factordb.com/index.php?query=25*36%5E%28n%2B2%29-289&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36O{5}(169×36n−1)/7 (n ≥ 1)difference-of-two-squares factorization
(169×36n−1)/7 = (13×6n−1) × (13×6n+1) / 7
http://factordb.com/index.php?query=%28169*36%5En-1%29%2F7&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36O{7}(121×36n−1)/5 (n ≥ 1)difference-of-two-squares factorization
(121×36n−1)/5 = (11×6n−1) × (11×6n+1) / 5
http://factordb.com/index.php?query=%28121*36%5En-1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36{9}1(9×36n+1−289)/35 (n ≥ 1)difference-of-two-squares factorization
(9×36n+1−289)/35 = (3×6n+1−17) × (3×6n+1+17) / 35
http://factordb.com/index.php?query=%289*36%5E%28n%2B1%29-289%29%2F35&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36T{9}1(1024×36n+1−289)/35 (n ≥ 0)difference-of-two-squares factorization
(1024×36n+1−289)/35 = (32×6n+1−17) × (32×6n+1+17) / 35
http://factordb.com/index.php?query=%281024*36%5E%28n%2B1%29-289%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36{G}D(16×36n+1−121)/35 (n ≥ 1)difference-of-two-squares factorization
(16×36n+1−121)/35 = (4×6n+1−11) × (4×6n+1+11) / 35
http://factordb.com/index.php?query=%2816*36%5E%28n%2B1%29-121%29%2F35&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36{G}8D(16×36n+2−10201)/35 (n ≥ 0)difference-of-two-squares factorization
(16×36n+2−10201)/35 = (4×6n+2−101) × (4×6n+2+101) / 35
http://factordb.com/index.php?query=%2816*36%5E%28n%2B2%29-10201%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36R{G}D(961×36n+1−121)/35 (n ≥ 0)difference-of-two-squares factorization
(961×36n+1−121)/35 = (31×6n+1−11) × (31×6n+1+11) / 35
http://factordb.com/index.php?query=%28961*36%5E%28n%2B2%29-121%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
363{G}8D(121×36n+2−10201)/35 (n ≥ 0)difference-of-two-squares factorization
(121×36n+2−10201)/35 = (11×6n+2−101) × (11×6n+2+101) / 35
http://factordb.com/index.php?query=%28121*36%5E%28n%2B2%29-10201%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36R{G}8D(961×36n+2−10201)/35 (n ≥ 0)difference-of-two-squares factorization
(961×36n+2−10201)/35 = (31×6n+2−101) × (31×6n+2+101) / 35
http://factordb.com/index.php?query=%28961*36%5E%28n%2B2%29-10201%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36{K}H(4×36n+1−25)/7 (n ≥ 1)difference-of-two-squares factorization
(4×36n+1−25)/7 = (2×6n+1−5) × (2×6n+1+5) / 7
http://factordb.com/index.php?query=%284*36%5E%28n%2B1%29-25%29%2F7&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36B{K}H(81×36n+1−25)/7 (n ≥ 0)difference-of-two-squares factorization
(81×36n+1−25)/7 = (9×6n+1−5) × (9×6n+1+5) / 7
http://factordb.com/index.php?query=%2881*36%5E%28n%2B1%29-25%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
363{K}IH(25×36n+2−529)/7 (n ≥ 0)difference-of-two-squares factorization
(25×36n+2−529)/7 = (5×6n+2−23) × (5×6n+2+23) / 7
http://factordb.com/index.php?query=%2825*36%5E%28n%2B2%29-529%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36B{K}IH(81×36n+2−529)/7 (n ≥ 0)difference-of-two-squares factorization
(81×36n+2−529)/7 = (9×6n+2−23) × (9×6n+2+23) / 7
http://factordb.com/index.php?query=%2881*36%5E%28n%2B2%29-529%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36{S}J(4×36n+1−49)/5 (n ≥ 1)difference-of-two-squares factorization
(4×36n+1−49)/5 = (2×6n+1−7) × (2×6n+1+7) / 5
http://factordb.com/index.php?query=%284*36%5E%28n%2B1%29-49%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36{S}IJ(4×36n+2−1849)/5 (n ≥ 0)difference-of-two-squares factorization
(4×36n+2−1849)/5 = (2×6n+2−43) × (2×6n+2+43) / 5
http://factordb.com/index.php?query=%284*36%5E%28n%2B2%29-1849%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
361{S}J(9×36n+1−49)/5 (n ≥ 0)difference-of-two-squares factorization
(9×36n+1−49)/5 = (3×6n+1−7) × (3×6n+1+7) / 5
http://factordb.com/index.php?query=%289*36%5E%28n%2B1%29-49%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36C{S}J(64×36n+1−49)/5 (n ≥ 0)difference-of-two-squares factorization
(64×36n+1−49)/5 = (8×6n+1−7) × (8×6n+1+7) / 5
http://factordb.com/index.php?query=%2864*36%5E%28n%2B1%29-49%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36X{S}J(169×36n+1−49)/5 (n ≥ 0)difference-of-two-squares factorization
(169×36n+1−49)/5 = (13×6n+1−7) × (13×6n+1+7) / 5
http://factordb.com/index.php?query=%28169*36%5E%28n%2B1%29-49%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
361{S}GJ(9×36n+2−2209)/5 (n ≥ 0)difference-of-two-squares factorization
(9×36n+2−2209)/5 = (3×6n+2−47) × (3×6n+2+47) / 5
http://factordb.com/index.php?query=%289*36%5E%28n%2B2%29-2209%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
369{S}GJ(49×36n+2−2209)/5 (n ≥ 0)difference-of-two-squares factorization
(49×36n+2−2209)/5 = (7×6n+2−47) × (7×6n+2+47) / 5
http://factordb.com/index.php?query=%2849*36%5E%28n%2B2%29-2209%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36C{S}GJ(64×36n+2−2209)/5 (n ≥ 0)difference-of-two-squares factorization
(64×36n+2−2209)/5 = (8×6n+2−47) × (8×6n+2+47) / 5
http://factordb.com/index.php?query=%2864*36%5E%28n%2B2%29-2209%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36X{S}GJ(169×36n+2−2209)/5 (n ≥ 0)difference-of-two-squares factorization
(169×36n+2−2209)/5 = (13×6n+2−47) × (13×6n+2+47) / 5
http://factordb.com/index.php?query=%28169*36%5E%28n%2B2%29-2209%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
361{S}IJ(9×36n+2−1849)/5 (n ≥ 0)difference-of-two-squares factorization
(9×36n+2−1849)/5 = (3×6n+2−43) × (3×6n+2+43) / 5
http://factordb.com/index.php?query=%289*36%5E%28n%2B2%29-1849%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
369{S}IJ(49×36n+2−1849)/5 (n ≥ 0)difference-of-two-squares factorization
(49×36n+2−1849)/5 = (7×6n+2−43) × (7×6n+2+43) / 5
http://factordb.com/index.php?query=%2849*36%5E%28n%2B2%29-1849%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
148{D}9×14n−1 (n ≥ 1)combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if n is odd, 9×14n−1 = (3×14n/2−1) × (3×14n/2+1) if n is even
http://factordb.com/index.php?query=9*14%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
12{B}9B12n+2−25 (n ≥ 0)combine of factor 13 and difference-of-two-squares factorization
divisible by 13 if n is odd, 12n+2−25 = (12(n+2)/2−5) × (12(n+2)/2+5) if n is even
http://factordb.com/index.php?query=12%5E%28n%2B2%29-25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
14{D}514n+1−9 (n ≥ 1)combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if n is even, 14n+1−9 = (14(n+1)/2−3) × (14(n+1)/2+3) if n is odd
http://factordb.com/index.php?query=14%5E%28n%2B1%29-9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
171{9}(25×17n−9)/16 (n ≥ 1)combine of factor 2 and difference-of-two-squares factorization
divisible by 2 if n is odd, (25×17n−9)/16 = (5×17n/2−3) × (5×17n/2+3) / 16 if n is even
http://factordb.com/index.php?query=%2825*17%5En-9%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
177{9}(121×17n−9)/16 (n ≥ 1)combine of factor 2 and difference-of-two-squares factorization
divisible by 2 if n is odd, (121×17n−9)/16 = (11×17n/2−3) × (11×17n/2+3) / 16 if n is even
http://factordb.com/index.php?query=%28121*17%5En-9%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17{9}2(9×17n+1−121)/16 (n ≥ 1)combine of factor 2 and difference-of-two-squares factorization
divisible by 2 if n is even, (9×17n+1−121)/16 = (3×17(n+1)/2−11) × (3×17(n+1)/2+11) / 16 if n is odd
http://factordb.com/index.php?query=%289*17%5E%28n%2B1%29-121%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17{9}8(9×17n+1−25)/16 (n ≥ 1)combine of factor 2 and difference-of-two-squares factorization
divisible by 2 if n is even, (9×17n+1−25)/16 = (3×17(n+1)/2−5) × (3×17(n+1)/2+5) / 16 if n is odd
http://factordb.com/index.php?query=%289*17%5E%28n%2B1%29-25%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
191{6}(4×19n−1)/3 (n ≥ 1)combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if n is odd, (4×19n−1)/3 = (2×19n/2−1) × (2×19n/2+1) / 3 if n is even
http://factordb.com/index.php?query=%284*19%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
19{6}5(19n+1−4)/3 (n ≥ 1)combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if n is even, (19n+1−4)/3 = (19(n+1)/2−2) × (19(n+1)/2+2) / 3 if n is odd
http://factordb.com/index.php?query=%2819%5E%28n%2B1%29-4%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
197{2}(64×19n−1)/9 (n ≥ 1)combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if n is odd, (64×19n−1)/9 = (8×19n/2−1) × (8×19n/2+1) / 9 if n is even
http://factordb.com/index.php?query=%2864*19%5En-1%29%2F9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1989{6}(484×19n−1)/3 (n ≥ 0)combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if n is odd, (484×19n−1)/3 = (22×19n/2−1) × (22×19n/2+1) / 3 if n is even
http://factordb.com/index.php?query=%28484*19%5En-1%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
243{N}4×24n−1 (n ≥ 1)combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if n is odd, 4×24n−1 = (2×24n/2−1) × (2×24n/2+1) if n is even
http://factordb.com/index.php?query=4*24%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
245{N}6×24n−1 (n ≥ 1)combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if n is even, 6×24n−1 = (12×24(n−1)/2−1) × (12×24(n−1)/2+1) if n is odd
http://factordb.com/index.php?query=6*24%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
248{N}9×24n−1 (n ≥ 1)combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if n is odd, 9×24n−1 = (3×24n/2−1) × (3×24n/2+1) if n is even
http://factordb.com/index.php?query=9*24%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
24{6}1(6×24n+1−121)/23 (n ≥ 1)combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if n is odd, (6×24n+1−121)/23 = (12×24n/2−11) × (12×24n/2+11) / 23 if n is even
http://factordb.com/index.php?query=%286*24%5E%28n%2B1%29-121%29%2F23&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
24{N}LN24n+2−49 (n ≥ 0)combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if n is odd, 24n+2−49 = (24(n+2)/2−7) × (24(n+2)/2+7) if n is even
http://factordb.com/index.php?query=24%5E%28n%2B2%29-49&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
33F{W}16×33n−1 (n ≥ 1)combine of factor 17 and difference-of-two-squares factorization
divisible by 17 if n is odd, 16×33n−1 = (4×33n/2−1) × (4×33n/2+1) if n is even
http://factordb.com/index.php?query=16*33%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
33{W}H33n+1−16 (n ≥ 1)combine of factor 17 and difference-of-two-squares factorization
divisible by 17 if n is even, 33n+1−16 = (33(n+1)/2−4) × (33(n+1)/2+4) if n is odd
http://factordb.com/index.php?query=33%5E%28n%2B1%29-16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
33L{4}(169×33n−1)/8 (n ≥ 1)combine of factor 17 and difference-of-two-squares factorization
divisible by 17 if n is odd, (169×33n−1)/8 = (13×33n/2−1) × (13×33n/2+1) / 8 if n is even
http://factordb.com/index.php?query=%28169*33%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
333{P}(121×33n−25)/32 (n ≥ 1)combine of factor 2 and difference-of-two-squares factorization
divisible by 2 if n is odd, (121×33n−25)/32 = (11×33n/2−5) × (11×33n/2+5) / 32 if n is even
http://factordb.com/index.php?query=%28121*33%5En-25%29%2F32&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
33D{P}(441×33n−25)/32 (n ≥ 1)combine of factor 2 and difference-of-two-squares factorization
divisible by 2 if n is odd, (441×33n−25)/32 = (21×33n/2−5) × (21×33n/2+5) / 32 if n is even
http://factordb.com/index.php?query=%28441*33%5En-25%29%2F32&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
33{9}4(9×33n+1−169)/32 (n ≥ 1)combine of factor 2 and difference-of-two-squares factorization
divisible by 2 if n is even, (9×33n+1−169)/32 = (3×33(n+1)/2−13) × (3×33(n+1)/2+13) / 32 if n is odd
http://factordb.com/index.php?query=%289*33%5E%28n%2B1%29-169%29%2F32&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
33{I}H(9×33n+1−25)/16 (n ≥ 1)combine of factor 17 and difference-of-two-squares factorization
divisible by 17 if n is even, (9×33n+1−25)/16 = (3×33(n+1)/2−5) × (3×33(n+1)/2+5) / 16 if n is odd
http://factordb.com/index.php?query=%289*33%5E%28n%2B1%29-25%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
341{B}(4×34n−1)/3 (n ≥ 1)combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if n is odd, (4×34n−1)/3 = (2×34n/2−1) × (2×34n/2+1) / 3 if n is even
http://factordb.com/index.php?query=%284*34%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
34G{B}(49×34n−1)/3 (n ≥ 1)combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if n is odd, (49×34n−1)/3 = (7×34n/2−1) × (7×34n/2+1) / 3 if n is even
http://factordb.com/index.php?query=%2849*34%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
34G{1}(529×34n−1)/33 (n ≥ 1)combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if n is odd, (529×34n−1)/33 = (23×34n/2−1) × (23×34n/2+1) / 33 if n is even
(special example, as the numbers with length ≥ 14 in this family contain "prime > b" subsequence, this prime is 113)
http://factordb.com/index.php?query=%28529*34%5En-1%29%2F33&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
34V{1}(1024×34n−1)/33 (n ≥ 1)combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if n is odd, (1024×34n−1)/33 = (32×34n/2−1) × (32×34n/2+1) / 33 if n is even
(special example, as the numbers with length ≥ 14 in this family contain "prime > b" subsequence, this prime is 113)
http://factordb.com/index.php?query=%281024*34%5En-1%29%2F33&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
34D{3}(144×34n−1)/11 (n ≥ 1)combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if n is odd, (144×34n−1)/11 = (12×34n/2−1) × (12×34n/2+1) / 11 if n is even
http://factordb.com/index.php?query=%28144*34%5En-1%29%2F11&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
348{X}9×34n−1 (n ≥ 1)combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if n is odd, 9×34n−1 = (3×34n/2−1) × (3×34n/2+1) if n is even
http://factordb.com/index.php?query=9*34%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
34{X}P34n+1−9 (n ≥ 1)combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if n is even, 34n+1−9 = (34(n+1)/2−3) × (34(n+1)/2+3) if n is odd
http://factordb.com/index.php?query=34%5E%28n%2B1%29-9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show

Also families which contain only one very small prime > b: (this is because: for the case of covering congruence, an element in S is indeed in the family and > b, to make the factorizations be trivial; for the case of algebraic factorization (if the numbers are factored as F × G / d), F (or G) is equal to d and G (or F) is a prime > b, to make the factorizations be trivial; for the case of combine of covering congruence and algebraic factorization (if the numbers are factored as F × G / d), an element in S is indeed in the family and > b or/and F (or G) is equal to d and G (or F) is prime > b, to make the factorizations be trivial)

b
(2 ≤ b ≤ 36)
familyalgebraic ((a×bn+c)/d) form of this family (n is the number of digits in the "{}", also the lower bound of n to make the numbers > b)
(note: d divides gcd(a+c,b−1), but d need not be gcd(a+c,b−1), d = gcd(a+c,b−1) if and only if the numbers in the family are not divisible by some prime factor of b−1, i.e. the numbers in the family are coprime to b−1, in fact, d = gcd(a+c,b−1)/gcd(a+c,b−1,(largest trivial factor of the family)))
the only prime > b in this familythis prime > b written in decimalwhy this family contains only this prime > bfactorization of the first 200 numbers in this family (n is the number of digits in the "{}", start with the smallest n making the number > b (if n = 0 already makes the number > b, then start with n = 0))
92{7}5(23×9n+1−23)/8 (n ≥ 0)2523always divisible by 23
(in fact, also difference-of-two-squares factorization)
(23×9n+1−23)/8 = 23 × (3n+1−1) × (3n+1+1) / 8
http://factordb.com/index.php?query=%2823*9%5E%28n%2B1%29-23%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
95{7}2(47×9n+1−47)/8 (n ≥ 0)5247always divisible by 47
(in fact, also difference-of-two-squares factorization)
(47×9n+1−47)/8 = 47 × (3n+1−1) × (3n+1+1) / 8
http://factordb.com/index.php?query=%2847*9%5E%28n%2B1%29-47%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
113{7}4(37×11n+1−37)/10 (n ≥ 0)3437always divisible by 37http://factordb.com/index.php?query=%2837*11%5E%28n%2B1%29-37%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
114{7}3(47×11n+1−47)/10 (n ≥ 0)4347always divisible by 47http://factordb.com/index.php?query=%2847*11%5E%28n%2B1%29-47%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
272{0}J2×27n+1+19 (n ≥ 0)2J73always divisible by some element of {5,7,73}
divisible by 7 if n is odd, divisible by 5 if n == 2 mod 4, divisible by 73 if n == 0 mod 4
http://factordb.com/index.php?query=2*27%5E%28n%2B1%29%2B19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
4{1}(4n−1)/3 (n ≥ 2)115difference-of-two-squares factorization
but 11 is prime, and 11 is the only prime > b in this family
(4n−1)/3 = (2n−1) × (2n+1) / 3
http://factordb.com/index.php?query=%284%5En-1%29%2F3&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
8{1}(8n−1)/7 (n ≥ 2)11173difference-of-two-cubes factorization
but 111 is prime, and 111 is the only prime > b in this family
(8n−1)/7 = (2n−1) × (4n+2n+1) / 7
http://factordb.com/index.php?query=%288%5En-1%29%2F7&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16{1}(16n−1)/15 (n ≥ 2)1117difference-of-two-squares factorization
but 11 is prime, and 11 is the only prime > b in this family
(16n−1)/15 = (4n−1) × (4n+1) / 15
(in fact, difference-of-two-4th-powers factorization)
(16n−1)/15 = (2n−1) × (2n+1) × (4n+1) / 15
(in fact, also combine of difference-of-8th-powers factorization and Aurifeuillean factorization of x8−16×y8)
(16n−1)/15 = (2n/2−1) × (2n/2+1) × (2n+1) × (4n+1) / 15 if n is even, (16n−1)/15 = (2n−1) × (2n+1) × (2n−2(n+1)/2+1) × (2n+2(n+1)/2+1) / 15 if n is odd
http://factordb.com/index.php?query=%2816%5En-1%29%2F15&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
27{1}(27n−1)/26 (n ≥ 2)111757difference-of-two-cubes factorization
but 111 is prime, and 111 is the only prime > b in this family
(27n−1)/26 = (3n−1) × (9n+3n+1) / 26
http://factordb.com/index.php?query=%2827%5En-1%29%2F26&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
27{2}7(27n+1+64)/13 (n ≥ 1)2761sum-of-two-cubes factorization
but 27 is prime, and 27 is the only prime > b in this family
(27n+1+64)/13 = (3n+1+4) × (9n+1−4×3n+1+16) / 13
(in fact, also combine of Aurifeuillean factorization of x4+4×y4 and Aurifeuillean factorization of x6+27×y6 and Aurifeuillean factorization of x12+46656×y12)
(27n+1+64)/13 = (3n+1+4) × (3n+1−2×3(n+2)/2+4) × (3n+1+2×3(n+2)/2+4) if n is even, (27n+1+64)/13 = (3(n+1)/2−2×3(n+1)/4+2) × (3(n+1)/2+2×3(n+1)/4+2) × (3n+1−2×3(3×n+3)/4+2×3(n+1)/2−4×3(n+1)/4+4) × (3n+1+2×33×(n+1)/4+2×3(n+1)/2+4×3(n+1)/4+4) / 13 if n == 3 mod 4, (27n+1+64)/13 = (3n+1+4) × (3n+1−2×3(3×n+5)/4+2×3(n+3)/2−4×3(n+3)/4+4) × (3n+1+2×3(3×n+5)/4+2×3(n+3)/2+4×3(n+3)/4+4) if n == 1 mod 4
http://factordb.com/index.php?query=%2827%5E%28n%2B1%29%2B64%29%2F13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
27{G}7(8×27n+1−125)/13 (n ≥ 1)G7439difference-of-two-cubes factorization
but G7 is prime, and G7 is the only prime > b in this family
(8×27n+1−125)/13 = (2×3n+1−5) × (4×9n+1+10×3n+1+25) / 13
http://factordb.com/index.php?query=%288*27%5E%28n%2B1%29-125%29%2F13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36{1}(36n−1)/35 (n ≥ 2)1137difference-of-two-squares factorization
but 11 is prime, and 11 is the only prime > b in this family
(36n−1)/35 = (6n−1) × (6n+1) / 35
http://factordb.com/index.php?query=%2836%5En-1%29%2F35&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show

It is interesting to note that all numbers in families {1}, 2{1}, 5{1}, 7{1}, C{1}, F{1}, M{1}, 1F{1}, 27{1}, 7C{1}, D5{1} in base b = 25 are generalized pentagonal numbers (https://en.wikipedia.org/wiki/Pentagonal_number, https://mathworld.wolfram.com/PentagonalNumber.html, https://www.numbersaplenty.com/set/pentagonal_number/, https://oeis.org/A001318), and all numbers in families {1}, 3{1}, 6{1}, 16{1} in base b = 9 and families 1{3}, A{3} in base b = 25 are triangular numbers (https://en.wikipedia.org/wiki/Triangular_number, https://mathworld.wolfram.com/TriangularNumber.html, https://www.numbersaplenty.com/set/triangular_number/, https://oeis.org/A000217), and all numbers in family {1} in base b = 4 and families 1{5}, 8{5}, 10{5}, A1{5} in base b = 16 and families 1{8}, 5{8}, L{8} in base b = 25 are generalized octagonal numbers (https://en.wikipedia.org/wiki/Octagonal_number, https://mathworld.wolfram.com/OctagonalNumber.html, https://www.numbersaplenty.com/set/octagonal_number/, https://oeis.org/A001082), since all generalized pentagonal numbers × 25 + 1 are also generalized pentagonal numbers, all triangular numbers × 9 + 1 and all triangular numbers × 25 + 3 are also triangular numbers, all generalized octagonal numbers × 4 + 1 and all generalized octagonal numbers × 16 + 5 and all generalized octagonal numbers × 25 + 8 are also generalized octagonal numbers (thus: In base b = 4, all numbers of the form (generalized octagonal numbers){1} are also generalized octagonal numbers; in base b = 9, all numbers of the form (triangular numbers){1} are also triangular numbers; in base b = 16, all numbers of the form (generalized octagonal numbers){5} are also generalized octagonal numbers; in base b = 25, all numbers of the form (generalized pentagonal numbers){1} are also generalized pentagonal numbers, all numbers of the form (triangular numbers){3} are also triangular numbers, all numbers of the form (generalized octagonal numbers){8} are also generalized octagonal numbers), and all generalized pentagonal numbers (except 0, 1, 2, 5, 7) and all triangular numbers (except 0, 1, 3) and all generalized octagonal numbers (except 0, 1, 5) are composite (since all generalized polygonal numbers (https://en.wikipedia.org/wiki/Polygonal_number, https://mathworld.wolfram.com/PolygonalNumber.html, https://oeis.org/A195152, https://oeis.org/A194801, https://oeis.org/A303301) are composite, with only possible exception of indices 0, 1, −1, 2, −2), thus these families contain no primes > b (except the family {1} in base 4, which contains a prime 11 (5 in decimal) > b).

Generalized pentagonal numbers (https://en.wikipedia.org/wiki/Pentagonal_number, https://mathworld.wolfram.com/PentagonalNumber.html, https://www.numbersaplenty.com/set/pentagonal_number/, https://oeis.org/A001318) are very important in number theory, since they appear in the pentagonal number theorem (https://en.wikipedia.org/wiki/Pentagonal_number_theorem, https://mathworld.wolfram.com/PentagonalNumberTheorem.html, https://oeis.org/A010815, https://oeis.org/A195310, https://oeis.org/A175003, https://oeis.org/A238442, https://arxiv.org/pdf/math/0505373.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_172.pdf), https://arxiv.org/pdf/math/0411587.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_173.pdf), http://eulerarchive.maa.org//docs/originals/E542.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_174.pdf), https://arxiv.org/pdf/math/0510054.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_175.pdf)) for both the partition function (https://en.wikipedia.org/wiki/Partition_function_(number_theory), https://mathworld.wolfram.com/PartitionFunctionP.html, https://t5k.org/top20/page.php?id=54, https://t5k.org/primes/search.php?Comment=Partitions&OnList=all&Number=1000000&Style=HTML, https://www.numbersaplenty.com/set/partition_number/, http://www.numericana.com/answer/numbers.htm#partitions, http://www.numericana.com/data/partition.htm, http://www.numbertheory.org/php/partition.html, https://web.archive.org/web/20120719080234/http://www.btinternet.com/~se16/js/partitions.htm, https://oeis.org/A000041) and the sum-of-divisors function (https://en.wikipedia.org/wiki/Divisor_function, https://t5k.org/glossary/xpage/SigmaFunction.html, https://mathworld.wolfram.com/DivisorFunction.html, http://www.javascripter.net/math/calculators/divisorscalculator.htm, https://oeis.org/A000203), also, for the pentagonal number theorem for the sum-of-divisors function, "if the last term is sigma(0), then change it to n" occurs exactly when n is a generalized pentagonal number (i.e. exactly when calculate the sum-of-divisors function of a generalized pentagonal number).

Also, for bases 2 ≤ b ≤ 18, the reasons for families can be ruled out as only containing composites (only count the numbers > b) which are needed to use to prove the "minimal prime problem" in base b are: (only consider the numbers coprime to b, i.e. only consider the number string with the first digit not 0 and the last digit coprime to b) (only listed the bases 2 ≤ b ≤ 18, since for 19 ≤ b ≤ 36 there may be reasons which are still not found by me, e.g. the family {D}GA in base b = 23 (which has a covering set {2,5,7,37,79}, also a covering set {3,5,7,37,79}) and the family L{5}L in base b = 23 (which has a covering set {2,5,7,13,37}, also a covering set {3,5,7,13,37}) are very hard to found)

breasons for families can be ruled out as only containing composites (only count the numbers > b) which are needed to use to prove the "minimal prime problem" in base b
2
3Divisible by 2
4Divisible by 3
5Divisible by 2
Divisible by 3
6Divisible by 5
7Divisible by 2
Divisible by 3
Divisible by 5
8Divisible by 3
Divisible by 5
Divisible by 7
Sum-of-two-cubes factorization
9Divisible by 2
Divisible by 5
Divisible by 7
Covering set {2,5}
Difference-of-two-squares factorization
10Divisible by 3
Divisible by 7
11Divisible by 2
Divisible by 3
Divisible by 5
Divisible by 7
Covering set {2,3}
12Divisible by 5
Divisible by 7
Divisible by 11
Combine of divisible by 13 and difference-of-two-squares factorization
13Divisible by 2
Divisible by 3
Divisible by 5
Divisible by 7
Divisible by 11
Covering set {2,7}
Covering set {2,5,17}
Covering set {5,7,17}
14Divisible by 3
Divisible by 5
Divisible by 11
Divisible by 13
Covering set {3,5}
Combine of divisible by 5 and difference-of-two-squares factorization
15Divisible by 2
Divisible by 7
Divisible by 11
Divisible by 13
16Divisible by 3
Divisible by 5
Divisible by 7
Divisible by 11
Divisible by 13
Covering set {3,7,13}
Difference-of-two-squares factorization
Aurifeuillean factorization of x4+4×y4
17Divisible by 2
Divisible by 3
Divisible by 5
Divisible by 7
Divisible by 11
Divisible by 13
Covering set {2,3}
Covering set {2,5,29}
Covering set {3,5,29}
Combine of divisible by 2 and difference-of-two-squares factorization
18Divisible by 5
Divisible by 7
Divisible by 11
Divisible by 13
Divisible by 17

Some x{y}z (where x and z are strings (may be empty) of digits in base b, y is a digit in base b) families could not be proven to contain no primes > b (by covering congruence, algebraic factorization, or combine of them) but no primes > b could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to test the primality of larger and larger numbers of such form and hope a prime is eventually discovered.

Many x{y}z (where x and z are strings (may be empty) of digits in base b, y is a digit in base b) families contain no small primes > b even though they do contain very large primes. e.g.

b
(2 ≤ b ≤ 36)
familyalgebraic formbase-b form of the smallest (probable) prime in this family (write "dn" if there are 5 or more (n) consecutive same digits d)length of the smallest (probable) prime in this familylength of the smallest (probable) prime in this family in decimalalgebraic ((a×bn+c)/gcd(a+c,b−1)) form of the smallest (probable) prime in this familyfactordb entry of the smallest (probable) prime in this familythe smallest (probable) prime in this family written in base b (use lower case letters instead of upper case letters)the smallest (probable) prime in this family written in decimalfactorization of the first 200 numbers in corresponding family (n is the number of digits in the "{}", start with the smallest n making the number > b (if n = 0 already makes the number > b, then start with n = 0))definitely prime or probable prime?the entry of the smallest (probable) prime in this family in the top definitely primes page or the top probable primes page
139{5}(113×13n−5)/1295197420197421219916(113×13197421−5)/12http://factordb.com/index.php?id=1100000003943359311&open=primehttp://factordb.com/index.php?showid=1100000003943359311&base=13http://factordb.com/index.php?showid=1100000003943359311http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showprobable primehttp://www.primenumbers.net/prptop/searchform.php?form=%28113*13%5E197420-5%29%2F12&action=Search
16{3}AF(16n+2+619)/53116137AF116139139845(16116139+619)/5http://factordb.com/index.php?id=1100000003851731988&open=primehttp://factordb.com/index.php?showid=1100000003851731988&base=16http://factordb.com/index.php?showid=1100000003851731988http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showprobable primehttp://www.primenumbers.net/prptop/searchform.php?form=%2816%5E116139%2B619%29%2F5&action=Search
239{E}(106×23n−7)/119E8008738008741090573(106×23800873−7)/11http://factordb.com/index.php?id=1100000000782858648&open=primehttp://factordb.com/index.php?showid=1100000000782858648&base=23http://factordb.com/index.php?showid=1100000000782858648http://factordb.com/index.php?query=%28106*23%5En-7%29%2F11&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showprobable primehttp://www.primenumbers.net/prptop/searchform.php?form=%28106*23%5E800873-7%29%2F11&action=Search
2571JD{0}1110488×25n+1+171JD04585491458554641031110488×25458550+1http://factordb.com/index.php?id=1100000002341496334&open=primehttp://factordb.com/index.php?showid=1100000002341496334&base=25http://factordb.com/index.php?showid=1100000002341496334http://factordb.com/index.php?query=110488*25%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showdefinitely prime, this number can be proven prime by N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since N−1 is trivially fully factoredhttps://t5k.org/primes/page.php?id=111834
32NU{0}1766×32n+1+1NU06618631661866996208766×32661864+1http://factordb.com/index.php?id=1100000003813355148&open=primehttp://factordb.com/index.php?showid=1100000003813355148&base=32http://factordb.com/index.php?showid=1100000003813355148http://factordb.com/index.php?query=766*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showdefinitely prime, this number can be proven prime by N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since N−1 is trivially fully factoredhttps://t5k.org/primes/page.php?id=134216
36{P}SZ(5×36n+2+821)/7P81993SZ81995127609(5×3681995+821)/7http://factordb.com/index.php?id=1100000002394962083&open=primehttp://factordb.com/index.php?showid=1100000002394962083&base=36http://factordb.com/index.php?showid=1100000002394962083http://factordb.com/index.php?query=%285*36%5E%28n%2B2%29%2B821%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showprobable primehttp://www.primenumbers.net/prptop/searchform.php?form=%285*36%5E81995%2B821%29%2F7&action=Search

(Technically, probable (https://en.wikipedia.org/wiki/Probabilistic_algorithm) primality tests (https://t5k.org/prove/prove2.html) were used to show these for the numbers which cannot be proven prime by N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) or N+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), i.e. for the ordinary primes (https://t5k.org/glossary/xpage/OrdinaryPrime.html) (which have a very small chance of making an error (https://t5k.org/notes/prp_prob.html, https://www.ams.org/journals/mcom/1989-53-188/S0025-5718-1989-0982368-4/S0025-5718-1989-0982368-4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_22.pdf))) because all known primality tests (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/prove3.html, https://t5k.org/prove/prove4.html) run far too slowly (the CPU time (https://en.wikipedia.org/wiki/CPU_time) is longer than the life expectancy of human (https://en.wikipedia.org/wiki/Life_expectancy) for numbers > 10100000, and longer than the age of the universe (https://en.wikipedia.org/wiki/Age_of_the_universe) for numbers > 10500000, and longer than one quettasecond (https://en.wikipedia.org/wiki/Quetta-) for numbers > 103000000, even if we can do 109 bitwise operations (https://en.wikipedia.org/wiki/Bitwise_operation) per second (https://en.wikipedia.org/wiki/Second) to run on these numbers, see https://www.mersenneforum.org/showpost.php?p=627117&postcount=1) to run on numbers of these sizes unless either N−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) or N+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) (or both) (unfortunately, none of Wikipedia, Prime Wiki, Mathworld has article for N−1 primality test or N+1 primality test, but a similar article for Pocklington primality test: https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, also see the article for the cyclotomy primality test: https://t5k.org/glossary/xpage/Cyclotomy.html) can be ≥ 1/3 factored (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) or can be ≥ 1/4 factored and the number is not very large (say not > 10100000), or Nn−1 can be ≥ 1/3 factored for a small n. If either N−1 or N+1 (or both) can be ≥ 1/2 factored, then we can use the Pocklington N−1 primality test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1) (the N−1 case) or the Morrison N+1 primality test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) (the N+1 case); if either N−1 or N+1 (or both) can be ≥ 1/3 factored, then we can use the Brillhart-Lehmer-Selfridge primality test (https://www.ams.org/journals/mcom/1975-29-130/S0025-5718-1975-0384673-1/S0025-5718-1975-0384673-1.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_23.pdf), https://en.wikipedia.org/wiki/Pocklington_primality_test#Extensions_and_variants); if either N−1 or N+1 (or both) can be ≥ 1/4 factored but neither can be ≥ 1/3 factored, then we need to use CHG (https://www.mersenneforum.org/attachment.php?attachmentid=21133&d=1571237465, https://t5k.org/bios/page.php?id=797, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/CHG) to prove its primality (see https://www.mersenneforum.org/showpost.php?p=528149&postcount=3 and https://www.mersenneforum.org/showpost.php?p=603181&postcount=438 and https://www.mersenneforum.org/showpost.php?p=277617&postcount=7), however, unlike Brillhart-Lehmer-Selfridge primality test for the numbers N such that N−1 or N+1 (or both) can be ≥ 1/3 factored can run for arbitrarily large numbers N (thus, there are no unproven probable primes N such that N−1 or N+1 (or both) can be ≥ 1/3 factored), CHG for the numbers N such that either N−1 or N+1 (or both) can be ≥ 1/4 factored but neither can be ≥ 1/3 factored cannot run for very large N (say > 10100000), for the examples of the numbers which are proven prime by CHG, see https://t5k.org/primes/page.php?id=134725, https://t5k.org/primes/page.php?id=126454, https://t5k.org/primes/page.php?id=131964, https://t5k.org/primes/page.php?id=132705, https://t5k.org/primes/page.php?id=132704, https://t5k.org/primes/page.php?id=134345, https://t5k.org/primes/page.php?id=135387, https://t5k.org/primes/page.php?id=121904, https://t5k.org/primes/page.php?id=118734, https://t5k.org/primes/page.php?id=137137, https://t5k.org/primes/page.php?id=136666, https://t5k.org/primes/page.php?id=123456, https://t5k.org/primes/page.php?id=130933, https://stdkmd.net/nrr/cert/1/ (search for "CHG"), https://stdkmd.net/nrr/cert/2/ (search for "CHG"), https://stdkmd.net/nrr/cert/3/ (search for "CHG"), https://stdkmd.net/nrr/cert/4/ (search for "CHG"), https://stdkmd.net/nrr/cert/5/ (search for "CHG"), https://stdkmd.net/nrr/cert/6/ (search for "CHG"), https://stdkmd.net/nrr/cert/7/ (search for "CHG"), https://stdkmd.net/nrr/cert/8/ (search for "CHG"), https://stdkmd.net/nrr/cert/9/ (search for "CHG"), http://xenon.stanford.edu/~tjw/pp/index.html (search for "CHG"), however, factordb (http://factordb.com/, https://www.rieselprime.de/ziki/Factoring_Database) lacks the ability to verify CHG proofs, see https://www.mersenneforum.org/showpost.php?p=608362&postcount=165; if neither N−1 nor N+1 can be ≥ 1/4 factored but Nn−1 can be ≥ 1/3 factored for a small n, then we can use the cyclotomy primality test (https://t5k.org/glossary/xpage/Cyclotomy.html, https://t5k.org/prove/prove3_3.html, https://t5k.org/primes/search.php?Comment=Cyclotomy&OnList=all&Number=1000000&Style=HTML, http://factordb.com/nmoverview.php?method=3)), i.e. it is too hard to prove primes for general numbers (https://t5k.org/glossary/xpage/OrdinaryPrime.html) of this size, but they are expected to be primes, since they are > 1025000 and the probability that they are in fact composite is < 10−2000, see https://t5k.org/notes/prp_prob.html and https://www.ams.org/journals/mcom/1989-53-188/S0025-5718-1989-0982368-4/S0025-5718-1989-0982368-4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_22.pdf).

The numbers in x{y}z (where x and z are strings (may be empty) of digits in base b, y is a digit in base b) families are of the form (a×bn+c)/gcd(a+c,b−1) for some fixed a, b, c such that a ≥ 1, b ≥ 2 (b is the base), c ≠ 0, gcd(a,c) = 1, gcd(b,c) = 1 (i.e. they are the near-Cunningham numbers (http://factordb.com/tables.php?open=4, https://oeis.org/wiki/OEIS_sequences_needing_factors#Near_powers.2C_factorials.2C_and_primorials (sections "near-powers with b = 2" and "near-powers with b = 3" and "near-powers with b = 5" and "near-powers with b = 6" and "near-powers with b = 7" and "near-powers with b = 10" and "near-powers with b > 10")) a×bn+c divided by the largest number which divides all of them (i.e. divides a×bn+c for all n)) and variable n (thus, all large minimal primes base b (but possible not all minimal primes base b if b and eulerphi(b) (where eulerphi is Euler's totient function (https://en.wikipedia.org/wiki/Euler%27s_totient_function, https://t5k.org/glossary/xpage/EulersPhi.html, https://mathworld.wolfram.com/TotientFunction.html, http://www.numericana.com/answer/modular.htm#phi, http://www.javascripter.net/math/calculators/eulertotientfunction.htm, https://oeis.org/A000010)) are both large, e.g. b = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35) have a nice short algebraic description (see https://t5k.org/lists/single_primes/36000bit.html and https://t5k.org/lists/single_primes/50005bit.html, the prime numbers in these two pages do not have nice short algebraic descriptions, also see http://primerecords.dk/primegaps/gaps20.htm) and have simple expression (expression (https://en.wikipedia.org/wiki/Expression_(mathematics), https://mathworld.wolfram.com/AlgebraicExpression.html) with ≤ 40 characters (https://en.wikipedia.org/wiki/Character_(computing), all "algebraic form" in this "README" file (as well as the "README" files in the "code", "primality-certificates", "unproven-probable-primes", i.e. https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/README.md and https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/primality-certificates/README.md and https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/unproven-probable-primes/README.md) are also simple expressions), all taken from "0" "1" "2" "3" "4" "5" "6" "7" "8" "9" "+" "-" "*" "/" "(" ")", i.e. all taken from the Arabic numerals (https://en.wikipedia.org/wiki/Arabic_numerals, https://mathworld.wolfram.com/ArabicNumeral.html) and addition (https://en.wikipedia.org/wiki/Addition, https://www.rieselprime.de/ziki/Addition, https://mathworld.wolfram.com/Addition.html) and subtraction (https://en.wikipedia.org/wiki/Subtraction, https://www.rieselprime.de/ziki/Subtraction, https://mathworld.wolfram.com/Subtraction.html) and multiplication (https://en.wikipedia.org/wiki/Multiplication, https://www.rieselprime.de/ziki/Multiplication, https://mathworld.wolfram.com/Multiplication.html) and division (https://en.wikipedia.org/wiki/Division_(mathematics), https://www.rieselprime.de/ziki/Division, https://mathworld.wolfram.com/Division.html) and exponentiation (https://en.wikipedia.org/wiki/Exponentiation, https://mathworld.wolfram.com/Exponentiation.html) and brackets (https://en.wikipedia.org/wiki/Bracket_(mathematics), https://mathworld.wolfram.com/Bracket.html)), factorial (!) (https://en.wikipedia.org/wiki/Factorial, https://t5k.org/glossary/xpage/Factorial.html, https://www.rieselprime.de/ziki/Factorial_number, https://mathworld.wolfram.com/Factorial.html, https://www.numbersaplenty.com/set/factorial/, https://oeis.org/A000142) and double factorial (!!) (https://en.wikipedia.org/wiki/Double_factorial, https://mathworld.wolfram.com/DoubleFactorial.html, https://www.numbersaplenty.com/set/double_factorial/, https://oeis.org/A006882) and primorial (#) (https://en.wikipedia.org/wiki/Primorial, https://t5k.org/glossary/xpage/Primorial.html, https://mathworld.wolfram.com/Primorial.html, https://www.numbersaplenty.com/set/primorial/, https://oeis.org/A002110) are not allowed since they can be used to ensure many small factors, see http://primerecords.dk/primegaps/gaps20.htm). Except in the special case (https://en.wikipedia.org/wiki/Special_case) c = ±1 and gcd(a+c,b−1) = 1 (since this special case c = ±1 and gcd(a+c,b−1) = 1 is the only case which N−1 or/and N+1 is trivially (https://en.wikipedia.org/wiki/Triviality_(mathematics), https://mathworld.wolfram.com/Trivial.html) fully factored (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) and is smooth (https://en.wikipedia.org/wiki/Smooth_number, https://mathworld.wolfram.com/SmoothNumber.html, https://oeis.org/A003586, https://oeis.org/A051037, https://oeis.org/A002473, https://oeis.org/A051038, https://oeis.org/A080197, https://oeis.org/A080681, https://oeis.org/A080682, https://oeis.org/A080683) (i.e. the greatest prime factor (http://mathworld.wolfram.com/GreatestPrimeFactor.html, https://oeis.org/A006530) of N−1 or/and N+1 is small) (c = 1 and gcd(a+c,b−1) = 1 if and only if the digit y is 0 and the string z is 1, and c = −1 and gcd(a+c,b−1) = 1 if and only if the digit y is b−1 and the string z is 𝜆 (the empty string (https://en.wikipedia.org/wiki/Empty_string)), if we reduce these families by removing all trailing digits y from x, and removing all leading digits y from z, to make the families be easier, e.g. family 12333{3}33345 in base b is reduced to family 12{3}45 in base b, since they are in fact the same family), i.e. the case c = 1 and gcd(a+c,b−1) = 1 (corresponding to generalized Proth prime (https://en.wikipedia.org/wiki/Proth_prime, https://t5k.org/glossary/xpage/ProthPrime.html, https://www.rieselprime.de/ziki/Proth_prime, https://mathworld.wolfram.com/ProthNumber.html, http://www.prothsearch.com/frequencies.html, http://www.prothsearch.com/history.html, https://www.rieselprime.de/Data/PStatistics.htm, https://www.rieselprime.de/Data/PRanges50.htm, https://www.rieselprime.de/Data/PRanges300.htm, https://www.rieselprime.de/Data/PRanges1200.htm, http://irvinemclean.com/maths/pfaq6.htm, https://www.numbersaplenty.com/set/Proth_number/, https://web.archive.org/web/20230706041914/https://pzktupel.de/Primetables/TableProthTOP10KK.php, https://web.archive.org/web/20231030081449/https://pzktupel.de/Primetables/ProthK.txt, https://pzktupel.de/Primetables/TableProthTOP10KS.php, https://pzktupel.de/Primetables/ProthS.txt, https://pzktupel.de/Primetables/TableProthGen.php, https://pzktupel.de/Primetables/TableProthGen.txt, https://sites.google.com/view/proth-primes, https://t5k.org/primes/search_proth.php, https://t5k.org/top20/page.php?id=66, https://www.primegrid.com/forum_thread.php?id=2665, https://www.primegrid.com/stats_321_llr.php, https://www.primegrid.com/stats_pps_llr.php, https://www.primegrid.com/stats_ppse_llr.php, https://www.primegrid.com/stats_mega_llr.php, https://www.primegrid.com/primes/primes.php?project=321&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=27&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=121&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=PPS&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=PPSE&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=MEG&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, http://boincvm.proxyma.ru:30080/test4vm/public/pps_dc_status.php, http://boincvm.proxyma.ru:30080/test4vm/user_profile/llr2_status.html, https://web.archive.org/web/20231223043356/https://www.mersenneforum.org/321search/index.html, https://web.archive.org/web/20110601231527/http://www.bodang.com/12121/, https://web.archive.org/web/20100518081012/http://www.bodang.com/12121/27k/, https://web.archive.org/web/20210415051133/http://prpnet.primegrid.com:12001/, https://web.archive.org/web/20220115151556/http://prpnet.primegrid.com:12006/, https://www.rieselprime.de/ziki/321_Search, https://www.rieselprime.de/ziki/12121_Search, https://www.rieselprime.de/ziki/27121_Search, https://www.rieselprime.de/ziki/PrimeGrid_321_Prime_Search, https://www.rieselprime.de/ziki/PrimeGrid_27121_Prime_Search, https://www.rieselprime.de/ziki/PrimeGrid_Proth_Prime_Search, https://www.rieselprime.de/ziki/PrimeGrid_Proth_Prime_Search_Extended, https://www.rieselprime.de/ziki/PrimeGrid_Proth_Mega_Prime_Search) base b: a×bn+1, they are related to generalized Sierpinski conjecture base b (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian))) can be easily proven prime using Pocklington N−1 method (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1), and the case c = −1 and gcd(a+c,b−1) = 1 (corresponding to generalized Riesel prime (https://www.rieselprime.de/ziki/Riesel_prime, https://www.rieselprime.de/Data/Statistics.htm, http://irvinemclean.com/maths/pfaq6.htm, https://web.archive.org/web/20230628151418/https://pzktupel.de/Primetables/TableRieselTOP10KK.php, https://web.archive.org/web/20231030081316/https://pzktupel.de/Primetables/RieselK.txt, https://pzktupel.de/Primetables/TableRieselTOP10KS.php, https://pzktupel.de/Primetables/RieselS.txt, https://pzktupel.de/Primetables/TableRieselGen.php, https://pzktupel.de/Primetables/TableRieselGen.txt, https://sites.google.com/view/proth-primes, https://www.primegrid.com/stats_321_llr.php, https://www.primegrid.com/primes/primes.php?project=321&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=27&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=121&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, http://www.noprimeleftbehind.net/stats/index.php?content=prime_list, https://t5k.org/primes/search_proth.php, http://www.noprimeleftbehind.net/prpnet/, http://www.noprimeleftbehind.net:9000/all.html, http://www.noprimeleftbehind.net:4000/all.html, http://www.noprimeleftbehind.net:2000/all.html, http://www.noprimeleftbehind.net:1468/all.html, http://www.noprimeleftbehind.net:1400/all.html, https://web.archive.org/web/20231223043356/https://www.mersenneforum.org/321search/index.html, https://web.archive.org/web/20110601231527/http://www.bodang.com/12121/, https://web.archive.org/web/20100518081012/http://www.bodang.com/12121/27k/, https://web.archive.org/web/20210415051133/http://prpnet.primegrid.com:12001/, https://web.archive.org/web/20220115151556/http://prpnet.primegrid.com:12006/, https://www.rieselprime.de/ziki/321_Search, https://www.rieselprime.de/ziki/12121_Search, https://www.rieselprime.de/ziki/27121_Search, https://www.rieselprime.de/ziki/PrimeGrid_321_Prime_Search, https://www.rieselprime.de/ziki/PrimeGrid_27121_Prime_Search, https://www.rieselprime.de/ziki/NPLB_Drive_17, https://www.rieselprime.de/ziki/NPLB_Drive_18, https://www.rieselprime.de/ziki/NPLB_Drive_19, https://www.rieselprime.de/ziki/NPLB_Drive_High-n) base b: a×bn−1, they are related to generalized Riesel conjecture base b (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.bitman.name/math/article/2005 (in Italian))) can be easily proven prime using Morrison N+1 method (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), these primes can be proven prime using Yves Gallot's Proth.exe (https://t5k.org/programs/gallot/, https://t5k.org/bios/page.php?id=411, https://www.rieselprime.de/ziki/Proth.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/proth), these primes can also be proven prime using Jean Penné's LLR (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64), see https://www.mersenneforum.org/showpost.php?p=611607&postcount=10 and https://www.mersenneforum.org/showpost.php?p=235113&postcount=8 and https://www.mersenneforum.org/showpost.php?p=541285&postcount=4 and https://www.mersenneforum.org/showpost.php?p=642861&postcount=2 and https://www.mersenneforum.org/showpost.php?p=538954&postcount=414 and https://www.mersenneforum.org/showpost.php?p=564758&postcount=428 and https://www.mersenneforum.org/showpost.php?p=586913&postcount=429 and https://www.mersenneforum.org/showpost.php?p=603149&postcount=437 and https://www.mersenneforum.org/showpost.php?p=605958&postcount=441 and http://csic.som.emory.edu/~lzhou/blogs/?p=186 and http://csic.som.emory.edu/~lzhou/blogs/?p=187 and http://csic.som.emory.edu/~lzhou/blogs/?p=190 and http://boincvm.proxyma.ru:30080/test4vm/user_profile/gfn12mega_hunt_status.html and http://boincvm.proxyma.ru:30080/test4vm/user_profile/gfn11mega_hunt_status.html, also see https://web.archive.org/web/20020809212051/http://www.users.globalnet.co.uk/~aads/C0034177.html and https://web.archive.org/web/20020701171455/http://www.users.globalnet.co.uk/~aads/C0066883.html and https://web.archive.org/web/20020809122706/http://www.users.globalnet.co.uk/~aads/C0071699.html and https://web.archive.org/web/20020809122635/http://www.users.globalnet.co.uk/~aads/C0101031.html and https://web.archive.org/web/20020809122237/http://www.users.globalnet.co.uk/~aads/C0114801.html and https://web.archive.org/web/20020809122947/http://www.users.globalnet.co.uk/~aads/C0130991.html and https://web.archive.org/web/20020809124216/http://www.users.globalnet.co.uk/~aads/C0131021.html and https://web.archive.org/web/20020809125049/http://www.users.globalnet.co.uk/~aads/C0131193.html and https://web.archive.org/web/20020809124458/http://www.users.globalnet.co.uk/~aads/C0152579.html and https://web.archive.org/web/20020809124537/http://www.users.globalnet.co.uk/~aads/C0220857.html and https://web.archive.org/web/20020809152611/http://www.users.globalnet.co.uk/~aads/C0315581.html and https://web.archive.org/web/20020809124929/http://www.users.globalnet.co.uk/~aads/C0351297.html and http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt (the bottom of the page, about the factorization of (7116384+1)/2−1 and (7116384+1)/2+1) and http://csic.som.emory.edu/~lzhou/blogs/?p=717 and http://csic.som.emory.edu/~lzhou/blogs/?p=263, also see the README file for LLR (https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/llr403win64/Readme.txt, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/llr403linux64/Readme.txt, http://jpenne.free.fr/index2.html), also see the (generalized) Proth/Riesel prime search page (https://t5k.org/primes/search_proth.php), i.e. there are no unproven probable primes N such that N−1 or/and N+1 is ≥ 1/3 factored (the Fermat number (https://en.wikipedia.org/wiki/Fermat_number, https://t5k.org/glossary/xpage/FermatNumber.html, https://www.rieselprime.de/ziki/Fermat_number, https://mathworld.wolfram.com/FermatNumber.html, https://mathworld.wolfram.com/FermatPrime.html, https://pzktupel.de/Primetables/TableFermat.php, http://www.prothsearch.com/fermat.html, https://t5k.org/top20/page.php?id=8, https://t5k.org/primes/search.php?Comment=Divides&OnList=all&Number=1000000&Style=HTML, http://www.fermatsearch.org/, https://64ordle.au/fermat/, https://64ordle.au/fermat/small/, https://www.primegrid.com/forum_thread.php?id=8778, https://www.primegrid.com/stats_div_llr.php, https://www.primegrid.com/primes/primes.php?project=DIV&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.rieselprime.de/ziki/PrimeGrid_Fermat_Divisor_Search, http://www.fermatsearch.org/factors/faclist.php, http://www.fermatsearch.org/factors/composite.php) F33 = 2233+1 (see http://www.prothsearch.com/fermat.html and http://www.fermatsearch.org/factors/faclist.php and http://www.fermatsearch.org/factors/composite.php and https://oeis.org/A093179 and https://oeis.org/A053576) and the double Mersenne number (https://en.wikipedia.org/wiki/Double_Mersenne_number, https://www.rieselprime.de/ziki/Double_Mersenne_number, https://mathworld.wolfram.com/DoubleMersenneNumber.html, http://www.doublemersennes.org/, https://oeis.org/A077586, https://oeis.org/A277634) MM61 = 2261−1−1 (see http://www.doublemersennes.org/mm61.php and http://www.hoegge.dk/mersenne/NMC.html and https://oeis.org/A263686 and https://oeis.org/A309130) are not "unproven probable primes" (http://factordb.com/listtype.php?t=1), they are "numbers with unknown status" (http://factordb.com/listtype.php?t=2), they are too large to be primality tested or probable-primality tested (just like the numbers 1010100+37 (see https://www.alpertron.com.ar/GOOGOL.HTM and https://oeis.org/A072288) and 1010100−57 (see https://www.alpertron.com.ar/GOOGOLM.HTM and https://oeis.org/A078814), although their N−1 and N+1 are not ≥ 1/3 factored (and they are only trial factored to 3.5×1014), unlike 2233+1 and 2261−1−1), and we can only do trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) to disprove their primality, F33 = 2233+1 is trial factored to (3.4×1017)×235+1 (see http://www.fermatsearch.org/stat/n.php) and MM61 = 2261−1−1 is trial factored to (2.8×1017)×(261−1)+1 (see http://www.doublemersennes.org/mm61.php)), you should know the difference of probable primes and definitely primes (see https://www.mersenneforum.org/showpost.php?p=651069&postcount=3 and https://www.mersenneforum.org/showpost.php?p=572047&postcount=239), you can compare the top definitely primes page (https://t5k.org/primes/lists/all.txt) and the top probable primes page (http://www.primenumbers.net/prptop/prptop.php), also you can compare the definitely primes with ≥ 100000 decimal digits in factordb (http://factordb.com/listtype.php?t=4&mindig=100000&perpage=5000&start=0) and the probable primes with ≥ 100000 decimal digits in factordb (http://factordb.com/listtype.php?t=1&mindig=100000&perpage=5000&start=0), http://factordb.com/nmoverview.php?method=1&digits=100000&perpage=500&skip=0 is the primes with ≥ 100000 decimal digits in factordb which are proven primes by the N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1), http://factordb.com/nmoverview.php?method=2&digits=100000&perpage=500&skip=0 is the primes with ≥ 100000 decimal digits in factordb which are proven primes by the N+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), also see https://web.archive.org/web/20240305200806/https://stdkmd.net/nrr/prime/primesize.txt and https://web.archive.org/web/20240305201054/https://stdkmd.net/nrr/prime/primesize.zip (see which numbers have "-proven" or "+proven" in the "note" column), also see https://stdkmd.net/nrr/prime/prime_all.htm and https://stdkmd.net/nrr/prime/prime_all.txt (see which numbers have "pr" in the "status" column), also see https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm (compare the sections "Prime numbers" and "Probable prime numbers")), when n is large the known primality tests (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/prove3.html, https://t5k.org/prove/prove4.html) for such a number are too inefficient to run (e.g. the prime 8957 in base b = 20 (although this prime is not minimal prime in base b = 20), its algebraic form is (161×2057−9)/19, neither N−1 nor N+1 is trivially fully factored, see https://www.mersenneforum.org/showthread.php?t=29763). In this case one must resort to a probable (https://en.wikipedia.org/wiki/Probabilistic_algorithm) primality test (https://t5k.org/prove/prove2.html) such as a Miller–Rabin primality test (https://t5k.org/prove/prove2_3.html, https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test, https://en.wikipedia.org/wiki/Strong_pseudoprime, https://t5k.org/glossary/xpage/MillersTest.html, https://t5k.org/glossary/xpage/StrongPRP.html, https://www.rieselprime.de/ziki/Miller-Rabin_pseudoprimality_test, https://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html, https://mathworld.wolfram.com/StrongPseudoprime.html, http://www.numericana.com/answer/pseudo.htm#rabin, http://www.numericana.com/answer/pseudo.htm#strong, http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm, http://ntheory.org/data/spsps.txt, https://faculty.lynchburg.edu/~nicely/misc/mpzspsp.html, https://miller-rabin.appspot.com/, http://www.pi-e.de/Miller-Rabin-Pseudoprimzahlen.htm (in German), http://factordb.com/prooffailed.php, https://sites.google.com/view/strong-pseudoprime, https://sites.google.com/view/bases-strong-pseudoprime, https://oeis.org/A001262, https://oeis.org/A020229, https://oeis.org/A020230, https://oeis.org/A020231, https://oeis.org/A020232, https://oeis.org/A020233, https://oeis.org/A020234, https://oeis.org/A020235, https://oeis.org/A020236, https://oeis.org/A020237, https://oeis.org/A020238, https://oeis.org/A020239, https://oeis.org/A020240, https://oeis.org/A020241, https://oeis.org/A020242, https://oeis.org/A020243, https://oeis.org/A020244, https://oeis.org/A020245, https://oeis.org/A020246, https://oeis.org/A020247, https://oeis.org/A020248, https://oeis.org/A020249, https://oeis.org/A020250, https://oeis.org/A020251, https://oeis.org/A020252, https://oeis.org/A020253, https://oeis.org/A020254, https://oeis.org/A020255, https://oeis.org/A020256, https://oeis.org/A020257, https://oeis.org/A020258, https://oeis.org/A020259, https://oeis.org/A020260, https://oeis.org/A020261, https://oeis.org/A020262, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773, https://oeis.org/A014233, https://oeis.org/A006945, https://oeis.org/A089825, https://oeis.org/A181782, https://oeis.org/A071294, https://oeis.org/A141768, https://oeis.org/A195328, https://oeis.org/A329759, https://oeis.org/A298756) or a Baillie–PSW primality test (https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test, https://mathworld.wolfram.com/Baillie-PSWPrimalityTest.html, http://pseudoprime.com/dopo.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_318.pdf)) (the Baillie–PSW primality test is the combine of the Miller–Rabin primality test (https://t5k.org/prove/prove2_3.html, https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test, https://en.wikipedia.org/wiki/Strong_pseudoprime, https://t5k.org/glossary/xpage/MillersTest.html, https://t5k.org/glossary/xpage/StrongPRP.html, https://www.rieselprime.de/ziki/Miller-Rabin_pseudoprimality_test, https://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html, https://mathworld.wolfram.com/StrongPseudoprime.html, http://www.numericana.com/answer/pseudo.htm#rabin, http://www.numericana.com/answer/pseudo.htm#strong, http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm, http://ntheory.org/data/spsps.txt, https://faculty.lynchburg.edu/~nicely/misc/mpzspsp.html, https://miller-rabin.appspot.com/, http://www.pi-e.de/Miller-Rabin-Pseudoprimzahlen.htm (in German), http://factordb.com/prooffailed.php, https://sites.google.com/view/strong-pseudoprime, https://sites.google.com/view/bases-strong-pseudoprime, https://oeis.org/A001262, https://oeis.org/A020229, https://oeis.org/A020230, https://oeis.org/A020231, https://oeis.org/A020232, https://oeis.org/A020233, https://oeis.org/A020234, https://oeis.org/A020235, https://oeis.org/A020236, https://oeis.org/A020237, https://oeis.org/A020238, https://oeis.org/A020239, https://oeis.org/A020240, https://oeis.org/A020241, https://oeis.org/A020242, https://oeis.org/A020243, https://oeis.org/A020244, https://oeis.org/A020245, https://oeis.org/A020246, https://oeis.org/A020247, https://oeis.org/A020248, https://oeis.org/A020249, https://oeis.org/A020250, https://oeis.org/A020251, https://oeis.org/A020252, https://oeis.org/A020253, https://oeis.org/A020254, https://oeis.org/A020255, https://oeis.org/A020256, https://oeis.org/A020257, https://oeis.org/A020258, https://oeis.org/A020259, https://oeis.org/A020260, https://oeis.org/A020261, https://oeis.org/A020262, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773, https://oeis.org/A014233, https://oeis.org/A006945, https://oeis.org/A089825, https://oeis.org/A181782, https://oeis.org/A071294, https://oeis.org/A141768, https://oeis.org/A195328, https://oeis.org/A329759, https://oeis.org/A298756) with base b = 2 and the strong Lucas primality test (https://en.wikipedia.org/wiki/Strong_Lucas_pseudoprime, https://mathworld.wolfram.com/StrongLucasPseudoprime.html, http://ntheory.org/data/slpsps-baillie.txt, http://www.hoegge.dk/lucasselfridgeprps.txt, https://oeis.org/A217255) with parameters P = 1 and Q = (1−D)/4, where D is the first number in the sequence 5, −7, 9, −11, 13, −15, 17, −19, ... such that the Jacobi symbol (https://en.wikipedia.org/wiki/Jacobi_symbol, https://t5k.org/glossary/xpage/JacobiSymbol.html, https://mathworld.wolfram.com/JacobiSymbol.html, http://www.numericana.com/answer/reciprocity.htm#legendre, http://math.fau.edu/richman/jacobi.htm, https://oeis.org/A110242, https://oeis.org/A110247, https://oeis.org/A157412) (D|N) = −1), unless a divisor of the number can be found. Since we are testing many numbers in an exponential sequence (https://en.wikipedia.org/wiki/Exponential_growth, https://mathworld.wolfram.com/ExponentialGrowth.html), it is possible to use a sieving process (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.primegrid.com/stats_psp_sieve.php, https://www.primegrid.com/stats_pps_sieve.php, https://www.primegrid.com/stats_sr5_sieve.php, https://www.primegrid.com/stats_cw_sieve.php, https://www.primegrid.com/sieving/rsp/) to find divisors rather than using trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172), the sieving process should remove the n such that (a×bn+c)/gcd(a+c,b−1) has small prime factors (say < 109) (i.e. the least prime factor (http://mathworld.wolfram.com/LeastPrimeFactor.html, https://oeis.org/A020639) of (a×bn+c)/gcd(a+c,b−1) is smaller than 109) (i.e. is not 109-rough (https://en.wikipedia.org/wiki/Rough_number, https://mathworld.wolfram.com/RoughNumber.html, https://oeis.org/A007310, https://oeis.org/A007775, https://oeis.org/A008364, https://oeis.org/A008365, https://oeis.org/A008366, https://oeis.org/A166061, https://oeis.org/A166063)) or/and has algebraic factors (e.g. difference-of-two-squares factorization (https://en.wikipedia.org/wiki/Difference_of_two_squares) and sum/difference-of-two-cubes factorization (https://en.wikipedia.org/wiki/Sum_of_two_cubes) and Sophie Germain's identity (https://en.wikipedia.org/wiki/Sophie_Germain%27s_identity, https://www.theoremoftheday.org/Binomial/GermainId/TotDGermainIdentity.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_478.pdf)) and difference-of-two-nth-powers factorization with n > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and sum/difference-of-two-nth-powers factorization with odd n > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and Aurifeuillean factorization (https://en.wikipedia.org/wiki/Aurifeuillean_factorization, https://www.rieselprime.de/ziki/Aurifeuillian_factor, https://mathworld.wolfram.com/AurifeuilleanFactorization.html, http://www.numericana.com/answer/numbers.htm#aurifeuille, https://web.archive.org/web/20231002141924/http://colin.barker.pagesperso-orange.fr/lpa/cycl_fac.htm, http://list.seqfan.eu/oldermail/seqfan/2017-March/017363.html, http://myfactorcollection.mooo.com:8090/source/cyclo.cpp, http://myfactorcollection.mooo.com:8090/LCD_2_199, http://myfactorcollection.mooo.com:8090/LCD_2_998, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/source/cyclo.cpp, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/LucasCD/LCD_2_199, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/LucasCD/LCD_2_998, https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean, https://www.unshlump.com/hcn/aurif.html, https://www.mersenneforum.org/showthread.php?t=10439, https://www.mersenneforum.org/showpost.php?p=515828&postcount=8, https://maths-people.anu.edu.au/~brent/pd/rpb135.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_97.pdf), https://www.ams.org/journals/mcom/2006-75-253/S0025-5718-05-01766-7/S0025-5718-05-01766-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_138.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb127.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_164.pdf), https://www.jams.jp/scm/contents/Vol-2-3/2-3-16.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_167.pdf), https://web.archive.org/web/20130702000532/http://xyyxf.at.tut.by/aurifeuillean.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_443.pdf))), like https://stdkmd.net/nrr/1/10003.htm#prime_period and https://stdkmd.net/nrr/3/30001.htm#prime_period and https://stdkmd.net/nrr/1/11113.htm#prime_period and https://stdkmd.net/nrr/3/31111.htm#prime_period and https://stdkmd.net/nrr/1/13333.htm#prime_period and https://stdkmd.net/nrr/3/33331.htm#prime_period and https://stdkmd.net/nrr/1/11131.htm#prime_period and https://stdkmd.net/nrr/1/13111.htm#prime_period and https://stdkmd.net/nrr/3/31333.htm#prime_period and https://stdkmd.net/nrr/3/33313.htm#prime_period and https://stdkmd.net/nrr/1/13331.htm#prime_period and https://stdkmd.net/nrr/3/31113.htm#prime_period (we should remove the n with these forms, e.g. 6×k+1, 6×k+4, 15×k+14, 16×k+3, 18×k+14, 21×k+19, 22×k+9, 28×k+13, 33×k+21, 34×k+2, ... for the family 1{0}3 in decimal (since the n with these forms will make the number either has small prime factors or has algebraic factors (or both)), and 20.74% of the n will remain in the sieve file of the family 1{0}3 in decimal (20.74% is the Nash weight (or difficulty) of the family 1{0}3 in decimal), if (and only if) the family can be proven to contain no primes > b (or only contain finitely many primes > b) (by covering congruence, algebraic factorization, or combine of them), then the sieve file will be empty, and the Nash weight (or difficulty) of such family is 0%, like https://stdkmd.net/nrr/9/91113.htm#prime_period and https://stdkmd.net/nrr/9/94449.htm#prime_period and https://stdkmd.net/nrr/9/95559.htm#prime_period).

To do this, we made use of Geoffrey Reynolds' SRSIEVE software (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, https://www.mersenneforum.org/showpost.php?p=631129&postcount=1, http://www.rieselprime.de/dl/CRUS_pack.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve_1.8.2, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve-other-programs, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2cl.exe). This program uses the baby-step giant-step (https://en.wikipedia.org/wiki/Baby-step_giant-step) algorithm (https://en.wikipedia.org/wiki/Algorithm, https://www.rieselprime.de/ziki/Algorithm, https://mathworld.wolfram.com/Algorithm.html) to find all primes p which divide a×bn+c where p and n lie in a specified range (https://en.wikipedia.org/wiki/Interval_(mathematics), https://mathworld.wolfram.com/Interval.html), by using discrete logarithm (https://en.wikipedia.org/wiki/Discrete_logarithm, https://mathworld.wolfram.com/DiscreteLogarithm.html, https://www.alpertron.com.ar/DILOG.HTM, http://www.numbertheory.org/php/discrete_log.html) to solve the equation (https://en.wikipedia.org/wiki/Equation, https://mathworld.wolfram.com/Equation.html) a×bn+c == 0 mod p (i.e. solve the equation (https://en.wikipedia.org/wiki/Equation, https://mathworld.wolfram.com/Equation.html) a×bn+c = 0 in the finite field (https://en.wikipedia.org/wiki/Finite_field, https://mathworld.wolfram.com/FiniteField.html) Zp) (also, this program was updated so that it also removes the n such that a×bn+c has algebraic factors (e.g. difference-of-two-squares factorization (https://en.wikipedia.org/wiki/Difference_of_two_squares) and sum/difference-of-two-cubes factorization (https://en.wikipedia.org/wiki/Sum_of_two_cubes) and Sophie Germain's identity (https://en.wikipedia.org/wiki/Sophie_Germain%27s_identity, https://www.theoremoftheday.org/Binomial/GermainId/TotDGermainIdentity.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_478.pdf)) and difference-of-two-nth-powers factorization with n > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and sum/difference-of-two-nth-powers factorization with odd n > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and Aurifeuillean factorization (https://en.wikipedia.org/wiki/Aurifeuillean_factorization, https://www.rieselprime.de/ziki/Aurifeuillian_factor, https://mathworld.wolfram.com/AurifeuilleanFactorization.html, http://www.numericana.com/answer/numbers.htm#aurifeuille, https://web.archive.org/web/20231002141924/http://colin.barker.pagesperso-orange.fr/lpa/cycl_fac.htm, http://list.seqfan.eu/oldermail/seqfan/2017-March/017363.html, http://myfactorcollection.mooo.com:8090/source/cyclo.cpp, http://myfactorcollection.mooo.com:8090/LCD_2_199, http://myfactorcollection.mooo.com:8090/LCD_2_998, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/source/cyclo.cpp, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/LucasCD/LCD_2_199, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/LucasCD/LCD_2_998, https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean, https://www.unshlump.com/hcn/aurif.html, https://www.mersenneforum.org/showthread.php?t=10439, https://www.mersenneforum.org/showpost.php?p=515828&postcount=8, https://maths-people.anu.edu.au/~brent/pd/rpb135.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_97.pdf), https://www.ams.org/journals/mcom/2006-75-253/S0025-5718-05-01766-7/S0025-5718-05-01766-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_138.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb127.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_164.pdf), https://www.jams.jp/scm/contents/Vol-2-3/2-3-16.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_167.pdf), https://web.archive.org/web/20130702000532/http://xyyxf.at.tut.by/aurifeuillean.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_443.pdf))), see https://www.mersenneforum.org/showpost.php?p=452132&postcount=66 and https://www.mersenneforum.org/showpost.php?p=451337&postcount=32 and https://www.mersenneforum.org/showpost.php?p=208852&postcount=227 and https://www.mersenneforum.org/showpost.php?p=232904&postcount=604 and https://www.mersenneforum.org/showthread.php?t=21916 and https://www.mersenneforum.org/showpost.php?p=383690&postcount=1 and https://www.mersenneforum.org/showpost.php?p=207886&postcount=253 and https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445 and https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/srsieve_1.1.4/algebraic.c (note: for the sequence (a×bn+c)/gcd(a+c,b−1), the case of "Mersenne number" in https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/srsieve_1.1.4/algebraic.c is the case which a is rational power of b, c = −1 and the case which a is rational power of b, c = 1, gcd(a+c,b−1) ≥ 3, and the case of "GFN" in https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/srsieve_1.1.4/algebraic.c is the case which a is rational power of b, c = 1, gcd(a+c,b−1) is either 1 or 2)). Since this program cannot handle the general case (a×bn+c)/gcd(a+c,b−1) when gcd(a+c,b−1) > 1 we only used it to sieve the sequence a×bn+c for primes p not dividing gcd(a+c,b−1), and initialized the list of candidates to not include n for which there is some prime p dividing gcd(a+c,b−1) for which p dividing (a×bn+c)/gcd(a+c,b−1). The program had to be modified slightly to remove a check which would prevent it from running in the case when a, b, and c were all odd (since then 2 divides a×bn+c, but 2 may not divide (a×bn+c)/gcd(a+c,b−1)) (see https://github.com/curtisbright/mepn-data/commit/1b55b353f46c707bbe52897573914128b3303960).

(Edit: Now the SRSIEVE in MTSIEVE (https://sourceforge.net/projects/mtsieve/, https://www.mersenneforum.org/rogue/mtsieve.html, https://download.mersenne.ca/mtsieve, https://t5k.org/bios/page.php?id=449, https://www.rieselprime.de/ziki/Mtsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/mtsieve_2.4.8) can handle the general case (a×bn+c)/gcd(a+c,b−1) when gcd(a+c,b−1) > 1, see https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/CHANGES.txt, thus now we can sieve the sequence (a×bn+c)/gcd(a+c,b−1) directly)

When sieving the sequence (a×bn+c)/gcd(a+c,b−1) (a ≥ 1, b ≥ 2, c ≠ 0, gcd(a,c) = 1, gcd(b,c) = 1), the sieve program should: (below, r is a linear function of n, m is a constant like a, b, c (m ≥ 2))

  1. General:
  1. Remove all n cases:
  • 2.1. If a, b, −c are all squares; remove all n.
  • 2.2. If a, b, c are all r-th powers for an odd r > 1; remove all n.
  • 2.3. If one of a and c is also a 4th power, and the other is of the form 4×m4, and b is also a 4th power; remove all n. These are Aurifeuillean factors.
  1. Remove partial n cases:
  • 3.1. If a and −c are both squares; remove all n == 0 mod 2.
  • 3.2. If a and c are both r-th powers for an odd r > 1; for each such r, remove all n == 0 mod r.
  • 3.3. If one of a and c is also a 4th power, and the other is of the form 4×m4; remove all n == 0 mod 4.
  • 3.4. If one of a and c is also a 4th power, and the other is of the form 4×m4, and b is a square; remove all n == 0 mod 2.
  • 3.5. If a, c, 4×b are all 4th powers; remove all n == 1 mod 2.
  • 3.6. If a and c are both 4th powers, and 2×b is a square; remove all n == 2 mod 4.

#1 and #2 should all be checked first before preceding, #3.3 and #3.4 and #3.5 are more Aurifeuillean factors.

The sequences (mr−1)/(m−1), (mr+1)/(m+1), mr+1 (with even m), (mr+1)/2 (with odd m) are special examples, since they are Lucas sequences (https://en.wikipedia.org/wiki/Lucas_sequence, https://mathworld.wolfram.com/LucasSequence.html, https://t5k.org/top20/page.php?id=23, https://t5k.org/primes/search.php?Comment=Generalized%20Lucas%20number&OnList=all&Number=1000000&Style=HTML) like the Fibonacci numbers (https://en.wikipedia.org/wiki/Fibonacci_number, https://t5k.org/glossary/xpage/FibonacciNumber.html, https://mathworld.wolfram.com/FibonacciNumber.html, https://www.numbersaplenty.com/set/Fibonacci_number/, https://t5k.org/top20/page.php?id=39, https://t5k.org/primes/search.php?Comment=^Fibonacci%20number&OnList=all&Number=1000000&Style=HTML, https://pzktupel.de/Primetables/TableFibonacci.php, https://oeis.org/A000045, https://oeis.org/A005478, https://oeis.org/A001605) and the Lucas numbers (https://en.wikipedia.org/wiki/Lucas_number, https://t5k.org/glossary/xpage/LucasNumber.html, https://mathworld.wolfram.com/LucasNumber.html, https://www.numbersaplenty.com/set/Lucas_number/, https://t5k.org/top20/page.php?id=48, https://t5k.org/primes/search.php?Comment=^Lucas%20number&OnList=all&Number=1000000&Style=HTML, https://pzktupel.de/Primetables/TableLucas.php, https://oeis.org/A000032, https://oeis.org/A000204, https://oeis.org/A005479, https://oeis.org/A001606), (mr−1)/(m−1) is the Lucas sequence Ur(m+1,m), (mr+1)/(m+1) is the Lucas sequence Ur(m−1,−m), mr+1 (with even m) is the Lucas sequence Vr(m+1,m), (mr+1)/2 (with odd m) is half of the Lucas sequence Vr(m+1,m) (this Lucas sequence V only contains even numbers), thus they have the divisibility properties of the Lucas sequences like the Fibonacci numbers and the Lucas numbers (which are the Lucas sequences Un(1,−1) and Vn(1,−1), respectively) (another example of "half of the Lucas sequence V" is the sequence https://oeis.org/A001333, which is half of the Lucas sequence Vr(2,−1) (this Lucas sequence V also only contains even numbers), it has the divisibility property as the Lucas sequences V), i.e. they are strong divisibility sequences (https://en.wikipedia.org/wiki/Divisibility_sequence), (mr−1)/(m−1) and (mr+1)/(m+1) can be prime if r is a prime, mr+1 (with even m) and (mr+1)/2 (with odd m) can be prime if r is a power of 2, since otherwise mr±1 is a binomial number (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) which can be factored algebraically (and this algebraic factorization is nontrivial), thus they cannot be ruled out as only containing composites (only count the numbers > b) (by covering congruence, algebraic factorization, or combine of them) unless there are other algebraic factorization of them, like the case of (9n−1)/8 with n ≥ 2 (the family {1} in base b = 9, which has difference-of-two-squares factorization) and 8n+1+1 with n ≥ 1 (the family 1{0}1 in base b = 8, which has sum-of-two-cubes factorization).

(these are exactly the n such that (a×bn+c)/gcd(a+c,b−1) has algebraic factorization, and (a×bn+c)/gcd(a+c,b−1) can be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them) if and only if there is a prime p such that there is no n satisfies these two conditions simultaneously: "(a×bn+c)/gcd(a+c,b−1) does not have algebraic factorization" and "(a×bn+c)/gcd(a+c,b−1) is a p-rough number" (if and only if (a×bn+c)/gcd(a+c,b−1) can be proven to only contain composites or only contain finitely many primes by covering congruence, then there is a prime p such that there is no n such that (a×bn+c)/gcd(a+c,b−1) is a p-rough number, if and only if (a×bn+c)/gcd(a+c,b−1) can be proven to only contain composites or only contain finitely many primes by algebraic factorization, then (a×bn+c)/gcd(a+c,b−1) has algebraic factorization for all n, and if and only if (a×bn+c)/gcd(a+c,b−1) can be proven to only contain composites or only contain finitely many primes by combine of covering congruence and algebraic factorization, then there is a prime p such that there is no n satisfies these two conditions simultaneously: "(a×bn+c)/gcd(a+c,b−1) does not have algebraic factorization" and "(a×bn+c)/gcd(a+c,b−1) is a p-rough number"), thus, if and only if (a×bn+c)/gcd(a+c,b−1) can be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them), then the sieve file of (a×bn+c)/gcd(a+c,b−1) can be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them) will be empty)

(for the factorization of the numbers in these families, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=snfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#smallpolynomial, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=gnfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) may be used, they have SNFS polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt and https://alfredreichlg.de/10w7/10w7.txt and https://web.archive.org/web/20020320010222/http://proth.cjb.net/ and http://web.archive.org/web/20111104173105/http://www.immortaltheory.com/NumberTheory/2nm3_db.txt and https://www.asahi-net.or.jp/~KC2H-MSM/mathland/aba/index.htm and https://www.alpertron.com.ar/BRILLIANT.HTM and https://www.alpertron.com.ar/BRILLIANT3.HTM and https://www.alpertron.com.ar/BRILLIANT4.HTM and https://www.alpertron.com.ar/BRILLIANT2.HTM and https://oeis.org/wiki/Factors_of_33*2%5En%2B1 and https://oeis.org/wiki/Factors_of_33*2%5En-1 and https://web.archive.org/web/20111018190410/http://www.sr5.psp-project.de/s5stats.html (section "Top ten factors:") and https://web.archive.org/web/20111018190339/http://www.sr5.psp-project.de/r5stats.html (section "Top ten factors:") and https://brnikat.com/nums/cullen_woodall/cw.html and https://oeis.org/wiki/OEIS_sequences_needing_factors#Near_powers.2C_factorials.2C_and_primorials (sections "near-powers with b = 2" and "near-powers with b = 3" and "near-powers with b = 5" and "near-powers with b = 6" and "near-powers with b = 7" and "near-powers with b = 10" and "near-powers with b > 10") and https://www.mersenneforum.org/showthread.php?t=9554 and https://www.mersenneforum.org/showthread.php?t=9167 and https://www.mersenneforum.org/showpost.php?p=644144&postcount=5, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm and https://www.mersenneforum.org/showthread.php?t=12962)

(link to the factorization of the first 200 numbers in these families (start with the smallest number in these families which is > b), to show the small prime factors (and their period) (i.e. show whether these numbers are rough numbers) and the algebraic factors (and their period) and let you feel the Nash weight (or difficulty) of these families (high or low (or zero if the sieve program should remove all n)), all small prime factors (< 1012, by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172)) and all algebraic factors (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/53335.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/83336.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=algebraic&maxrows=10000, https://sites.google.com/view/algebraic-factors-of-xn-kyn, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://www.mersenneforum.org/showpost.php?p=96560&postcount=99, https://www.mersenneforum.org/showpost.php?p=96651&postcount=101, https://www.mersenneforum.org/showthread.php?t=21916, https://www.mersenneforum.org/showpost.php?p=196598&postcount=492, https://www.mersenneforum.org/showpost.php?p=203083&postcount=149, https://www.mersenneforum.org/showpost.php?p=206065&postcount=192, https://www.mersenneforum.org/showpost.php?p=208044&postcount=260, https://www.mersenneforum.org/showpost.php?p=210533&postcount=336, https://www.mersenneforum.org/showpost.php?p=452132&postcount=66, https://www.mersenneforum.org/showpost.php?p=451337&postcount=32, https://www.mersenneforum.org/showpost.php?p=208852&postcount=227, https://www.mersenneforum.org/showpost.php?p=232904&postcount=604, https://www.mersenneforum.org/showpost.php?p=383690&postcount=1, https://www.mersenneforum.org/showpost.php?p=207886&postcount=253, https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php, https://www.alpertron.com.ar/POLFACT.HTM, https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) of the first 200 numbers (start with the smallest n making the number > b (if n = 0 already makes the number > b, then start with n = 0)) in these families were added to factordb)

For examples:

b
(2 ≤ b ≤ 36)
familyalgebraic ((a×bn+c)/gcd(a+c,b−1)) form of the familythe sieve program shouldreasonthis family corresponding tofactorization of the first 200 numbers in this family (n is the number of digits in the "{}", start with the smallest n making the number > b (if n = 0 already makes the number > b, then start with n = 0))
35{1}(35n−1)/34display a warning message on the screen that this form is a generalized repunit number and could better be factored algebraically or sieved with another program (remove all n such that n is composite, and only sieve with the primes p == 1 mod n)this form can be written as (35n−1)/341313 (the ?th minimal prime in base 35)http://factordb.com/index.php?query=%2835%5En-1%29%2F34&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
274{D}(9×27n−1)/2display a warning message on the screen that this form is a generalized repunit number and could better be factored algebraically or sieved with another program (remove all n such that 3×n+2 is composite, and only sieve with the primes p == 1 mod 3×n+2)this form can be written as (3n+2−1)/24D23 (the 99696th minimal prime in base 27)http://factordb.com/index.php?query=%289*27%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
321{V}2×32n−1display a warning message on the screen that this form is a generalized repunit number and could better be factored algebraically or sieved with another program (remove all n such that 5×n+1 is composite, and only sieve with the primes p == 1 mod 5×n+1)this form can be written as 2n+1−11V6 (the 72855th minimal prime in base 32)http://factordb.com/index.php?query=2*32%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36{U}V(6×36n+1)/7display a warning message on the screen that this form is a generalized Wagstaff number and could better be factored algebraically or sieved with another program (remove all n such that 2×n+3 is composite, and only sieve with the primes p == 1 mod 2×n+3)this form can be written as (6n+3+1)/7U4V (the 12765th minimal prime in base 36)http://factordb.com/index.php?query=%286*36%5E%28n%2B1%29%2B1%29%2F7&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
324{0}14×32n+1+1display a warning message on the screen that this form is a generalized Fermat number and could better be factored algebraically or sieved with another program (remove all n such that 5×n+7 is not power of 2, and no need to sieve, just look for n = (2s+1−7)/5 (2r == 7 mod 5 if and only if r == 1 mod 4), and instead use trial division with the primes p == 1 mod 10×n+14)this form can be written as 2n+7+1unsolved familyhttp://factordb.com/index.php?query=4*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
32G{0}116×32n+1+1display a warning message on the screen that this form is a generalized Fermat number and could better be factored algebraically or sieved with another program (remove all n such that 5×n+9 is not power of 2, and no need to sieve, just look for n = (2s+2−9)/5 (2r == 9 mod 5 if and only if r == 2 mod 4), and instead use trial division with the primes p == 1 mod 10×n+18)this form can be written as 2n+9+1unsolved familyhttp://factordb.com/index.php?query=16*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
31{F}G(31n+1+1)/2display a warning message on the screen that this form is a generalized half Fermat number and could better be factored algebraically or sieved with another program (remove all n such that n+1 is not power of 2, and no need to sieve, just look for n = 2s−1, and instead use trial division with the primes p == 1 mod 2×n+2)this form can be written as (31n+1+1)/2unsolved familyhttp://factordb.com/index.php?query=%2831%5E%28n%2B1%29%2B1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
274{D}E(9×27n+1+1)/2display a warning message on the screen that this form is a generalized half Fermat number and could better be factored algebraically or sieved with another program (remove all n such that 3×n+5 is not power of 2, and no need to sieve, just look for n = (2s+1−5)/3 (2r == 5 mod 3 if and only if r == 1 mod 2), and instead use trial division with the primes p == 1 mod 6×n+10)this form can be written as (3n+5+1)/24D10E (the 88466th minimal prime in base 27)http://factordb.com/index.php?query=%289*27%5E%28n%2B1%29%2B1%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
93{8}4×9n−1remove all n4, 9, 1 are all squaresruled out as only contain composites (only count the numbers > b)http://factordb.com/index.php?query=4*9%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
93{1}(25×9n−1)/8remove all n25, 9, 1 are all squaresruled out as only contain composites (only count the numbers > b)http://factordb.com/index.php?query=%2825*9%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
168{F}9×16n−1remove all n9, 16, 1 are all squaresruled out as only contain composites (only count the numbers > b)http://factordb.com/index.php?query=9*16%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
278{0}18×27n+1+1remove all n8, 27, 1 are all cubesruled out as only contain composites (only count the numbers > b)http://factordb.com/index.php?query=8*27%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
279{G}(125×27n−8)/13remove all n125, 27, −8 are all cubesruled out as only contain composites (only count the numbers > b)http://factordb.com/index.php?query=%28125*27%5En-8%29%2F13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16{C}DD(4×16n+2+81)/5remove all n16 and 4×4×81 are both 4th powersruled out as only contain composites (only count the numbers > b)http://factordb.com/index.php?query=%284*16%5E%28n%2B2%29%2B81%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
8{4}1(4×8n+1−25)/7remove all n such that n+1 == 0 mod 2 (i.e. remove all n == 1 mod 2)4 and 25 are both squares481 (the 70th minimal prime in base 8)http://factordb.com/index.php?query=%284*8%5E%28n%2B1%29-25%29%2F7&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
20{G}99(16×20n+2−2809)/19remove all n such that n+2 == 0 mod 2 (i.e. remove all n == 0 mod 2)16 and 2809 are both squaresG44799 (the 3307th minimal prime in base 20)http://factordb.com/index.php?query=%2816*20%5E%28n%2B2%29-2809%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
22{7}2L(22n+2−289)/3remove all n such that n+2 == 0 mod 2 (i.e. remove all n == 0 mod 2)1 and 289 are both squares738152L (the 8002nd minimal prime in base 22)http://factordb.com/index.php?query=%2822%5E%28n%2B2%29-289%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
232E{B}(121×23n−1)/2remove all n == 0 mod 2121 and 1 are both squares2EB29583 (the 65156th minimal prime in base 23)http://factordb.com/index.php?query=%28121*23%5En-1%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26O{5}(121×26n−1)/5remove all n == 0 mod 2121 and 1 are both squaresO51509 (the 25235th minimal prime in base 26)http://factordb.com/index.php?query=%28121*26%5En-1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
35{Y}V35n+1−4remove all n such that n+1 == 0 mod 2 (i.e. remove all n == 1 mod 2)1 and 4 are both squaresY12V (the ?th minimal prime in base 35)http://factordb.com/index.php?query=35%5E%28n%2B1%29-4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
51{0}135n+2+8remove all n such that n+2 == 0 mod 3 (i.e. remove all n == 1 mod 3)1 and 8 are both cubes109313 (the 22nd minimal prime in base 5)http://factordb.com/index.php?query=5%5E%28n%2B2%29%2B8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
238{0}18×23n+1+1remove all n such that n+1 == 0 mod 3 (i.e. remove all n == 2 mod 3)8 and 1 are both cubesunsolved familyhttp://factordb.com/index.php?query=8*23%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
171{0}1F17n+2+32remove all n such that n+2 == 0 mod 5 (i.e. remove all n == 3 mod 5)1 and 32 are both 5th powers1090191F (the 10400th minimal prime in base 17)http://factordb.com/index.php?query=17%5E%28n%2B2%29%2B32&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1779{0}1128×17n+1+1remove all n such that n+1 == 0 mod 7 (i.e. remove all n == 6 mod 7)128 and 1 are both 7th powers7902241 (the 10307th minimal prime in base 17)http://factordb.com/index.php?query=128*17%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
234{0}14×23n+1+1remove all n such that n+1 == 0 mod 4 (i.e. remove all n == 3 mod 4)4×4×1 is a 4th power403411 (the 64770th minimal prime in base 23)http://factordb.com/index.php?query=4*23%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36{S}T(4×36n+1+1)/5remove all n such that n+1 == 0 mod 2 (i.e. remove all n == 1 mod 2)4×4×1 is a 4th power and 36 is a squareS44T (the 35018th minimal prime in base 36)http://factordb.com/index.php?query=%284*36%5E%28n%2B1%29%2B1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show

Also the forms satisfying both "General" and "Remove all n cases", in this case the program should not display a warning message on the screen and instead stop immediately (just like the forms only satisfying "Remove all n cases"):

b
(2 ≤ b ≤ 36)
familyalgebraic ((a×bn+c)/gcd(a+c,b−1)) form of the familythe sieve program shouldreasonthis family corresponding tofactorization of the first 200 numbers in this family (n is the number of digits in the "{}", start with the smallest n making the number > b (if n = 0 already makes the number > b, then start with n = 0))
9{1}(9n−1)/8remove all n1, 9, 1 are all squaresruled out as only contain composites (only count the numbers > b)http://factordb.com/index.php?query=%289%5En-1%29%2F8&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
161{5}(4×16n−1)/3remove all n4, 16, 1 are all squaresruled out as only contain composites (only count the numbers > b)http://factordb.com/index.php?query=%284*16%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25{1}(25n−1)/24remove all n1, 25, 1 are all squaresruled out as only contain composites (only count the numbers > b)http://factordb.com/index.php?query=%2825%5En-1%29%2F24&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
32{1}(32n−1)/31remove all n1, 32, 1 are all 5th powersruled out as only contain composites (only count the numbers > b)http://factordb.com/index.php?query=%2832%5En-1%29%2F31&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16{C}D(4×16n+1)/5remove all n16 and 4×4×1 are both 4th powersruled out as only contain composites (only count the numbers > b)http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29%2B1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
81{0}18n+1+1remove all n1, 8, 1 are all cubesruled out as only contain composites (only count the numbers > b)http://factordb.com/index.php?query=8%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
27{D}E(27n+1+1)/2remove all n1, 27, 1 are all cubesruled out as only contain composites (only count the numbers > b)http://factordb.com/index.php?query=%2827%5E%28n%2B1%29%2B1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
321{0}132n+1+1remove all n1, 32, 1 are all 5th powersruled out as only contain composites (only count the numbers > b)http://factordb.com/index.php?query=32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show

Coordination with existing code:

  • If all n are removed by algebraic factors for sequence (a×bn+c)/gcd(a+c,b−1), program should stop immediately. This means that this family can be proven to only contain composites.
  • If some n are removed by algebraic factors for sequence (a×bn+c)/gcd(a+c,b−1), program continues sieving for removing the numbers with small prime factors.
  • Program should be able to handle input of one or multiple sequences (a×bn+c)/gcd(a+c,b−1) at the screen or in a file. Some sequences (a×bn+c)/gcd(a+c,b−1) could have algebraic factors while others do not.
  • Program should be able to handle an already sieved file as input, check the file for algebraic factors, remove them, and then continue sieving more deeply. Once again some sequences (a×bn+c)/gcd(a+c,b−1) could have algebraic factors while others do not.

Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software LLR by Jean Penné (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64) or PFGW (https://sourceforge.net/projects/openpfgw/, https://t5k.org/bios/page.php?id=175, https://www.rieselprime.de/ziki/PFGW, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/pfgw_win_4.0.3). Although undocumented, it is possible to run these two programs on numbers of the form (a×bn+c)/d when d > 1 (however, of course, the numbers (a×bn+c)/d with |c| ≠ 1 or/and d ≠ 1 or/and a > bn can only be probable primes (https://en.wikipedia.org/wiki/Probable_prime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Probable_prime, https://mathworld.wolfram.com/ProbablePrime.html, http://www.primenumbers.net/prptop/prptop.php, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#probableprimenumbers, https://stdkmd.net/nrr/repunit/prpfactors.htm, https://www.mersenne.ca/prp.php?show=1, https://www.alfredreichlg.de/10w7/prp.html, http://factordb.com/listtype.php?t=1) tested; the numbers a×2n±1 (with a < 2n) are the fastest to test, a×2n+1 numbers are tested using the Proth algorithm (https://en.wikipedia.org/wiki/Proth%27s_theorem, https://www.rieselprime.de/ziki/Proth%27s_theorem, https://mathworld.wolfram.com/ProthsTheorem.html, http://www.numericana.com/answer/primes.htm#proth), a×2n−1 numbers are tested using the Lucas-Lehmer-Riesel algorithm (https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer%E2%80%93Riesel_test); the numbers a×bn±1 (with b > 2, a < bn) can also be definitely prime (https://en.wikipedia.org/wiki/Provable_prime, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#primenumbers, http://factordb.com/listtype.php?t=4) tested, a×bn+1 numbers are tested using the N−1 Pocklington algorithm (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1), a×bn−1 numbers are tested using the N+1 Morrison algorithm (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2)), so this program required no modifications. A script was also written which allowed one to run srsieve while LLR or PFGW was testing the remaining candidates, so that when a divisor was found by srsieve on a number which had not yet been tested by LLR or PFGW it would be removed from the list of candidates.

For the primes < 1025000 for the "easy" bases (bases b with ≤ 150 minimal primes > 10299 (base b = 26 has 82 known minimal (probable) primes > 10299 and 4 unsolved families, base b = 36 has 75 known minimal (probable) primes > 10299 and 4 unsolved families, base b = 17 has 99 known minimal (probable) primes > 10299 and 18 unsolved families, base b = 21 has 80 known minimal (probable) primes > 10299 and 12 unsolved families, base b = 19 has 201 known minimal (probable) primes > 10299 and 23 unsolved families), i.e. bases b = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36), we employed PRIMO by Marcel Martin (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309) or CM by Andreas Enge (https://www.multiprecision.org/cm/index.html, https://t5k.org/bios/page.php?id=5485, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/cm), two different elliptic curve primality proving (https://t5k.org/prove/prove4_2.html, https://en.wikipedia.org/wiki/Elliptic_curve_primality, https://t5k.org/glossary/xpage/ECPP.html, https://mathworld.wolfram.com/EllipticCurvePrimalityProving.html, http://irvinemclean.com/maths/pfaq7.htm, https://t5k.org/top20/page.php?id=27, https://t5k.org/primes/search.php?Comment=ECPP&OnList=all&Number=1000000&Style=HTML, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/~morain/Prgms/ecpp.english.html, https://www.multiprecision.org/cm/ecpp.html, https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1199989-X/S0025-5718-1993-1199989-X.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_256.pdf), https://arxiv.org/pdf/2404.05506.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_428.pdf)) implementations, to compute primality certificates (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://alfredreichlg.de/cert/certificates.tpm.html, https://alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php, http://5.199.134.130/certificates.tar.xz, http://5.199.134.130/certificates.tar.xz.SHA256SUM, http://5.199.134.130/certificates.tar.xz.par2, http://5.199.134.130/certificates.tar.xz.vol00+10.par2, http://5.199.134.130/certificates/) for the candidates for minimal prime base b which are > 10299 and neither N−1 nor N+1 can be ≥ 1/3 factored (need CHG proof if either N−1 or N+1 (or both) can be ≥ 1/4 factored but neither can be ≥ 1/3 factored, but factordb (http://factordb.com/, https://www.rieselprime.de/ziki/Factoring_Database) lacks the ability to verify CHG proofs, see https://www.mersenneforum.org/showpost.php?p=608362&postcount=165).

(sorry, I do not give the sieve files of this problem in the pages, for the examples of the sieve files, see http://www.noprimeleftbehind.net/crus/sieve-sierp-base3-30G-40G-50K-100K.zip (Sierpinski problem base 3, k = 30000000000 to 40000000000, n = 50000 to 100000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base10-5M-10M.txt (Sierpinski problem base 10, n = 5000000 to 10000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base16-1M-2M.zip (Sierpinski problem base 16, n = 1000000 to 2000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base22-2M-5M.txt (Sierpinski problem base 22, n = 2000000 to 5000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base24-400K-1M.zip (Sierpinski problem base 24, n = 400000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base25-600K-1M.zip (Sierpinski problem base 25, n = 600000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base26-2M-3M.txt (Sierpinski problem base 26, n = 2000000 to 3000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base27-3M-5M.txt (Sierpinski problem base 27, n = 3000000 to 5000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base28-1M-3M.txt (Sierpinski problem base 28, n = 1000000 to 3000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base35-100K-500K.zip (Sierpinski problem base 35, n = 100000 to 500000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base37-1M-3M.txt (Sierpinski problem base 37, n = 1000000 to 3000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base43-1M-3M.txt (Sierpinski problem base 43, n = 1000000 to 3000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base45-400K-1M.zip (Sierpinski problem base 45, n = 400000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base46-700K-1M.txt (Sierpinski problem base 46, n = 700000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base48-700K-1M.txt (Sierpinski problem base 48, n = 700000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base52-500K-1M.zip (Sierpinski problem base 52, n = 500000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base53-k4-3M-5M.txt (Sierpinski problem base 53, k = 4, n = 3000000 to 5000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base53-700K-1M.txt (Sierpinski problem base 53, all k except k = 4, n = 700000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base55-1M-3M.zip (Sierpinski problem base 55, n = 1000000 to 3000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base60-500K-1M.zip (Sierpinski problem base 60, n = 500000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-sierp-base61-500K-1M.txt (Sierpinski problem base 61, n = 500000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-R2-2nd-conj-8M-10M.zip (2nd Riesel problem base 2, n = 8000000 to 10000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base3-k3677878-0M-50M.zip (Riesel problem base 3, k = 3677878, n = 0 to 50000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base3-11G-20G-100K-200K.zip (Riesel problem base 3, k = 11000000000 to 20000000000, n = 100000 to 200000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base6-8.013M-15M.txt (Riesel problem base 6, n = 8013000 to 15000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base7-300M-400M-25K-100K.zip (Riesel problem base 7, k = 300000000 to 400000000, n = 25000 to 100000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base7-400M-500M-25K-100K.zip (Riesel problem base 7, k = 400000000 to 500000000, n = 25000 to 100000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base10-4M-5M.txt (Riesel problem base 10, n = 4000000 to 5000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base23-3M-5M.txt (Riesel problem base 23, n = 3000000 to 5000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base24-600K-1M.zip (Riesel problem base 24, n = 600000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base25-600K-1M.zip (Riesel problem base 25, n = 600000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base28-1M-3M.txt (Riesel problem base 28, n = 1000000 to 3000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base36-465K-2M.zip (Riesel problem base 36, n = 465000 to 2000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base37-500K-1M.txt (Riesel problem base 37, n = 500000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base42-500K-1M.zip (Riesel problem base 42, n = 500000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base45-700K-1M.txt (Riesel problem base 45, n = 700000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base46-700K-1M.txt (Riesel problem base 46, n = 700000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base48-500K-1M.zip (Riesel problem base 48, n = 500000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base55-500K-1M.txt (Riesel problem base 55, n = 500000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-riesel-base60-600K-1M.zip (Riesel problem base 60, n = 600000 to 1000000) and http://www.noprimeleftbehind.net/crus/sieve-real-riesel-base2-1M-2M.txt (1st real Riesel conjecture, n = 1000000 to 2000000) and http://www.noprimeleftbehind.net/crus/sieve-real-riesel-base2-2nd-conj-1M-2M.txt (2nd real Riesel conjecture, n = 1000000 to 2000000) and http://www.noprimeleftbehind.net/crus/sieve-real-riesel-base2-3rd-conj-1M-2M.txt (3rd real Riesel conjecture, n = 1000000 to 2000000) and http://www.noprimeleftbehind.net/crus/sieve-real-riesel-base2-4th-conj-1M-2M.txt (4th real Riesel conjecture, n = 1000000 to 2000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.17.txt (original minimal prime problem base 17, n = 1000000 to 2000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.19.txt (original minimal prime problem base 19, n = 707348 to 1000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.21.txt (original minimal prime problem base 21, n = 506720 to 1000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.25.txt (original minimal prime problem base 25, n = 300000 to 1000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.26.txt (original minimal prime problem base 26, n = 486721 to 1000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.27.txt (original minimal prime problem base 27, n = 360000 to 1000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.28.txt (original minimal prime problem base 28 (in fact also this new minimal prime problem base 28), n = 543202 to 1000000) and https://raw.githubusercontent.com/curtisbright/mepn-data/master/data/sieve.29.txt (original minimal prime problem base 29, n = 240000 to 1000000) and https://kurtbeschorner.de/db-details-3-1M.htm (family {1} in decimal) and https://www.alfredreichlg.de/10w7/prp/ProofFile.200001-1000000.txt (family 1{0}7 in decimal) and https://www.alfredreichlg.de/10w7/prp/ProofFile.1000001-1075000.txt (family 1{0}7 in decimal) and https://oeis.org/A076336/a076336d.html (4847×2n+1) and http://web.archive.org/web/20050929031631/http://robin.mathi.com/28433/ (28433×2n+1) and https://www.primegrid.com/sieving/rsp/rsp1M_20150702.7z (k×2n−1, 5 ≤ k ≤ 9999, k odd, 1 ≤ n ≤ 1000000) and https://www.primegrid.com/sieving/rsp/rsp2M_20150702.7z (k×2n−1, 5 ≤ k ≤ 9999, k odd, 1000001 ≤ n ≤ 2000000) and https://www.primegrid.com/sieving/rsp/rsp3M_20150702.7z (k×2n−1, 5 ≤ k ≤ 9999, k odd, 2000001 ≤ n ≤ 3000000) and https://www.primegrid.com/sieving/rsp/rsp4M_20150702.7z (k×2n−1, 5 ≤ k ≤ 9999, k odd, 3000001 ≤ n ≤ 4000000) and https://www.primegrid.com/sieving/rsp/rsp5M_20150702.7z (k×2n−1, 5 ≤ k ≤ 9999, k odd, 4000001 ≤ n ≤ 5000000) and https://www.primegrid.com/sieving/rsp/rsp6M_20150702.7z (k×2n−1, 5 ≤ k ≤ 9999, k odd, 5000001 ≤ n ≤ 6000000) and https://www.primegrid.com/sieving/rsp/rsp7M_20230827.7z (k×2n−1, 5 ≤ k ≤ 9999, k odd, 6000001 ≤ n ≤ 7000000) and https://www.primegrid.com/sieving/rsp/rsp8M_20230827.7z (k×2n−1, 5 ≤ k ≤ 9999, k odd, 7000001 ≤ n ≤ 8000000) and https://www.primegrid.com/sieving/rsp/rsp9M_20230827.7z (k×2n−1, 5 ≤ k ≤ 9999, k odd, 8000001 ≤ n ≤ 9000000) and https://www.primegrid.com/sieving/rsp/rsp10M_20240504.7z (k×2n−1, 5 ≤ k ≤ 9999, k odd, 9000001 ≤ n ≤ 10000000) and https://www.primegrid.com/sieving/rsp/rsp11M_20240504.7z (k×2n−1, 5 ≤ k ≤ 9999, k odd, 10000001 ≤ n ≤ 11000000) and https://www.primegrid.com/sieving/rsp/rsp12M_20240504.7z (k×2n−1, 5 ≤ k ≤ 9999, k odd, 11000001 ≤ n ≤ 12000000) and https://www.primegrid.com/sieving/rsp/rsp1M_20150702_k_lt_300.7z (k×2n−1, 5 ≤ k ≤ 299, k odd, 1 ≤ n ≤ 1000000) and https://www.primegrid.com/sieving/rsp/rsp2M_20150702_k_lt_300.7z (k×2n−1, 5 ≤ k ≤ 299, k odd, 1000001 ≤ n ≤ 2000000) and https://www.primegrid.com/sieving/rsp/rsp3M_20150702_k_lt_300.7z (k×2n−1, 5 ≤ k ≤ 299, k odd, 2000001 ≤ n ≤ 3000000) and https://www.primegrid.com/sieving/rsp/rsp4M_20150702_k_lt_300.7z (k×2n−1, 5 ≤ k ≤ 299, k odd, 3000001 ≤ n ≤ 4000000) and https://www.primegrid.com/sieving/rsp/rsp5M_20150702_k_lt_300.7z (k×2n−1, 5 ≤ k ≤ 299, k odd, 4000001 ≤ n ≤ 5000000) and https://www.primegrid.com/sieving/rsp/rsp6M_20150702_k_lt_300.7z (k×2n−1, 5 ≤ k ≤ 299, k odd, 5000001 ≤ n ≤ 6000000) and https://www.primegrid.com/sieving/rsp/rsp7M_20230827_k_lt_300.7z (k×2n−1, 5 ≤ k ≤ 299, k odd, 6000001 ≤ n ≤ 7000000) and https://www.primegrid.com/sieving/rsp/rsp8M_20230827_k_lt_300.7z (k×2n−1, 5 ≤ k ≤ 299, k odd, 7000001 ≤ n ≤ 8000000) and https://www.primegrid.com/sieving/rsp/rsp9M_20230827_k_lt_300.7z (k×2n−1, 5 ≤ k ≤ 299, k odd, 8000001 ≤ n ≤ 9000000) and https://www.primegrid.com/sieving/rsp/rsp10M_20240504_k_lt_300.7z (k×2n−1, 5 ≤ k ≤ 299, k odd, 9000001 ≤ n ≤ 10000000) and https://www.primegrid.com/sieving/rsp/rsp11M_20240504_k_lt_300.7z (k×2n−1, 5 ≤ k ≤ 299, k odd, 10000001 ≤ n ≤ 11000000) and https://www.primegrid.com/sieving/rsp/rsp12M_20240504_k_lt_300.7z (k×2n−1, 5 ≤ k ≤ 299, k odd, 11000001 ≤ n ≤ 12000000))

We have completely solved this problem for bases b = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24 (i.e. we have found all minimal primes in these bases and proved that they are all such primes and proved that they are definitely primes (https://en.wikipedia.org/wiki/Provable_prime, http://factordb.com/listtype.php?t=4) (i.e. not merely probable primes)) (thus, currently we can complete the classification of the minimal primes in these bases, and the "minimal prime problem" for these bases are theorems (https://en.wikipedia.org/wiki/Theorem, https://mathworld.wolfram.com/Theorem.html, https://t5k.org/notes/proofs/), they can be called "minimal prime theorems", only bases b = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24 currently have "minimal prime theorems", and like the four color theorem (https://en.wikipedia.org/wiki/Four_color_theorem, https://mathworld.wolfram.com/Four-ColorTheorem.html) and the theorem that the Ramsey number (https://en.wikipedia.org/wiki/Ramsey%27s_theorem, https://mathworld.wolfram.com/RamseyNumber.html, https://oeis.org/A212954) R(4,5) = 25, the harder parts of the proof (https://en.wikipedia.org/wiki/Mathematical_proof, https://mathworld.wolfram.com/Proof.html, https://t5k.org/notes/proofs/) are completed by computers instead of humans, such as the proof for base b = 24 (the largest base b which is currently completely solved, including the primality proving (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/prove3.html, https://t5k.org/prove/prove4.html) for all primes in the set), computing data up to linear families (i.e. only linear families left) (see https://github.com/curtisbright/mepn-data/commit/7acfa0656d3c6b759f95a031f475a30f7664a122 for the original minimal prime problem in bases 2 ≤ b ≤ 26) and computing the primality certificate (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://alfredreichlg.de/cert/certificates.tpm.html, https://alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php, http://5.199.134.130/certificates.tar.xz, http://5.199.134.130/certificates.tar.xz.SHA256SUM, http://5.199.134.130/certificates.tar.xz.par2, http://5.199.134.130/certificates.tar.xz.vol00+10.par2, http://5.199.134.130/certificates/) for the largest minimal prime in base b = 24 (i.e. N00N8129LN, its algebraic form is 13249×248131−49) (see http://factordb.com/cert.php?id=1100000003593391606 and https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/primality-certificates/certificate24_3409 for its primality certificate) are completed by computers instead of humans (I am very glad that the problem in base b = 24 can be completely solved, since the number 24 is an important number in number theory, see https://sites.google.com/view/24-important-number-theory and https://oeis.org/A018253 and https://math.ucr.edu/home/baez/numbers/24.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_176.pdf) and https://arxiv.org/pdf/1104.5052.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_177.pdf)), the fully proof for base b = 24 is almost impossible to be written by hand (only bases b = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12 is possible to write the fully proof by hand), also we have completely solved this problem for bases b = 11, 13, 16, 22, 30 if we allow probable primes (https://en.wikipedia.org/wiki/Probable_prime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Probable_prime, https://mathworld.wolfram.com/ProbablePrime.html, http://www.primenumbers.net/prptop/prptop.php, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#probableprimenumbers, https://stdkmd.net/nrr/repunit/prpfactors.htm, https://www.mersenne.ca/prp.php?show=1, https://www.alfredreichlg.de/10w7/prp.html, http://factordb.com/listtype.php?t=1) > 1025000 in place of proven primes (thus, we have completely solved this problem for bases b = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30 if we allow probable primes > 1025000 in place of proven primes), besides, we have completely solved this problem for bases b = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36 (if we allow strong probable primes in place of proven primes) except the 793+? families x{y}z (where x and z are strings (may be empty) of digits in base b, y is a digit in base b) listed in the "left b" files (see the condensed table below for the searching limit of these families) (thus, currently the "minimal prime problem" for these bases are still unsolved problems (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://t5k.org/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, https://t5k.org/notes/conjectures/)).

There are totally 2028110+? ~ 2028875+? minimal primes in bases 2 ≤ b ≤ 36 (counted with multiplicity, e.g. the prime p = 41 is 221 in base b = 4, 131 in base b = 5, 56 in base b = 7, 51 in base b = 8, 45 in base b = 9, 41 in base b = 10, 38 in base b = 11, 35 in base b = 12, 32 in base b = 13, 2D in base b = 14, 2B in base b = 15, 29 in base b = 16, 27 in base b = 17, 25 in base b = 18, 23 in base b = 19, 21 in base b = 20, 1K in base b = 21, 1J in base b = 22, 1I in base b = 23, 1H in base b = 24, 1G in base b = 25, 1F in base b = 26, 1E in base b = 27, 1D in base b = 28, 1C in base b = 29, 1B in base b = 30, 1A in base b = 31, 19 in base b = 32, 18 in base b = 33, 17 in base b = 34, 16 in base b = 35, 15 in base b = 36, all are minimal primes and counted with 32 primes; and the prime p = 577 is 711 in base b = 9, 577 in base b = 10, 485 in base b = 11, 401 in base b = 12, 355 in base b = 13, 241 in base b = 16, 1GG in base b = 17, 1B7 in base b = 19, 145 in base b = 22, 122 in base b = 23, 101 in base b = 24, N2 in base b = 25, M5 in base b = 26, LA in base b = 27, KH in base b = 28, JQ in base b = 29, J7 in base b = 30, IJ in base b = 31, I1 in base b = 32, HG in base b = 33, GX in base b = 34, GH in base b = 35, G1 in base b = 36, all are minimal primes and counted with 23 primes; and the prime p = 1063 is 887 in base b = 11, 747 in base b = 12, 63A in base b = 13, 3B9 in base b = 17, 2HI in base b = 19, 247 in base b = 22, 205 in base b = 23, 1HD in base b = 25, 1EN in base b = 26, 17J in base b = 29, 139 in base b = 31, 117 in base b = 32, W7 in base b = 33, V9 in base b = 34, UD in base b = 35, TJ in base b = 36, all are minimal primes and counted with 16 primes, in fact, by definition, every prime p is a minimal prime in every base sqrt(p) < b < p, and I conjectured that all primes p other than 2, 3, 5, 7, 11, 17, 19, 23, 37, 47, 53, 67, 167, 233 are minimal primes in at least one base b < sqrt(p), and all primes p other than 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 79, 83, 89, 101, 103, 107, 137, 139, 163, 167, 191, 199, 233, 239 are minimal primes in at least two bases b < sqrt(p), and all primes p other than 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 191, 199, 211, 223, 229, 233, 239, 263, 317, 353, 461, 479 are minimal primes in at least three bases b < sqrt(p), and all primes p other than 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 229, 233, 239, 251, 263, 269, 283, 293, 317, 331, 347, 349, 353, 359, 367, 373, 383, 389, 461, 479, 503, 509, 523, 563, 593, 1039 are minimal primes in at least four bases b < sqrt(p), etc. also, for every n, all sufficiently large (https://en.wikipedia.org/wiki/Sufficiently_large, https://mathworld.wolfram.com/SufficientlyLarge.html) primes p are minimal primes in at least n bases b < sqrt(p) (of course, there are bases b > 36 (which are not in this project) mentioned)) and totally 765+? families x{y}z (where x and z are strings (may be empty) of digits in base b, y is a digit in base b) for which we cannot determine whether they contain a prime (only count the numbers > b) or not (even if we allow strong probable primes).

The largest probable prime we found was the number A3592197A (expressed as a base b = 13 string) or (41×13592198+27)/4 in standard notation. It contains 659677 decimal digits and at the time of discovery (Nov. 24, 2023) was the 183rd largest known probable prime according to Henri and Renaud Lifchitz's ranking.

We are unable to determine if the 765+? families x{y}z (where x and z are strings (may be empty) of digits in base b, y is a digit in base b) listed in the "left b" files (see the condensed table below for the searching limit of these families) contain a prime (only count the numbers > b) or not (even if we allow strong probable primes), i.e. these families have no known prime (or strong probable prime) members > b, nor can they be ruled out as only containing composites (only count the numbers > b) (by covering congruence, algebraic factorization, or combine of them), i.e. whether these families contain a prime or a strong probable prime (only count the numbers > b) are open problems (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://t5k.org/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, http://www.numericana.com/answer/open.htm, https://t5k.org/notes/conjectures/), and all of these families are expected to contain a prime > b (in fact, expected to contain infinitely many primes), since there is a heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, https://web.archive.org/web/20230911032453/https://www.utm.edu/staff/caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf), https://arxiv.org/pdf/2103.04483.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_113.pdf)) that all families which cannot be ruled out as only containing composites or only containing finitely many primes (by covering congruence, algebraic factorization, or combine of them) contain infinitely many primes (references: https://t5k.org/mersenne/heuristic.html, https://t5k.org/notes/faq/NextMersenne.html, https://web.archive.org/web/20100628035147/http://www.math.niu.edu/~rusin/known-math/98/exp_primes, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, https://mathoverflow.net/questions/268918/density-of-primes-in-sequences-of-the-form-anb, https://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Mathematics/2023_September_25#Are_there_infinitely_many_primes_of_the_form_1000%E2%80%A60007,_333%E2%80%A63331,_7111%E2%80%A6111,_or_3444%E2%80%A64447_in_base_10?, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf), https://www.mersenneforum.org/showthread.php?t=12327, https://oeis.org/A234285 (the comment by Farideh Firoozbakht, although this comment is not true, there is no prime for s = 509203 and s = −78557, s = 509203 has a covering set of {3, 5, 7, 13, 17, 241}, and s = −78557 has a covering set of {3, 5, 7, 13, 19, 37, 73}), https://www.mersenneforum.org/showpost.php?p=564786&postcount=3, https://www.mersenneforum.org/showpost.php?p=461665&postcount=7, https://www.mersenneforum.org/showpost.php?p=354505&postcount=5, https://www.mersenneforum.org/showpost.php?p=344985&postcount=293, https://www.mersenneforum.org/showpost.php?p=625978&postcount=1027, https://www.primegrid.com/forum_thread.php?id=5093&nowrap=true#66471, https://www.primegrid.com/forum_thread.php?id=4935&nowrap=true#63813), since by the prime number theorem (https://en.wikipedia.org/wiki/Prime_number_theorem, https://t5k.org/glossary/xpage/PrimeNumberThm.html, https://mathworld.wolfram.com/PrimeNumberTheorem.html, https://t5k.org/howmany.html, http://www.numericana.com/answer/primes.htm#pnt, https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Tables_of_special_primes) the chance (https://en.wikipedia.org/wiki/Probability, https://mathworld.wolfram.com/Probability.html) that a random (https://en.wikipedia.org/wiki/Random_number, https://mathworld.wolfram.com/RandomNumber.html) n-digit base b number is prime is approximately (https://en.wikipedia.org/wiki/Asymptotic_analysis, https://t5k.org/glossary/xpage/AsymptoticallyEqual.html, https://mathworld.wolfram.com/Asymptotic.html) 1/n (more accurately, the chance is approximately 1/(n×ln(b)), where ln is the natural logarithm (https://en.wikipedia.org/wiki/Natural_logarithm, https://t5k.org/glossary/xpage/Log.html, https://mathworld.wolfram.com/NaturalLogarithm.html), i.e. the logarithm with base e = 2.718281828459... (https://en.wikipedia.org/wiki/E_(mathematical_constant), https://mathworld.wolfram.com/e.html, https://oeis.org/A001113)). If one conjectures the numbers x{y}z behave similarly (i.e. the numbers x{y}z is a pseudorandom sequence (https://en.wikipedia.org/wiki/Pseudorandomness, https://mathworld.wolfram.com/PseudorandomNumber.html, https://people.seas.harvard.edu/~salil/pseudorandomness/pseudorandomness-Aug12.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_197.pdf))) you would expect 1/1 + 1/2 + 1/3 + 1/4 + ... = ∞ (https://en.wikipedia.org/wiki/Harmonic_series_(mathematics), https://mathworld.wolfram.com/HarmonicSeries.html) primes of the form x{y}z (of course, this does not always happen, since some x{y}z families can be ruled out as only containing composites (only count the numbers > b) (by covering congruence, algebraic factorization, or combine of them), and every family has its own Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=50442&postcount=1, https://www.mersenneforum.org/showpost.php?p=50444&postcount=1, https://www.mersenneforum.org/showpost.php?p=201642&postcount=1, https://www.mersenneforum.org/attachment.php?attachmentid=13663&d=1451910741, https://github.com/happy5214/nash, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, https://web.archive.org/web/20230928115952/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, https://web.archive.org/web/20230928120009/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, https://web.archive.org/web/20230928120025/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, https://web.archive.org/web/20230928120047/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)), see https://www.mersenneforum.org/showpost.php?p=564786&postcount=3 and https://www.primegrid.com/forum_thread.php?id=4935&nowrap=true#63813, families which can be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them) have Nash weight (or difficulty) 0, and families which cannot be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them) have positive Nash weight (or difficulty), but it is at least a reasonable conjecture in the absence of evidence to the contrary).

Some of the left families may cover another left family, e.g. the base 19 left family 5{H}5 covers another base 19 left family 5{H}05, and if the smallest prime in family 5{H}5 in base 19 has length n, and the family 5{H}05 in base 19 has no prime with length ≤ n, then family 5{H}05 in base 19 can be removed from the unsolved families for base 19, however, if the smallest prime in family 5{H}5 in base 19 has length n, but the family 5{H}05 in base 19 is not tested to length n or more, then family 5{H}05 in base 19 should not be removed from the unsolved families for base 19, since a number in family 5{H}05 covers the prime in family 5{H}5 with length n if and only if the length of this number is ≥ n+1; besides, the base 19 left family FH0{H} covers another base 19 left family FHHH0{H}, and if the smallest prime in family FH0{H} in base 19 has length n, and the family FHHH0{H} in base 19 has no prime with length ≤ n+1, then family FHHH0{H} in base 19 can be removed from the unsolved families for base 19, however, if the smallest prime in family FH0{H} in base 19 has length n, but the family FHHH0{H} in base 19 is not tested to length n+1 or more, then family FHHH0{H} in base 19 should not be removed from the unsolved families for base 19, since a number in family FHHH0{H} covers the prime in family FH0{H} with length n if and only if the length of this number is ≥ n+2; besides, the base 21 left family {9}D covers another base 21 left family F{9}D, and if the smallest prime in family {9}D in base 21 has length n, and the family F{9}D in base 21 has no prime with length ≤ n, then family F{9}D in base 21 can be removed from the unsolved families for base 21, however, if the smallest prime in family {9}D in base 21 has length n, but the family F{9}D in base 21 is not tested to length n or more, then family F{9}D in base 21 should not be removed from the unsolved families for base 21, since a number in family F{9}D covers the prime in family {9}D with length n if and only if the length of this number is ≥ n+1 (if a family has no primes, then we say "the smallest prime in this family has length ∞ (https://en.wikipedia.org/wiki/Infinity, https://t5k.org/glossary/xpage/Infinite.html, https://mathworld.wolfram.com/Infinity.html) (instead of 0 or −1)", see http://gladhoboexpress.blogspot.com/2019/05/prime-sandwiches-made-with-one-derbread.html and http://chesswanks.com/seq/a306861.txt (for the OEIS sequence https://oeis.org/A306861) and http://chesswanks.com/seq/a269254.txt (for the OEIS sequence https://oeis.org/A269254) (since this is more convenient, e.g. the n of the smallest prime in the base 21 family 9nD, this family has been searched to n = 100000 with no prime or probable prime found, we can use ">100000" for the n of the smallest prime in the base 21 family 9nD (while for the n of the smallest prime in the base 13 family A3nA, it is 592199), ">100000" includes infinity (since infinity is > 100000) but does not includes 0 or −1, it is still possible that there is no prime in the base 21 family 9nD, although by the heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, https://web.archive.org/web/20230911032453/https://www.utm.edu/staff/caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf), https://arxiv.org/pdf/2103.04483.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_113.pdf)) above, this is very impossible, also "the smallest n ≥ 1 such that (a×bn+c)/gcd(a+c,b−1) is prime" should be the infimum (https://en.wikipedia.org/wiki/Infimum, https://mathworld.wolfram.com/Infimum.html) of the set S of the numbers n ≥ 1 such that (a×bn+c)/gcd(a+c,b−1) is prime, and if there is no n ≥ 1 such that (a×bn+c)/gcd(a+c,b−1) is prime, then this set S is the empty set (https://en.wikipedia.org/wiki/Empty_set, https://mathworld.wolfram.com/EmptySet.html), and by the definition of "inf", the infimum of the empty set is ∞), ∞ is > any finite number, e.g. "the smallest n ≥ 1 such that k×2n+1 is prime" is ∞ for k = 78557, 157114, 271129, 271577, 314228, 322523, 327739, 482719, ..., while it is 31172165 for k = 10223 and 13018586 for k = 19249, another example is "the smallest n such that (bn−1)/(b−1) is prime" is ∞ for b = 9, 25, 32, 49, 64, 81, 121, 125, 144, 169, 216, 225, 243, 289, 324, 343, 361, 441, 484, 529, 625, 729, 784, 841, 900, 961, 1000, ..., while it is 62903 for b = 691 and 41189 for b = 693).

The final goal of this project is finding all minimal primes and proving that these are all such primes (including the primality proving for the probable primes) in all bases 2 ≤ b ≤ 36, i.e. solving all families in all bases 2 ≤ b ≤ 36. Solving all (left) families in all bases 2 ≤ b ≤ 36 (and proving the primality of all probable primes in the sets of all bases 2 ≤ b ≤ 36) is not possible but we aim to solve many of them (and proving the primality of many of them), at least find a probable prime for many of them (since the smallest prime in a family may be too large (> 1025000) to be proved primality, unless its N−1 or/and N+1 can be ≥ 25% factored).

These sets of minimal primes are computed (https://en.wikipedia.org/wiki/Computing) by: make data up to linear families (i.e. only linear families left) (https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/kGMP.cc, https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/searchpp.cc) (see https://github.com/curtisbright/mepn-data/commit/7acfa0656d3c6b759f95a031f475a30f7664a122 for the original minimal prime problem) → search the left linear families to length 1000 (https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/searchpm.cc) (just like the new base script for Conjectures 'R Us Sierpinski/Riesel conjectures base b (http://www.noprimeleftbehind.net/crus/, http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-stats.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-top20.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-proven.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, https://t5k.org/bios/page.php?id=1372, https://www.rieselprime.de/ziki/Conjectures_%27R_Us, https://srbase.my-firewall.org/sr5/, https://srbase.my-firewall.org/sr5/stats.php, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), http://www.bitman.name/math/article/2005 (in Italian)), see http://www.noprimeleftbehind.net/crus/new-bases-5.1.txt and https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/CRUS_pack/scripts/new-bases-5.1.txt, also see https://github.com/curtisbright/mepn-data/commit/4e524f26e39cc3df98f017e8106720ba4588e981 and https://github.com/curtisbright/mepn-data/commit/f238288fac40d97a85d7cc707367cc91cdf75ec9 and https://github.com/curtisbright/mepn-data/commit/e6b2b806f341e9dc5b96662edba2caf3220c98b7 for the original minimal prime problem) → use a program like SRSIEVE (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, https://www.mersenneforum.org/showpost.php?p=631129&postcount=1, http://www.rieselprime.de/dl/CRUS_pack.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve_1.8.2, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve-other-programs, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2cl.exe) to sieve (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.primegrid.com/stats_psp_sieve.php, https://www.primegrid.com/stats_pps_sieve.php, https://www.primegrid.com/stats_sr5_sieve.php, https://www.primegrid.com/stats_cw_sieve.php, https://www.primegrid.com/sieving/rsp/) the left linear families with primes p < 109 (https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/searchLLR.cc) → use LLR (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64) to primality test (or probable-primality test) the numbers in the sieve files (LLR will do the Miller–Rabin primality test (https://t5k.org/prove/prove2_3.html, https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test, https://en.wikipedia.org/wiki/Strong_pseudoprime, https://t5k.org/glossary/xpage/MillersTest.html, https://t5k.org/glossary/xpage/StrongPRP.html, https://www.rieselprime.de/ziki/Miller-Rabin_pseudoprimality_test, https://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html, https://mathworld.wolfram.com/StrongPseudoprime.html, http://www.numericana.com/answer/pseudo.htm#rabin, http://www.numericana.com/answer/pseudo.htm#strong, http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm, http://ntheory.org/data/spsps.txt, https://faculty.lynchburg.edu/~nicely/misc/mpzspsp.html, https://miller-rabin.appspot.com/, http://www.pi-e.de/Miller-Rabin-Pseudoprimzahlen.htm (in German), http://factordb.com/prooffailed.php, https://sites.google.com/view/strong-pseudoprime, https://sites.google.com/view/bases-strong-pseudoprime, https://oeis.org/A001262, https://oeis.org/A020229, https://oeis.org/A020230, https://oeis.org/A020231, https://oeis.org/A020232, https://oeis.org/A020233, https://oeis.org/A020234, https://oeis.org/A020235, https://oeis.org/A020236, https://oeis.org/A020237, https://oeis.org/A020238, https://oeis.org/A020239, https://oeis.org/A020240, https://oeis.org/A020241, https://oeis.org/A020242, https://oeis.org/A020243, https://oeis.org/A020244, https://oeis.org/A020245, https://oeis.org/A020246, https://oeis.org/A020247, https://oeis.org/A020248, https://oeis.org/A020249, https://oeis.org/A020250, https://oeis.org/A020251, https://oeis.org/A020252, https://oeis.org/A020253, https://oeis.org/A020254, https://oeis.org/A020255, https://oeis.org/A020256, https://oeis.org/A020257, https://oeis.org/A020258, https://oeis.org/A020259, https://oeis.org/A020260, https://oeis.org/A020261, https://oeis.org/A020262, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773, https://oeis.org/A014233, https://oeis.org/A006945, https://oeis.org/A089825, https://oeis.org/A181782, https://oeis.org/A071294, https://oeis.org/A141768, https://oeis.org/A195328, https://oeis.org/A329759, https://oeis.org/A298756) with first 50 prime bases, the strong Lucas primality test (https://en.wikipedia.org/wiki/Strong_Lucas_pseudoprime, https://mathworld.wolfram.com/StrongLucasPseudoprime.html, http://ntheory.org/data/slpsps-baillie.txt, http://www.hoegge.dk/lucasselfridgeprps.txt, https://oeis.org/A217255), and the strong Frobenius primality test (https://en.wikipedia.org/wiki/Strong_Frobenius_pseudoprime, https://t5k.org/glossary/xpage/FrobeniusPseudoprime.html, https://mathworld.wolfram.com/FrobeniusPseudoprime.html, https://mathworld.wolfram.com/StrongFrobeniusPseudoprime.html, https://ntheory.org/data/a212424.txt, https://ntheory.org/data/frob-3-5-psps.txt, https://oeis.org/A212424, http://pseudoprime.com/pseudo1.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_316.pdf)), also for a×bn+1 numbers with a < bn, LLR will do the N−1 Pocklington primality test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1) and can prove that these numbers are primes, also for a×bn−1 numbers a < bn, LLR will do the N+1 Morrison primality test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) and can prove that these numbers are primes) → trial factor (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) from 109 to 1016 → use PRIMO (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309) to elliptic curve primality prove (https://t5k.org/prove/prove4_2.html, https://en.wikipedia.org/wiki/Elliptic_curve_primality, https://t5k.org/glossary/xpage/ECPP.html, https://mathworld.wolfram.com/EllipticCurvePrimalityProving.html, http://irvinemclean.com/maths/pfaq7.htm, https://t5k.org/top20/page.php?id=27, https://t5k.org/primes/search.php?Comment=ECPP&OnList=all&Number=1000000&Style=HTML, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/~morain/Prgms/ecpp.english.html, https://www.multiprecision.org/cm/ecpp.html, https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1199989-X/S0025-5718-1993-1199989-X.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_256.pdf), https://arxiv.org/pdf/2404.05506.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_428.pdf)) the numbers < 1025000.

There are also unproven probable primes (however, in this project our results assume that they are in fact primes, since they are > 1025000 and the probability that they are in fact composite is < 10−2000, see https://t5k.org/notes/prp_prob.html and https://www.ams.org/journals/mcom/1989-53-188/S0025-5718-1989-0982368-4/S0025-5718-1989-0982368-4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_22.pdf)), the unproven probable primes for bases b = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36 (the "easy" bases (bases b with ≤ 150 minimal primes > 10299 (base b = 26 has 82 known minimal (probable) primes > 10299 and 4 unsolved families, base b = 36 has 75 known minimal (probable) primes > 10299 and 4 unsolved families, base b = 17 has 99 known minimal (probable) primes > 10299 and 18 unsolved families, base b = 21 has 80 known minimal (probable) primes > 10299 and 12 unsolved families, base b = 19 has 201 known minimal (probable) primes > 10299 and 23 unsolved families))) (all of them are Fermat probable primes (https://t5k.org/prove/prove2_2.html, https://en.wikipedia.org/wiki/Fermat_primality_test, https://en.wikipedia.org/wiki/Fermat_pseudoprime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Fermat_pseudoprimality_test, https://mathworld.wolfram.com/FermatPseudoprime.html, https://www.numbersaplenty.com/set/Poulet_number/, http://www.numericana.com/answer/pseudo.htm#pseudoprime, http://ntheory.org/data/psps.txt, https://www.cecm.sfu.ca/Pseudoprimes/psps-below-2-to-64.txt.bz2, https://www.cecm.sfu.ca/Pseudoprimes/factored-psps-below-2-to-64.txt.bz2, https://www.cecm.sfu.ca/Pseudoprimes/annotated-psps-below-2-to-64.txt.bz2, https://sites.google.com/view/fermat-pseudoprime, https://sites.google.com/view/bases-fermat-pseudoprime, https://oeis.org/A001567, https://oeis.org/A005935, https://oeis.org/A020136, https://oeis.org/A005936, https://oeis.org/A005937, https://oeis.org/A005938, https://oeis.org/A020137, https://oeis.org/A020138, https://oeis.org/A005939, https://oeis.org/A020139, https://oeis.org/A020140, https://oeis.org/A020141, https://oeis.org/A020142, https://oeis.org/A020143, https://oeis.org/A020144, https://oeis.org/A020145, https://oeis.org/A020146, https://oeis.org/A020147, https://oeis.org/A020148, https://oeis.org/A020149, https://oeis.org/A020150, https://oeis.org/A020151, https://oeis.org/A020152, https://oeis.org/A020153, https://oeis.org/A020154, https://oeis.org/A020155, https://oeis.org/A020156, https://oeis.org/A020157, https://oeis.org/A020158, https://oeis.org/A020159, https://oeis.org/A020160, https://oeis.org/A020161, https://oeis.org/A020162, https://oeis.org/A020163, https://oeis.org/A020164, https://oeis.org/A000864, https://oeis.org/A052155, https://oeis.org/A083737, https://oeis.org/A083739, https://oeis.org/A083876, https://oeis.org/A271221, https://oeis.org/A348258, https://oeis.org/A181780, https://oeis.org/A211455, https://oeis.org/A211456, https://oeis.org/A211457, https://oeis.org/A211458, https://oeis.org/A063994, https://oeis.org/A105222, https://oeis.org/A194946, https://oeis.org/A195327, https://oeis.org/A002997, https://oeis.org/A191311, https://oeis.org/A090086, https://oeis.org/A007535, https://oeis.org/A090087, https://oeis.org/A090085, https://oeis.org/A090088, https://oeis.org/A090089, https://oeis.org/A253233, https://oeis.org/A271801) to all prime bases ≤ 64 (i.e. bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61), also for the unproven probable primes for bases 2 ≤ b ≤ 16 I also ran the bases 6, 10, 12 although they are not prime bases (running perfect power (i.e. of the form mr with r > 1) (https://oeis.org/A001597, https://en.wikipedia.org/wiki/Perfect_power, https://mathworld.wolfram.com/PerfectPower.html, https://www.numbersaplenty.com/set/perfect_power/) bases is useless since a Fermat probable prime to base m must also be a Fermat probable prime to base mr for all r, thus do not run these bases, in fact, this is also true for Miller–Rabin primality tests, a strong probable prime to base m must also be a strong probable prime to base mr for all r, thus running perfect power (i.e. of the form mr with r > 1) (https://oeis.org/A001597, https://en.wikipedia.org/wiki/Perfect_power, https://mathworld.wolfram.com/PerfectPower.html, https://www.numbersaplenty.com/set/perfect_power/) bases is also useless for the Miller–Rabin primality tests), see the "Bases checked" section of the "Primality proving" box in the factordb entries of these probable primes, the "Bases checked" section of the "Primality proving" box in the factordb entries of these probable primes contain all of the bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, also contain the bases 6, 10, 12 for the unproven probable primes for bases 2 ≤ b ≤ 16, unfortunately, the "Primality proving" box in factordb only runs the Fermat primality test instead of the Miller–Rabin primality test) are (together with the factorization of the numbers in their corresponding families):

(In progess to add bases b = 17 and b = 21)

(for the factorization of the numbers in these families and the N−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) and N+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) of these probable primes, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=snfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#smallpolynomial, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=gnfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) may be used, they have SNFS polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt and https://alfredreichlg.de/10w7/10w7.txt and https://web.archive.org/web/20020320010222/http://proth.cjb.net/ and http://web.archive.org/web/20111104173105/http://www.immortaltheory.com/NumberTheory/2nm3_db.txt and https://www.asahi-net.or.jp/~KC2H-MSM/mathland/aba/index.htm and https://www.alpertron.com.ar/BRILLIANT.HTM and https://www.alpertron.com.ar/BRILLIANT3.HTM and https://www.alpertron.com.ar/BRILLIANT4.HTM and https://www.alpertron.com.ar/BRILLIANT2.HTM and https://oeis.org/wiki/Factors_of_33*2%5En%2B1 and https://oeis.org/wiki/Factors_of_33*2%5En-1 and https://web.archive.org/web/20111018190410/http://www.sr5.psp-project.de/s5stats.html (section "Top ten factors:") and https://web.archive.org/web/20111018190339/http://www.sr5.psp-project.de/r5stats.html (section "Top ten factors:") and https://brnikat.com/nums/cullen_woodall/cw.html and https://oeis.org/wiki/OEIS_sequences_needing_factors#Near_powers.2C_factorials.2C_and_primorials (sections "near-powers with b = 2" and "near-powers with b = 3" and "near-powers with b = 5" and "near-powers with b = 6" and "near-powers with b = 7" and "near-powers with b = 10" and "near-powers with b > 10") and https://www.mersenneforum.org/showthread.php?t=9554 and https://www.mersenneforum.org/showthread.php?t=9167 and https://www.mersenneforum.org/showpost.php?p=644144&postcount=5, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm and https://www.mersenneforum.org/showthread.php?t=12962)

(For the Primo input files of more unproven probable primes, see http://factordb.com/primobatch.php and http://factordb.com/primobatch.php?digits=300&files=10&parts=1&start=Generate+zip and http://factordb.com/primobatch.php?digits=300&files=32000&parts=32&start=Generate+zip)

(link to the factorization of the first 200 numbers in corresponding families of these minimal primes (start with the smallest number in these families which is > b), to show the small prime factors (and their period) (i.e. show whether these numbers are rough numbers) and the algebraic factors (and their period) and let you feel the Nash weight (or difficulty) of these families (high or low), all small prime factors (< 1012, by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172)) and all algebraic factors (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/53335.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/83336.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=algebraic&maxrows=10000, https://sites.google.com/view/algebraic-factors-of-xn-kyn, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://www.mersenneforum.org/showpost.php?p=96560&postcount=99, https://www.mersenneforum.org/showpost.php?p=96651&postcount=101, https://www.mersenneforum.org/showthread.php?t=21916, https://www.mersenneforum.org/showpost.php?p=196598&postcount=492, https://www.mersenneforum.org/showpost.php?p=203083&postcount=149, https://www.mersenneforum.org/showpost.php?p=206065&postcount=192, https://www.mersenneforum.org/showpost.php?p=208044&postcount=260, https://www.mersenneforum.org/showpost.php?p=210533&postcount=336, https://www.mersenneforum.org/showpost.php?p=452132&postcount=66, https://www.mersenneforum.org/showpost.php?p=451337&postcount=32, https://www.mersenneforum.org/showpost.php?p=208852&postcount=227, https://www.mersenneforum.org/showpost.php?p=232904&postcount=604, https://www.mersenneforum.org/showpost.php?p=383690&postcount=1, https://www.mersenneforum.org/showpost.php?p=207886&postcount=253, https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php, https://www.alpertron.com.ar/POLFACT.HTM, https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) of the N−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) and N+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) of these probable primes and the first 200 numbers (start with the smallest n making the number > b (if n = 0 already makes the number > b, then start with n = 0)) in corresponding families of these probable primes were added to factordb (for the examples of the large primes whose N−1 or/and N+1 has algebraic factors, see https://stdkmd.net/nrr/cert/1/#CERT_11101_4809 and https://stdkmd.net/nrr/cert/1/#CERT_15551_2197 and https://stdkmd.net/nrr/cert/1/#CERT_16667_4296 and https://stdkmd.net/nrr/cert/2/#CERT_20111_2692 and https://stdkmd.net/nrr/cert/2/#CERT_23309_10029 and https://stdkmd.net/nrr/cert/3/#CERT_37773_15768 and https://stdkmd.net/nrr/cert/6/#CERT_6805W7_3739 and https://stdkmd.net/nrr/cert/6/#CERT_68883_5132 and https://stdkmd.net/nrr/cert/7/#CERT_79921_11629 and https://stdkmd.net/nrr/cert/8/#CERT_80081_5736 and https://stdkmd.net/nrr/cert/8/#CERT_83W16W7_543 and https://stdkmd.net/nrr/cert/9/#CERT_93307_2197 for the related numbers (although some of them are related to Cunningham numbers, and some of them has N−1 and N+1 does not have algebraic factors but has a large prime factor), e.g. "11101_4809" (decimal (base b = 10) form: 1480701, algebraic form: (104809−91)/9) is related to "Phi_4807_10" (the number Φ4807(10), where Φ is the cyclotomic polynomial), "15551_2197" (decimal (base b = 10) form: 1521961, algebraic form: (14×102197−41)/9, the prime is a cofactor of it (divided it by 11×23×167)) is related to "93307_2197" (decimal (base b = 10) form: 93219507, algebraic form: (28×102197−79)/3), "16667_4296" (decimal (base b = 10) form: 1642957, algebraic form: (5×104296+1)/3, the prime is a cofactor of it (divided it by 347×821×140235709×806209146522749)) is related to "33337_12891" (decimal (base b = 10) form: 3128907, algebraic form: (1012891+11)/3), "20111_2692" (decimal (base b = 10) form: 2012692, algebraic form: (181×102692−1)/9, the prime is a cofactor of it (divided it by 3×43)) is related to "20111_2693" (decimal (base b = 10) form: 2012693, algebraic form: (181×102693−1)/9), "23309_10029" (decimal (base b = 10) form: 231002709, algebraic form: (7×1010029−73)/3) is related to "Phi_5014_10" (the number Φ5014(10), where Φ is the cyclotomic polynomial), "37773_15768" (decimal (base b = 10) form: 37157673, algebraic form: (34×1015768−43)/9) is related to "Phi_7884_10" (the number Φ7884(10), where Φ is the cyclotomic polynomial), "6805w7_3739" (decimal (base b = 10) form: 680537387, algebraic form: (6125×103739+13)/9, the prime is a cofactor of it (divided it by 32)) is related to "27227_3741" (decimal (base b = 10) form: 27237407, algebraic form: (245×103741+43)/9), "68883_5132" (decimal (base b = 10) form: 6851313, algebraic form: (62×105132−53)/9) is related to "Phi_1283_10" (the number Φ1283(10), where Φ is the cyclotomic polynomial), "79921_11629" (decimal (base b = 10) form: 791162721, algebraic form: 8×1011629−79) is related to "Phi_2907_10" (the number Φ2907(10), where Φ is the cyclotomic polynomial), "80081_5736" (decimal (base b = 10) form: 80573481, algebraic form: 8×105736+81) is related to "Phi_11470_10" (the number Φ11470(10), where Φ is the cyclotomic polynomial), "83w16w7_543" (decimal (base b = 10) form: 83542165427, algebraic form: (25×101086−5×10543+1)/3, the prime is a cofactor of it (divided it by 7×109×563041×869047141×147372142447)) is related to "11103_3258" (decimal (base b = 10) form: 1325603, algebraic form: (103258−73)/9), etc. the N−1 of "11101_4809" is 100 × R4807(10) (which is equivalent to the Cunningham number 104807−1) and Φ4807(10) is an algebraic factor of the Cunningham number 104807−1, the N−1 of "93307_2197" is 6 × "15551_2197", the N−1 of "33337_12891" has sum-of-two-cubes factorization and an algebraic factor is 2 × "16667_4296", the N−1 of "20111_2693" is 10 × "20111_2692", the N+1 of "23309_10029" is 210 × R10028(10) (which is equivalent to the Cunningham number 1010028−1) and Φ5014(10) is an algebraic factor of the Cunningham number 1010028−1, the N+1 of "37773_15768" is 34 × R15768(10) (which is equivalent to the Cunningham number 1015768−1) and Φ7884(10) is an algebraic factor of the Cunningham number 1015768−1, the N+1 of "27227_3741" is 4 × "6805w7_3739", the N−1 of "68883_5132" is 62 × R5132(10) (which is equivalent to the Cunningham number 105132−1) and Φ1283(10) is an algebraic factor of the Cunningham number 105132−1, the N−1 of "79921_11629" is 720 × R11628(10) (which is equivalent to the Cunningham number 1011628−1) and Φ2907(10) is an algebraic factor of the Cunningham number 1011628−1, the N−1 of "80081_5736" is 80 × S5735(10) (which is equivalent to the Cunningham number 105735+1) and Φ11470(10) is an algebraic factor of the Cunningham number 105735+1, the N+1 of "11103_3258" has difference-of-two-6th-powers factorization and an algebraic factor is 4 × "83w16w7_543", etc.), unfortunately, none of these numbers have algebraic factors)

(if the prime is (a×bn+c)/gcd(a+c,b−1) (with a ≥ 1, b ≥ 2, c ≠ 0, gcd(a,c) = 1, gcd(b,c) = 1), then its N−1 is (a×bn+cgcd(a+c,b−1))/gcd(a+c,b−1), and its N+1 is (a×bn+c+gcd(a+c,b−1))/gcd(a+c,b−1), although (a×bn+c)/gcd(a+c,b−1) (with a ≥ 1, b ≥ 2, c ≠ 0, gcd(a,c) = 1, gcd(b,c) = 1) has algebraic factorization if and only if either "there is an integer r > 1 such that a×bn and −c are both rth powers" (in this case, (a×bn+c)/gcd(a+c,b−1) has sum-of-two-rth-powers factorization if c > 0, or difference-of-two-rth-powers factorization if c < 0, although there is no "sum-of-two-rth-powers factorization" for even r, but no such situation (i.e. c > 0 and r is even) exists, since if c > 0 then −c < 0, but negative numbers cannot be squares, however, if r is even then all rth powers are squares (since if s divides r, then all rth powers are sth powers), thus, −c cannot be an rth power if c > 0 and r is even) or "one of a×bn and c is a 4th power, and the other is of the form 4×m4" (in this case, (a×bn+c)/gcd(a+c,b−1) has Aurifeuillean factorization of x4+4×y4), however, since cgcd(a+c,b−1) and c+gcd(a+c,b−1) may not coprime to a and b, it is not that simple to known whether (a×bn+cgcd(a+c,b−1))/gcd(a+c,b−1) and (a×bn+c+gcd(a+c,b−1))/gcd(a+c,b−1) have algebraic factorization, (a×bn+cgcd(a+c,b−1))/gcd(a+c,b−1) has algebraic factorization if and only if either "cgcd(a+c,b−1) = 0" (if and only if cgcd(a+c,b−1) = 0, then (a×bn+cgcd(a+c,b−1))/gcd(a+c,b−1) is trivially fully factored) or "there is an integer r > 1 such that a×bn/gcd(a×bn,cgcd(a+c,b−1)) and −(cgcd(a+c,b−1))/gcd(a×bn,cgcd(a+c,b−1)) are both rth powers" (in this case, (a×bn+cgcd(a+c,b−1))/gcd(a+c,b−1) has sum-of-two-rth-powers factorization if (cgcd(a+c,b−1))/gcd(a×bn,cgcd(a+c,b−1)) > 0, or difference-of-two-rth-powers factorization if (cgcd(a+c,b−1))/gcd(a×bn,cgcd(a+c,b−1)) < 0, although there is no "sum-of-two-rth-powers factorization" for even r, but no such situation (i.e. (cgcd(a+c,b−1))/gcd(a×bn,cgcd(a+c,b−1)) > 0 and r is even) exists, since if (cgcd(a+c,b−1))/gcd(a×bn,cgcd(a+c,b−1)) > 0 then −(cgcd(a+c,b−1))/gcd(a×bn,cgcd(a+c,b−1)) < 0, but negative numbers cannot be squares, however, if r is even then all rth powers are squares (since if s divides r, then all rth powers are sth powers), thus, −(cgcd(a+c,b−1))/gcd(a×bn,cgcd(a+c,b−1)) cannot be an rth power if (cgcd(a+c,b−1))/gcd(a×bn,cgcd(a+c,b−1)) > 0 and r is even) or "one of a×bn/gcd(a×bn,cgcd(a+c,b−1)) and (cgcd(a+c,b−1))/gcd(a×bn,cgcd(a+c,b−1)) is a 4th power, and the other is of the form 4×m4" (in this case, (a×bn+cgcd(a+c,b−1))/gcd(a+c,b−1) has Aurifeuillean factorization of x4+4×y4), (a×bn+c+gcd(a+c,b−1))/gcd(a+c,b−1) has algebraic factorization if and only if either "c+gcd(a+c,b−1) = 0" (if and only if c+gcd(a+c,b−1) = 0, then (a×bn+c+gcd(a+c,b−1))/gcd(a+c,b−1) is trivially fully factored) or "there is an integer r > 1 such that a×bn/gcd(a×bn,c+gcd(a+c,b−1)) and −(c+gcd(a+c,b−1))/gcd(a×bn,c+gcd(a+c,b−1)) are both rth powers" (in this case, (a×bn+c+gcd(a+c,b−1))/gcd(a+c,b−1) has sum-of-two-rth-powers factorization if (c+gcd(a+c,b−1))/gcd(a×bn,c+gcd(a+c,b−1)) > 0, or difference-of-two-rth-powers factorization if (c+gcd(a+c,b−1))/gcd(a×bn,c+gcd(a+c,b−1)) < 0, although there is no "sum-of-two-rth-powers factorization" for even r, but no such situation (i.e. (c+gcd(a+c,b−1))/gcd(a×bn,c+gcd(a+c,b−1)) > 0 and r is even) exists, since if (c+gcd(a+c,b−1))/gcd(a×bn,c+gcd(a+c,b−1)) > 0 then −(c+gcd(a+c,b−1))/gcd(a×bn,c+gcd(a+c,b−1)) < 0, but negative numbers cannot be squares, however, if r is even then all rth powers are squares (since if s divides r, then all rth powers are sth powers), thus, −(c+gcd(a+c,b−1))/gcd(a×bn,c+gcd(a+c,b−1)) cannot be an rth power if (c+gcd(a+c,b−1))/gcd(a×bn,c+gcd(a+c,b−1)) > 0 and r is even) or "one of a×bn/gcd(a×bn,c+gcd(a+c,b−1)) and (c+gcd(a+c,b−1))/gcd(a×bn,c+gcd(a+c,b−1)) is a 4th power, and the other is of the form 4×m4" (in this case, (a×bn+c+gcd(a+c,b−1))/gcd(a+c,b−1) has Aurifeuillean factorization of x4+4×y4))

b
(2 ≤ b ≤ 36)
index of this minimal prime in base b (assuming the primality of all probable primes in base b)base-b form of this unproven probable prime (which is a minimal prime assuming its primality)algebraic ((a×bn+c)/gcd(a+c,b−1)) form of this unproven probable prime (which is a minimal prime assuming its primality)length of this unproven probable prime (which is a minimal prime assuming its primality) written in base blength of this unproven probable prime (which is a minimal prime assuming its primality) written in decimalfactordb entry of this unproven probable prime (which is a minimal prime assuming its primality)this unproven probable prime (which is a minimal prime assuming its primality) written in base b (use lower case letters instead of upper case letters)this unproven probable prime (which is a minimal prime assuming its primality) written in decimalhelper file of this unproven probable prime (which is a minimal prime assuming its primality)Primo input file of this unproven probable prime (which is a minimal prime assuming its primality)factorization of the first 200 numbers in corresponding family (n is the number of digits in the "{}", start with the smallest n making the number > b (if n = 0 already makes the number > b, then start with n = 0) of this unproven probable prime (which is a minimal prime assuming its primality))
1110685762668(57×1162668−7)/106266965263http://factordb.com/index.php?id=1100000003573679860&open=primehttp://factordb.com/index.php?showid=1100000003573679860&base=11http://factordb.com/index.php?showid=1100000003573679860http://factordb.com/helper.php?id=1100000003573679860http://factordb.com/cert.php?id=1100000003573679860&inputfilehttp://factordb.com/index.php?query=%2857*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
133194C523755C(149×1323756+79)/122375726464http://factordb.com/index.php?id=1100000003590647776&open=primehttp://factordb.com/index.php?showid=1100000003590647776&base=13http://factordb.com/index.php?showid=1100000003590647776http://factordb.com/helper.php?id=1100000003590647776http://factordb.com/cert.php?id=1100000003590647776&inputfilehttp://factordb.com/index.php?query=%28149*13%5E%28n%2B1%29%2B79%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13319580320171118×1332020+1833202135670http://factordb.com/index.php?id=1100000000490878060&open=primehttp://factordb.com/index.php?showid=1100000000490878060&base=13http://factordb.com/index.php?showid=1100000000490878060http://factordb.com/helper.php?id=1100000000490878060http://factordb.com/cert.php?id=1100000000490878060&inputfilehttp://factordb.com/index.php?query=8*13%5E%28n%2B3%29%2B183&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13319695197420(113×13197420−5)/12197421219916http://factordb.com/index.php?id=1100000003943359311&open=primehttp://factordb.com/index.php?showid=1100000003943359311&base=13http://factordb.com/index.php?showid=1100000003943359311(no helper file, since this unproven probable prime is too large (> 10199999) to be PRP-tested in factordb, and factordb does not have helper file for numbers with status (http://factordb.com/status.html, http://factordb.com/distribution.php) "U" (i.e. in http://factordb.com/listtype.php?t=2), factordb only has helper file for numbers with status "P" (i.e. in http://factordb.com/listtype.php?t=4) or "PRP" (i.e. in http://factordb.com/listtype.php?t=1))(no Primo input file, since this unproven probable prime is too large (> 10199999) to be PRP-tested in factordb, and factordb does not have Primo input file for numbers with status (http://factordb.com/status.html, http://factordb.com/distribution.php) "U" (i.e. in http://factordb.com/listtype.php?t=2), factordb only has Primo input file for numbers with status "PRP" (i.e. in http://factordb.com/listtype.php?t=1))http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
133197A3592197A(41×13592198+27)/4592199659677http://factordb.com/index.php?id=1100000005489162806&open=primehttp://factordb.com/index.php?showid=1100000005489162806&base=13http://factordb.com/index.php?showid=1100000005489162806(no helper file, since this unproven probable prime is too large (> 10199999) to be PRP-tested in factordb, and factordb does not have helper file for numbers with status (http://factordb.com/status.html, http://factordb.com/distribution.php) "U" (i.e. in http://factordb.com/listtype.php?t=2), factordb only has helper file for numbers with status "P" (i.e. in http://factordb.com/listtype.php?t=4) or "PRP" (i.e. in http://factordb.com/listtype.php?t=1))(no Primo input file, since this unproven probable prime is too large (> 10199999) to be PRP-tested in factordb, and factordb does not have Primo input file for numbers with status (http://factordb.com/status.html, http://factordb.com/distribution.php) "U" (i.e. in http://factordb.com/listtype.php?t=2), factordb only has Primo input file for numbers with status "PRP" (i.e. in http://factordb.com/listtype.php?t=1))http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
162345DB32234(206×1632234−11)/153223538815http://factordb.com/index.php?id=1100000002383583629&open=primehttp://factordb.com/index.php?showid=1100000002383583629&base=16http://factordb.com/index.php?showid=1100000002383583629http://factordb.com/helper.php?id=1100000002383583629http://factordb.com/cert.php?id=1100000002383583629&inputfilehttp://factordb.com/index.php?query=%28206*16%5En-11%29%2F15&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
162346472785DD(4×1672787+2291)/157278787644http://factordb.com/index.php?id=1100000003615909841&open=primehttp://factordb.com/index.php?showid=1100000003615909841&base=16http://factordb.com/index.php?showid=1100000003615909841http://factordb.com/helper.php?id=1100000003615909841http://factordb.com/cert.php?id=1100000003615909841&inputfilehttp://factordb.com/index.php?query=%284*16%5E%28n%2B2%29%2B2291%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1623473116137AF(16116139+619)/5116139139845http://factordb.com/index.php?id=1100000003851731988&open=primehttp://factordb.com/index.php?showid=1100000003851731988&base=16http://factordb.com/index.php?showid=1100000003851731988http://factordb.com/helper.php?id=1100000003851731988http://factordb.com/cert.php?id=1100000003851731988&inputfilehttp://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
228003BK220015(251×2222002−335)/212200329538http://factordb.com/index.php?id=1100000003594696838&open=primehttp://factordb.com/index.php?showid=1100000003594696838&base=22http://factordb.com/index.php?showid=1100000003594696838http://factordb.com/helper.php?id=1100000003594696838http://factordb.com/cert.php?id=1100000003594696838&inputfilehttp://factordb.com/index.php?query=%28251*22%5E%28n%2B1%29-335%29%2F21&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26252505193916F(2619393+179)/51939327440http://factordb.com/index.php?id=1100000003850151202&open=primehttp://factordb.com/index.php?showid=1100000003850151202&base=26http://factordb.com/index.php?showid=1100000003850151202http://factordb.com/helper.php?id=1100000003850151202http://factordb.com/cert.php?id=1100000003850151202&inputfilehttp://factordb.com/index.php?query=%2826%5E%28n%2B2%29%2B179%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625251720279OL(7×2620281+11393)/252028128697http://factordb.com/index.php?id=1100000003892628605&open=primehttp://factordb.com/index.php?showid=1100000003892628605&base=26http://factordb.com/index.php?showid=1100000003892628605http://factordb.com/helper.php?id=1100000003892628605http://factordb.com/cert.php?id=1100000003892628605&inputfilehttp://factordb.com/index.php?query=%287*26%5E%28n%2B2%29%2B11393%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625252LD0209757559×2620976+72097829684http://factordb.com/index.php?id=1100000003892628658&open=primehttp://factordb.com/index.php?showid=1100000003892628658&base=26http://factordb.com/index.php?showid=1100000003892628658http://factordb.com/helper.php?id=1100000003892628658http://factordb.com/cert.php?id=1100000003892628658&inputfilehttp://factordb.com/index.php?query=559*26%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26252536K233005(34×2623301−79)/52330232972http://factordb.com/index.php?id=1100000003892628745&open=primehttp://factordb.com/index.php?showid=1100000003892628745&base=26http://factordb.com/index.php?showid=1100000003892628745http://factordb.com/helper.php?id=1100000003892628745http://factordb.com/cert.php?id=1100000003892628745&inputfilehttp://factordb.com/index.php?query=%2834*26%5E%28n%2B1%29-79%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625254J044303KCB19×2644306+138434430762694http://factordb.com/index.php?id=1100000003968156595&open=primehttp://factordb.com/index.php?showid=1100000003968156595&base=26http://factordb.com/index.php?showid=1100000003968156595http://factordb.com/helper.php?id=1100000003968156595http://factordb.com/cert.php?id=1100000003968156595&inputfilehttp://factordb.com/index.php?query=19*26%5E%28n%2B3%29%2B13843&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625255M0611862BB22×2661189+16496119086583http://factordb.com/index.php?id=1100000003968169875&open=primehttp://factordb.com/index.php?showid=1100000003968169875&base=26http://factordb.com/index.php?showid=1100000003968169875http://factordb.com/helper.php?id=1100000003968169875http://factordb.com/cert.php?id=1100000003968169875&inputfilehttp://factordb.com/index.php?query=22*26%5E%28n%2B3%29%2B1649&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825526N624051LR(209×2824053+3967)/92405434810http://factordb.com/index.php?id=1100000003879667576&open=primehttp://factordb.com/index.php?showid=1100000003879667576&base=28http://factordb.com/index.php?showid=1100000003879667576http://factordb.com/helper.php?id=1100000003879667576http://factordb.com/cert.php?id=1100000003879667576&inputfilehttp://factordb.com/index.php?query=%28209*28%5E%28n%2B2%29%2B3967%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28255275OA31238F(4438×2831239+125)/273124145210http://factordb.com/index.php?id=1100000003880455200&open=primehttp://factordb.com/index.php?showid=1100000003880455200&base=28http://factordb.com/index.php?showid=1100000003880455200http://factordb.com/helper.php?id=1100000003880455200http://factordb.com/cert.php?id=1100000003880455200&inputfilehttp://factordb.com/index.php?query=%284438*28%5E%28n%2B1%29%2B125%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825528O4O945359(6092×2894536−143)/994538136812http://factordb.com/index.php?id=1100000000808118231&open=primehttp://factordb.com/index.php?showid=1100000000808118231&base=28http://factordb.com/index.php?showid=1100000000808118231http://factordb.com/helper.php?id=1100000000808118231http://factordb.com/cert.php?id=1100000000808118231&inputfilehttp://factordb.com/index.php?query=%286092*28%5E%28n%2B1%29-143%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
302618I024608D18×3024609+132461036352http://factordb.com/index.php?id=1100000003593967511&open=primehttp://factordb.com/index.php?showid=1100000003593967511&base=30http://factordb.com/index.php?showid=1100000003593967511http://factordb.com/helper.php?id=1100000003593967511http://factordb.com/cert.php?id=1100000003593967511&inputfilehttp://factordb.com/index.php?query=18*30%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36352847K26567Z(53×3626568+101)/72656941349http://factordb.com/index.php?id=1100000003896952461&open=primehttp://factordb.com/index.php?showid=1100000003896952461&base=36http://factordb.com/index.php?showid=1100000003896952461http://factordb.com/helper.php?id=1100000003896952461http://factordb.com/cert.php?id=1100000003896952461&inputfilehttp://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635285S0750078H28×3675009+30575010116739http://factordb.com/index.php?id=1100000004020085177&open=primehttp://factordb.com/index.php?showid=1100000004020085177&base=36http://factordb.com/index.php?showid=1100000004020085177http://factordb.com/helper.php?id=1100000004020085177http://factordb.com/cert.php?id=1100000004020085177&inputfilehttp://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635286P81993SZ(5×3681995+821)/781995127609http://factordb.com/index.php?id=1100000002394962083&open=primehttp://factordb.com/index.php?showid=1100000002394962083&base=36http://factordb.com/index.php?showid=1100000002394962083http://factordb.com/helper.php?id=1100000002394962083http://factordb.com/cert.php?id=1100000002394962083&inputfilehttp://factordb.com/index.php?query=%285*36%5E%28n%2B2%29%2B821%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show

All these numbers are strong probable primes (https://t5k.org/prove/prove2_3.html, https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test, https://en.wikipedia.org/wiki/Strong_pseudoprime, https://t5k.org/glossary/xpage/MillersTest.html, https://t5k.org/glossary/xpage/StrongPRP.html, https://www.rieselprime.de/ziki/Miller-Rabin_pseudoprimality_test, https://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html, https://mathworld.wolfram.com/StrongPseudoprime.html, http://www.numericana.com/answer/pseudo.htm#rabin, http://www.numericana.com/answer/pseudo.htm#strong, http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm, http://ntheory.org/data/spsps.txt, https://faculty.lynchburg.edu/~nicely/misc/mpzspsp.html, https://miller-rabin.appspot.com/, http://www.pi-e.de/Miller-Rabin-Pseudoprimzahlen.htm (in German), http://factordb.com/prooffailed.php, https://sites.google.com/view/strong-pseudoprime, https://sites.google.com/view/bases-strong-pseudoprime, https://oeis.org/A001262, https://oeis.org/A020229, https://oeis.org/A020230, https://oeis.org/A020231, https://oeis.org/A020232, https://oeis.org/A020233, https://oeis.org/A020234, https://oeis.org/A020235, https://oeis.org/A020236, https://oeis.org/A020237, https://oeis.org/A020238, https://oeis.org/A020239, https://oeis.org/A020240, https://oeis.org/A020241, https://oeis.org/A020242, https://oeis.org/A020243, https://oeis.org/A020244, https://oeis.org/A020245, https://oeis.org/A020246, https://oeis.org/A020247, https://oeis.org/A020248, https://oeis.org/A020249, https://oeis.org/A020250, https://oeis.org/A020251, https://oeis.org/A020252, https://oeis.org/A020253, https://oeis.org/A020254, https://oeis.org/A020255, https://oeis.org/A020256, https://oeis.org/A020257, https://oeis.org/A020258, https://oeis.org/A020259, https://oeis.org/A020260, https://oeis.org/A020261, https://oeis.org/A020262, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773, https://oeis.org/A014233, https://oeis.org/A006945, https://oeis.org/A089825, https://oeis.org/A181782, https://oeis.org/A071294, https://oeis.org/A141768, https://oeis.org/A195328, https://oeis.org/A329759, https://oeis.org/A298756) to all prime bases ≤ 64 (i.e. bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61) (see https://oeis.org/A014233 and https://oeis.org/A141768 and https://oeis.org/A001262 and https://oeis.org/A074773 and http://ntheory.org/data/spsps.txt), and strong Lucas probable primes (https://en.wikipedia.org/wiki/Strong_Lucas_pseudoprime, https://mathworld.wolfram.com/StrongLucasPseudoprime.html) with parameters (P, Q) defined by Selfridge's Method A (see https://oeis.org/A217255 and http://ntheory.org/data/slpsps-baillie.txt and http://www.hoegge.dk/lucasselfridgeprps.txt), and trial factored (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) to 1016 (i.e. the least prime factors (http://mathworld.wolfram.com/LeastPrimeFactor.html, https://oeis.org/A020639) of all these numbers are larger than 1016) (i.e. all these numbers are 1016-rough numbers (https://en.wikipedia.org/wiki/Rough_number, https://mathworld.wolfram.com/RoughNumber.html, https://oeis.org/A007310, https://oeis.org/A007775, https://oeis.org/A008364, https://oeis.org/A008365, https://oeis.org/A008366, https://oeis.org/A166061, https://oeis.org/A166063)), thus, all these numbers are Baillie–PSW probable primes (https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test, https://mathworld.wolfram.com/Baillie-PSWPrimalityTest.html, http://pseudoprime.com/dopo.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_318.pdf)), and no composites < 264 pass the Baillie–PSW probable prime test (see http://ntheory.org/pseudoprimes.html (the box "#BPSW") and https://faculty.lynchburg.edu/~nicely/misc/bpsw.html), thus if one of these numbers is in fact composite, it will be a pseudoprime to the Baillie–PSW probable prime test, which currently no single example is known!

The unsolved families for bases b = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36 (the "easy" bases (bases b with ≤ 150 minimal primes > 10299 (base b = 26 has 82 known minimal (probable) primes > 10299 and 4 unsolved families, base b = 36 has 75 known minimal (probable) primes > 10299 and 4 unsolved families, base b = 17 has 99 known minimal (probable) primes > 10299 and 18 unsolved families, base b = 21 has 80 known minimal (probable) primes > 10299 and 12 unsolved families, base b = 19 has 201 known minimal (probable) primes > 10299 and 23 unsolved families))) and the factorization of the numbers in these families:

(you can calculate "equivalent searching limit of length in decimal" by: "current searching limit of length of this family" × log(b), where log is the common logarithm (https://en.wikipedia.org/wiki/Common_logarithm, https://mathworld.wolfram.com/CommonLogarithm.html), i.e. the logarithm with base 10)

(for the factorization of the numbers in these families, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=snfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#smallpolynomial, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=gnfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) may be used, they have SNFS polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt and https://alfredreichlg.de/10w7/10w7.txt and https://web.archive.org/web/20020320010222/http://proth.cjb.net/ and http://web.archive.org/web/20111104173105/http://www.immortaltheory.com/NumberTheory/2nm3_db.txt and https://www.asahi-net.or.jp/~KC2H-MSM/mathland/aba/index.htm and https://www.alpertron.com.ar/BRILLIANT.HTM and https://www.alpertron.com.ar/BRILLIANT3.HTM and https://www.alpertron.com.ar/BRILLIANT4.HTM and https://www.alpertron.com.ar/BRILLIANT2.HTM and https://oeis.org/wiki/Factors_of_33*2%5En%2B1 and https://oeis.org/wiki/Factors_of_33*2%5En-1 and https://web.archive.org/web/20111018190410/http://www.sr5.psp-project.de/s5stats.html (section "Top ten factors:") and https://web.archive.org/web/20111018190339/http://www.sr5.psp-project.de/r5stats.html (section "Top ten factors:") and https://brnikat.com/nums/cullen_woodall/cw.html and https://oeis.org/wiki/OEIS_sequences_needing_factors#Near_powers.2C_factorials.2C_and_primorials (sections "near-powers with b = 2" and "near-powers with b = 3" and "near-powers with b = 5" and "near-powers with b = 6" and "near-powers with b = 7" and "near-powers with b = 10" and "near-powers with b > 10") and https://www.mersenneforum.org/showthread.php?t=9554 and https://www.mersenneforum.org/showthread.php?t=9167 and https://www.mersenneforum.org/showpost.php?p=644144&postcount=5, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm and https://www.mersenneforum.org/showthread.php?t=12962)

(link to the factorization of the first 200 numbers in these unsolved families (start with the smallest number in these families which is > b), to show the small prime factors (and their period) (i.e. show whether these numbers are rough numbers) and the algebraic factors (and their period) and let you feel the Nash weight (or difficulty) of these families (high or low), all small prime factors (< 1012, by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172)) and all algebraic factors (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/53335.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/83336.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=algebraic&maxrows=10000, https://sites.google.com/view/algebraic-factors-of-xn-kyn, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://www.mersenneforum.org/showpost.php?p=96560&postcount=99, https://www.mersenneforum.org/showpost.php?p=96651&postcount=101, https://www.mersenneforum.org/showthread.php?t=21916, https://www.mersenneforum.org/showpost.php?p=196598&postcount=492, https://www.mersenneforum.org/showpost.php?p=203083&postcount=149, https://www.mersenneforum.org/showpost.php?p=206065&postcount=192, https://www.mersenneforum.org/showpost.php?p=208044&postcount=260, https://www.mersenneforum.org/showpost.php?p=210533&postcount=336, https://www.mersenneforum.org/showpost.php?p=452132&postcount=66, https://www.mersenneforum.org/showpost.php?p=451337&postcount=32, https://www.mersenneforum.org/showpost.php?p=208852&postcount=227, https://www.mersenneforum.org/showpost.php?p=232904&postcount=604, https://www.mersenneforum.org/showpost.php?p=383690&postcount=1, https://www.mersenneforum.org/showpost.php?p=207886&postcount=253, https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php, https://www.alpertron.com.ar/POLFACT.HTM, https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) of the first 200 numbers (start with the smallest n making the number > b (if n = 0 already makes the number > b, then start with n = 0)) in these unsolved families were added to factordb, unfortunately, none of these numbers have algebraic factors)

b
(2 ≤ b ≤ 36)
base-b form of the unsolved familyalgebraic ((a×bn+c)/gcd(a+c,b−1)) form of the unsolved familycurrent searching limit of length of this familyfactorization of the first 200 numbers in this family (n is the number of digits in the "{}", start with the smallest n making the number > b (if n = 0 already makes the number > b, then start with n = 0))
171{7}(23×17n−7)/16100000http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
171F{0}732×17n+1+7100000http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
174{7}A(71×17n+1+41)/16100000http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
174{9}(73×17n−9)/16100000http://factordb.com/index.php?query=%2873*17%5En-9%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1751{0}D86×17n+1+13100000http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1770F{0}D2038×17n+1+13100000http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
178{B}9(139×17n+1−43)/16100000http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
179{5}9(149×17n+1+59)/16100000http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1795{F}(2543×17n−15)/16100000http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1797{0}1160×17n+1+1100000http://factordb.com/index.php?query=160*17%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17A{D}F(173×17n+1+19)/16100000http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17B{0}B311×17n+2+190100000http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17B{0}DB11×17n+2+232100000http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17{B}E9(11×17n+2+773)/16100000http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B773%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17{B}EE(11×17n+2+853)/16100000http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B853%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17F1{9}(4105×17n−9)/16100000http://factordb.com/index.php?query=%284105*17%5En-9%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17F7{0}1262×17n+1+1100000http://factordb.com/index.php?query=262*17%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17FD0{D}(72909×17n−13)/16100000http://factordb.com/index.php?query=%2872909*17%5En-13%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
212{7}9D(47×21n+2+953)/20100000http://factordb.com/index.php?query=%2847*21%5E%28n%2B2%29%2B953%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
212F{C}A(288×21n+1−13)/5100000http://factordb.com/index.php?query=%28288*21%5E%28n%2B1%29-13%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
215{0}DJ5×21n+2+292100000http://factordb.com/index.php?query=5*21%5E%28n%2B2%29%2B292&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
21{7}ID(7×21n+2+4733)/20100000http://factordb.com/index.php?query=%287*21%5E%28n%2B2%29%2B4733%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
21{9}D(9×21n+1+71)/20100000http://factordb.com/index.php?query=%289*21%5E%28n%2B1%29%2B71%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
21B3{0}EB234×21n+2+305100000http://factordb.com/index.php?query=234*21%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
21B{H}6H(237×21n+2−4637)/20100000http://factordb.com/index.php?query=%28237*21%5E%28n%2B2%29-4637%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
21C{F}0K(51×21n+2−1243)/4100000http://factordb.com/index.php?query=%2851*21%5E%28n%2B2%29-1243%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
21F{9}D(309×21n+1+71)/20100000http://factordb.com/index.php?query=%28309*21%5E%28n%2B1%29%2B71%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
21{F}35(3×21n+2−1051)/4100000http://factordb.com/index.php?query=%283*21%5E%28n%2B2%29-1051%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
21G{0}FK16×21n+2+335100000http://factordb.com/index.php?query=16*21%5E%28n%2B2%29%2B335&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
21H{0}777117×21n+4+68062100000http://factordb.com/index.php?query=17*21%5E%28n%2B4%29%2B68062&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2685{M}B(5347×26n+1−297)/25100000http://factordb.com/index.php?query=%285347*26%5E%28n%2B1%29-297%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26{A}6F(2×26n+2−497)/5486722http://factordb.com/index.php?query=%282*26%5E%28n%2B2%29-497%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26{H}MH(17×26n+2+3233)/25100000http://factordb.com/index.php?query=%2817*26%5E%28n%2B2%29%2B3233%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26{I}GL(18×26n+2−1243)/25497679http://factordb.com/index.php?query=%2818*26%5E%28n%2B2%29-1243%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28O{A}F(658×28n+1+125)/27543203http://factordb.com/index.php?query=(658*28^(n%2B1)%2B125)/27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36B{0}EUV11×36n+3+19255100000http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36HM{0}N634×36n+1+23100000http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36N{0}YYN23×36n+3+45311100000http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36O{L}Z(123×36n+1+67)/5100000http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show

The large proven primes (> 10299) for bases b = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36 (the "easy" bases (bases b with ≤ 150 minimal primes > 10299 (base b = 26 has 82 known minimal (probable) primes > 10299 and 4 unsolved families, base b = 36 has 75 known minimal (probable) primes > 10299 and 4 unsolved families, base b = 17 has 99 known minimal (probable) primes > 10299 and 18 unsolved families, base b = 21 has 80 known minimal (probable) primes > 10299 and 12 unsolved families, base b = 19 has 201 known minimal (probable) primes > 10299 and 23 unsolved families))) and their primality certificates (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://alfredreichlg.de/cert/certificates.tpm.html, https://alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php, http://5.199.134.130/certificates.tar.xz, http://5.199.134.130/certificates.tar.xz.SHA256SUM, http://5.199.134.130/certificates.tar.xz.par2, http://5.199.134.130/certificates.tar.xz.vol00+10.par2, http://5.199.134.130/certificates/) and the factorization of the numbers in their corresponding families:

(In progess to add bases b = 17 and b = 21)

(for the factorization of the numbers in these families and the N−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) and N+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) of these primes, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=snfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#smallpolynomial, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=gnfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) may be used, they have SNFS polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt and https://alfredreichlg.de/10w7/10w7.txt and https://web.archive.org/web/20020320010222/http://proth.cjb.net/ and http://web.archive.org/web/20111104173105/http://www.immortaltheory.com/NumberTheory/2nm3_db.txt and https://www.asahi-net.or.jp/~KC2H-MSM/mathland/aba/index.htm and https://www.alpertron.com.ar/BRILLIANT.HTM and https://www.alpertron.com.ar/BRILLIANT3.HTM and https://www.alpertron.com.ar/BRILLIANT4.HTM and https://www.alpertron.com.ar/BRILLIANT2.HTM and https://oeis.org/wiki/Factors_of_33*2%5En%2B1 and https://oeis.org/wiki/Factors_of_33*2%5En-1 and https://web.archive.org/web/20111018190410/http://www.sr5.psp-project.de/s5stats.html (section "Top ten factors:") and https://web.archive.org/web/20111018190339/http://www.sr5.psp-project.de/r5stats.html (section "Top ten factors:") and https://brnikat.com/nums/cullen_woodall/cw.html and https://oeis.org/wiki/OEIS_sequences_needing_factors#Near_powers.2C_factorials.2C_and_primorials (sections "near-powers with b = 2" and "near-powers with b = 3" and "near-powers with b = 5" and "near-powers with b = 6" and "near-powers with b = 7" and "near-powers with b = 10" and "near-powers with b > 10") and https://www.mersenneforum.org/showthread.php?t=9554 and https://www.mersenneforum.org/showthread.php?t=9167 and https://www.mersenneforum.org/showpost.php?p=644144&postcount=5, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm and https://www.mersenneforum.org/showthread.php?t=12962)

(link to the factorization of the first 200 numbers in corresponding families of these minimal primes (start with the smallest number in these families which is > b), to show the small prime factors (and their period) (i.e. show whether these numbers are rough numbers) and the algebraic factors (and their period) and let you feel the Nash weight (or difficulty) of these families (high or low), all small prime factors (< 1012, by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172)) and all algebraic factors (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/53335.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/83336.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=algebraic&maxrows=10000, https://sites.google.com/view/algebraic-factors-of-xn-kyn, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://www.mersenneforum.org/showpost.php?p=96560&postcount=99, https://www.mersenneforum.org/showpost.php?p=96651&postcount=101, https://www.mersenneforum.org/showthread.php?t=21916, https://www.mersenneforum.org/showpost.php?p=196598&postcount=492, https://www.mersenneforum.org/showpost.php?p=203083&postcount=149, https://www.mersenneforum.org/showpost.php?p=206065&postcount=192, https://www.mersenneforum.org/showpost.php?p=208044&postcount=260, https://www.mersenneforum.org/showpost.php?p=210533&postcount=336, https://www.mersenneforum.org/showpost.php?p=452132&postcount=66, https://www.mersenneforum.org/showpost.php?p=451337&postcount=32, https://www.mersenneforum.org/showpost.php?p=208852&postcount=227, https://www.mersenneforum.org/showpost.php?p=232904&postcount=604, https://www.mersenneforum.org/showpost.php?p=383690&postcount=1, https://www.mersenneforum.org/showpost.php?p=207886&postcount=253, https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php, https://www.alpertron.com.ar/POLFACT.HTM, https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) of the N−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) and N+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) of these primes and the first 200 numbers (start with the smallest n making the number > b (if n = 0 already makes the number > b, then start with n = 0)) in corresponding families of these primes were added to factordb (for the examples of the large primes whose N−1 or/and N+1 has algebraic factors, see https://stdkmd.net/nrr/cert/1/#CERT_11101_4809 and https://stdkmd.net/nrr/cert/1/#CERT_15551_2197 and https://stdkmd.net/nrr/cert/1/#CERT_16667_4296 and https://stdkmd.net/nrr/cert/2/#CERT_20111_2692 and https://stdkmd.net/nrr/cert/2/#CERT_23309_10029 and https://stdkmd.net/nrr/cert/3/#CERT_37773_15768 and https://stdkmd.net/nrr/cert/6/#CERT_6805W7_3739 and https://stdkmd.net/nrr/cert/6/#CERT_68883_5132 and https://stdkmd.net/nrr/cert/7/#CERT_79921_11629 and https://stdkmd.net/nrr/cert/8/#CERT_80081_5736 and https://stdkmd.net/nrr/cert/8/#CERT_83W16W7_543 and https://stdkmd.net/nrr/cert/9/#CERT_93307_2197 for the related numbers (although some of them are related to Cunningham numbers, and some of them has N−1 and N+1 does not have algebraic factors but has a large prime factor), e.g. "11101_4809" (decimal (base b = 10) form: 1480701, algebraic form: (104809−91)/9) is related to "Phi_4807_10" (the number Φ4807(10), where Φ is the cyclotomic polynomial), "15551_2197" (decimal (base b = 10) form: 1521961, algebraic form: (14×102197−41)/9, the prime is a cofactor of it (divided it by 11×23×167)) is related to "93307_2197" (decimal (base b = 10) form: 93219507, algebraic form: (28×102197−79)/3), "16667_4296" (decimal (base b = 10) form: 1642957, algebraic form: (5×104296+1)/3, the prime is a cofactor of it (divided it by 347×821×140235709×806209146522749)) is related to "33337_12891" (decimal (base b = 10) form: 3128907, algebraic form: (1012891+11)/3), "20111_2692" (decimal (base b = 10) form: 2012692, algebraic form: (181×102692−1)/9, the prime is a cofactor of it (divided it by 3×43)) is related to "20111_2693" (decimal (base b = 10) form: 2012693, algebraic form: (181×102693−1)/9), "23309_10029" (decimal (base b = 10) form: 231002709, algebraic form: (7×1010029−73)/3) is related to "Phi_5014_10" (the number Φ5014(10), where Φ is the cyclotomic polynomial), "37773_15768" (decimal (base b = 10) form: 37157673, algebraic form: (34×1015768−43)/9) is related to "Phi_7884_10" (the number Φ7884(10), where Φ is the cyclotomic polynomial), "6805w7_3739" (decimal (base b = 10) form: 680537387, algebraic form: (6125×103739+13)/9, the prime is a cofactor of it (divided it by 32)) is related to "27227_3741" (decimal (base b = 10) form: 27237407, algebraic form: (245×103741+43)/9), "68883_5132" (decimal (base b = 10) form: 6851313, algebraic form: (62×105132−53)/9) is related to "Phi_1283_10" (the number Φ1283(10), where Φ is the cyclotomic polynomial), "79921_11629" (decimal (base b = 10) form: 791162721, algebraic form: 8×1011629−79) is related to "Phi_2907_10" (the number Φ2907(10), where Φ is the cyclotomic polynomial), "80081_5736" (decimal (base b = 10) form: 80573481, algebraic form: 8×105736+81) is related to "Phi_11470_10" (the number Φ11470(10), where Φ is the cyclotomic polynomial), "83w16w7_543" (decimal (base b = 10) form: 83542165427, algebraic form: (25×101086−5×10543+1)/3, the prime is a cofactor of it (divided it by 7×109×563041×869047141×147372142447)) is related to "11103_3258" (decimal (base b = 10) form: 1325603, algebraic form: (103258−73)/9), etc. the N−1 of "11101_4809" is 100 × R4807(10) (which is equivalent to the Cunningham number 104807−1) and Φ4807(10) is an algebraic factor of the Cunningham number 104807−1, the N−1 of "93307_2197" is 6 × "15551_2197", the N−1 of "33337_12891" has sum-of-two-cubes factorization and an algebraic factor is 2 × "16667_4296", the N−1 of "20111_2693" is 10 × "20111_2692", the N+1 of "23309_10029" is 210 × R10028(10) (which is equivalent to the Cunningham number 1010028−1) and Φ5014(10) is an algebraic factor of the Cunningham number 1010028−1, the N+1 of "37773_15768" is 34 × R15768(10) (which is equivalent to the Cunningham number 1015768−1) and Φ7884(10) is an algebraic factor of the Cunningham number 1015768−1, the N+1 of "27227_3741" is 4 × "6805w7_3739", the N−1 of "68883_5132" is 62 × R5132(10) (which is equivalent to the Cunningham number 105132−1) and Φ1283(10) is an algebraic factor of the Cunningham number 105132−1, the N−1 of "79921_11629" is 720 × R11628(10) (which is equivalent to the Cunningham number 1011628−1) and Φ2907(10) is an algebraic factor of the Cunningham number 1011628−1, the N−1 of "80081_5736" is 80 × S5735(10) (which is equivalent to the Cunningham number 105735+1) and Φ11470(10) is an algebraic factor of the Cunningham number 105735+1, the N+1 of "11103_3258" has difference-of-two-6th-powers factorization and an algebraic factor is 4 × "83w16w7_543", etc.), unfortunately, the only numbers having algebraic factors (other than trivially (https://en.wikipedia.org/wiki/Triviality_(mathematics), https://mathworld.wolfram.com/Trivial.html) fully factored (i.e. primes of the form k×bn±1, with small k) and Cunningham numbers (of the form bn±1, see https://en.wikipedia.org/wiki/Cunningham_number, https://mathworld.wolfram.com/CunninghamNumber.html, https://www.numbersaplenty.com/set/Cunningham_number/, https://en.wikipedia.org/wiki/Cunningham_Project, https://t5k.org/glossary/xpage/CunninghamProject.html, https://www.rieselprime.de/ziki/Cunningham_project, https://oeis.org/wiki/OEIS_sequences_needing_factors#Cunningham_numbers (sections "b = 2" and "b = 3" and "b = 10" and "other integer b"), https://homes.cerias.purdue.edu/~ssw/cun/index.html, https://maths-people.anu.edu.au/~brent/factors.html, https://web.archive.org/web/20190315214330/http://cage.ugent.be/~jdemeyer/cunningham/, http://myfactors.mooo.com/, https://doi.org/10.1090/conm/022, https://www.mersenneforum.org/attachment.php?attachmentid=7727&d=1330555980 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_7.pdf), https://homes.cerias.purdue.edu/~ssw/cun/mine.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_6.pdf), http://homes.cerias.purdue.edu/~ssw/cun1.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_71.pdf)), which are in the "README" file of the "primality-certificates" folder: https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/primality-certificates/README.md) are 17n+2+32 (the corresponding family of the 10400th minimal prime in base b = 17, 1090191F = 179021+32) with n == 3 mod 5 (which has sum-of-two-5th-powers factorization and can be factored to (17(n+2)/5+2) × (174×(n+2)/5−2×173×(n+2)/5+4×172×(n+2)/5−8×17(n+2)/5+16)) and 8×18299+12 (the N+1 of the 547th minimal prime in base b = 18, 80298B = 8×18299+11) (which has sum-of-two-cubes factorization and can be factored to 12 × (6×1899+1) × (2×18199−6×1899+1)) and 18768−36 (the N+1 of the 548th minimal prime in base b = 18, H766FH = 18768−37) (which has difference-of-two-squares factorization and can be factored to (18384−6) × (18384+6)) and (16×20n+2−2809)/19 (the corresponding family of the 3307th minimal prime in base b = 20, G44799 = (16×20449−2809)/19) with even n (which has difference-of-two-squares factorization and can be factored to (4×20(n+2)/2−53) × (4×20(n+2)/2+53) / 19) and 9×21297+243 (the N−1 of the 13316th minimal prime in base b = 21, 90295BD = 9×21297+244) (which has sum-of-two-cubes factorization and can be factored to 9 × (2199+3) × (21198−3×2199+9)) and (22n+2−289)/3 (the corresponding family of the 8002nd minimal prime in base b = 22, 738152L = (223817−289)/3) with even n (which has difference-of-two-squares factorization and can be factored to (22(n+2)/2−17) × (22(n+2)/2+17) / 3) and (121×26n−1)/5 (the corresponding family of the 25235th minimal prime in base b = 26, O51509 = (121×261509−1)/5) with even n (which has difference-of-two-squares factorization and can be factored to (11×26n/2−1) × (11×26n/2+1) / 5) and 25×30n−1 (the corresponding family of the 2619th minimal prime in base b = 30, OT34205 = 25×3034205−1) with even n (which has difference-of-two-squares factorization and can be factored to (5×30n/2−1) × (5×30n/2+1)) and 27×36459+8 (the N+1 of the 35244th minimal prime in base b = 36, R04587 = 27×36459+7) (which has sum-of-two-cubes factorization and can be factored to (3×36153+2) × (9×36306−6×36153+4)) (of course, 13n−49 (the N+1 of the 3193rd minimal prime in base b = 13, C1063192 = 1310633−50, is 1310633−49) has difference-of-two-squares factorization (factored to (13n/2−7) × (13n/2+7)) if n is even, but 10633 is odd) (of course, (9×17n−121)/16 (the N+1 of the 10320th minimal prime in base b = 17, 92921 = (9×17293−137)/16, is (9×17293−121)/16) has difference-of-two-squares factorization (factored to (3×17n/2−11) × (3×17n/2+11) / 16) if n is even, but 293 is odd) (of course, 22n−128 (the N+1 of the 7995th minimal prime in base b = 22, L483G3 = 22485−129, is 22485−128) has difference-of-two-7th-powers factorization (factored to (22n/7−2) × (22n/7+2×22n/7+4×22n/7+8×22n/7+16×22n/7+32×22n/7+64)) if n is divisible by 7, but 485 is not divisible by 7) (of course, 2×24n+6 (the N−1 of the 3403rd minimal prime in base b = 24, 203137 = 2×24314+7, is 2×24314+6) has sum-of-two-cubes factorization (factored to 6 × (2×24(n−1)/3+1) × (4×242×(n−1)/3−2×24(n−1)/3+1)) if n == 1 mod 3, but 314 is not == 1 mod 3) (of course, 2×24n+8 (the N+1 of the 3403rd minimal prime in base b = 24, 203137 = 2×24314+7, is 2×24314+8) has Aurifeuillean factorization of x4+4×y4 (factored to 2 × (24n/2−2×24n/4+2) × (24n/2+2×24n/4+2)) if n is divisible by 4, but 314 is not divisible by 4) (of course, 4×13n+1 has Aurifeuillean factorization of x4+4×y4 (factored to (2×13n/2−2×13n/4+1) × (2×13n/2+2×13n/4+1)) if n is divisible by 4, but 16×13n+1 (which is a factor of 128×13n+8, the N−1 of the 3173rd minimal prime in base b = 13, 9B03919 = 128×13392+9, is 128×13392+8) has no algebraic factors for any n, xn+16×yn has no algebraic factors for any n (while xn+4×yn has algebraic factors for n divisible by 4, xn+8×yn has algebraic factors for n divisible by 3, xn+32×yn has algebraic factors for n divisible by 5, xn+64×yn has algebraic factors for n divisible by 3 or/and 4), for the references see https://stdkmd.net/nrr/5/50008.htm and https://stdkmd.net/nrr/8/80005.htm, neither of them has the "Algebraic factorization" section) (of course, 22n+4 has Aurifeuillean factorization of x4+4×y4 (factored to (22n/2−2×22n/4+2) × (22n/2+2×22n/4+2)) if n is divisible by 4, but 22n+16 (which is a factor of 20×22n+1+7040, the N−1 of the 7998th minimal prime in base b = 22, K0760EC1 = 20×22763+7041, is 20×22763+7040) has no algebraic factors for any n, xn+16×yn has no algebraic factors for any n (while xn+4×yn has algebraic factors for n divisible by 4, xn+8×yn has algebraic factors for n divisible by 3, xn+32×yn has algebraic factors for n divisible by 5, xn+64×yn has algebraic factors for n divisible by 3 or/and 4), for the references see https://stdkmd.net/nrr/5/50008.htm and https://stdkmd.net/nrr/8/80005.htm, neither of them has the "Algebraic factorization" section))

(if the prime is (a×bn+c)/gcd(a+c,b−1) (with a ≥ 1, b ≥ 2, c ≠ 0, gcd(a,c) = 1, gcd(b,c) = 1), then its N−1 is (a×bn+cgcd(a+c,b−1))/gcd(a+c,b−1), and its N+1 is (a×bn+c+gcd(a+c,b−1))/gcd(a+c,b−1), although (a×bn+c)/gcd(a+c,b−1) (with a ≥ 1, b ≥ 2, c ≠ 0, gcd(a,c) = 1, gcd(b,c) = 1) has algebraic factorization if and only if either "there is an integer r > 1 such that a×bn and −c are both rth powers" (in this case, (a×bn+c)/gcd(a+c,b−1) has sum-of-two-rth-powers factorization if c > 0, or difference-of-two-rth-powers factorization if c < 0, although there is no "sum-of-two-rth-powers factorization" for even r, but no such situation (i.e. c > 0 and r is even) exists, since if c > 0 then −c < 0, but negative numbers cannot be squares, however, if r is even then all rth powers are squares (since if s divides r, then all rth powers are sth powers), thus, −c cannot be an rth power if c > 0 and r is even) or "one of a×bn and c is a 4th power, and the other is of the form 4×m4" (in this case, (a×bn+c)/gcd(a+c,b−1) has Aurifeuillean factorization of x4+4×y4), however, since cgcd(a+c,b−1) and c+gcd(a+c,b−1) may not coprime to a and b, it is not that simple to known whether (a×bn+cgcd(a+c,b−1))/gcd(a+c,b−1) and (a×bn+c+gcd(a+c,b−1))/gcd(a+c,b−1) have algebraic factorization (however, if there is a prime p divides a but not divides b or cgcd(a+c,b−1), and p2 does not divide a, then (a×bn+cgcd(a+c,b−1))/gcd(a+c,b−1) does not have algebraic factorization, also, if there is a prime p divides a but not divides either b or c+gcd(a+c,b−1), and p2 does not divide a, then (a×bn+c+gcd(a+c,b−1))/gcd(a+c,b−1) does not have algebraic factorization, also, if there is a prime p divides cgcd(a+c,b−1) but not divides a or b, and p2 does not divide cgcd(a+c,b−1), then (a×bn+cgcd(a+c,b−1))/gcd(a+c,b−1) does not have algebraic factorization, also, if there is a prime p divides c+gcd(a+c,b−1) but not divides either a or b, and p2 does not divide c+gcd(a+c,b−1), then (a×bn+c+gcd(a+c,b−1))/gcd(a+c,b−1) does not have algebraic factorization), (a×bn+cgcd(a+c,b−1))/gcd(a+c,b−1) has algebraic factorization if and only if either "cgcd(a+c,b−1) = 0" (if and only if cgcd(a+c,b−1) = 0, then (a×bn+cgcd(a+c,b−1))/gcd(a+c,b−1) is trivially fully factored) or "there is an integer r > 1 such that a×bn/gcd(a×bn,cgcd(a+c,b−1)) and −(cgcd(a+c,b−1))/gcd(a×bn,cgcd(a+c,b−1)) are both rth powers" (in this case, (a×bn+cgcd(a+c,b−1))/gcd(a+c,b−1) has sum-of-two-rth-powers factorization if (cgcd(a+c,b−1))/gcd(a×bn,cgcd(a+c,b−1)) > 0, or difference-of-two-rth-powers factorization if (cgcd(a+c,b−1))/gcd(a×bn,cgcd(a+c,b−1)) < 0, although there is no "sum-of-two-rth-powers factorization" for even r, but no such situation (i.e. (cgcd(a+c,b−1))/gcd(a×bn,cgcd(a+c,b−1)) > 0 and r is even) exists, since if (cgcd(a+c,b−1))/gcd(a×bn,cgcd(a+c,b−1)) > 0 then −(cgcd(a+c,b−1))/gcd(a×bn,cgcd(a+c,b−1)) < 0, but negative numbers cannot be squares, however, if r is even then all rth powers are squares (since if s divides r, then all rth powers are sth powers), thus, −(cgcd(a+c,b−1))/gcd(a×bn,cgcd(a+c,b−1)) cannot be an rth power if (cgcd(a+c,b−1))/gcd(a×bn,cgcd(a+c,b−1)) > 0 and r is even) or "one of a×bn/gcd(a×bn,cgcd(a+c,b−1)) and (cgcd(a+c,b−1))/gcd(a×bn,cgcd(a+c,b−1)) is a 4th power, and the other is of the form 4×m4" (in this case, (a×bn+cgcd(a+c,b−1))/gcd(a+c,b−1) has Aurifeuillean factorization of x4+4×y4), (a×bn+c+gcd(a+c,b−1))/gcd(a+c,b−1) has algebraic factorization if and only if either "c+gcd(a+c,b−1) = 0" (if and only if c+gcd(a+c,b−1) = 0, then (a×bn+c+gcd(a+c,b−1))/gcd(a+c,b−1) is trivially fully factored) or "there is an integer r > 1 such that a×bn/gcd(a×bn,c+gcd(a+c,b−1)) and −(c+gcd(a+c,b−1))/gcd(a×bn,c+gcd(a+c,b−1)) are both rth powers" (in this case, (a×bn+c+gcd(a+c,b−1))/gcd(a+c,b−1) has sum-of-two-rth-powers factorization if (c+gcd(a+c,b−1))/gcd(a×bn,c+gcd(a+c,b−1)) > 0, or difference-of-two-rth-powers factorization if (c+gcd(a+c,b−1))/gcd(a×bn,c+gcd(a+c,b−1)) < 0, although there is no "sum-of-two-rth-powers factorization" for even r, but no such situation (i.e. (c+gcd(a+c,b−1))/gcd(a×bn,c+gcd(a+c,b−1)) > 0 and r is even) exists, since if (c+gcd(a+c,b−1))/gcd(a×bn,c+gcd(a+c,b−1)) > 0 then −(c+gcd(a+c,b−1))/gcd(a×bn,c+gcd(a+c,b−1)) < 0, but negative numbers cannot be squares, however, if r is even then all rth powers are squares (since if s divides r, then all rth powers are sth powers), thus, −(c+gcd(a+c,b−1))/gcd(a×bn,c+gcd(a+c,b−1)) cannot be an rth power if (c+gcd(a+c,b−1))/gcd(a×bn,c+gcd(a+c,b−1)) > 0 and r is even) or "one of a×bn/gcd(a×bn,c+gcd(a+c,b−1)) and (c+gcd(a+c,b−1))/gcd(a×bn,c+gcd(a+c,b−1)) is a 4th power, and the other is of the form 4×m4" (in this case, (a×bn+c+gcd(a+c,b−1))/gcd(a+c,b−1) has Aurifeuillean factorization of x4+4×y4))

b
(2 ≤ b ≤ 36)
index of this minimal prime in base bbase-b form of this minimal primealgebraic ((a×bn+c)/gcd(a+c,b−1)) form of this minimal primelength of this minimal prime written in base blength of this minimal prime written in decimalfactordb entry of this minimal primethis minimal prime written in base b (use lower case letters instead of upper case letters)this minimal prime written in decimalhelper file of this minimal primeprimality certificate of this minimal primecertificate chain of this minimal prime (for the minimal primes which are proven prime by primality certificate) or factorization of N−1 or N+1 (for the minimal primes which are proven prime by N−1 test or N+1 test)factorization of the first 200 numbers in corresponding family (n is the number of digits in the "{}", start with the smallest n making the number > b (if n = 0 already makes the number > b, then start with n = 0) of this minimal prime)
9149763292(31×9330−19)/4331316http://factordb.com/index.php?id=1100000002359003642&open=primehttp://factordb.com/index.php?showid=1100000002359003642&base=9http://factordb.com/index.php?showid=1100000002359003642http://factordb.com/helper.php?id=1100000002359003642http://factordb.com/cert.php?id=1100000002359003642http://factordb.com/certchain.php?fid=1100000002359003642&action=all&fr=0&to=100http://factordb.com/index.php?query=%2831*9%5E%28n%2B1%29-19%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
91502768607(23×9688−511)/8689657http://factordb.com/index.php?id=1100000002495467486&open=primehttp://factordb.com/index.php?showid=1100000002495467486&base=9http://factordb.com/index.php?showid=1100000002495467486http://factordb.com/helper.php?id=1100000002495467486http://factordb.com/cert.php?id=1100000002495467486http://factordb.com/certchain.php?fid=1100000002495467486&action=all&fr=0&to=100http://factordb.com/index.php?query=%2823*9%5E%28n%2B2%29-511%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
9151301158113×91160+1011611108http://factordb.com/index.php?id=1100000002376318423&open=primehttp://factordb.com/index.php?showid=1100000002376318423&base=9http://factordb.com/index.php?showid=1100000002376318423http://factordb.com/helper.php?id=1100000002376318423http://factordb.com/cert.php?id=1100000002376318423http://factordb.com/certchain.php?fid=1100000002376318423&action=all&fr=0&to=100http://factordb.com/index.php?query=3*9%5E%28n%2B2%29%2B10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
111065A7135811715−58715745http://factordb.com/index.php?id=1100000003576826487&open=primehttp://factordb.com/index.php?showid=1100000003576826487&base=11http://factordb.com/index.php?showid=1100000003576826487http://factordb.com/helper.php?id=1100000003576826487http://factordb.com/cert.php?id=1100000003576826487http://factordb.com/certchain.php?fid=1100000003576826487&action=all&fr=0&to=100http://factordb.com/index.php?query=11%5E%28n%2B2%29-58&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
111066775944(7×11761−367)/10761793http://factordb.com/index.php?id=1100000002505568840&open=primehttp://factordb.com/index.php?showid=1100000002505568840&base=11http://factordb.com/index.php?showid=1100000002505568840http://factordb.com/helper.php?id=1100000002505568840http://factordb.com/cert.php?id=1100000002505568840http://factordb.com/certchain.php?fid=1100000002505568840&action=all&fr=0&to=100http://factordb.com/index.php?query=%287*11%5E%28n%2B2%29-367%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1110675571011(607×111011−7)/1010131055http://factordb.com/index.php?id=1100000002361376522&open=primehttp://factordb.com/index.php?showid=1100000002361376522&base=11http://factordb.com/index.php?showid=1100000002361376522http://factordb.com/helper.php?id=1100000002361376522http://factordb.com/cert.php?id=1100000002361376522http://factordb.com/certchain.php?fid=1100000002361376522&action=all&fr=0&to=100http://factordb.com/index.php?query=%28607*11%5En-7%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13316550270445×13272+56273304http://factordb.com/index.php?id=1100000002632397005&open=primehttp://factordb.com/index.php?showid=1100000002632397005&base=13http://factordb.com/index.php?showid=1100000002632397005http://factordb.com/helper.php?id=1100000002632397005http://factordb.com/cert.php?id=1100000002632397005http://factordb.com/certchain.php?fid=1100000002632397005&action=all&fr=0&to=100http://factordb.com/index.php?query=5*13%5E%28n%2B2%29%2B56&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1331669271095(3×13274−6103)/4274306http://factordb.com/index.php?id=1100000003590431654&open=primehttp://factordb.com/index.php?showid=1100000003590431654&base=13http://factordb.com/index.php?showid=1100000003590431654http://factordb.com/helper.php?id=1100000003590431654http://factordb.com/cert.php?id=1100000003590431654http://factordb.com/certchain.php?fid=1100000003590431654&action=all&fr=0&to=100http://factordb.com/index.php?query=%283*13%5E%28n%2B3%29-6103%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13316710286777113290+16654291324http://factordb.com/index.php?id=1100000003590431633&open=primehttp://factordb.com/index.php?showid=1100000003590431633&base=13http://factordb.com/index.php?showid=1100000003590431633http://factordb.com/helper.php?id=1100000003590431633http://factordb.com/cert.php?id=1100000003590431633http://factordb.com/certchain.php?fid=1100000003590431633&action=all&fr=0&to=100http://factordb.com/index.php?query=13%5E%28n%2B4%29%2B16654&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13316893081(3×13309−35)/4309345http://factordb.com/index.php?id=1100000000840126705&open=primehttp://factordb.com/index.php?showid=1100000000840126705&base=13http://factordb.com/index.php?showid=1100000000840126705http://factordb.com/helper.php?id=1100000000840126705proven prime by N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since N−1 is 39/4×(13308−1), thus factor N−1 is equivalent to factor 13308−1, and for the factorization of 13308−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN=http://factordb.com/index.php?id=1100000000840126706&open=ecmhttp://factordb.com/index.php?query=%283*13%5E%28n%2B1%29-35%29%2F4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
133169B341C4(11×13343+61)/12343383http://factordb.com/index.php?id=1100000003590431618&open=primehttp://factordb.com/index.php?showid=1100000003590431618&base=13http://factordb.com/index.php?showid=1100000003590431618http://factordb.com/helper.php?id=1100000003590431618http://factordb.com/cert.php?id=1100000003590431618http://factordb.com/certchain.php?fid=1100000003590431618&action=all&fr=0&to=100http://factordb.com/index.php?query=%2811*13%5E%28n%2B2%29%2B61%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1331708B343(107×13343−11)/12344384http://factordb.com/index.php?id=1100000002321018736&open=primehttp://factordb.com/index.php?showid=1100000002321018736&base=13http://factordb.com/index.php?showid=1100000002321018736http://factordb.com/helper.php?id=1100000002321018736http://factordb.com/cert.php?id=1100000002321018736http://factordb.com/certchain.php?fid=1100000002321018736&action=all&fr=0&to=100http://factordb.com/index.php?query=%28107*13%5En-11%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13317171037111192×13374+183376419http://factordb.com/index.php?id=1100000003590431609&open=primehttp://factordb.com/index.php?showid=1100000003590431609&base=13http://factordb.com/index.php?showid=1100000003590431609http://factordb.com/helper.php?id=1100000003590431609http://factordb.com/cert.php?id=1100000003590431609http://factordb.com/certchain.php?fid=1100000003590431609&action=all&fr=0&to=100http://factordb.com/index.php?query=92*13%5E%28n%2B3%29%2B183&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
133172753757(89×13376+19)/12377420http://factordb.com/index.php?id=1100000003590431596&open=primehttp://factordb.com/index.php?showid=1100000003590431596&base=13http://factordb.com/index.php?showid=1100000003590431596http://factordb.com/helper.php?id=1100000003590431596http://factordb.com/cert.php?id=1100000003590431596http://factordb.com/certchain.php?fid=1100000003590431596&action=all&fr=0&to=100http://factordb.com/index.php?query=%2889*13%5E%28n%2B1%29%2B19%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1331739B03919128×13392+9394439http://factordb.com/index.php?id=1100000002632396790&open=primehttp://factordb.com/index.php?showid=1100000002632396790&base=13http://factordb.com/index.php?showid=1100000002632396790http://factordb.com/helper.php?id=1100000002632396790http://factordb.com/cert.php?id=1100000002632396790http://factordb.com/certchain.php?fid=1100000002632396790&action=all&fr=0&to=100http://factordb.com/index.php?query=128*13%5E%28n%2B1%29%2B9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1331747B0B397(15923×13397−11)/12400446http://factordb.com/index.php?id=1100000003590431574&open=primehttp://factordb.com/index.php?showid=1100000003590431574&base=13http://factordb.com/index.php?showid=1100000003590431574http://factordb.com/helper.php?id=1100000003590431574http://factordb.com/cert.php?id=1100000003590431574http://factordb.com/certchain.php?fid=1100000003590431574&action=all&fr=0&to=100http://factordb.com/index.php?query=%2815923*13%5En-11%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
133175104149313416+120417464http://factordb.com/index.php?id=1100000002523249240&open=primehttp://factordb.com/index.php?showid=1100000002523249240&base=13http://factordb.com/index.php?showid=1100000002523249240http://factordb.com/helper.php?id=1100000002523249240http://factordb.com/cert.php?id=1100000002523249240http://factordb.com/certchain.php?fid=1100000002523249240&action=all&fr=0&to=100http://factordb.com/index.php?query=13%5E%28n%2B2%29%2B120&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13317681010415117746×13416+1420468http://factordb.com/index.php?id=1100000003590431555&open=primehttp://factordb.com/index.php?showid=1100000003590431555&base=13http://factordb.com/index.php?showid=1100000003590431555http://factordb.com/helper.php?id=1100000003590431555proven prime by N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since N−1 is trivially fully factoredhttp://factordb.com/index.php?id=1100000003590431556&open=ecmhttp://factordb.com/index.php?query=17746*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
133177811043511366×13436+1439489http://factordb.com/index.php?id=1100000002373259109&open=primehttp://factordb.com/index.php?showid=1100000002373259109&base=13http://factordb.com/index.php?showid=1100000002373259109http://factordb.com/helper.php?id=1100000002373259109proven prime by N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since N−1 is trivially fully factoredhttp://factordb.com/index.php?id=1100000002373259124&open=ecmhttp://factordb.com/index.php?query=1366*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
133178B7486(139×13486−7)/12487543http://factordb.com/index.php?id=1100000002321015892&open=primehttp://factordb.com/index.php?showid=1100000002321015892&base=13http://factordb.com/index.php?showid=1100000002321015892http://factordb.com/helper.php?id=1100000002321015892http://factordb.com/cert.php?id=1100000002321015892http://factordb.com/certchain.php?fid=1100000002321015892&action=all&fr=0&to=100http://factordb.com/index.php?query=%28139*13%5En-7%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
133179B563C(11×13564+1)/12564629http://factordb.com/index.php?id=1100000000000217927&open=primehttp://factordb.com/index.php?showid=1100000000000217927&base=13http://factordb.com/index.php?showid=1100000000000217927http://factordb.com/helper.php?id=1100000000000217927proven prime by N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since N−1 is 11/12×(13564−1), thus factor N−1 is equivalent to factor 13564−1, and for the factorization of 13564−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN=http://factordb.com/index.php?id=1100000000271764311&open=ecmhttp://factordb.com/index.php?query=%2811*13%5E%28n%2B1%29%2B1%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1331801B576(23×13576−11)/12577642http://factordb.com/index.php?id=1100000002321021456&open=primehttp://factordb.com/index.php?showid=1100000002321021456&base=13http://factordb.com/index.php?showid=1100000002321021456http://factordb.com/helper.php?id=1100000002321021456proven prime by N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since N−1 is 23/12×(13576−1), thus factor N−1 is equivalent to factor 13576−1, and for the factorization of 13576−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN=http://factordb.com/index.php?id=1100000002321021531&open=ecmhttp://factordb.com/index.php?query=%2823*13%5En-11%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13318180693878×13695+111696776http://factordb.com/index.php?id=1100000002615636527&open=primehttp://factordb.com/index.php?showid=1100000002615636527&base=13http://factordb.com/index.php?showid=1100000002615636527http://factordb.com/helper.php?id=1100000002615636527http://factordb.com/cert.php?id=1100000002615636527http://factordb.com/certchain.php?fid=1100000002615636527&action=all&fr=0&to=100http://factordb.com/index.php?query=8*13%5E%28n%2B2%29%2B111&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
133182CC5713(2021×13713−5)/12715797http://factordb.com/index.php?id=1100000002615627353&open=primehttp://factordb.com/index.php?showid=1100000002615627353&base=13http://factordb.com/index.php?showid=1100000002615627353http://factordb.com/helper.php?id=1100000002615627353http://factordb.com/cert.php?id=1100000002615627353http://factordb.com/certchain.php?fid=1100000002615627353&action=all&fr=0&to=100http://factordb.com/index.php?query=%282021*13%5En-5%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
133183B83474(11×13836−719)/12836932http://factordb.com/index.php?id=1100000003590430871&open=primehttp://factordb.com/index.php?showid=1100000003590430871&base=13http://factordb.com/index.php?showid=1100000003590430871http://factordb.com/helper.php?id=1100000003590430871http://factordb.com/cert.php?id=1100000003590430871http://factordb.com/certchain.php?fid=1100000003590430871&action=all&fr=0&to=100http://factordb.com/index.php?query=%2811*13%5E%28n%2B2%29-719%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1331849968B(3×13969+5)/49691080http://factordb.com/index.php?id=1100000000258566244&open=primehttp://factordb.com/index.php?showid=1100000000258566244&base=13http://factordb.com/index.php?showid=1100000000258566244http://factordb.com/helper.php?id=1100000000258566244http://factordb.com/cert.php?id=1100000000258566244http://factordb.com/certchain.php?fid=1100000000258566244&action=all&fr=0&to=100http://factordb.com/index.php?query=%283*13%5E%28n%2B1%29%2B5%29%2F4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
133185101295181131298+27412991446http://factordb.com/index.php?id=1100000002615445013&open=primehttp://factordb.com/index.php?showid=1100000002615445013&base=13http://factordb.com/index.php?showid=1100000002615445013http://factordb.com/helper.php?id=1100000002615445013http://factordb.com/cert.php?id=1100000002615445013http://factordb.com/certchain.php?fid=1100000002615445013&action=all&fr=0&to=100http://factordb.com/index.php?query=13%5E%28n%2B3%29%2B274&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
133186913625(3×131363−19)/413631519http://factordb.com/index.php?id=1100000002321017776&open=primehttp://factordb.com/index.php?showid=1100000002321017776&base=13http://factordb.com/index.php?showid=1100000002321017776http://factordb.com/helper.php?id=1100000002321017776http://factordb.com/cert.php?id=1100000002321017776http://factordb.com/certchain.php?fid=1100000002321017776&action=all&fr=0&to=100http://factordb.com/index.php?query=%283*13%5E%28n%2B1%29-19%29%2F4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
133187715041(7×131505−79)/1215051677http://factordb.com/index.php?id=1100000002320890755&open=primehttp://factordb.com/index.php?showid=1100000002320890755&base=13http://factordb.com/index.php?showid=1100000002320890755http://factordb.com/helper.php?id=1100000002320890755http://factordb.com/cert.php?id=1100000002320890755http://factordb.com/certchain.php?fid=1100000002320890755&action=all&fr=0&to=100http://factordb.com/index.php?query=%287*13%5E%28n%2B1%29-79%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13318893015511120×131552+115541731http://factordb.com/index.php?id=1100000000765961452&open=primehttp://factordb.com/index.php?showid=1100000000765961452&base=13http://factordb.com/index.php?showid=1100000000765961452http://factordb.com/helper.php?id=1100000000765961452proven prime by N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since N−1 is trivially fully factoredhttp://factordb.com/index.php?id=1100000000765961453&open=ecmhttp://factordb.com/index.php?query=120*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1331897202297293×132298+223002562http://factordb.com/index.php?id=1100000002632396910&open=primehttp://factordb.com/index.php?showid=1100000002632396910&base=13http://factordb.com/index.php?showid=1100000002632396910http://factordb.com/helper.php?id=1100000002632396910http://factordb.com/cert.php?id=1100000002632396910http://factordb.com/certchain.php?fid=1100000002632396910&action=all&fr=0&to=100http://factordb.com/index.php?query=93*13%5E%28n%2B1%29%2B2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1331901770270317267×132705+2027083016http://factordb.com/index.php?id=1100000003590430825&open=primehttp://factordb.com/index.php?showid=1100000003590430825&base=13http://factordb.com/index.php?showid=1100000003590430825http://factordb.com/helper.php?id=1100000003590430825http://factordb.com/cert.php?id=1100000003590430825http://factordb.com/certchain.php?fid=1100000003590430825&action=all&fr=0&to=100http://factordb.com/index.php?query=267*13%5E%28n%2B2%29%2B20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1331913906266148×136267+162696983http://factordb.com/index.php?id=1100000000765961441&open=primehttp://factordb.com/index.php?showid=1100000000765961441&base=13http://factordb.com/index.php?showid=1100000000765961441http://factordb.com/helper.php?id=1100000000765961441proven prime by N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since N−1 is trivially fully factoredhttp://factordb.com/index.php?id=1100000000765961451&open=ecmhttp://factordb.com/index.php?query=48*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
133192B06540BBA11×136543+201265447290http://factordb.com/index.php?id=1100000002616382906&open=primehttp://factordb.com/index.php?showid=1100000002616382906&base=13http://factordb.com/index.php?showid=1100000002616382906http://factordb.com/helper.php?id=1100000002616382906http://factordb.com/cert.php?id=1100000002616382906http://factordb.com/certchain.php?fid=1100000002616382906&action=all&fr=0&to=100http://factordb.com/index.php?query=11*13%5E%28n%2B3%29%2B2012&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
133193C10631921310633−501063311845http://factordb.com/index.php?id=1100000003590493750&open=primehttp://factordb.com/index.php?showid=1100000003590493750&base=13http://factordb.com/index.php?showid=1100000003590493750http://factordb.com/helper.php?id=1100000003590493750http://factordb.com/cert.php?id=1100000003590493750http://factordb.com/certchain.php?fid=1100000003590493750&action=all&fr=0&to=100http://factordb.com/index.php?query=13%5E%28n%2B2%29-50&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1464934D70847×14708−1710814http://factordb.com/index.php?id=1100000001540144903&open=primehttp://factordb.com/index.php?showid=1100000001540144903&base=14http://factordb.com/index.php?showid=1100000001540144903http://factordb.com/helper.php?id=1100000001540144903proven prime by N+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), since N+1 is trivially fully factoredhttp://factordb.com/index.php?id=1100000001540144907&open=ecmhttp://factordb.com/index.php?query=47*14%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
146504D196985×1419698−11969922578http://factordb.com/index.php?id=1100000000884560233&open=primehttp://factordb.com/index.php?showid=1100000000884560233&base=14http://factordb.com/index.php?showid=1100000000884560233http://factordb.com/helper.php?id=1100000000884560233proven prime by N+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), since N+1 is trivially fully factoredhttp://factordb.com/index.php?id=1100000000884560625&open=ecmhttp://factordb.com/index.php?query=5*14%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1623288802467136×16247+7249300http://factordb.com/index.php?id=1100000002468140199&open=primehttp://factordb.com/index.php?showid=1100000002468140199&base=16http://factordb.com/index.php?showid=1100000002468140199http://factordb.com/helper.php?id=1100000002468140199proven prime by N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), N−1 is 23×3×7×13×25703261×(289-digit prime)http://factordb.com/index.php?id=1100000002468140641&open=ecmhttp://factordb.com/index.php?query=136*16%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
162329D4263D(199×16264+131)/15265320http://factordb.com/index.php?id=1100000002468170238&open=primehttp://factordb.com/index.php?showid=1100000002468170238&base=16http://factordb.com/index.php?showid=1100000002468170238http://factordb.com/helper.php?id=1100000002468170238http://factordb.com/cert.php?id=1100000002468170238http://factordb.com/certchain.php?fid=1100000002468170238&action=all&fr=0&to=100http://factordb.com/index.php?query=%28199*16%5E%28n%2B1%29%2B131%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
162330E02614DD14×16264+1245265320http://factordb.com/index.php?id=1100000003588388352&open=primehttp://factordb.com/index.php?showid=1100000003588388352&base=16http://factordb.com/index.php?showid=1100000003588388352http://factordb.com/helper.php?id=1100000003588388352http://factordb.com/cert.php?id=1100000003588388352http://factordb.com/certchain.php?fid=1100000003588388352&action=all&fr=0&to=100http://factordb.com/index.php?query=14*16%5E%28n%2B3%29%2B1245&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1623318C0290ED140×16292+237294354http://factordb.com/index.php?id=1100000003588388307&open=primehttp://factordb.com/index.php?showid=1100000003588388307&base=16http://factordb.com/index.php?showid=1100000003588388307http://factordb.com/helper.php?id=1100000003588388307http://factordb.com/cert.php?id=1100000003588388307http://factordb.com/certchain.php?fid=1100000003588388307&action=all&fr=0&to=100http://factordb.com/index.php?query=140*16%5E%28n%2B2%29%2B237&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
162332DA3055(41×16306−17)/3307370http://factordb.com/index.php?id=1100000003588388284&open=primehttp://factordb.com/index.php?showid=1100000003588388284&base=16http://factordb.com/index.php?showid=1100000003588388284http://factordb.com/helper.php?id=1100000003588388284http://factordb.com/cert.php?id=1100000003588388284http://factordb.com/certchain.php?fid=1100000003588388284&action=all&fr=0&to=100http://factordb.com/index.php?query=%2841*16%5E%28n%2B1%29-17%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
162333CE80422D3304×16423+13426513http://factordb.com/index.php?id=1100000003588388257&open=primehttp://factordb.com/index.php?showid=1100000003588388257&base=16http://factordb.com/index.php?showid=1100000003588388257http://factordb.com/helper.php?id=1100000003588388257http://factordb.com/cert.php?id=1100000003588388257http://factordb.com/certchain.php?fid=1100000003588388257&action=all&fr=0&to=100http://factordb.com/index.php?query=3304*16%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1623345F5446F6×16546−145547659http://factordb.com/index.php?id=1100000002604723967&open=primehttp://factordb.com/index.php?showid=1100000002604723967&base=16http://factordb.com/index.php?showid=1100000002604723967http://factordb.com/helper.php?id=1100000002604723967http://factordb.com/cert.php?id=1100000002604723967http://factordb.com/certchain.php?fid=1100000002604723967&action=all&fr=0&to=100http://factordb.com/index.php?query=6*16%5E%28n%2B2%29-145&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16233588F545137×16545−1547659http://factordb.com/index.php?id=1100000000413679658&open=primehttp://factordb.com/index.php?showid=1100000000413679658&base=16http://factordb.com/index.php?showid=1100000000413679658http://factordb.com/helper.php?id=1100000000413679658proven prime by N+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), since N+1 is trivially fully factoredhttp://factordb.com/index.php?id=1100000000413877337&open=ecmhttp://factordb.com/index.php?query=137*16%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
162336BE0792BB190×16794+187796959http://factordb.com/index.php?id=1100000003588387938&open=primehttp://factordb.com/index.php?showid=1100000003588387938&base=16http://factordb.com/index.php?showid=1100000003588387938http://factordb.com/helper.php?id=1100000003588387938http://factordb.com/cert.php?id=1100000003588387938http://factordb.com/certchain.php?fid=1100000003588387938&action=all&fr=0&to=100http://factordb.com/index.php?query=190*16%5E%28n%2B2%29%2B187&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
162337D91052(68×161052−3)/510531268http://factordb.com/index.php?id=1100000002321036020&open=primehttp://factordb.com/index.php?showid=1100000002321036020&base=16http://factordb.com/index.php?showid=1100000002321036020http://factordb.com/helper.php?id=1100000002321036020http://factordb.com/cert.php?id=1100000002321036020http://factordb.com/certchain.php?fid=1100000002321036020&action=all&fr=0&to=100http://factordb.com/index.php?query=%2868*16%5En-3%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
162338FAF106245251×161064−18710661284http://factordb.com/index.php?id=1100000003588387610&open=primehttp://factordb.com/index.php?showid=1100000003588387610&base=16http://factordb.com/index.php?showid=1100000003588387610http://factordb.com/helper.php?id=1100000003588387610http://factordb.com/cert.php?id=1100000003588387610http://factordb.com/certchain.php?fid=1100000003588387610&action=all&fr=0&to=100http://factordb.com/index.php?query=251*16%5E%28n%2B2%29-187&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
162339F81517F(233×161518+97)/1515191830http://factordb.com/index.php?id=1100000000633744824&open=primehttp://factordb.com/index.php?showid=1100000000633744824&base=16http://factordb.com/index.php?showid=1100000000633744824http://factordb.com/helper.php?id=1100000000633744824http://factordb.com/cert.php?id=1100000000633744824http://factordb.com/certchain.php?fid=1100000000633744824&action=all&fr=0&to=100http://factordb.com/index.php?query=%28233*16%5E%28n%2B1%29%2B97%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1623402017133212×161716+80117172067http://factordb.com/index.php?id=1100000003588386735&open=primehttp://factordb.com/index.php?showid=1100000003588386735&base=16http://factordb.com/index.php?showid=1100000003588386735http://factordb.com/helper.php?id=1100000003588386735http://factordb.com/cert.php?id=1100000003588386735http://factordb.com/certchain.php?fid=1100000003588386735&action=all&fr=0&to=100http://factordb.com/index.php?query=2*16%5E%28n%2B3%29%2B801&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
162341300F1960AF769×161962−8119652366http://factordb.com/index.php?id=1100000003588368750&open=primehttp://factordb.com/index.php?showid=1100000003588368750&base=16http://factordb.com/index.php?showid=1100000003588368750http://factordb.com/helper.php?id=1100000003588368750http://factordb.com/cert.php?id=1100000003588368750http://factordb.com/certchain.php?fid=1100000003588368750&action=all&fr=0&to=100http://factordb.com/index.php?query=769*16%5E%28n%2B2%29-81&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
162342903542919×163544+14535454269http://factordb.com/index.php?id=1100000000633424191&open=primehttp://factordb.com/index.php?showid=1100000000633424191&base=16http://factordb.com/index.php?showid=1100000000633424191http://factordb.com/helper.php?id=1100000000633424191http://factordb.com/cert.php?id=1100000000633424191http://factordb.com/certchain.php?fid=1100000000633424191&action=all&fr=0&to=100http://factordb.com/index.php?query=9*16%5E%28n%2B2%29%2B145&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1623435BC3700D(459×163701+1)/537034459http://factordb.com/index.php?id=1100000000993764322&open=primehttp://factordb.com/index.php?showid=1100000000993764322&base=16http://factordb.com/index.php?showid=1100000000993764322http://factordb.com/helper.php?id=1100000000993764322http://factordb.com/cert.php?id=1100000000993764322http://factordb.com/certchain.php?fid=1100000000993764322&action=all&fr=0&to=100http://factordb.com/index.php?query=%28459*16%5E%28n%2B1%29%2B1%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
162344D0B17804(3131×1617804−11)/151780621441http://factordb.com/index.php?id=1100000003589278511&open=primehttp://factordb.com/index.php?showid=1100000003589278511&base=16http://factordb.com/index.php?showid=1100000003589278511http://factordb.com/helper.php?id=1100000003589278511http://factordb.com/cert.php?id=1100000003589278511http://factordb.com/certchain.php?fid=1100000003589278511&action=all&fr=0&to=100http://factordb.com/index.php?query=%283131*16%5En-11%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
171031185A24155(1133×17243−725)/8245302http://factordb.com/index.php?id=1100000003782940703&open=primehttp://factordb.com/index.php?showid=1100000003782940703&base=17http://factordb.com/index.php?showid=1100000003782940703http://factordb.com/helper.php?id=1100000003782940703proven prime by N+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), N+1 is 2×1291×942385161439×(286-digit prime)http://factordb.com/index.php?id=1100000003782944423&open=ecmhttp://factordb.com/index.php?query=%281133*17%5E%28n%2B2%29-725%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1710312902427019×17245+2024246303http://factordb.com/index.php?id=1100000003782940704&open=primehttp://factordb.com/index.php?showid=1100000003782940704&base=17http://factordb.com/index.php?showid=1100000003782940704http://factordb.com/helper.php?id=1100000003782940704proven prime by N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), N−1 is 26×172×1773259×4348181×603217519×(277-digit prime)http://factordb.com/index.php?id=1100000003782941925&open=ecmhttp://factordb.com/index.php?query=9*17%5E%28n%2B3%29%2B2024&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
171031376E246(1007×17246−7)/8248305http://factordb.com/index.php?id=1100000003782940705&open=primehttp://factordb.com/index.php?showid=1100000003782940705&base=17http://factordb.com/index.php?showid=1100000003782940705http://factordb.com/helper.php?id=1100000003782940705http://factordb.com/cert.php?id=1100000003782940705http://factordb.com/certchain.php?fid=1100000003782940705&action=all&fr=0&to=100http://factordb.com/index.php?query=%281007*17%5En-7%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1710314AAE5A5246(14215701×17246−5)/16251309http://factordb.com/index.php?id=1100000003782940706&open=primehttp://factordb.com/index.php?showid=1100000003782940706&base=17http://factordb.com/index.php?showid=1100000003782940706http://factordb.com/helper.php?id=1100000003782940706http://factordb.com/cert.php?id=1100000003782940706http://factordb.com/certchain.php?fid=1100000003782940706&action=all&fr=0&to=100http://factordb.com/index.php?query=%2814215701*17%5En-5%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1710315E7255A(231×17256+41)/16257317http://factordb.com/index.php?id=1100000003782940707&open=primehttp://factordb.com/index.php?showid=1100000003782940707&base=17http://factordb.com/index.php?showid=1100000003782940707http://factordb.com/helper.php?id=1100000003782940707proven prime by N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), N−1 is 24×283×619471×62754967151×(296-digit prime)http://factordb.com/index.php?id=1100000003782941928&open=ecmhttp://factordb.com/index.php?query=%28231*17%5E%28n%2B1%29%2B41%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
171031685A5270(38517×17270−5)/16273336http://factordb.com/index.php?id=1100000003782940708&open=primehttp://factordb.com/index.php?showid=1100000003782940708&base=17http://factordb.com/index.php?showid=1100000003782940708http://factordb.com/helper.php?id=1100000003782940708http://factordb.com/cert.php?id=1100000003782940708http://factordb.com/certchain.php?fid=1100000003782940708&action=all&fr=0&to=100http://factordb.com/index.php?query=%2838517*17%5En-5%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17103175A7027411622×17275+1278342http://factordb.com/index.php?id=1100000003782940709&open=primehttp://factordb.com/index.php?showid=1100000003782940709&base=17http://factordb.com/index.php?showid=1100000003782940709http://factordb.com/helper.php?id=1100000003782940709proven prime by N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since N−1 is trivially fully factoredhttp://factordb.com/index.php?id=1100000003782941930&open=ecmhttp://factordb.com/index.php?query=1622*17%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1710318D0D275F(3549×17276+19)/16278342http://factordb.com/index.php?id=1100000003782940710&open=primehttp://factordb.com/index.php?showid=1100000003782940710&base=17http://factordb.com/index.php?showid=1100000003782940710http://factordb.com/helper.php?id=1100000003782940710http://factordb.com/cert.php?id=1100000003782940710http://factordb.com/certchain.php?fid=1100000003782940710&action=all&fr=0&to=100http://factordb.com/index.php?query=%283549*17%5E%28n%2B1%29%2B19%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
171031970D0287B2036×17288+11291358http://factordb.com/index.php?id=1100000003782940711&open=primehttp://factordb.com/index.php?showid=1100000003782940711&base=17http://factordb.com/index.php?showid=1100000003782940711http://factordb.com/helper.php?id=1100000003782940711http://factordb.com/cert.php?id=1100000003782940711http://factordb.com/certchain.php?fid=1100000003782940711&action=all&fr=0&to=100http://factordb.com/index.php?query=2036*17%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
171032092921(9×17293−137)/16293361http://factordb.com/index.php?id=1100000000840355814&open=primehttp://factordb.com/index.php?showid=1100000000840355814&base=17http://factordb.com/index.php?showid=1100000000840355814http://factordb.com/helper.php?id=1100000000840355814proven prime by N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since N−1 is 153/16×(17292−1), thus factor N−1 is equivalent to factor 17292−1, and for the factorization of 17292−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=17&Exp=292&c0=-&EN=http://factordb.com/index.php?id=1100000000840355817&open=ecmhttp://factordb.com/index.php?query=%289*17%5E%28n%2B1%29-137%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1710321A7010298551154×17299+5303373http://factordb.com/index.php?id=1100000003782940712&open=primehttp://factordb.com/index.php?showid=1100000003782940712&base=17http://factordb.com/index.php?showid=1100000003782940712http://factordb.com/helper.php?id=1100000003782940712http://factordb.com/cert.php?id=1100000003782940712http://factordb.com/certchain.php?fid=1100000003782940712&action=all&fr=0&to=100http://factordb.com/index.php?query=51154*17%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1710322973093(151×17310−71)/16311383http://factordb.com/index.php?id=1100000002668315056&open=primehttp://factordb.com/index.php?showid=1100000002668315056&base=17http://factordb.com/index.php?showid=1100000002668315056http://factordb.com/helper.php?id=1100000002668315056http://factordb.com/cert.php?id=1100000002668315056http://factordb.com/certchain.php?fid=1100000002668315056&action=all&fr=0&to=100http://factordb.com/index.php?query=%28151*17%5E%28n%2B1%29-71%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17103237F03095134×17310+5312384http://factordb.com/index.php?id=1100000002748150540&open=primehttp://factordb.com/index.php?showid=1100000002748150540&base=17http://factordb.com/index.php?showid=1100000002748150540http://factordb.com/helper.php?id=1100000002748150540http://factordb.com/cert.php?id=1100000002748150540http://factordb.com/certchain.php?fid=1100000002748150540&action=all&fr=0&to=100http://factordb.com/index.php?query=134*17%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1710324BFB323(3243×17323−11)/16325400http://factordb.com/index.php?id=1100000003782940713&open=primehttp://factordb.com/index.php?showid=1100000003782940713&base=17http://factordb.com/index.php?showid=1100000003782940713http://factordb.com/helper.php?id=1100000003782940713http://factordb.com/cert.php?id=1100000003782940713http://factordb.com/certchain.php?fid=1100000003782940713&action=all&fr=0&to=100http://factordb.com/index.php?query=%283243*17%5En-11%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1710325A03285210×17330+87331408http://factordb.com/index.php?id=1100000002748151228&open=primehttp://factordb.com/index.php?showid=1100000002748151228&base=17http://factordb.com/index.php?showid=1100000002748151228http://factordb.com/helper.php?id=1100000002748151228http://factordb.com/cert.php?id=1100000002748151228http://factordb.com/certchain.php?fid=1100000002748151228&action=all&fr=0&to=100http://factordb.com/index.php?query=10*17%5E%28n%2B2%29%2B87&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1710326GD3343(269×17335−173)/16336414http://factordb.com/index.php?id=1100000003782940714&open=primehttp://factordb.com/index.php?showid=1100000003782940714&base=17http://factordb.com/index.php?showid=1100000003782940714http://factordb.com/helper.php?id=1100000003782940714http://factordb.com/cert.php?id=1100000003782940714http://factordb.com/certchain.php?fid=1100000003782940714&action=all&fr=0&to=100http://factordb.com/index.php?query=%28269*17%5E%28n%2B1%29-173%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1710327534795(5×17349+1083)/16349429http://factordb.com/index.php?id=1100000003782940715&open=primehttp://factordb.com/index.php?showid=1100000003782940715&base=17http://factordb.com/index.php?showid=1100000003782940715http://factordb.com/helper.php?id=1100000003782940715http://factordb.com/cert.php?id=1100000003782940715http://factordb.com/certchain.php?fid=1100000003782940715&action=all&fr=0&to=100http://factordb.com/index.php?query=%285*17%5E%28n%2B2%29%2B1083%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1710328E5A5348(66261×17348−5)/16351432http://factordb.com/index.php?id=1100000003782940716&open=primehttp://factordb.com/index.php?showid=1100000003782940716&base=17http://factordb.com/index.php?showid=1100000003782940716http://factordb.com/helper.php?id=1100000003782940716http://factordb.com/cert.php?id=1100000003782940716http://factordb.com/certchain.php?fid=1100000003782940716&action=all&fr=0&to=100http://factordb.com/index.php?query=%2866261*17%5En-5%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1710329FB03595266×17360+5362446http://factordb.com/index.php?id=1100000002632835926&open=primehttp://factordb.com/index.php?showid=1100000002632835926&base=17http://factordb.com/index.php?showid=1100000002632835926http://factordb.com/helper.php?id=1100000002632835926http://factordb.com/cert.php?id=1100000002632835926http://factordb.com/certchain.php?fid=1100000002632835926&action=all&fr=0&to=100http://factordb.com/index.php?query=266*17%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
171033010703633296×17364+3367451http://factordb.com/index.php?id=1100000003782940717&open=primehttp://factordb.com/index.php?showid=1100000003782940717&base=17http://factordb.com/index.php?showid=1100000003782940717http://factordb.com/helper.php?id=1100000003782940717http://factordb.com/cert.php?id=1100000003782940717http://factordb.com/certchain.php?fid=1100000003782940717&action=all&fr=0&to=100http://factordb.com/index.php?query=296*17%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1710331AE5371(2949×17371−5)/16373459http://factordb.com/index.php?id=1100000003782940718&open=primehttp://factordb.com/index.php?showid=1100000003782940718&base=17http://factordb.com/index.php?showid=1100000003782940718http://factordb.com/helper.php?id=1100000003782940718http://factordb.com/cert.php?id=1100000003782940718http://factordb.com/certchain.php?fid=1100000003782940718&action=all&fr=0&to=100http://factordb.com/index.php?query=%282949*17%5En-5%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1710332F53819(245×17382+59)/16383472http://factordb.com/index.php?id=1100000003782940719&open=primehttp://factordb.com/index.php?showid=1100000003782940719&base=17http://factordb.com/index.php?showid=1100000003782940719http://factordb.com/helper.php?id=1100000003782940719http://factordb.com/cert.php?id=1100000003782940719http://factordb.com/certchain.php?fid=1100000003782940719&action=all&fr=0&to=100http://factordb.com/index.php?query=%28245*17%5E%28n%2B1%29%2B59%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
171033310383F517385+260386474http://factordb.com/index.php?id=1100000002632399724&open=primehttp://factordb.com/index.php?showid=1100000002632399724&base=17http://factordb.com/index.php?showid=1100000002632399724http://factordb.com/helper.php?id=1100000002632399724http://factordb.com/cert.php?id=1100000002632399724http://factordb.com/certchain.php?fid=1100000002632399724&action=all&fr=0&to=100http://factordb.com/index.php?query=17%5E%28n%2B2%29%2B260&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17103346389GF(3×17391+1429)/8391481http://factordb.com/index.php?id=1100000000840355826&open=primehttp://factordb.com/index.php?showid=1100000000840355826&base=17http://factordb.com/index.php?showid=1100000000840355826http://factordb.com/helper.php?id=1100000000840355826http://factordb.com/cert.php?id=1100000000840355826http://factordb.com/certchain.php?fid=1100000000840355826&action=all&fr=0&to=100http://factordb.com/index.php?query=%283*17%5E%28n%2B2%29%2B1429%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1710335B39880B(11×17401−16875)/16401494http://factordb.com/index.php?id=1100000003782940720&open=primehttp://factordb.com/index.php?showid=1100000003782940720&base=17http://factordb.com/index.php?showid=1100000003782940720http://factordb.com/helper.php?id=1100000003782940720http://factordb.com/cert.php?id=1100000003782940720http://factordb.com/certchain.php?fid=1100000003782940720&action=all&fr=0&to=100http://factordb.com/index.php?query=%2811*17%5E%28n%2B3%29-16875%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1710336B704067194×17407+7409504http://factordb.com/index.php?id=1100000002748151387&open=primehttp://factordb.com/index.php?showid=1100000002748151387&base=17http://factordb.com/index.php?showid=1100000002748151387http://factordb.com/helper.php?id=1100000002748151387http://factordb.com/cert.php?id=1100000002748151387http://factordb.com/certchain.php?fid=1100000002748151387&action=all&fr=0&to=100http://factordb.com/index.php?query=194*17%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17103372B4172(43×17418−155)/16419515http://factordb.com/index.php?id=1100000003782940721&open=primehttp://factordb.com/index.php?showid=1100000003782940721&base=17http://factordb.com/index.php?showid=1100000003782940721http://factordb.com/helper.php?id=1100000003782940721http://factordb.com/cert.php?id=1100000003782940721http://factordb.com/certchain.php?fid=1100000003782940721&action=all&fr=0&to=100http://factordb.com/index.php?query=%2843*17%5E%28n%2B1%29-155%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1710338BF04165202×17417+5419516http://factordb.com/index.php?id=1100000002632836900&open=primehttp://factordb.com/index.php?showid=1100000002632836900&base=17http://factordb.com/index.php?showid=1100000002632836900http://factordb.com/helper.php?id=1100000002632836900http://factordb.com/cert.php?id=1100000002632836900http://factordb.com/certchain.php?fid=1100000002632836900&action=all&fr=0&to=100http://factordb.com/index.php?query=202*17%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17103395D0417598×17418+5420517http://factordb.com/index.php?id=1100000002632463013&open=primehttp://factordb.com/index.php?showid=1100000002632463013&base=17http://factordb.com/index.php?showid=1100000002632463013http://factordb.com/helper.php?id=1100000002632463013http://factordb.com/cert.php?id=1100000002632463013http://factordb.com/certchain.php?fid=1100000002632463013&action=all&fr=0&to=100http://factordb.com/index.php?query=98*17%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1710340D0439FB13×17441+266442544http://factordb.com/index.php?id=1100000002632836180&open=primehttp://factordb.com/index.php?showid=1100000002632836180&base=17http://factordb.com/index.php?showid=1100000002632836180http://factordb.com/helper.php?id=1100000002632836180http://factordb.com/cert.php?id=1100000002632836180http://factordb.com/certchain.php?fid=1100000002632836180&action=all&fr=0&to=100http://factordb.com/index.php?query=13*17%5E%28n%2B2%29%2B266&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17103417A1044652194×17447+5450554http://factordb.com/index.php?id=1100000003782940722&open=primehttp://factordb.com/index.php?showid=1100000003782940722&base=17http://factordb.com/index.php?showid=1100000003782940722http://factordb.com/helper.php?id=1100000003782940722http://factordb.com/cert.php?id=1100000003782940722http://factordb.com/certchain.php?fid=1100000003782940722&action=all&fr=0&to=100http://factordb.com/index.php?query=2194*17%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1710342D0D44909(3549×17451−3613)/16453558http://factordb.com/index.php?id=1100000003782940723&open=primehttp://factordb.com/index.php?showid=1100000003782940723&base=17http://factordb.com/index.php?showid=1100000003782940723http://factordb.com/helper.php?id=1100000003782940723http://factordb.com/cert.php?id=1100000003782940723http://factordb.com/certchain.php?fid=1100000003782940723&action=all&fr=0&to=100http://factordb.com/index.php?query=%283549*17%5E%28n%2B2%29-3613%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1710343794546(121×17455−57)/16456561http://factordb.com/index.php?id=1100000003782940724&open=primehttp://factordb.com/index.php?showid=1100000003782940724&base=17http://factordb.com/index.php?showid=1100000003782940724http://factordb.com/helper.php?id=1100000003782940724http://factordb.com/cert.php?id=1100000003782940724http://factordb.com/certchain.php?fid=1100000003782940724&action=all&fr=0&to=100http://factordb.com/index.php?query=%28121*17%5E%28n%2B1%29-57%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17103449760473F2726×17474+15477587http://factordb.com/index.php?id=1100000003782940725&open=primehttp://factordb.com/index.php?showid=1100000003782940725&base=17http://factordb.com/index.php?showid=1100000003782940725http://factordb.com/helper.php?id=1100000003782940725http://factordb.com/cert.php?id=1100000003782940725http://factordb.com/certchain.php?fid=1100000003782940725&action=all&fr=0&to=100http://factordb.com/index.php?query=2726*17%5E%28n%2B1%29%2B15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
171034580C4929(547×17493−15)/4495609http://factordb.com/index.php?id=1100000000840355847&open=primehttp://factordb.com/index.php?showid=1100000000840355847&base=17http://factordb.com/index.php?showid=1100000000840355847http://factordb.com/helper.php?id=1100000000840355847http://factordb.com/cert.php?id=1100000000840355847http://factordb.com/certchain.php?fid=1100000000840355847&action=all&fr=0&to=100http://factordb.com/index.php?query=%28547*17%5E%28n%2B1%29-15%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1710346D500F4(13×17502+387)/16502618http://factordb.com/index.php?id=1100000003782940726&open=primehttp://factordb.com/index.php?showid=1100000003782940726&base=17http://factordb.com/index.php?showid=1100000003782940726http://factordb.com/helper.php?id=1100000003782940726http://factordb.com/cert.php?id=1100000003782940726http://factordb.com/certchain.php?fid=1100000003782940726&action=all&fr=0&to=100http://factordb.com/index.php?query=%2813*17%5E%28n%2B2%29%2B387%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17103477495082(1977×17509−121)/16511629http://factordb.com/index.php?id=1100000003782940727&open=primehttp://factordb.com/index.php?showid=1100000003782940727&base=17http://factordb.com/index.php?showid=1100000003782940727http://factordb.com/helper.php?id=1100000003782940727http://factordb.com/cert.php?id=1100000003782940727http://factordb.com/certchain.php?fid=1100000003782940727&action=all&fr=0&to=100http://factordb.com/index.php?query=%281977*17%5E%28n%2B1%29-121%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1710348A55372(165×17538−53)/16539663http://factordb.com/index.php?id=1100000002999318480&open=primehttp://factordb.com/index.php?showid=1100000002999318480&base=17http://factordb.com/index.php?showid=1100000002999318480http://factordb.com/helper.php?id=1100000002999318480http://factordb.com/cert.php?id=1100000002999318480http://factordb.com/certchain.php?fid=1100000002999318480&action=all&fr=0&to=100http://factordb.com/index.php?query=%28165*17%5E%28n%2B1%29-53%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1710349B0542FB11×17544+266545671http://factordb.com/index.php?id=1100000002632836856&open=primehttp://factordb.com/index.php?showid=1100000002632836856&base=17http://factordb.com/index.php?showid=1100000002632836856http://factordb.com/helper.php?id=1100000002632836856http://factordb.com/cert.php?id=1100000002632836856http://factordb.com/certchain.php?fid=1100000002632836856&action=all&fr=0&to=100http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B266&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17103509604B(9×17605+23)/16605745http://factordb.com/index.php?id=1100000002321188365&open=primehttp://factordb.com/index.php?showid=1100000002321188365&base=17http://factordb.com/index.php?showid=1100000002321188365http://factordb.com/helper.php?id=1100000002321188365http://factordb.com/cert.php?id=1100000002321188365http://factordb.com/certchain.php?fid=1100000002321188365&action=all&fr=0&to=100http://factordb.com/index.php?query=%289*17%5E%28n%2B1%29%2B23%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
1854780298B8×18299+11300377http://factordb.com/index.php?id=1100000002355574745&open=primehttp://factordb.com/index.php?showid=1100000002355574745&base=18http://factordb.com/index.php?showid=1100000002355574745http://factordb.com/helper.php?id=1100000002355574745http://factordb.com/cert.php?id=1100000002355574745http://factordb.com/certchain.php?fid=1100000002355574745&action=all&fr=0&to=100http://factordb.com/index.php?query=8*18%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
18548H766FH18768−37768965http://factordb.com/index.php?id=1100000003590430490&open=primehttp://factordb.com/index.php?showid=1100000003590430490&base=18http://factordb.com/index.php?showid=1100000003590430490http://factordb.com/helper.php?id=1100000003590430490http://factordb.com/cert.php?id=1100000003590430490http://factordb.com/certchain.php?fid=1100000003590430490&action=all&fr=0&to=100http://factordb.com/index.php?query=18%5E%28n%2B2%29-37&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
18549C06268C512×186270+22162717872http://factordb.com/index.php?id=1100000003590442437&open=primehttp://factordb.com/index.php?showid=1100000003590442437&base=18http://factordb.com/index.php?showid=1100000003590442437http://factordb.com/helper.php?id=1100000003590442437http://factordb.com/cert.php?id=1100000003590442437http://factordb.com/certchain.php?fid=1100000003590442437&action=all&fr=0&to=100http://factordb.com/index.php?query=12*18%5E%28n%2B2%29%2B221&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
203301H247A0H(17×20250−59677)/19250326http://factordb.com/index.php?id=1100000003590502619&open=primehttp://factordb.com/index.php?showid=1100000003590502619&base=20http://factordb.com/index.php?showid=1100000003590502619http://factordb.com/helper.php?id=1100000003590502619http://factordb.com/cert.php?id=1100000003590502619http://factordb.com/certchain.php?fid=1100000003590502619&action=all&fr=0&to=100http://factordb.com/index.php?query=%2817*20%5E%28n%2B3%29-59677%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2033027249A7(7×20251+1133)/19251327http://factordb.com/index.php?id=1100000003590502602&open=primehttp://factordb.com/index.php?showid=1100000003590502602&base=20http://factordb.com/index.php?showid=1100000003590502602http://factordb.com/helper.php?id=1100000003590502602http://factordb.com/cert.php?id=1100000003590502602http://factordb.com/certchain.php?fid=1100000003590502602&action=all&fr=0&to=100http://factordb.com/index.php?query=%287*20%5E%28n%2B2%29%2B1133%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
203303J7270(368×20270−7)/19271353http://factordb.com/index.php?id=1100000002325395462&open=primehttp://factordb.com/index.php?showid=1100000002325395462&base=20http://factordb.com/index.php?showid=1100000002325395462http://factordb.com/helper.php?id=1100000002325395462http://factordb.com/cert.php?id=1100000002325395462http://factordb.com/certchain.php?fid=1100000002325395462&action=all&fr=0&to=100http://factordb.com/index.php?query=%28368*20%5En-7%29%2F19&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
203304J330CCC720334−58953334435http://factordb.com/index.php?id=1100000003590502572&open=primehttp://factordb.com/index.php?showid=1100000003590502572&base=20http://factordb.com/index.php?showid=1100000003590502572http://factordb.com/helper.php?id=1100000003590502572http://factordb.com/cert.php?id=1100000003590502572http://factordb.com/certchain.php?fid=1100000003590502572&action=all&fr=0&to=100http://factordb.com/index.php?query=20%5E%28n%2B4%29-58953&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
20330540387404B4×20391+32091392510http://factordb.com/index.php?id=1100000003590502563&open=primehttp://factordb.com/index.php?showid=1100000003590502563&base=20http://factordb.com/index.php?showid=1100000003590502563http://factordb.com/helper.php?id=1100000003590502563http://factordb.com/cert.php?id=1100000003590502563http://factordb.com/certchain.php?fid=1100000003590502563&action=all&fr=0&to=100http://factordb.com/index.php?query=4*20%5E%28n%2B4%29%2B32091&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
203306EC04297292×20430+7432562http://factordb.com/index.php?id=1100000002633348702&open=primehttp://factordb.com/index.php?showid=1100000002633348702&base=20http://factordb.com/index.php?showid=1100000002633348702http://factordb.com/helper.php?id=1100000002633348702http://factordb.com/cert.php?id=1100000002633348702http://factordb.com/certchain.php?fid=1100000002633348702&action=all&fr=0&to=100http://factordb.com/index.php?query=292*20%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
203307G44799(16×20449−2809)/19449585http://factordb.com/index.php?id=1100000000840126753&open=primehttp://factordb.com/index.php?showid=1100000000840126753&base=20http://factordb.com/index.php?showid=1100000000840126753http://factordb.com/helper.php?id=1100000000840126753http://factordb.com/cert.php?id=1100000000840126753http://factordb.com/certchain.php?fid=1100000000840126753&action=all&fr=0&to=100http://factordb.com/index.php?query=%2816*20%5E%28n%2B2%29-2809%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2033083A5273(67×20528−143)/19529688http://factordb.com/index.php?id=1100000003590502531&open=primehttp://factordb.com/index.php?showid=1100000003590502531&base=20http://factordb.com/index.php?showid=1100000003590502531http://factordb.com/helper.php?id=1100000003590502531http://factordb.com/cert.php?id=1100000003590502531http://factordb.com/certchain.php?fid=1100000003590502531&action=all&fr=0&to=100http://factordb.com/index.php?query=%2867*20%5E%28n%2B1%29-143%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
203309E566C7(14×20568−907)/19568739http://factordb.com/index.php?id=1100000003590502516&open=primehttp://factordb.com/index.php?showid=1100000003590502516&base=20http://factordb.com/index.php?showid=1100000003590502516http://factordb.com/helper.php?id=1100000003590502516http://factordb.com/cert.php?id=1100000003590502516http://factordb.com/certchain.php?fid=1100000003590502516&action=all&fr=0&to=100http://factordb.com/index.php?query=%2814*20%5E%28n%2B2%29-907%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
203310JCJ629393×20629−1631821http://factordb.com/index.php?id=1100000001559454258&open=primehttp://factordb.com/index.php?showid=1100000001559454258&base=20http://factordb.com/index.php?showid=1100000001559454258http://factordb.com/helper.php?id=1100000001559454258proven prime by N+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), since N+1 is trivially fully factoredhttp://factordb.com/index.php?id=1100000001559454271&open=ecmhttp://factordb.com/index.php?query=393*20%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
203311J65505J20658−7881658857http://factordb.com/index.php?id=1100000003590502490&open=primehttp://factordb.com/index.php?showid=1100000003590502490&base=20http://factordb.com/index.php?showid=1100000003590502490http://factordb.com/helper.php?id=1100000003590502490http://factordb.com/cert.php?id=1100000003590502490http://factordb.com/certchain.php?fid=1100000003590502490&action=all&fr=0&to=100http://factordb.com/index.php?query=20%5E%28n%2B3%29-7881&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
203312501163AJ5×201165+21911661517http://factordb.com/index.php?id=1100000003590502412&open=primehttp://factordb.com/index.php?showid=1100000003590502412&base=20http://factordb.com/index.php?showid=1100000003590502412http://factordb.com/helper.php?id=1100000003590502412http://factordb.com/cert.php?id=1100000003590502412http://factordb.com/certchain.php?fid=1100000003590502412&action=all&fr=0&to=100http://factordb.com/index.php?query=5*20%5E%28n%2B2%29%2B219&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
203313CD2449(241×202449−13)/1924503188http://factordb.com/index.php?id=1100000002325393915&open=primehttp://factordb.com/index.php?showid=1100000002325393915&base=20http://factordb.com/index.php?showid=1100000002325393915http://factordb.com/helper.php?id=1100000002325393915http://factordb.com/cert.php?id=1100000002325393915http://factordb.com/certchain.php?fid=1100000002325393915&action=all&fr=0&to=100http://factordb.com/index.php?query=%28241*20%5En-13%29%2F19&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
203314G06269D16×206270+1362718159http://factordb.com/index.php?id=1100000003590539457&open=primehttp://factordb.com/index.php?showid=1100000003590539457&base=20http://factordb.com/index.php?showid=1100000003590539457http://factordb.com/helper.php?id=1100000003590539457http://factordb.com/cert.php?id=1100000003590539457http://factordb.com/certchain.php?fid=1100000003590539457&action=all&fr=0&to=100http://factordb.com/index.php?query=16*20%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2113303E9227K(289×21228+211)/20229303http://factordb.com/index.php?id=1100000000633439877&open=primehttp://factordb.com/index.php?showid=1100000000633439877&base=21http://factordb.com/index.php?showid=1100000000633439877http://factordb.com/helper.php?id=1100000000633439877http://factordb.com/cert.php?id=1100000000633439877http://factordb.com/certchain.php?fid=1100000000633439877&action=all&fr=0&to=100http://factordb.com/index.php?query=%28289*21%5E%28n%2B1%29%2B211%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
211330472301(7×21231−127)/20231305http://factordb.com/index.php?id=1100000002325398836&open=primehttp://factordb.com/index.php?showid=1100000002325398836&base=21http://factordb.com/index.php?showid=1100000002325398836http://factordb.com/helper.php?id=1100000002325398836proven prime by N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since N−1 is 147/20×(21230−1), thus factor N−1 is equivalent to factor 21230−1, and for the factorization of 21230−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=21&Exp=230&c0=-&EN=http://factordb.com/index.php?id=1100000002325398854&open=ecmhttp://factordb.com/index.php?query=%287*21%5E%28n%2B1%29-127%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2113305E7236B(287×21237+73)/20238315http://factordb.com/index.php?id=1100000003815102868&open=primehttp://factordb.com/index.php?showid=1100000003815102868&base=21http://factordb.com/index.php?showid=1100000003815102868http://factordb.com/helper.php?id=1100000003815102868http://factordb.com/cert.php?id=1100000003815102868http://factordb.com/certchain.php?fid=1100000003815102868&action=all&fr=0&to=100http://factordb.com/index.php?query=%28287*21%5E%28n%2B1%29%2B73%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2113306A5238B(41×21239+23)/4240318http://factordb.com/index.php?id=1100000003815102869&open=primehttp://factordb.com/index.php?showid=1100000003815102869&base=21http://factordb.com/index.php?showid=1100000003815102869http://factordb.com/helper.php?id=1100000003815102869http://factordb.com/cert.php?id=1100000003815102869http://factordb.com/certchain.php?fid=1100000003815102869&action=all&fr=0&to=100http://factordb.com/index.php?query=%2841*21%5E%28n%2B1%29%2B23%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2113307C24632(3×21248−998)/5248328http://factordb.com/index.php?id=1100000003815102870&open=primehttp://factordb.com/index.php?showid=1100000003815102870&base=21http://factordb.com/index.php?showid=1100000003815102870http://factordb.com/helper.php?id=1100000003815102870http://factordb.com/cert.php?id=1100000003815102870http://factordb.com/certchain.php?fid=1100000003815102870&action=all&fr=0&to=100http://factordb.com/index.php?query=%283*21%5E%28n%2B2%29-998%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2113308D02487E513×21251+3386252333http://factordb.com/index.php?id=1100000003815102871&open=primehttp://factordb.com/index.php?showid=1100000003815102871&base=21http://factordb.com/index.php?showid=1100000003815102871http://factordb.com/helper.php?id=1100000003815102871http://factordb.com/cert.php?id=1100000003815102871http://factordb.com/certchain.php?fid=1100000003815102871&action=all&fr=0&to=100http://factordb.com/index.php?query=13*21%5E%28n%2B3%29%2B3386&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2113309E2499IK(7×21252−21157)/10252334http://factordb.com/index.php?id=1100000000633439878&open=primehttp://factordb.com/index.php?showid=1100000000633439878&base=21http://factordb.com/index.php?showid=1100000000633439878http://factordb.com/helper.php?id=1100000000633439878http://factordb.com/cert.php?id=1100000000633439878http://factordb.com/certchain.php?fid=1100000000633439878&action=all&fr=0&to=100http://factordb.com/index.php?query=%287*21%5E%28n%2B3%29-21157%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
21133101HHH0248H17132×21249+17253334http://factordb.com/index.php?id=1100000003815102872&open=primehttp://factordb.com/index.php?showid=1100000003815102872&base=21http://factordb.com/index.php?showid=1100000003815102872http://factordb.com/helper.php?id=1100000003815102872http://factordb.com/cert.php?id=1100000003815102872http://factordb.com/certchain.php?fid=1100000003815102872&action=all&fr=0&to=100http://factordb.com/index.php?query=17132*21%5E%28n%2B1%29%2B17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2113311D2517B(13×21253−2573)/20253335http://factordb.com/index.php?id=1100000003815102873&open=primehttp://factordb.com/index.php?showid=1100000003815102873&base=21http://factordb.com/index.php?showid=1100000003815102873http://factordb.com/helper.php?id=1100000003815102873http://factordb.com/cert.php?id=1100000003815102873http://factordb.com/certchain.php?fid=1100000003815102873&action=all&fr=0&to=100http://factordb.com/index.php?query=%2813*21%5E%28n%2B2%29-2573%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
21133123F0273IH78×21275+395277366http://factordb.com/index.php?id=1100000003815102874&open=primehttp://factordb.com/index.php?showid=1100000003815102874&base=21http://factordb.com/index.php?showid=1100000003815102874http://factordb.com/helper.php?id=1100000003815102874http://factordb.com/cert.php?id=1100000003815102874http://factordb.com/certchain.php?fid=1100000003815102874&action=all&fr=0&to=100http://factordb.com/index.php?query=78*21%5E%28n%2B2%29%2B395&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
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2113339E0500B3B14×21503+4925504667http://factordb.com/index.php?id=1100000003815102893&open=primehttp://factordb.com/index.php?showid=1100000003815102893&base=21http://factordb.com/index.php?showid=1100000003815102893http://factordb.com/helper.php?id=1100000003815102893http://factordb.com/cert.php?id=1100000003815102893http://factordb.com/certchain.php?fid=1100000003815102893&action=all&fr=0&to=100http://factordb.com/index.php?query=14*21%5E%28n%2B3%29%2B4925&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2113340J5199H(19×21521−4259)/20521689http://factordb.com/index.php?id=1100000003815102894&open=primehttp://factordb.com/index.php?showid=1100000003815102894&base=21http://factordb.com/index.php?showid=1100000003815102894http://factordb.com/helper.php?id=1100000003815102894http://factordb.com/cert.php?id=1100000003815102894http://factordb.com/certchain.php?fid=1100000003815102894&action=all&fr=0&to=100http://factordb.com/index.php?query=%2819*21%5E%28n%2B2%29-4259%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
227984I7G0254H8882×22255+17258347http://factordb.com/index.php?id=1100000003591372788&open=primehttp://factordb.com/index.php?showid=1100000003591372788&base=22http://factordb.com/index.php?showid=1100000003591372788http://factordb.com/helper.php?id=1100000003591372788http://factordb.com/cert.php?id=1100000003591372788http://factordb.com/certchain.php?fid=1100000003591372788&action=all&fr=0&to=100http://factordb.com/index.php?query=8882*22%5E%28n%2B1%29%2B17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
227985D02555EEF13×22259+60339260349http://factordb.com/index.php?id=1100000003591371932&open=primehttp://factordb.com/index.php?showid=1100000003591371932&base=22http://factordb.com/index.php?showid=1100000003591371932http://factordb.com/helper.php?id=1100000003591371932http://factordb.com/cert.php?id=1100000003591371932http://factordb.com/certchain.php?fid=1100000003591371932&action=all&fr=0&to=100http://factordb.com/index.php?query=13*22%5E%28n%2B4%29%2B60339&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
227986IK322F(398×22323−125)/21324435http://factordb.com/index.php?id=1100000000840384145&open=primehttp://factordb.com/index.php?showid=1100000000840384145&base=22http://factordb.com/index.php?showid=1100000000840384145http://factordb.com/helper.php?id=1100000000840384145http://factordb.com/cert.php?id=1100000000840384145http://factordb.com/certchain.php?fid=1100000000840384145&action=all&fr=0&to=100http://factordb.com/index.php?query=%28398*22%5E%28n%2B1%29-125%29%2F21&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
227987C0340G912×22342+361343461http://factordb.com/index.php?id=1100000000840384159&open=primehttp://factordb.com/index.php?showid=1100000000840384159&base=22http://factordb.com/index.php?showid=1100000000840384159http://factordb.com/helper.php?id=1100000000840384159http://factordb.com/cert.php?id=1100000000840384159http://factordb.com/certchain.php?fid=1100000000840384159&action=all&fr=0&to=100http://factordb.com/index.php?query=12*22%5E%28n%2B2%29%2B361&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
22798877E348K7(485×22350+373)/3352473http://factordb.com/index.php?id=1100000003591369779&open=primehttp://factordb.com/index.php?showid=1100000003591369779&base=22http://factordb.com/index.php?showid=1100000003591369779http://factordb.com/helper.php?id=1100000003591369779http://factordb.com/cert.php?id=1100000003591369779http://factordb.com/certchain.php?fid=1100000003591369779&action=all&fr=0&to=100http://factordb.com/index.php?query=%28485*22%5E%28n%2B2%29%2B373%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
227989J379KJ(19×22381+443)/21381512http://factordb.com/index.php?id=1100000003591369027&open=primehttp://factordb.com/index.php?showid=1100000003591369027&base=22http://factordb.com/index.php?showid=1100000003591369027http://factordb.com/helper.php?id=1100000003591369027http://factordb.com/cert.php?id=1100000003591369027http://factordb.com/certchain.php?fid=1100000003591369027&action=all&fr=0&to=100http://factordb.com/index.php?query=%2819*22%5E%28n%2B2%29%2B443%29%2F21&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
227990J388EJ(19×22390−2329)/21390524http://factordb.com/index.php?id=1100000003591367729&open=primehttp://factordb.com/index.php?showid=1100000003591367729&base=22http://factordb.com/index.php?showid=1100000003591367729http://factordb.com/helper.php?id=1100000003591367729http://factordb.com/cert.php?id=1100000003591367729http://factordb.com/certchain.php?fid=1100000003591367729&action=all&fr=0&to=100http://factordb.com/index.php?query=%2819*22%5E%28n%2B2%29-2329%29%2F21&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
227991DJ400(292×22400−19)/21401539http://factordb.com/index.php?id=1100000002325880110&open=primehttp://factordb.com/index.php?showid=1100000002325880110&base=22http://factordb.com/index.php?showid=1100000002325880110http://factordb.com/helper.php?id=1100000002325880110http://factordb.com/cert.php?id=1100000002325880110http://factordb.com/certchain.php?fid=1100000002325880110&action=all&fr=0&to=100http://factordb.com/index.php?query=%28292*22%5En-19%29%2F21&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
227992E404K7(2×22406+373)/3406545http://factordb.com/index.php?id=1100000003591366298&open=primehttp://factordb.com/index.php?showid=1100000003591366298&base=22http://factordb.com/index.php?showid=1100000003591366298http://factordb.com/helper.php?id=1100000003591366298http://factordb.com/cert.php?id=1100000003591366298http://factordb.com/certchain.php?fid=1100000003591366298&action=all&fr=0&to=100http://factordb.com/index.php?query=%282*22%5E%28n%2B2%29%2B373%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
22799366F453B3(971×22455−705)/7457613http://factordb.com/index.php?id=1100000003591365809&open=primehttp://factordb.com/index.php?showid=1100000003591365809&base=22http://factordb.com/index.php?showid=1100000003591365809http://factordb.com/helper.php?id=1100000003591365809http://factordb.com/cert.php?id=1100000003591365809http://factordb.com/certchain.php?fid=1100000003591365809&action=all&fr=0&to=100http://factordb.com/index.php?query=%28971*22%5E%28n%2B2%29-705%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
227994L0454B6321×22457+5459458615http://factordb.com/index.php?id=1100000003591365331&open=primehttp://factordb.com/index.php?showid=1100000003591365331&base=22http://factordb.com/index.php?showid=1100000003591365331http://factordb.com/helper.php?id=1100000003591365331http://factordb.com/cert.php?id=1100000003591365331http://factordb.com/certchain.php?fid=1100000003591365331&action=all&fr=0&to=100http://factordb.com/index.php?query=21*22%5E%28n%2B3%29%2B5459&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
227995L483G322485−129485652http://factordb.com/index.php?id=1100000003591364730&open=primehttp://factordb.com/index.php?showid=1100000003591364730&base=22http://factordb.com/index.php?showid=1100000003591364730http://factordb.com/helper.php?id=1100000003591364730http://factordb.com/cert.php?id=1100000003591364730http://factordb.com/certchain.php?fid=1100000003591364730&action=all&fr=0&to=100http://factordb.com/index.php?query=22%5E%28n%2B2%29-129&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
227996E60496L314×22497+21499670http://factordb.com/index.php?id=1100000000632703239&open=primehttp://factordb.com/index.php?showid=1100000000632703239&base=22http://factordb.com/index.php?showid=1100000000632703239http://factordb.com/helper.php?id=1100000000632703239http://factordb.com/cert.php?id=1100000000632703239http://factordb.com/certchain.php?fid=1100000000632703239&action=all&fr=0&to=100http://factordb.com/index.php?query=314*22%5E%28n%2B1%29%2B21&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
227997I626AF(6×22628−1259)/7628843http://factordb.com/index.php?id=1100000000632724334&open=primehttp://factordb.com/index.php?showid=1100000000632724334&base=22http://factordb.com/index.php?showid=1100000000632724334http://factordb.com/helper.php?id=1100000000632724334http://factordb.com/cert.php?id=1100000000632724334http://factordb.com/certchain.php?fid=1100000000632724334&action=all&fr=0&to=100http://factordb.com/index.php?query=%286*22%5E%28n%2B2%29-1259%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
227998K0760EC120×22763+70417641026http://factordb.com/index.php?id=1100000000632724415&open=primehttp://factordb.com/index.php?showid=1100000000632724415&base=22http://factordb.com/index.php?showid=1100000000632724415http://factordb.com/helper.php?id=1100000000632724415http://factordb.com/cert.php?id=1100000000632724415http://factordb.com/certchain.php?fid=1100000000632724415&action=all&fr=0&to=100http://factordb.com/index.php?query=20*22%5E%28n%2B3%29%2B7041&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
227999J0767IGGJ19×22771+1997797721037http://factordb.com/index.php?id=1100000003591362567&open=primehttp://factordb.com/index.php?showid=1100000003591362567&base=22http://factordb.com/index.php?showid=1100000003591362567http://factordb.com/helper.php?id=1100000003591362567http://factordb.com/cert.php?id=1100000003591362567http://factordb.com/certchain.php?fid=1100000003591362567&action=all&fr=0&to=100http://factordb.com/index.php?query=19*22%5E%28n%2B4%29%2B199779&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2280007959K7(22961+857)/39611290http://factordb.com/index.php?id=1100000003591361817&open=primehttp://factordb.com/index.php?showid=1100000003591361817&base=22http://factordb.com/index.php?showid=1100000003591361817http://factordb.com/helper.php?id=1100000003591361817http://factordb.com/cert.php?id=1100000003591361817http://factordb.com/certchain.php?fid=1100000003591361817&action=all&fr=0&to=100http://factordb.com/index.php?query=%2822%5E%28n%2B2%29%2B857%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
228001L2385KE7222388−65323883206http://factordb.com/index.php?id=1100000003591360774&open=primehttp://factordb.com/index.php?showid=1100000003591360774&base=22http://factordb.com/index.php?showid=1100000003591360774http://factordb.com/helper.php?id=1100000003591360774http://factordb.com/cert.php?id=1100000003591360774http://factordb.com/certchain.php?fid=1100000003591360774&action=all&fr=0&to=100http://factordb.com/index.php?query=22%5E%28n%2B3%29-653&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
228002738152L(223817−289)/338175124http://factordb.com/index.php?id=1100000003591359839&open=primehttp://factordb.com/index.php?showid=1100000003591359839&base=22http://factordb.com/index.php?showid=1100000003591359839http://factordb.com/helper.php?id=1100000003591359839http://factordb.com/cert.php?id=1100000003591359839http://factordb.com/certchain.php?fid=1100000003591359839&action=all&fr=0&to=100http://factordb.com/index.php?query=%2822%5E%28n%2B2%29-289%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
243400I0241I518×24243+437244337http://factordb.com/index.php?id=1100000002633360037&open=primehttp://factordb.com/index.php?showid=1100000002633360037&base=24http://factordb.com/index.php?showid=1100000002633360037http://factordb.com/helper.php?id=1100000002633360037http://factordb.com/cert.php?id=1100000002633360037http://factordb.com/certchain.php?fid=1100000002633360037&action=all&fr=0&to=100http://factordb.com/index.php?query=18*24%5E%28n%2B2%29%2B437&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
243401D0259KKD13×24262+12013263363http://factordb.com/index.php?id=1100000003593270725&open=primehttp://factordb.com/index.php?showid=1100000003593270725&base=24http://factordb.com/index.php?showid=1100000003593270725http://factordb.com/helper.php?id=1100000003593270725http://factordb.com/cert.php?id=1100000003593270725http://factordb.com/certchain.php?fid=1100000003593270725&action=all&fr=0&to=100http://factordb.com/index.php?query=13*24%5E%28n%2B3%29%2B12013&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
243402C7298(283×24298−7)/23299413http://factordb.com/index.php?id=1100000002326181235&open=primehttp://factordb.com/index.php?showid=1100000002326181235&base=24http://factordb.com/index.php?showid=1100000002326181235http://factordb.com/helper.php?id=1100000002326181235http://factordb.com/cert.php?id=1100000002326181235http://factordb.com/certchain.php?fid=1100000002326181235&action=all&fr=0&to=100http://factordb.com/index.php?query=%28283*24%5En-7%29%2F23&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2434032031372×24314+7315434http://factordb.com/index.php?id=1100000002355610241&open=primehttp://factordb.com/index.php?showid=1100000002355610241&base=24http://factordb.com/index.php?showid=1100000002355610241http://factordb.com/helper.php?id=1100000002355610241http://factordb.com/cert.php?id=1100000002355610241http://factordb.com/certchain.php?fid=1100000002355610241&action=all&fr=0&to=100http://factordb.com/index.php?query=2*24%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
243404BC0331B276×24332+11334461http://factordb.com/index.php?id=1100000002633359842&open=primehttp://factordb.com/index.php?showid=1100000002633359842&base=24http://factordb.com/index.php?showid=1100000002633359842http://factordb.com/helper.php?id=1100000002633359842http://factordb.com/cert.php?id=1100000002633359842http://factordb.com/certchain.php?fid=1100000002633359842&action=all&fr=0&to=100http://factordb.com/index.php?query=276*24%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
243405N2644LLN242647−120126473654http://factordb.com/index.php?id=1100000003593270089&open=primehttp://factordb.com/index.php?showid=1100000003593270089&base=24http://factordb.com/index.php?showid=1100000003593270089http://factordb.com/helper.php?id=1100000003593270089http://factordb.com/cert.php?id=1100000003593270089http://factordb.com/certchain.php?fid=1100000003593270089&action=all&fr=0&to=100http://factordb.com/index.php?query=24%5E%28n%2B3%29-1201&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
243406D2698LD(13×242700+4403)/2327003727http://factordb.com/index.php?id=1100000003593269876&open=primehttp://factordb.com/index.php?showid=1100000003593269876&base=24http://factordb.com/index.php?showid=1100000003593269876http://factordb.com/helper.php?id=1100000003593269876http://factordb.com/cert.php?id=1100000003593269876http://factordb.com/certchain.php?fid=1100000003593269876&action=all&fr=0&to=100http://factordb.com/index.php?query=%2813*24%5E%28n%2B2%29%2B4403%29%2F23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
243407A029518ID10×242954+505329554079http://factordb.com/index.php?id=1100000003593269654&open=primehttp://factordb.com/index.php?showid=1100000003593269654&base=24http://factordb.com/index.php?showid=1100000003593269654http://factordb.com/helper.php?id=1100000003593269654http://factordb.com/cert.php?id=1100000003593269654http://factordb.com/certchain.php?fid=1100000003593269654&action=all&fr=0&to=100http://factordb.com/index.php?query=10*24%5E%28n%2B3%29%2B5053&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
24340888N5951201×245951−159538216http://factordb.com/index.php?id=1100000003593275880&open=primehttp://factordb.com/index.php?showid=1100000003593275880&base=24http://factordb.com/index.php?showid=1100000003593275880http://factordb.com/helper.php?id=1100000003593275880proven prime by N+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), since N+1 is trivially fully factoredhttp://factordb.com/index.php?id=1100000003593373246&open=ecmhttp://factordb.com/index.php?query=201*24%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
243409N00N8129LN13249×248131−49813411227http://factordb.com/index.php?id=1100000003593391606&open=primehttp://factordb.com/index.php?showid=1100000003593391606&base=24http://factordb.com/index.php?showid=1100000003593391606http://factordb.com/helper.php?id=1100000003593391606http://factordb.com/cert.php?id=1100000003593391606http://factordb.com/certchain.php?fid=1100000003593391606&action=all&fr=0&to=100http://factordb.com/index.php?query=13249*24%5E%28n%2B2%29-49&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625174OL0214M9645×26216+581218309http://factordb.com/index.php?id=1100000000840631576&open=primehttp://factordb.com/index.php?showid=1100000000840631576&base=26http://factordb.com/index.php?showid=1100000000840631576http://factordb.com/helper.php?id=1100000000840631576proven prime by N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), N−1 is 22×52×7×223×42849349×(296-digit prime)http://factordb.com/index.php?id=1100000000840631577&open=ecmhttp://factordb.com/index.php?query=645*26%5E%28n%2B2%29%2B581&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26251751A219P(7×26220+73)/5221312http://factordb.com/index.php?id=1100000000840631595&open=primehttp://factordb.com/index.php?showid=1100000000840631595&base=26http://factordb.com/index.php?showid=1100000000840631595http://factordb.com/helper.php?id=1100000000840631595http://factordb.com/cert.php?id=1100000000840631595http://factordb.com/certchain.php?fid=1100000000840631595&action=all&fr=0&to=100http://factordb.com/index.php?query=%287*26%5E%28n%2B1%29%2B73%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625176A223DP(2×26225+463)/5225318http://factordb.com/index.php?id=1100000003850155262&open=primehttp://factordb.com/index.php?showid=1100000003850155262&base=26http://factordb.com/index.php?showid=1100000003850155262http://factordb.com/helper.php?id=1100000003850155262http://factordb.com/cert.php?id=1100000003850155262http://factordb.com/certchain.php?fid=1100000003850155262&action=all&fr=0&to=100http://factordb.com/index.php?query=%282*26%5E%28n%2B2%29%2B463%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26251776J225(169×26225−19)/25226320http://factordb.com/index.php?id=1100000002328050895&open=primehttp://factordb.com/index.php?showid=1100000002328050895&base=26http://factordb.com/index.php?showid=1100000002328050895http://factordb.com/helper.php?id=1100000002328050895http://factordb.com/cert.php?id=1100000002328050895http://factordb.com/certchain.php?fid=1100000002328050895&action=all&fr=0&to=100http://factordb.com/index.php?query=%28169*26%5En-19%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625178O2285(24×26229−499)/25229325http://factordb.com/index.php?id=1100000002328059255&open=primehttp://factordb.com/index.php?showid=1100000002328059255&base=26http://factordb.com/index.php?showid=1100000002328059255http://factordb.com/helper.php?id=1100000002328059255http://factordb.com/cert.php?id=1100000002328059255http://factordb.com/certchain.php?fid=1100000002328059255&action=all&fr=0&to=100http://factordb.com/index.php?query=%2824*26%5E%28n%2B1%29-499%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625179K0230K0IP20×26234+352013235333http://factordb.com/index.php?id=1100000000840631669&open=primehttp://factordb.com/index.php?showid=1100000000840631669&base=26http://factordb.com/index.php?showid=1100000000840631669http://factordb.com/helper.php?id=1100000000840631669http://factordb.com/cert.php?id=1100000000840631669http://factordb.com/certchain.php?fid=1100000000840631669&action=all&fr=0&to=100http://factordb.com/index.php?query=20*26%5E%28n%2B4%29%2B352013&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625180B0236OB11×26238+635239338http://factordb.com/index.php?id=1100000002634136234&open=primehttp://factordb.com/index.php?showid=1100000002634136234&base=26http://factordb.com/index.php?showid=1100000002634136234http://factordb.com/helper.php?id=1100000002634136234http://factordb.com/cert.php?id=1100000002634136234http://factordb.com/certchain.php?fid=1100000002634136234&action=all&fr=0&to=100http://factordb.com/index.php?query=11*26%5E%28n%2B2%29%2B635&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
262518111G02399718×26240+9243343http://factordb.com/index.php?id=1100000000840631687&open=primehttp://factordb.com/index.php?showid=1100000000840631687&base=26http://factordb.com/index.php?showid=1100000000840631687http://factordb.com/helper.php?id=1100000000840631687http://factordb.com/cert.php?id=1100000000840631687http://factordb.com/certchain.php?fid=1100000000840631687&action=all&fr=0&to=100http://factordb.com/index.php?query=718*26%5E%28n%2B1%29%2B9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625182K0241E520×26243+369244346http://factordb.com/index.php?id=1100000002634136479&open=primehttp://factordb.com/index.php?showid=1100000002634136479&base=26http://factordb.com/index.php?showid=1100000002634136479http://factordb.com/helper.php?id=1100000002634136479http://factordb.com/cert.php?id=1100000002634136479http://factordb.com/certchain.php?fid=1100000002634136479&action=all&fr=0&to=100http://factordb.com/index.php?query=20*26%5E%28n%2B2%29%2B369&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625183J2430L(19×26245−12319)/25245347http://factordb.com/index.php?id=1100000003850155263&open=primehttp://factordb.com/index.php?showid=1100000003850155263&base=26http://factordb.com/index.php?showid=1100000003850155263http://factordb.com/helper.php?id=1100000003850155263http://factordb.com/cert.php?id=1100000003850155263http://factordb.com/certchain.php?fid=1100000003850155263&action=all&fr=0&to=100http://factordb.com/index.php?query=%2819*26%5E%28n%2B2%29-12319%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625184B251I9(11×26253+4489)/25253358http://factordb.com/index.php?id=1100000003850155264&open=primehttp://factordb.com/index.php?showid=1100000003850155264&base=26http://factordb.com/index.php?showid=1100000003850155264http://factordb.com/helper.php?id=1100000003850155264http://factordb.com/cert.php?id=1100000003850155264http://factordb.com/certchain.php?fid=1100000003850155264&action=all&fr=0&to=100http://factordb.com/index.php?query=%2811*26%5E%28n%2B2%29%2B4489%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625185F2500PCF(3×26254−1284793)/5254360http://factordb.com/index.php?id=1100000000840631708&open=primehttp://factordb.com/index.php?showid=1100000000840631708&base=26http://factordb.com/index.php?showid=1100000000840631708http://factordb.com/helper.php?id=1100000000840631708http://factordb.com/cert.php?id=1100000000840631708http://factordb.com/certchain.php?fid=1100000000840631708&action=all&fr=0&to=100http://factordb.com/index.php?query=%283*26%5E%28n%2B4%29-1284793%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26251864E7262(2957×26262−7)/25264373http://factordb.com/index.php?id=1100000003850155265&open=primehttp://factordb.com/index.php?showid=1100000003850155265&base=26http://factordb.com/index.php?showid=1100000003850155265http://factordb.com/helper.php?id=1100000003850155265http://factordb.com/cert.php?id=1100000003850155265http://factordb.com/certchain.php?fid=1100000003850155265&action=all&fr=0&to=100http://factordb.com/index.php?query=%282957*26%5En-7%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625187E7264OL(357×26266+11393)/25267378http://factordb.com/index.php?id=1100000003850155266&open=primehttp://factordb.com/index.php?showid=1100000003850155266&base=26http://factordb.com/index.php?showid=1100000003850155266http://factordb.com/helper.php?id=1100000003850155266http://factordb.com/cert.php?id=1100000003850155266http://factordb.com/certchain.php?fid=1100000003850155266&action=all&fr=0&to=100http://factordb.com/index.php?query=%28357*26%5E%28n%2B2%29%2B11393%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625188EIL267(9571×26267−21)/25269381http://factordb.com/index.php?id=1100000000840631801&open=primehttp://factordb.com/index.php?showid=1100000000840631801&base=26http://factordb.com/index.php?showid=1100000000840631801http://factordb.com/helper.php?id=1100000000840631801http://factordb.com/cert.php?id=1100000000840631801http://factordb.com/certchain.php?fid=1100000000840631801&action=all&fr=0&to=100http://factordb.com/index.php?query=%289571*26%5En-21%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
262518962684F(6×26270−1081)/25270382http://factordb.com/index.php?id=1100000000840631976&open=primehttp://factordb.com/index.php?showid=1100000000840631976&base=26http://factordb.com/index.php?showid=1100000000840631976http://factordb.com/helper.php?id=1100000000840631976http://factordb.com/cert.php?id=1100000000840631976http://factordb.com/certchain.php?fid=1100000000840631976&action=all&fr=0&to=100http://factordb.com/index.php?query=%286*26%5E%28n%2B2%29-1081%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625190D020273H8790×26274+17277392http://factordb.com/index.php?id=1100000003850155267&open=primehttp://factordb.com/index.php?showid=1100000003850155267&base=26http://factordb.com/index.php?showid=1100000003850155267http://factordb.com/helper.php?id=1100000003850155267http://factordb.com/cert.php?id=1100000003850155267http://factordb.com/certchain.php?fid=1100000003850155267&action=all&fr=0&to=100http://factordb.com/index.php?query=8790*26%5E%28n%2B1%29%2B17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625191B291KB(11×26293+5839)/25293415http://factordb.com/index.php?id=1100000003850155268&open=primehttp://factordb.com/index.php?showid=1100000003850155268&base=26http://factordb.com/index.php?showid=1100000003850155268http://factordb.com/helper.php?id=1100000003850155268http://factordb.com/cert.php?id=1100000003850155268http://factordb.com/certchain.php?fid=1100000003850155268&action=all&fr=0&to=100http://factordb.com/index.php?query=%2811*26%5E%28n%2B2%29%2B5839%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26251925293O5(26295+2469)/5295417http://factordb.com/index.php?id=1100000003850155269&open=primehttp://factordb.com/index.php?showid=1100000003850155269&base=26http://factordb.com/index.php?showid=1100000003850155269http://factordb.com/helper.php?id=1100000003850155269http://factordb.com/cert.php?id=1100000003850155269http://factordb.com/certchain.php?fid=1100000003850155269&action=all&fr=0&to=100http://factordb.com/index.php?query=%2826%5E%28n%2B2%29%2B2469%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625193D7300(332×26300−7)/25301426http://factordb.com/index.php?id=1100000002328053362&open=primehttp://factordb.com/index.php?showid=1100000002328053362&base=26http://factordb.com/index.php?showid=1100000002328053362http://factordb.com/helper.php?id=1100000002328053362http://factordb.com/cert.php?id=1100000002328053362http://factordb.com/certchain.php?fid=1100000002328053362&action=all&fr=0&to=100http://factordb.com/index.php?query=%28332*26%5En-7%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625194E305IL(14×26307+2761)/25307435http://factordb.com/index.php?id=1100000000840632032&open=primehttp://factordb.com/index.php?showid=1100000000840632032&base=26http://factordb.com/index.php?showid=1100000000840632032http://factordb.com/helper.php?id=1100000000840632032http://factordb.com/cert.php?id=1100000000840632032http://factordb.com/certchain.php?fid=1100000000840632032&action=all&fr=0&to=100http://factordb.com/index.php?query=%2814*26%5E%28n%2B2%29%2B2761%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625195PO5312(3371×26312−1)/5314445http://factordb.com/index.php?id=1100000003850155270&open=primehttp://factordb.com/index.php?showid=1100000003850155270&base=26http://factordb.com/index.php?showid=1100000003850155270http://factordb.com/helper.php?id=1100000003850155270http://factordb.com/cert.php?id=1100000003850155270http://factordb.com/certchain.php?fid=1100000003850155270&action=all&fr=0&to=100http://factordb.com/index.php?query=%283371*26%5En-1%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
262519647314(107×26314−7)/25315445http://factordb.com/index.php?id=1100000002328050727&open=primehttp://factordb.com/index.php?showid=1100000002328050727&base=26http://factordb.com/index.php?showid=1100000002328050727http://factordb.com/helper.php?id=1100000002328050727http://factordb.com/cert.php?id=1100000002328050727http://factordb.com/certchain.php?fid=1100000002328050727&action=all&fr=0&to=100http://factordb.com/index.php?query=%28107*26%5En-7%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625197A33560F(2×26338−14797)/5338478http://factordb.com/index.php?id=1100000000840632163&open=primehttp://factordb.com/index.php?showid=1100000000840632163&base=26http://factordb.com/index.php?showid=1100000000840632163http://factordb.com/helper.php?id=1100000000840632163http://factordb.com/cert.php?id=1100000000840632163http://factordb.com/certchain.php?fid=1100000000840632163&action=all&fr=0&to=100http://factordb.com/index.php?query=%282*26%5E%28n%2B3%29-14797%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625198O5K5341(81871×26341−1)/5344487http://factordb.com/index.php?id=1100000003850155271&open=primehttp://factordb.com/index.php?showid=1100000003850155271&base=26http://factordb.com/index.php?showid=1100000003850155271http://factordb.com/helper.php?id=1100000003850155271http://factordb.com/cert.php?id=1100000003850155271http://factordb.com/certchain.php?fid=1100000003850155271&action=all&fr=0&to=100http://factordb.com/index.php?query=%2887871*26%5En-1%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26251999K343AP(49×26345−1279)/5346490http://factordb.com/index.php?id=1100000000840632228&open=primehttp://factordb.com/index.php?showid=1100000000840632228&base=26http://factordb.com/index.php?showid=1100000000840632228http://factordb.com/helper.php?id=1100000000840632228proven prime by N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since N−1 is 1274/5×(26344−1), thus factor N−1 is equivalent to factor 26344−1, and for the factorization of 26344−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&Exp=344&c0=-&EN=http://factordb.com/index.php?id=1100000000840632232&open=ecmhttp://factordb.com/index.php?query=%2849*26%5E%28n%2B2%29-1279%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
262520083541(8×26355−183)/25355502http://factordb.com/index.php?id=1100000000840632517&open=primehttp://factordb.com/index.php?showid=1100000000840632517&base=26http://factordb.com/index.php?showid=1100000000840632517http://factordb.com/helper.php?id=1100000000840632517proven prime by N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since N−1 is 208/25×(26354−1), thus factor N−1 is equivalent to factor 26354−1, and for the factorization of 26354−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&Exp=354&c0=-&EN=http://factordb.com/index.php?id=1100000000840632623&open=ecmhttp://factordb.com/index.php?query=%288*26%5E%28n%2B1%29-183%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625201L035666K921×26360+110041361511http://factordb.com/index.php?id=1100000000840632748&open=primehttp://factordb.com/index.php?showid=1100000000840632748&base=26http://factordb.com/index.php?showid=1100000000840632748http://factordb.com/helper.php?id=1100000000840632748http://factordb.com/cert.php?id=1100000000840632748http://factordb.com/certchain.php?fid=1100000000840632748&action=all&fr=0&to=100http://factordb.com/index.php?query=21*26%5E%28n%2B4%29%2B110041&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625202K0358KIP20×26361+14013362513http://factordb.com/index.php?id=1100000000840632880&open=primehttp://factordb.com/index.php?showid=1100000000840632880&base=26http://factordb.com/index.php?showid=1100000000840632880http://factordb.com/helper.php?id=1100000000840632880http://factordb.com/cert.php?id=1100000000840632880http://factordb.com/certchain.php?fid=1100000000840632880&action=all&fr=0&to=100http://factordb.com/index.php?query=20*26%5E%28n%2B3%29%2B14013&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625203J0360A0P19×26363+6785364515http://factordb.com/index.php?id=1100000003850155272&open=primehttp://factordb.com/index.php?showid=1100000003850155272&base=26http://factordb.com/index.php?showid=1100000003850155272http://factordb.com/helper.php?id=1100000003850155272http://factordb.com/cert.php?id=1100000003850155272http://factordb.com/certchain.php?fid=1100000003850155272&action=all&fr=0&to=100http://factordb.com/index.php?query=19*26%5E%28n%2B3%29%2B6785&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625204OK63769(16106×26377+69)/25379537http://factordb.com/index.php?id=1100000000840633320&open=primehttp://factordb.com/index.php?showid=1100000000840633320&base=26http://factordb.com/index.php?showid=1100000000840633320http://factordb.com/helper.php?id=1100000000840633320http://factordb.com/cert.php?id=1100000000840633320http://factordb.com/certchain.php?fid=1100000000840633320&action=all&fr=0&to=100http://factordb.com/index.php?query=%2816106*26%5E%28n%2B1%29%2B69%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625205J4067(19×26407−319)/25407576http://factordb.com/index.php?id=1100000002328055467&open=primehttp://factordb.com/index.php?showid=1100000002328055467&base=26http://factordb.com/index.php?showid=1100000002328055467http://factordb.com/helper.php?id=1100000002328055467http://factordb.com/cert.php?id=1100000002328055467http://factordb.com/certchain.php?fid=1100000002328055467&action=all&fr=0&to=100http://factordb.com/index.php?query=%2819*26%5E%28n%2B1%29-319%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26252069B437(236×26437−11)/25438620http://factordb.com/index.php?id=1100000002328051905&open=primehttp://factordb.com/index.php?showid=1100000002328051905&base=26http://factordb.com/index.php?showid=1100000002328051905http://factordb.com/helper.php?id=1100000002328051905http://factordb.com/cert.php?id=1100000002328051905http://factordb.com/certchain.php?fid=1100000002328051905&action=all&fr=0&to=100http://factordb.com/index.php?query=%28236*26%5En-11%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26252073442GL(3×26444+8897)/25444628http://factordb.com/index.php?id=1100000003850155273&open=primehttp://factordb.com/index.php?showid=1100000003850155273&base=26http://factordb.com/index.php?showid=1100000003850155273http://factordb.com/helper.php?id=1100000003850155273http://factordb.com/cert.php?id=1100000003850155273http://factordb.com/certchain.php?fid=1100000003850155273&action=all&fr=0&to=100http://factordb.com/index.php?query=%283*26%5E%28n%2B2%29%2B8897%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26252081M491P(47×26492+53)/25493697http://factordb.com/index.php?id=1100000000840633390&open=primehttp://factordb.com/index.php?showid=1100000000840633390&base=26http://factordb.com/index.php?showid=1100000000840633390http://factordb.com/helper.php?id=1100000000840633390http://factordb.com/cert.php?id=1100000000840633390http://factordb.com/certchain.php?fid=1100000000840633390&action=all&fr=0&to=100http://factordb.com/index.php?query=%2847*26%5E%28n%2B1%29%2B53%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
262520940509GL4×26511+437512724http://factordb.com/index.php?id=1100000000840633483&open=primehttp://factordb.com/index.php?showid=1100000000840633483&base=26http://factordb.com/index.php?showid=1100000000840633483http://factordb.com/helper.php?id=1100000000840633483http://factordb.com/cert.php?id=1100000000840633483http://factordb.com/certchain.php?fid=1100000000840633483&action=all&fr=0&to=100http://factordb.com/index.php?query=4*26%5E%28n%2B2%29%2B437&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625210BFA5115(1507×26512−27)/5514727http://factordb.com/index.php?id=1100000003850155274&open=primehttp://factordb.com/index.php?showid=1100000003850155274&base=26http://factordb.com/index.php?showid=1100000003850155274http://factordb.com/helper.php?id=1100000003850155274http://factordb.com/cert.php?id=1100000003850155274http://factordb.com/certchain.php?fid=1100000003850155274&action=all&fr=0&to=100http://factordb.com/index.php?query=%281507*26%5E%28n%2B1%29-27%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625211LK5185(109×26519−79)/5520736http://factordb.com/index.php?id=1100000003850155276&open=primehttp://factordb.com/index.php?showid=1100000003850155276&base=26http://factordb.com/index.php?showid=1100000003850155276http://factordb.com/helper.php?id=1100000003850155276http://factordb.com/cert.php?id=1100000003850155276http://factordb.com/certchain.php?fid=1100000003850155276&action=all&fr=0&to=100http://factordb.com/index.php?query=%28109*26%5E%28n%2B1%29-79%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625212MI5433(568×26544−393)/25545772http://factordb.com/index.php?id=1100000003850155277&open=primehttp://factordb.com/index.php?showid=1100000003850155277&base=26http://factordb.com/index.php?showid=1100000003850155277http://factordb.com/helper.php?id=1100000003850155277http://factordb.com/cert.php?id=1100000003850155277http://factordb.com/certchain.php?fid=1100000003850155277&action=all&fr=0&to=100http://factordb.com/index.php?query=%28568*26%5E%28n%2B1%29-393%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625213E4E5597(9214×26560−189)/25562795http://factordb.com/index.php?id=1100000003850155278&open=primehttp://factordb.com/index.php?showid=1100000003850155278&base=26http://factordb.com/index.php?showid=1100000003850155278http://factordb.com/helper.php?id=1100000003850155278http://factordb.com/cert.php?id=1100000003850155278http://factordb.com/certchain.php?fid=1100000003850155278&action=all&fr=0&to=100http://factordb.com/index.php?query=%289214*26%5E%28n%2B1%29-189%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
262521480577C78×26579+319580821http://factordb.com/index.php?id=1100000002634136160&open=primehttp://factordb.com/index.php?showid=1100000002634136160&base=26http://factordb.com/index.php?showid=1100000002634136160http://factordb.com/helper.php?id=1100000002634136160http://factordb.com/cert.php?id=1100000002634136160http://factordb.com/certchain.php?fid=1100000002634136160&action=all&fr=0&to=100http://factordb.com/index.php?query=8*26%5E%28n%2B2%29%2B319&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26252159E0619B248×26620+11622880http://factordb.com/index.php?id=1100000002634136193&open=primehttp://factordb.com/index.php?showid=1100000002634136193&base=26http://factordb.com/index.php?showid=1100000002634136193http://factordb.com/helper.php?id=1100000002634136193http://factordb.com/cert.php?id=1100000002634136193http://factordb.com/certchain.php?fid=1100000002634136193&action=all&fr=0&to=100http://factordb.com/index.php?query=248*26%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625216G60618KJ422×26620+539622880http://factordb.com/index.php?id=1100000003850155283&open=primehttp://factordb.com/index.php?showid=1100000003850155283&base=26http://factordb.com/index.php?showid=1100000003850155283http://factordb.com/helper.php?id=1100000003850155283http://factordb.com/cert.php?id=1100000003850155283http://factordb.com/certchain.php?fid=1100000003850155283&action=all&fr=0&to=100http://factordb.com/index.php?query=422*26%5E%28n%2B2%29%2B539&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625217OO0620D3648×26622+341624883http://factordb.com/index.php?id=1100000003850155285&open=primehttp://factordb.com/index.php?showid=1100000003850155285&base=26http://factordb.com/index.php?showid=1100000003850155285http://factordb.com/helper.php?id=1100000003850155285http://factordb.com/cert.php?id=1100000003850155285http://factordb.com/certchain.php?fid=1100000003850155285&action=all&fr=0&to=100http://factordb.com/index.php?query=648*26%5E%28n%2B2%29%2B341&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625218K0653IP20×26655+493656929http://factordb.com/index.php?id=1100000000840633594&open=primehttp://factordb.com/index.php?showid=1100000000840633594&base=26http://factordb.com/index.php?showid=1100000000840633594http://factordb.com/helper.php?id=1100000000840633594http://factordb.com/cert.php?id=1100000000840633594http://factordb.com/certchain.php?fid=1100000000840633594&action=all&fr=0&to=100http://factordb.com/index.php?query=20*26%5E%28n%2B2%29%2B493&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625219J00J698L(321119×26699+31)/25702994http://factordb.com/index.php?id=1100000003850155288&open=primehttp://factordb.com/index.php?showid=1100000003850155288&base=26http://factordb.com/index.php?showid=1100000003850155288http://factordb.com/helper.php?id=1100000003850155288http://factordb.com/cert.php?id=1100000003850155288http://factordb.com/certchain.php?fid=1100000003850155288&action=all&fr=0&to=100http://factordb.com/index.php?query=%28321119*26%5E%28n%2B1%29%2B31%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625220B077290J11×26775+61037761098http://factordb.com/index.php?id=1100000003850155290&open=primehttp://factordb.com/index.php?showid=1100000003850155290&base=26http://factordb.com/index.php?showid=1100000003850155290http://factordb.com/helper.php?id=1100000003850155290http://factordb.com/cert.php?id=1100000003850155290http://factordb.com/certchain.php?fid=1100000003850155290&action=all&fr=0&to=100http://factordb.com/index.php?query=11*26%5E%28n%2B3%29%2B6103&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625221J844B(19×26845−219)/258451196http://factordb.com/index.php?id=1100000002328055693&open=primehttp://factordb.com/index.php?showid=1100000002328055693&base=26http://factordb.com/index.php?showid=1100000002328055693http://factordb.com/helper.php?id=1100000002328055693http://factordb.com/cert.php?id=1100000002328055693http://factordb.com/certchain.php?fid=1100000002328055693&action=all&fr=0&to=100http://factordb.com/index.php?query=%2819*26%5E%28n%2B1%29-219%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625222H855M0H(17×26858+73433)/258581214http://factordb.com/index.php?id=1100000003850155291&open=primehttp://factordb.com/index.php?showid=1100000003850155291&base=26http://factordb.com/index.php?showid=1100000003850155291http://factordb.com/helper.php?id=1100000003850155291http://factordb.com/cert.php?id=1100000003850155291http://factordb.com/certchain.php?fid=1100000003850155291&action=all&fr=0&to=100http://factordb.com/index.php?query=%2817*26%5E%28n%2B3%29%2B73433%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625223J861OOL(19×26864+87781)/258641223http://factordb.com/index.php?id=1100000003850155296&open=primehttp://factordb.com/index.php?showid=1100000003850155296&base=26http://factordb.com/index.php?showid=1100000003850155296http://factordb.com/helper.php?id=1100000003850155296http://factordb.com/cert.php?id=1100000003850155296http://factordb.com/certchain.php?fid=1100000003850155296&action=all&fr=0&to=100http://factordb.com/index.php?query=%2819*26%5E%28n%2B3%29%2B87781%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625224B0979H11×26980+179811388http://factordb.com/index.php?id=1100000002355639467&open=primehttp://factordb.com/index.php?showid=1100000002355639467&base=26http://factordb.com/index.php?showid=1100000002355639467http://factordb.com/helper.php?id=1100000002355639467http://factordb.com/cert.php?id=1100000002355639467http://factordb.com/certchain.php?fid=1100000002355639467&action=all&fr=0&to=100http://factordb.com/index.php?query=11*26%5E%28n%2B1%29%2B17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625225L09914000J21×26996+18279239971411http://factordb.com/index.php?id=1100000003850155301&open=primehttp://factordb.com/index.php?showid=1100000003850155301&base=26http://factordb.com/index.php?showid=1100000003850155301http://factordb.com/helper.php?id=1100000003850155301http://factordb.com/cert.php?id=1100000003850155301http://factordb.com/certchain.php?fid=1100000003850155301&action=all&fr=0&to=100http://factordb.com/index.php?query=21*26%5E%28n%2B5%29%2B1827923&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625226E01101K2B14×261104+1358311051564http://factordb.com/index.php?id=1100000003850155305&open=primehttp://factordb.com/index.php?showid=1100000003850155305&base=26http://factordb.com/index.php?showid=1100000003850155305http://factordb.com/helper.php?id=1100000003850155305http://factordb.com/cert.php?id=1100000003850155305http://factordb.com/certchain.php?fid=1100000003850155305&action=all&fr=0&to=100http://factordb.com/index.php?query=14*26%5E%28n%2B3%29%2B13583&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625227G1105OO9(16×261108+140209)/2511081568http://factordb.com/index.php?id=1100000000840633717&open=primehttp://factordb.com/index.php?showid=1100000000840633717&base=26http://factordb.com/index.php?showid=1100000000840633717http://factordb.com/helper.php?id=1100000000840633717http://factordb.com/cert.php?id=1100000000840633717http://factordb.com/certchain.php?fid=1100000000840633717&action=all&fr=0&to=100http://factordb.com/index.php?query=%2816*26%5E%28n%2B3%29%2B140209%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625228MC01109N584×261110+2311121574http://factordb.com/index.php?id=1100000002634136576&open=primehttp://factordb.com/index.php?showid=1100000002634136576&base=26http://factordb.com/index.php?showid=1100000002634136576http://factordb.com/helper.php?id=1100000002634136576http://factordb.com/cert.php?id=1100000002634136576http://factordb.com/certchain.php?fid=1100000002634136576&action=all&fr=0&to=100http://factordb.com/index.php?query=584*26%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625229G11599(16×261160−191)/2511601642http://factordb.com/index.php?id=1100000000840633844&open=primehttp://factordb.com/index.php?showid=1100000000840633844&base=26http://factordb.com/index.php?showid=1100000000840633844http://factordb.com/helper.php?id=1100000000840633844http://factordb.com/cert.php?id=1100000000840633844http://factordb.com/certchain.php?fid=1100000000840633844&action=all&fr=0&to=100http://factordb.com/index.php?query=%2816*26%5E%28n%2B1%29-191%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625230A129506F(2×261298−34297)/512981837http://factordb.com/index.php?id=1100000000840633998&open=primehttp://factordb.com/index.php?showid=1100000000840633998&base=26http://factordb.com/index.php?showid=1100000000840633998http://factordb.com/helper.php?id=1100000000840633998http://factordb.com/cert.php?id=1100000000840633998http://factordb.com/certchain.php?fid=1100000000840633998&action=all&fr=0&to=100http://factordb.com/index.php?query=%282*26%5E%28n%2B3%29-34297%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625231KIA1298F(2692×261299+23)/513011841http://factordb.com/index.php?id=1100000000840634108&open=primehttp://factordb.com/index.php?showid=1100000000840634108&base=26http://factordb.com/index.php?showid=1100000000840634108http://factordb.com/helper.php?id=1100000000840634108http://factordb.com/cert.php?id=1100000000840634108http://factordb.com/certchain.php?fid=1100000000840634108&action=all&fr=0&to=100http://factordb.com/index.php?query=%282692*26%5E%28n%2B1%29%2B23%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625232L71319OL(532×261321+11393)/2513221871http://factordb.com/index.php?id=1100000003850155311&open=primehttp://factordb.com/index.php?showid=1100000003850155311&base=26http://factordb.com/index.php?showid=1100000003850155311http://factordb.com/helper.php?id=1100000003850155311http://factordb.com/cert.php?id=1100000003850155311http://factordb.com/certchain.php?fid=1100000003850155311&action=all&fr=0&to=100http://factordb.com/index.php?query=%28532*26%5E%28n%2B2%29%2B11393%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625233J1365L(19×261366+31)/2513661933http://factordb.com/index.php?id=1100000002328055922&open=primehttp://factordb.com/index.php?showid=1100000002328055922&base=26http://factordb.com/index.php?showid=1100000002328055922http://factordb.com/helper.php?id=1100000002328055922http://factordb.com/cert.php?id=1100000002328055922http://factordb.com/certchain.php?fid=1100000002328055922&action=all&fr=0&to=100http://factordb.com/index.php?query=%2819*26%5E%28n%2B1%29%2B31%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
262523421498H(2×261499+373)/2514992120http://factordb.com/index.php?id=1100000002328050300&open=primehttp://factordb.com/index.php?showid=1100000002328050300&base=26http://factordb.com/index.php?showid=1100000002328050300http://factordb.com/helper.php?id=1100000002328050300http://factordb.com/cert.php?id=1100000002328050300http://factordb.com/certchain.php?fid=1100000002328050300&action=all&fr=0&to=100http://factordb.com/index.php?query=%282*26%5E%28n%2B1%29%2B373%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625235O51509(121×261509−1)/515102137http://factordb.com/index.php?id=1100000000894500022&open=primehttp://factordb.com/index.php?showid=1100000000894500022&base=26http://factordb.com/index.php?showid=1100000000894500022http://factordb.com/helper.php?id=1100000000894500022http://factordb.com/cert.php?id=1100000000894500022http://factordb.com/certchain.php?fid=1100000000894500022&action=all&fr=0&to=100http://factordb.com/index.php?query=%28121*26%5En-1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625236DM1519P(347×261520+53)/2515212152http://factordb.com/index.php?id=1100000003850155312&open=primehttp://factordb.com/index.php?showid=1100000003850155312&base=26http://factordb.com/index.php?showid=1100000003850155312http://factordb.com/helper.php?id=1100000003850155312http://factordb.com/cert.php?id=1100000003850155312http://factordb.com/certchain.php?fid=1100000003850155312&action=all&fr=0&to=100http://factordb.com/index.php?query=%28347*26%5E%28n%2B1%29%2B53%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625237J01523P19×261524+2515252158http://factordb.com/index.php?id=1100000002355640604&open=primehttp://factordb.com/index.php?showid=1100000002355640604&base=26http://factordb.com/index.php?showid=1100000002355640604http://factordb.com/helper.php?id=1100000002355640604http://factordb.com/cert.php?id=1100000002355640604http://factordb.com/certchain.php?fid=1100000002355640604&action=all&fr=0&to=100http://factordb.com/index.php?query=19*26%5E%28n%2B1%29%2B25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625238F1569PCF(3×261572+33407)/515722225http://factordb.com/index.php?id=1100000000840634210&open=primehttp://factordb.com/index.php?showid=1100000000840634210&base=26http://factordb.com/index.php?showid=1100000000840634210http://factordb.com/helper.php?id=1100000000840634210http://factordb.com/cert.php?id=1100000000840634210http://factordb.com/certchain.php?fid=1100000000840634210&action=all&fr=0&to=100http://factordb.com/index.php?query=%283*26%5E%28n%2B3%29%2B33407%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625239N01647NEN23×261650+1593516512337http://factordb.com/index.php?id=1100000003850155313&open=primehttp://factordb.com/index.php?showid=1100000003850155313&base=26http://factordb.com/index.php?showid=1100000003850155313http://factordb.com/helper.php?id=1100000003850155313http://factordb.com/cert.php?id=1100000003850155313http://factordb.com/certchain.php?fid=1100000003850155313&action=all&fr=0&to=100http://factordb.com/index.php?query=23*26%5E%28n%2B3%29%2B15935&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625240518854P(261887−31)/518872670http://factordb.com/index.php?id=1100000003850155314&open=primehttp://factordb.com/index.php?showid=1100000003850155314&base=26http://factordb.com/index.php?showid=1100000003850155314http://factordb.com/helper.php?id=1100000003850155314http://factordb.com/cert.php?id=1100000003850155314http://factordb.com/certchain.php?fid=1100000003850155314&action=all&fr=0&to=100http://factordb.com/index.php?query=%2826%5E%28n%2B2%29-31%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26252416K2556A5(34×262558−1379)/525593621http://factordb.com/index.php?id=1100000003850155315&open=primehttp://factordb.com/index.php?showid=1100000003850155315&base=26http://factordb.com/index.php?showid=1100000003850155315http://factordb.com/helper.php?id=1100000003850155315http://factordb.com/cert.php?id=1100000003850155315http://factordb.com/certchain.php?fid=1100000003850155315&action=all&fr=0&to=100http://factordb.com/index.php?query=%2834*26%5E%28n%2B2%29-1379%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625242702613CN7×262615+33526163702http://factordb.com/index.php?id=1100000002634136105&open=primehttp://factordb.com/index.php?showid=1100000002634136105&base=26http://factordb.com/index.php?showid=1100000002634136105http://factordb.com/helper.php?id=1100000002634136105http://factordb.com/cert.php?id=1100000002634136105http://factordb.com/certchain.php?fid=1100000002634136105&action=all&fr=0&to=100http://factordb.com/index.php?query=7*26%5E%28n%2B2%29%2B335&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625243E02673H14×262674+1726753785http://factordb.com/index.php?id=1100000002355640062&open=primehttp://factordb.com/index.php?showid=1100000002355640062&base=26http://factordb.com/index.php?showid=1100000002355640062http://factordb.com/helper.php?id=1100000002355640062http://factordb.com/cert.php?id=1100000002355640062http://factordb.com/certchain.php?fid=1100000002355640062&action=all&fr=0&to=100http://factordb.com/index.php?query=14*26%5E%28n%2B1%29%2B17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625244G602740J422×262741+1927433882http://factordb.com/index.php?id=1100000002634136363&open=primehttp://factordb.com/index.php?showid=1100000002634136363&base=26http://factordb.com/index.php?showid=1100000002634136363http://factordb.com/helper.php?id=1100000002634136363http://factordb.com/cert.php?id=1100000002634136363http://factordb.com/certchain.php?fid=1100000002634136363&action=all&fr=0&to=100http://factordb.com/index.php?query=422*26%5E%28n%2B1%29%2B19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625245B29159(11×262916−61)/2529164126http://factordb.com/index.php?id=1100000002328052611&open=primehttp://factordb.com/index.php?showid=1100000002328052611&base=26http://factordb.com/index.php?showid=1100000002328052611http://factordb.com/helper.php?id=1100000002328052611http://factordb.com/cert.php?id=1100000002328052611http://factordb.com/certchain.php?fid=1100000002328052611&action=all&fr=0&to=100http://factordb.com/index.php?query=%2811*26%5E%28n%2B1%29-61%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625246J4222P(19×264223+131)/2542235976http://factordb.com/index.php?id=1100000002328056865&open=primehttp://factordb.com/index.php?showid=1100000002328056865&base=26http://factordb.com/index.php?showid=1100000002328056865http://factordb.com/helper.php?id=1100000002328056865http://factordb.com/cert.php?id=1100000002328056865http://factordb.com/certchain.php?fid=1100000002328056865&action=all&fr=0&to=100http://factordb.com/index.php?query=%2819*26%5E%28n%2B1%29%2B131%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625247K04364I520×264366+47343676180http://factordb.com/index.php?id=1100000002634136508&open=primehttp://factordb.com/index.php?showid=1100000002634136508&base=26http://factordb.com/index.php?showid=1100000002634136508http://factordb.com/helper.php?id=1100000002634136508http://factordb.com/cert.php?id=1100000002634136508http://factordb.com/certchain.php?fid=1100000002634136508&action=all&fr=0&to=100http://factordb.com/index.php?query=20*26%5E%28n%2B2%29%2B473&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2625248M8772P(22×268773+53)/25877312414http://factordb.com/index.php?id=1100000000758011195&open=primehttp://factordb.com/index.php?showid=1100000000758011195&base=26http://factordb.com/index.php?showid=1100000000758011195http://factordb.com/helper.php?id=1100000000758011195http://factordb.com/cert.php?id=1100000000758011195http://factordb.com/certchain.php?fid=1100000000758011195&action=all&fr=0&to=100http://factordb.com/index.php?query=%2822*26%5E%28n%2B1%29%2B53%29%2F25&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26252499GDK15920P(32569×2615921+21)/51592422532http://factordb.com/index.php?id=1100000003850155316&open=primehttp://factordb.com/index.php?showid=1100000003850155316&base=26http://factordb.com/index.php?showid=1100000003850155316http://factordb.com/helper.php?id=1100000003850155316http://factordb.com/cert.php?id=1100000003850155316http://factordb.com/certchain.php?fid=1100000003850155316&action=all&fr=0&to=100http://factordb.com/index.php?query=%2832569*26%5E%28n%2B1%29%2B21%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825485JN206(536×28206−23)/27207300http://factordb.com/index.php?id=1100000002611724435&open=primehttp://factordb.com/index.php?showid=1100000002611724435&base=28http://factordb.com/index.php?showid=1100000002611724435http://factordb.com/helper.php?id=1100000002611724435proven prime by N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), N−1 is 2×1061×1171×74311×(289-digit prime)http://factordb.com/index.php?id=1100000002611724440&open=ecmhttp://factordb.com/index.php?query=%28536*28%5En-23%29%2F27&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28254863211M9(28213+4841)/9213308http://factordb.com/index.php?id=1100000003850161936&open=primehttp://factordb.com/index.php?showid=1100000003850161936&base=28http://factordb.com/index.php?showid=1100000003850161936http://factordb.com/helper.php?id=1100000003850161936http://factordb.com/cert.php?id=1100000003850161936http://factordb.com/certchain.php?fid=1100000003850161936&action=all&fr=0&to=100http://factordb.com/index.php?query=%2828%5E%28n%2B2%29%2B4841%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825487HD0213D489×28214+13216313http://factordb.com/index.php?id=1100000003850161937&open=primehttp://factordb.com/index.php?showid=1100000003850161937&base=28http://factordb.com/index.php?showid=1100000003850161937http://factordb.com/helper.php?id=1100000003850161937http://factordb.com/cert.php?id=1100000003850161937http://factordb.com/certchain.php?fid=1100000003850161937&action=all&fr=0&to=100http://factordb.com/index.php?query=489*28%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
282548864O2179(1556×28218−143)/9220318http://factordb.com/index.php?id=1100000000840840215&open=primehttp://factordb.com/index.php?showid=1100000000840840215&base=28http://factordb.com/index.php?showid=1100000000840840215http://factordb.com/helper.php?id=1100000000840840215http://factordb.com/cert.php?id=1100000000840840215http://factordb.com/certchain.php?fid=1100000000840840215&action=all&fr=0&to=100http://factordb.com/index.php?query=%281556*28%5E%28n%2B1%29-143%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825489G0217A0N16×28220+7863221320http://factordb.com/index.php?id=1100000003850161938&open=primehttp://factordb.com/index.php?showid=1100000003850161938&base=28http://factordb.com/index.php?showid=1100000003850161938http://factordb.com/helper.php?id=1100000003850161938http://factordb.com/cert.php?id=1100000003850161938http://factordb.com/certchain.php?fid=1100000003850161938&action=all&fr=0&to=100http://factordb.com/index.php?query=16*28%5E%28n%2B3%29%2B7863&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
282549055OA226F(110278×28227+125)/27230333http://factordb.com/index.php?id=1100000003850161939&open=primehttp://factordb.com/index.php?showid=1100000003850161939&base=28http://factordb.com/index.php?showid=1100000003850161939http://factordb.com/helper.php?id=1100000003850161939http://factordb.com/cert.php?id=1100000003850161939http://factordb.com/certchain.php?fid=1100000003850161939&action=all&fr=0&to=100http://factordb.com/index.php?query=%28110278*28%5E%28n%2B1%29%2B125%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825491L0229Q321×28231+731232336http://factordb.com/index.php?id=1100000003850161940&open=primehttp://factordb.com/index.php?showid=1100000003850161940&base=28http://factordb.com/index.php?showid=1100000003850161940http://factordb.com/helper.php?id=1100000003850161940http://factordb.com/cert.php?id=1100000003850161940http://factordb.com/certchain.php?fid=1100000003850161940&action=all&fr=0&to=100http://factordb.com/index.php?query=21*28%5E%28n%2B2%29%2B731&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825492B02317ID11×28234+6005235340http://factordb.com/index.php?id=1100000003850161941&open=primehttp://factordb.com/index.php?showid=1100000003850161941&base=28http://factordb.com/index.php?showid=1100000003850161941http://factordb.com/helper.php?id=1100000003850161941http://factordb.com/cert.php?id=1100000003850161941http://factordb.com/certchain.php?fid=1100000003850161941&action=all&fr=0&to=100http://factordb.com/index.php?query=11*28%5E%28n%2B3%29%2B6005&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825493PM233B(697×28234−319)/27235341http://factordb.com/index.php?id=1100000003850161942&open=primehttp://factordb.com/index.php?showid=1100000003850161942&base=28http://factordb.com/index.php?showid=1100000003850161942http://factordb.com/helper.php?id=1100000003850161942http://factordb.com/cert.php?id=1100000003850161942http://factordb.com/certchain.php?fid=1100000003850161942&action=all&fr=0&to=100http://factordb.com/index.php?query=%28697*28%5E%28n%2B1%29-319%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825494K0238OF20×28240+687241349http://factordb.com/index.php?id=1100000000840840142&open=primehttp://factordb.com/index.php?showid=1100000000840840142&base=28http://factordb.com/index.php?showid=1100000000840840142http://factordb.com/helper.php?id=1100000000840840142http://factordb.com/cert.php?id=1100000000840840142http://factordb.com/certchain.php?fid=1100000000840840142&action=all&fr=0&to=100http://factordb.com/index.php?query=20*28%5E%28n%2B2%29%2B687&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825495I262E3(2×28264−383)/3264382http://factordb.com/index.php?id=1100000003850161943&open=primehttp://factordb.com/index.php?showid=1100000003850161943&base=28http://factordb.com/index.php?showid=1100000003850161943http://factordb.com/helper.php?id=1100000003850161943http://factordb.com/cert.php?id=1100000003850161943http://factordb.com/certchain.php?fid=1100000003850161943&action=all&fr=0&to=100http://factordb.com/index.php?query=%282*28%5E%28n%2B2%29-383%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825496C5A273F(9217×28274+125)/27276400http://factordb.com/index.php?id=1100000003850161944&open=primehttp://factordb.com/index.php?showid=1100000003850161944&base=28http://factordb.com/index.php?showid=1100000003850161944http://factordb.com/helper.php?id=1100000003850161944http://factordb.com/cert.php?id=1100000003850161944http://factordb.com/certchain.php?fid=1100000003850161944&action=all&fr=0&to=100http://factordb.com/index.php?query=%289217*28%5E%28n%2B1%29%2B125%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825497J0276IMB19×28279+14739280406http://factordb.com/index.php?id=1100000003850161945&open=primehttp://factordb.com/index.php?showid=1100000003850161945&base=28http://factordb.com/index.php?showid=1100000003850161945http://factordb.com/helper.php?id=1100000003850161945http://factordb.com/cert.php?id=1100000003850161945http://factordb.com/certchain.php?fid=1100000003850161945&action=all&fr=0&to=100http://factordb.com/index.php?query=19*28%5E%28n%2B3%29%2B14739&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825498F0282QAP15×28285+20689286414http://factordb.com/index.php?id=1100000000840840006&open=primehttp://factordb.com/index.php?showid=1100000000840840006&base=28http://factordb.com/index.php?showid=1100000000840840006http://factordb.com/helper.php?id=1100000000840840006http://factordb.com/cert.php?id=1100000000840840006http://factordb.com/certchain.php?fid=1100000000840840006&action=all&fr=0&to=100http://factordb.com/index.php?query=15*28%5E%28n%2B3%29%2B20689&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825499M0296KKN22×28299+16263300435http://factordb.com/index.php?id=1100000003850161946&open=primehttp://factordb.com/index.php?showid=1100000003850161946&base=28http://factordb.com/index.php?showid=1100000003850161946http://factordb.com/helper.php?id=1100000003850161946http://factordb.com/cert.php?id=1100000003850161946http://factordb.com/certchain.php?fid=1100000003850161946&action=all&fr=0&to=100http://factordb.com/index.php?query=22*28%5E%28n%2B3%29%2B16263&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825500C31043(4×28312−2101)/9312452http://factordb.com/index.php?id=1100000003850161947&open=primehttp://factordb.com/index.php?showid=1100000003850161947&base=28http://factordb.com/index.php?showid=1100000003850161947http://factordb.com/helper.php?id=1100000003850161947http://factordb.com/cert.php?id=1100000003850161947http://factordb.com/certchain.php?fid=1100000003850161947&action=all&fr=0&to=100http://factordb.com/index.php?query=%284*28%5E%28n%2B2%29-2101%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825501RN319(752×28319−23)/27320464http://factordb.com/index.php?id=1100000002611723967&open=primehttp://factordb.com/index.php?showid=1100000002611723967&base=28http://factordb.com/index.php?showid=1100000002611723967http://factordb.com/helper.php?id=1100000002611723967http://factordb.com/cert.php?id=1100000002611723967http://factordb.com/certchain.php?fid=1100000002611723967&action=all&fr=0&to=100http://factordb.com/index.php?query=%28752*28%5En-23%29%2F27&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825502CA320F(334×28321+125)/27322466http://factordb.com/index.php?id=1100000000840839995&open=primehttp://factordb.com/index.php?showid=1100000000840839995&base=28http://factordb.com/index.php?showid=1100000000840839995http://factordb.com/helper.php?id=1100000000840839995http://factordb.com/cert.php?id=1100000000840839995http://factordb.com/certchain.php?fid=1100000000840839995&action=all&fr=0&to=100http://factordb.com/index.php?query=%28334*28%5E%28n%2B1%29%2B125%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825503D6326LR(119×28328+3967)/9329476http://factordb.com/index.php?id=1100000003850161948&open=primehttp://factordb.com/index.php?showid=1100000003850161948&base=28http://factordb.com/index.php?showid=1100000003850161948http://factordb.com/helper.php?id=1100000003850161948http://factordb.com/cert.php?id=1100000003850161948http://factordb.com/certchain.php?fid=1100000003850161948&action=all&fr=0&to=100http://factordb.com/index.php?query=%28119*28%5E%28n%2B2%29%2B3967%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825504B350AB(11×28352−767)/27352510http://factordb.com/index.php?id=1100000003850161949&open=primehttp://factordb.com/index.php?showid=1100000003850161949&base=28http://factordb.com/index.php?showid=1100000003850161949http://factordb.com/helper.php?id=1100000003850161949http://factordb.com/cert.php?id=1100000003850161949http://factordb.com/certchain.php?fid=1100000003850161949&action=all&fr=0&to=100http://factordb.com/index.php?query=%2811*28%5E%28n%2B2%29-767%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825505GA0355N458×28356+23358518http://factordb.com/index.php?id=1100000003850161950&open=primehttp://factordb.com/index.php?showid=1100000003850161950&base=28http://factordb.com/index.php?showid=1100000003850161950http://factordb.com/helper.php?id=1100000003850161950http://factordb.com/cert.php?id=1100000003850161950http://factordb.com/certchain.php?fid=1100000003850161950&action=all&fr=0&to=100http://factordb.com/index.php?query=458*28%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825506A0356P7P10×28359+19821360521http://factordb.com/index.php?id=1100000003850161951&open=primehttp://factordb.com/index.php?showid=1100000003850161951&base=28http://factordb.com/index.php?showid=1100000003850161951http://factordb.com/helper.php?id=1100000003850161951http://factordb.com/cert.php?id=1100000003850161951http://factordb.com/certchain.php?fid=1100000003850161951&action=all&fr=0&to=100http://factordb.com/index.php?query=10*28%5E%28n%2B3%29%2B19821&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825507J363H(19×28364−73)/27364527http://factordb.com/index.php?id=1100000002611724460&open=primehttp://factordb.com/index.php?showid=1100000002611724460&base=28http://factordb.com/index.php?showid=1100000002611724460http://factordb.com/helper.php?id=1100000002611724460http://factordb.com/cert.php?id=1100000002611724460http://factordb.com/certchain.php?fid=1100000002611724460&action=all&fr=0&to=100http://factordb.com/index.php?query=%2819*28%5E%28n%2B1%29-73%29%2F27&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28255084B381(119×28381−11)/27382553http://factordb.com/index.php?id=1100000002611724588&open=primehttp://factordb.com/index.php?showid=1100000002611724588&base=28http://factordb.com/index.php?showid=1100000002611724588http://factordb.com/helper.php?id=1100000002611724588http://factordb.com/cert.php?id=1100000002611724588http://factordb.com/certchain.php?fid=1100000002611724588&action=all&fr=0&to=100http://factordb.com/index.php?query=%28119*28%5En-11%29%2F27&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825509EB04051403×28406+1408591http://factordb.com/index.php?id=1100000001534442374&open=primehttp://factordb.com/index.php?showid=1100000001534442374&base=28http://factordb.com/index.php?showid=1100000001534442374http://factordb.com/helper.php?id=1100000001534442374proven prime by N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since N−1 is trivially fully factoredhttp://factordb.com/index.php?id=1100000001534442380&open=ecmhttp://factordb.com/index.php?query=403*28%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825510AN461(293×28461−23)/27462669http://factordb.com/index.php?id=1100000002611724556&open=primehttp://factordb.com/index.php?showid=1100000002611724556&base=28http://factordb.com/index.php?showid=1100000002611724556http://factordb.com/helper.php?id=1100000002611724556http://factordb.com/cert.php?id=1100000002611724556http://factordb.com/certchain.php?fid=1100000002611724556&action=all&fr=0&to=100http://factordb.com/index.php?query=%28293*28%5En-23%29%2F27&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28255114O61409(44×28616−6191)/9617893http://factordb.com/index.php?id=1100000000840839989&open=primehttp://factordb.com/index.php?showid=1100000000840839989&base=28http://factordb.com/index.php?showid=1100000000840839989http://factordb.com/helper.php?id=1100000000840839989http://factordb.com/cert.php?id=1100000000840839989http://factordb.com/certchain.php?fid=1100000000840839989&action=all&fr=0&to=100http://factordb.com/index.php?query=%2844*28%5E%28n%2B2%29-6191%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28255122D641(67×28641−13)/27642929http://factordb.com/index.php?id=1100000002611725341&open=primehttp://factordb.com/index.php?showid=1100000002611725341&base=28http://factordb.com/index.php?showid=1100000002611725341http://factordb.com/helper.php?id=1100000002611725341http://factordb.com/cert.php?id=1100000002611725341http://factordb.com/certchain.php?fid=1100000002611725341&action=all&fr=0&to=100http://factordb.com/index.php?query=%2867*28%5En-13%29%2F27&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
282551370748M57×28750+6217511087http://factordb.com/index.php?id=1100000003850161956&open=primehttp://factordb.com/index.php?showid=1100000003850161956&base=28http://factordb.com/index.php?showid=1100000003850161956http://factordb.com/helper.php?id=1100000003850161956http://factordb.com/cert.php?id=1100000003850161956http://factordb.com/certchain.php?fid=1100000003850161956&action=all&fr=0&to=100http://factordb.com/index.php?query=7*28%5E%28n%2B2%29%2B621&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28255144A0804B122×28805+118071168http://factordb.com/index.php?id=1100000003850161957&open=primehttp://factordb.com/index.php?showid=1100000003850161957&base=28http://factordb.com/index.php?showid=1100000003850161957http://factordb.com/helper.php?id=1100000003850161957http://factordb.com/cert.php?id=1100000003850161957http://factordb.com/certchain.php?fid=1100000003850161957&action=all&fr=0&to=100http://factordb.com/index.php?query=122*28%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825515LK925F(587×28926−155)/279271342http://factordb.com/index.php?id=1100000000840839978&open=primehttp://factordb.com/index.php?showid=1100000000840839978&base=28http://factordb.com/index.php?showid=1100000000840839978http://factordb.com/helper.php?id=1100000000840839978http://factordb.com/cert.php?id=1100000000840839978http://factordb.com/certchain.php?fid=1100000000840839978&action=all&fr=0&to=100http://factordb.com/index.php?query=%28587*28%5E%28n%2B1%29-155%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825516J01071AC519×281074+818110751556http://factordb.com/index.php?id=1100000003850161959&open=primehttp://factordb.com/index.php?showid=1100000003850161959&base=28http://factordb.com/index.php?showid=1100000003850161959http://factordb.com/helper.php?id=1100000003850161959http://factordb.com/cert.php?id=1100000003850161959http://factordb.com/certchain.php?fid=1100000003850161959&action=all&fr=0&to=100http://factordb.com/index.php?query=19*28%5E%28n%2B3%29%2B8181&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825517J01252J519×281254+53712551817http://factordb.com/index.php?id=1100000003850161963&open=primehttp://factordb.com/index.php?showid=1100000003850161963&base=28http://factordb.com/index.php?showid=1100000003850161963http://factordb.com/helper.php?id=1100000003850161963http://factordb.com/cert.php?id=1100000003850161963http://factordb.com/certchain.php?fid=1100000003850161963&action=all&fr=0&to=100http://factordb.com/index.php?query=19*28%5E%28n%2B2%29%2B537&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825518513046F(5×281306+1021)/2713061890http://factordb.com/index.php?id=1100000003850161964&open=primehttp://factordb.com/index.php?showid=1100000003850161964&base=28http://factordb.com/index.php?showid=1100000003850161964http://factordb.com/helper.php?id=1100000003850161964http://factordb.com/cert.php?id=1100000003850161964http://factordb.com/certchain.php?fid=1100000003850161964&action=all&fr=0&to=100http://factordb.com/index.php?query=%285*28%5E%28n%2B2%29%2B1021%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
282551951332P8P(5×281335+426163)/2713351932http://factordb.com/index.php?id=1100000003850161965&open=primehttp://factordb.com/index.php?showid=1100000003850161965&base=28http://factordb.com/index.php?showid=1100000003850161965http://factordb.com/helper.php?id=1100000003850161965http://factordb.com/cert.php?id=1100000003850161965http://factordb.com/certchain.php?fid=1100000003850161965&action=all&fr=0&to=100http://factordb.com/index.php?query=%285*28%5E%28n%2B3%29%2B426163%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28255205I1370F(17×281371−11)/313721985http://factordb.com/index.php?id=1100000003850161972&open=primehttp://factordb.com/index.php?showid=1100000003850161972&base=28http://factordb.com/index.php?showid=1100000003850161972http://factordb.com/helper.php?id=1100000003850161972http://factordb.com/cert.php?id=1100000003850161972http://factordb.com/certchain.php?fid=1100000003850161972&action=all&fr=0&to=100http://factordb.com/index.php?query=%2817*28%5E%28n%2B1%29-11%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825521A14236F(10×281425−2899)/2714252062http://factordb.com/index.php?id=1100000000840839947&open=primehttp://factordb.com/index.php?showid=1100000000840839947&base=28http://factordb.com/index.php?showid=1100000000840839947http://factordb.com/helper.php?id=1100000000840839947http://factordb.com/cert.php?id=1100000000840839947http://factordb.com/certchain.php?fid=1100000000840839947&action=all&fr=0&to=100http://factordb.com/index.php?query=%2810*28%5E%28n%2B2%29-2899%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825522G01899AN16×281901+30319022753http://factordb.com/index.php?id=1100000003850161973&open=primehttp://factordb.com/index.php?showid=1100000003850161973&base=28http://factordb.com/index.php?showid=1100000003850161973http://factordb.com/helper.php?id=1100000003850161973http://factordb.com/cert.php?id=1100000003850161973http://factordb.com/certchain.php?fid=1100000003850161973&action=all&fr=0&to=100http://factordb.com/index.php?query=16*28%5E%28n%2B2%29%2B303&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825523537468P(5×283748+2803)/2737485424http://factordb.com/index.php?id=1100000003850161974&open=primehttp://factordb.com/index.php?showid=1100000003850161974&base=28http://factordb.com/index.php?showid=1100000003850161974http://factordb.com/helper.php?id=1100000003850161974http://factordb.com/cert.php?id=1100000003850161974http://factordb.com/certchain.php?fid=1100000003850161974&action=all&fr=0&to=100http://factordb.com/index.php?query=%285*28%5E%28n%2B2%29%2B2803%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825524QO423969(242×284241−4679)/942426139http://factordb.com/index.php?id=1100000000840839934&open=primehttp://factordb.com/index.php?showid=1100000000840839934&base=28http://factordb.com/index.php?showid=1100000000840839934http://factordb.com/helper.php?id=1100000000840839934http://factordb.com/cert.php?id=1100000000840839934http://factordb.com/certchain.php?fid=1100000000840839934&action=all&fr=0&to=100http://factordb.com/index.php?query=%28242*28%5E%28n%2B2%29-4679%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
2825525D0526777D13×285270+569752717628http://factordb.com/index.php?id=1100000003850151420&open=primehttp://factordb.com/index.php?showid=1100000003850151420&base=28http://factordb.com/index.php?showid=1100000003850151420http://factordb.com/helper.php?id=1100000003850151420http://factordb.com/cert.php?id=1100000003850151420http://factordb.com/certchain.php?fid=1100000003850151420&action=all&fr=0&to=100http://factordb.com/index.php?query=13*28%5E%28n%2B3%29%2B5697&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
302613AN206(313×30206−23)/29207306http://factordb.com/index.php?id=1100000002327651073&open=primehttp://factordb.com/index.php?showid=1100000002327651073&base=30http://factordb.com/index.php?showid=1100000002327651073http://factordb.com/helper.php?id=1100000002327651073http://factordb.com/cert.php?id=1100000002327651073http://factordb.com/certchain.php?fid=1100000002327651073&action=all&fr=0&to=100http://factordb.com/index.php?query=%28313*30%5En-23%29%2F29&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
302614M241QB(22×30243+3139)/29243359http://factordb.com/index.php?id=1100000003593408295&open=primehttp://factordb.com/index.php?showid=1100000003593408295&base=30http://factordb.com/index.php?showid=1100000003593408295http://factordb.com/helper.php?id=1100000003593408295http://factordb.com/cert.php?id=1100000003593408295http://factordb.com/certchain.php?fid=1100000003593408295&action=all&fr=0&to=100http://factordb.com/index.php?query=%2822*30%5E%28n%2B2%29%2B3139%29%2F29&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
302615M0547SS722×30550+26047551814http://factordb.com/index.php?id=1100000003593407988&open=primehttp://factordb.com/index.php?showid=1100000003593407988&base=30http://factordb.com/index.php?showid=1100000003593407988http://factordb.com/helper.php?id=1100000003593407988http://factordb.com/cert.php?id=1100000003593407988http://factordb.com/certchain.php?fid=1100000003593407988&action=all&fr=0&to=100http://factordb.com/index.php?query=22*30%5E%28n%2B3%29%2B26047&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
302616C01022112×301023+110241513http://factordb.com/index.php?id=1100000000785448736&open=primehttp://factordb.com/index.php?showid=1100000000785448736&base=30http://factordb.com/index.php?showid=1100000000785448736http://factordb.com/helper.php?id=1100000000785448736proven prime by N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1), since N−1 is trivially fully factoredhttp://factordb.com/index.php?id=1100000000785448737&open=ecmhttp://factordb.com/index.php?query=12*30%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
30261754882J(5×304883+401)/2948837213http://factordb.com/index.php?id=1100000002327649423&open=primehttp://factordb.com/index.php?showid=1100000002327649423&base=30http://factordb.com/index.php?showid=1100000002327649423http://factordb.com/helper.php?id=1100000002327649423http://factordb.com/cert.php?id=1100000002327649423http://factordb.com/certchain.php?fid=1100000002327649423&action=all&fr=0&to=100http://factordb.com/index.php?query=%285*30%5E%28n%2B1%29%2B401%29%2F29&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
302619OT3420525×3034205−13420650527http://factordb.com/index.php?id=1100000000800812865&open=primehttp://factordb.com/index.php?showid=1100000000800812865&base=30http://factordb.com/index.php?showid=1100000000800812865http://factordb.com/helper.php?id=1100000000800812865proven prime by N+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), since N+1 is trivially fully factoredhttp://factordb.com/index.php?id=1100000000819405041&open=ecmhttp://factordb.com/index.php?query=25*30%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635212Q19577(26×36197−24631)/35197307http://factordb.com/index.php?id=1100000003807362350&open=primehttp://factordb.com/index.php?showid=1100000003807362350&base=36http://factordb.com/index.php?showid=1100000003807362350http://factordb.com/helper.php?id=1100000003807362350http://factordb.com/cert.php?id=1100000003807362350http://factordb.com/certchain.php?fid=1100000003807362350&action=all&fr=0&to=100http://factordb.com/index.php?query=%2826*36%5E%28n%2B2%29-24631%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635213W0199ND32×36201+841202315http://factordb.com/index.php?id=1100000002634136732&open=primehttp://factordb.com/index.php?showid=1100000002634136732&base=36http://factordb.com/index.php?showid=1100000002634136732http://factordb.com/helper.php?id=1100000002634136732http://factordb.com/cert.php?id=1100000002634136732http://factordb.com/certchain.php?fid=1100000002634136732&action=all&fr=0&to=100http://factordb.com/index.php?query=32*36%5E%28n%2B2%29%2B841&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635214G0204YT16×36206+1253207322http://factordb.com/index.php?id=1100000002634137789&open=primehttp://factordb.com/index.php?showid=1100000002634137789&base=36http://factordb.com/index.php?showid=1100000002634137789http://factordb.com/helper.php?id=1100000002634137789http://factordb.com/cert.php?id=1100000002634137789http://factordb.com/certchain.php?fid=1100000002634137789&action=all&fr=0&to=100http://factordb.com/index.php?query=16*36%5E%28n%2B2%29%2B1253&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635215RHY223H(34649×36224−629)/35226352http://factordb.com/index.php?id=1100000003807362353&open=primehttp://factordb.com/index.php?showid=1100000003807362353&base=36http://factordb.com/index.php?showid=1100000003807362353http://factordb.com/helper.php?id=1100000003807362353http://factordb.com/cert.php?id=1100000003807362353http://factordb.com/certchain.php?fid=1100000003807362353&action=all&fr=0&to=100http://factordb.com/index.php?query=%2834649*36%5E%28n%2B1%29-629%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635216T0223ST29×36225+1037226352http://factordb.com/index.php?id=1100000002634136882&open=primehttp://factordb.com/index.php?showid=1100000002634136882&base=36http://factordb.com/index.php?showid=1100000002634136882http://factordb.com/helper.php?id=1100000002634136882http://factordb.com/cert.php?id=1100000002634136882http://factordb.com/certchain.php?fid=1100000002634136882&action=all&fr=0&to=100http://factordb.com/index.php?query=29*36%5E%28n%2B2%29%2B1037&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635217J0224U6V19×36227+39127228355http://factordb.com/index.php?id=1100000003807362355&open=primehttp://factordb.com/index.php?showid=1100000003807362355&base=36http://factordb.com/index.php?showid=1100000003807362355http://factordb.com/helper.php?id=1100000003807362355http://factordb.com/cert.php?id=1100000003807362355http://factordb.com/certchain.php?fid=1100000003807362355&action=all&fr=0&to=100http://factordb.com/index.php?query=19*36%5E%28n%2B3%29%2B39127&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635218BE0235IV410×36237+679239372http://factordb.com/index.php?id=1100000003807362356&open=primehttp://factordb.com/index.php?showid=1100000003807362356&base=36http://factordb.com/index.php?showid=1100000003807362356http://factordb.com/helper.php?id=1100000003807362356http://factordb.com/cert.php?id=1100000003807362356http://factordb.com/certchain.php?fid=1100000003807362356&action=all&fr=0&to=100http://factordb.com/index.php?query=410*36%5E%28n%2B2%29%2B679&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635219E0236KY114×36239+27145240374http://factordb.com/index.php?id=1100000000840634520&open=primehttp://factordb.com/index.php?showid=1100000000840634520&base=36http://factordb.com/index.php?showid=1100000000840634520http://factordb.com/helper.php?id=1100000000840634520http://factordb.com/cert.php?id=1100000000840634520http://factordb.com/certchain.php?fid=1100000000840634520&action=all&fr=0&to=100http://factordb.com/index.php?query=14*36%5E%28n%2B3%29%2B27145&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635220JXV0244B25843×36245+11248386http://factordb.com/index.php?id=1100000003807362357&open=primehttp://factordb.com/index.php?showid=1100000003807362357&base=36http://factordb.com/index.php?showid=1100000003807362357http://factordb.com/helper.php?id=1100000003807362357http://factordb.com/cert.php?id=1100000003807362357http://factordb.com/certchain.php?fid=1100000003807362357&action=all&fr=0&to=100http://factordb.com/index.php?query=25843*36%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36352215Q2497(201×36250−691)/35251390http://factordb.com/index.php?id=1100000003807362359&open=primehttp://factordb.com/index.php?showid=1100000003807362359&base=36http://factordb.com/index.php?showid=1100000003807362359http://factordb.com/helper.php?id=1100000003807362359http://factordb.com/cert.php?id=1100000003807362359http://factordb.com/certchain.php?fid=1100000003807362359&action=all&fr=0&to=100http://factordb.com/index.php?query=%28201*36%5E%28n%2B1%29-691%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635222N30253H831×36254+17256399http://factordb.com/index.php?id=1100000002634137359&open=primehttp://factordb.com/index.php?showid=1100000002634137359&base=36http://factordb.com/index.php?showid=1100000002634137359http://factordb.com/helper.php?id=1100000002634137359http://factordb.com/cert.php?id=1100000002634137359http://factordb.com/certchain.php?fid=1100000002634137359&action=all&fr=0&to=100http://factordb.com/index.php?query=831*36%5E%28n%2B1%29%2B17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635223Y261AH(34×36263−30869)/35263410http://factordb.com/index.php?id=1100000003807362360&open=primehttp://factordb.com/index.php?showid=1100000003807362360&base=36http://factordb.com/index.php?showid=1100000003807362360http://factordb.com/helper.php?id=1100000003807362360http://factordb.com/cert.php?id=1100000003807362360http://factordb.com/certchain.php?fid=1100000003807362360&action=all&fr=0&to=100http://factordb.com/index.php?query=%2834*36%5E%28n%2B2%29-30869%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
363522490277VV9×36279+1147280436http://factordb.com/index.php?id=1100000002634138388&open=primehttp://factordb.com/index.php?showid=1100000002634138388&base=36http://factordb.com/index.php?showid=1100000002634138388http://factordb.com/helper.php?id=1100000002634138388http://factordb.com/cert.php?id=1100000002634138388http://factordb.com/certchain.php?fid=1100000002634138388&action=all&fr=0&to=100http://factordb.com/index.php?query=9*36%5E%28n%2B2%29%2B1147&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635225J0281VB19×36283+1127284442http://factordb.com/index.php?id=1100000002634137683&open=primehttp://factordb.com/index.php?showid=1100000002634137683&base=36http://factordb.com/index.php?showid=1100000002634137683http://factordb.com/helper.php?id=1100000002634137683http://factordb.com/cert.php?id=1100000002634137683http://factordb.com/certchain.php?fid=1100000002634137683&action=all&fr=0&to=100http://factordb.com/index.php?query=19*36%5E%28n%2B2%29%2B1127&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635226J0281WV19×36283+1183284442http://factordb.com/index.php?id=1100000002634137660&open=primehttp://factordb.com/index.php?showid=1100000002634137660&base=36http://factordb.com/index.php?showid=1100000002634137660http://factordb.com/helper.php?id=1100000002634137660http://factordb.com/cert.php?id=1100000002634137660http://factordb.com/certchain.php?fid=1100000002634137660&action=all&fr=0&to=100http://factordb.com/index.php?query=19*36%5E%28n%2B2%29%2B1183&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635227DE028161482×36283+217285444http://factordb.com/index.php?id=1100000003807362361&open=primehttp://factordb.com/index.php?showid=1100000003807362361&base=36http://factordb.com/index.php?showid=1100000003807362361http://factordb.com/helper.php?id=1100000003807362361http://factordb.com/cert.php?id=1100000003807362361http://factordb.com/certchain.php?fid=1100000003807362361&action=all&fr=0&to=100http://factordb.com/index.php?query=482*36%5E%28n%2B2%29%2B217&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36352289H297(332×36297−17)/35298464http://factordb.com/index.php?id=1100000002332535884&open=primehttp://factordb.com/index.php?showid=1100000002332535884&base=36http://factordb.com/index.php?showid=1100000002332535884http://factordb.com/helper.php?id=1100000002332535884http://factordb.com/cert.php?id=1100000002332535884http://factordb.com/certchain.php?fid=1100000002332535884&action=all&fr=0&to=100http://factordb.com/index.php?query=%28332*36%5En-17%29%2F35&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635229M70297FD799×36299+553301469http://factordb.com/index.php?id=1100000003807362363&open=primehttp://factordb.com/index.php?showid=1100000003807362363&base=36http://factordb.com/index.php?showid=1100000003807362363http://factordb.com/helper.php?id=1100000003807362363http://factordb.com/cert.php?id=1100000003807362363http://factordb.com/certchain.php?fid=1100000003807362363&action=all&fr=0&to=100http://factordb.com/index.php?query=799*36%5E%28n%2B2%29%2B553&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36352309X301B(348×36302−803)/35303472http://factordb.com/index.php?id=1100000003807362364&open=primehttp://factordb.com/index.php?showid=1100000003807362364&base=36http://factordb.com/index.php?showid=1100000003807362364http://factordb.com/helper.php?id=1100000003807362364http://factordb.com/cert.php?id=1100000003807362364http://factordb.com/certchain.php?fid=1100000003807362364&action=all&fr=0&to=100http://factordb.com/index.php?query=%28348*36%5E%28n%2B1%29-803%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635231XE032571202×36326+7328511http://factordb.com/index.php?id=1100000002634136674&open=primehttp://factordb.com/index.php?showid=1100000002634136674&base=36http://factordb.com/index.php?showid=1100000002634136674http://factordb.com/helper.php?id=1100000002634136674http://factordb.com/cert.php?id=1100000002634136674http://factordb.com/certchain.php?fid=1100000002634136674&action=all&fr=0&to=100http://factordb.com/index.php?query=1202*36%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635232KP330SZ(145×36332+821)/7333519http://factordb.com/index.php?id=1100000000840634515&open=primehttp://factordb.com/index.php?showid=1100000000840634515&base=36http://factordb.com/index.php?showid=1100000000840634515http://factordb.com/helper.php?id=1100000000840634515http://factordb.com/cert.php?id=1100000000840634515http://factordb.com/certchain.php?fid=1100000000840634515&action=all&fr=0&to=100http://factordb.com/index.php?query=%28145*36%5E%28n%2B2%29%2B821%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36352335347QP(36349+5431)/7349543http://factordb.com/index.php?id=1100000003807362365&open=primehttp://factordb.com/index.php?showid=1100000003807362365&base=36http://factordb.com/index.php?showid=1100000003807362365http://factordb.com/helper.php?id=1100000003807362365http://factordb.com/cert.php?id=1100000003807362365http://factordb.com/certchain.php?fid=1100000003807362365&action=all&fr=0&to=100http://factordb.com/index.php?query=%2836%5E%28n%2B2%29%2B5431%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635234E03496U114×36352+8857353549http://factordb.com/index.php?id=1100000000840634509&open=primehttp://factordb.com/index.php?showid=1100000000840634509&base=36http://factordb.com/index.php?showid=1100000000840634509http://factordb.com/helper.php?id=1100000000840634509http://factordb.com/cert.php?id=1100000000840634509http://factordb.com/certchain.php?fid=1100000000840634509&action=all&fr=0&to=100http://factordb.com/index.php?query=14*36%5E%28n%2B3%29%2B8857&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635235K0367E6T20×36370+18389371578http://factordb.com/index.php?id=1100000003807362367&open=primehttp://factordb.com/index.php?showid=1100000003807362367&base=36http://factordb.com/index.php?showid=1100000003807362367http://factordb.com/helper.php?id=1100000003807362367http://factordb.com/cert.php?id=1100000003807362367http://factordb.com/certchain.php?fid=1100000003807362367&action=all&fr=0&to=100http://factordb.com/index.php?query=20*36%5E%28n%2B3%29%2B18389&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635236U0370WP30×36372+1177373581http://factordb.com/index.php?id=1100000000840634503&open=primehttp://factordb.com/index.php?showid=1100000000840634503&base=36http://factordb.com/index.php?showid=1100000000840634503http://factordb.com/helper.php?id=1100000000840634503http://factordb.com/cert.php?id=1100000000840634503http://factordb.com/certchain.php?fid=1100000000840634503&action=all&fr=0&to=100http://factordb.com/index.php?query=30*36%5E%28n%2B2%29%2B1177&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635237P8Z390909×36390−1392610http://factordb.com/index.php?id=1100000000764100228&open=primehttp://factordb.com/index.php?showid=1100000000764100228&base=36http://factordb.com/index.php?showid=1100000000764100228http://factordb.com/helper.php?id=1100000000764100228proven prime by N+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), since N+1 is trivially fully factoredhttp://factordb.com/index.php?id=1100000000764100231&open=ecmhttp://factordb.com/index.php?query=909*36%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635238903974B9×36399+155400622http://factordb.com/index.php?id=1100000002634138490&open=primehttp://factordb.com/index.php?showid=1100000002634138490&base=36http://factordb.com/index.php?showid=1100000002634138490http://factordb.com/helper.php?id=1100000002634138490http://factordb.com/cert.php?id=1100000002634138490http://factordb.com/certchain.php?fid=1100000002634138490&action=all&fr=0&to=100http://factordb.com/index.php?query=9*36%5E%28n%2B2%29%2B155&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
363523950405WW55×36408+42629409636http://factordb.com/index.php?id=1100000003807362369&open=primehttp://factordb.com/index.php?showid=1100000003807362369&base=36http://factordb.com/index.php?showid=1100000003807362369http://factordb.com/helper.php?id=1100000003807362369http://factordb.com/cert.php?id=1100000003807362369http://factordb.com/certchain.php?fid=1100000003807362369&action=all&fr=0&to=100http://factordb.com/index.php?query=5*36%5E%28n%2B3%29%2B42629&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635240V0405EE4B31×36409+671483410639http://factordb.com/index.php?id=1100000003807362370&open=primehttp://factordb.com/index.php?showid=1100000003807362370&base=36http://factordb.com/index.php?showid=1100000003807362370http://factordb.com/helper.php?id=1100000003807362370http://factordb.com/cert.php?id=1100000003807362370http://factordb.com/certchain.php?fid=1100000003807362370&action=all&fr=0&to=100http://factordb.com/index.php?query=31*36%5E%28n%2B4%29%2B671483&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635241TTR0434T38655×36435+29438682http://factordb.com/index.php?id=1100000003807362372&open=primehttp://factordb.com/index.php?showid=1100000003807362372&base=36http://factordb.com/index.php?showid=1100000003807362372http://factordb.com/helper.php?id=1100000003807362372http://factordb.com/cert.php?id=1100000003807362372http://factordb.com/certchain.php?fid=1100000003807362372&action=all&fr=0&to=100http://factordb.com/index.php?query=38655*36%5E%28n%2B1%29%2B29&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635242LK443Z(151×36444+101)/7445693http://factordb.com/index.php?id=1100000000840634496&open=primehttp://factordb.com/index.php?showid=1100000000840634496&base=36http://factordb.com/index.php?showid=1100000000840634496http://factordb.com/helper.php?id=1100000000840634496http://factordb.com/cert.php?id=1100000000840634496http://factordb.com/certchain.php?fid=1100000000840634496&action=all&fr=0&to=100http://factordb.com/index.php?query=%28151*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635243Q4547(26×36455−691)/35455708http://factordb.com/index.php?id=1100000002332534290&open=primehttp://factordb.com/index.php?showid=1100000002332534290&base=36http://factordb.com/index.php?showid=1100000002332534290http://factordb.com/helper.php?id=1100000002332534290http://factordb.com/cert.php?id=1100000002332534290http://factordb.com/certchain.php?fid=1100000002332534290&action=all&fr=0&to=100http://factordb.com/index.php?query=%2826*36%5E%28n%2B1%29-691%29%2F35&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635244R0458727×36459+7460716http://factordb.com/index.php?id=1100000002356257765&open=primehttp://factordb.com/index.php?showid=1100000002356257765&base=36http://factordb.com/index.php?showid=1100000002356257765http://factordb.com/helper.php?id=1100000002356257765http://factordb.com/cert.php?id=1100000002356257765http://factordb.com/certchain.php?fid=1100000002356257765&action=all&fr=0&to=100http://factordb.com/index.php?query=27*36%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
363524550460KGZ5×36463+26531464722http://factordb.com/index.php?id=1100000003807362374&open=primehttp://factordb.com/index.php?showid=1100000003807362374&base=36http://factordb.com/index.php?showid=1100000003807362374http://factordb.com/helper.php?id=1100000003807362374http://factordb.com/cert.php?id=1100000003807362374http://factordb.com/certchain.php?fid=1100000003807362374&action=all&fr=0&to=100http://factordb.com/index.php?query=5*36%5E%28n%2B3%29%2B26531&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635246K0474OY120×36477+32329478744http://factordb.com/index.php?id=1100000000840634488&open=primehttp://factordb.com/index.php?showid=1100000000840634488&base=36http://factordb.com/index.php?showid=1100000000840634488http://factordb.com/helper.php?id=1100000000840634488http://factordb.com/cert.php?id=1100000000840634488http://factordb.com/certchain.php?fid=1100000000840634488&action=all&fr=0&to=100http://factordb.com/index.php?query=20*36%5E%28n%2B3%29%2B32329&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635247B478KB(11×36480+11329)/35480747http://factordb.com/index.php?id=1100000003807362381&open=primehttp://factordb.com/index.php?showid=1100000003807362381&base=36http://factordb.com/index.php?showid=1100000003807362381http://factordb.com/helper.php?id=1100000003807362381http://factordb.com/cert.php?id=1100000003807362381http://factordb.com/certchain.php?fid=1100000003807362381&action=all&fr=0&to=100http://factordb.com/index.php?query=%2811*36%5E%28n%2B2%29%2B11329%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635248WY507H(1154×36508−629)/35509793http://factordb.com/index.php?id=1100000003807362386&open=primehttp://factordb.com/index.php?showid=1100000003807362386&base=36http://factordb.com/index.php?showid=1100000003807362386http://factordb.com/helper.php?id=1100000003807362386http://factordb.com/cert.php?id=1100000003807362386http://factordb.com/certchain.php?fid=1100000003807362386&action=all&fr=0&to=100http://factordb.com/index.php?query=%281154*36%5E%28n%2B1%29-629%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635249G0510USJ16×36513+39907514800http://factordb.com/index.php?id=1100000003807362389&open=primehttp://factordb.com/index.php?showid=1100000003807362389&base=36http://factordb.com/index.php?showid=1100000003807362389http://factordb.com/helper.php?id=1100000003807362389http://factordb.com/cert.php?id=1100000003807362389http://factordb.com/certchain.php?fid=1100000003807362389&action=all&fr=0&to=100http://factordb.com/index.php?query=16*36%5E%28n%2B3%29%2B39907&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635250Z527EX736530−27317530825http://factordb.com/index.php?id=1100000003807362391&open=primehttp://factordb.com/index.php?showid=1100000003807362391&base=36http://factordb.com/index.php?showid=1100000003807362391http://factordb.com/helper.php?id=1100000003807362391http://factordb.com/cert.php?id=1100000003807362391http://factordb.com/certchain.php?fid=1100000003807362391&action=all&fr=0&to=100http://factordb.com/index.php?query=36%5E%28n%2B3%29-27317&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635251EY0534A1538×36536+361538837http://factordb.com/index.php?id=1100000000840634482&open=primehttp://factordb.com/index.php?showid=1100000000840634482&base=36http://factordb.com/index.php?showid=1100000000840634482http://factordb.com/helper.php?id=1100000000840634482http://factordb.com/cert.php?id=1100000000840634482http://factordb.com/certchain.php?fid=1100000000840634482&action=all&fr=0&to=100http://factordb.com/index.php?query=538*36%5E%28n%2B2%29%2B361&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635252Z056399535×36566+11993567883http://factordb.com/index.php?id=1100000003807362394&open=primehttp://factordb.com/index.php?showid=1100000003807362394&base=36http://factordb.com/index.php?showid=1100000003807362394http://factordb.com/helper.php?id=1100000003807362394http://factordb.com/cert.php?id=1100000003807362394http://factordb.com/certchain.php?fid=1100000003807362394&action=all&fr=0&to=100http://factordb.com/index.php?query=35*36%5E%28n%2B3%29%2B11993&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635253F59095(3×36592−1585)/7592921http://factordb.com/index.php?id=1100000003807362398&open=primehttp://factordb.com/index.php?showid=1100000003807362398&base=36http://factordb.com/index.php?showid=1100000003807362398http://factordb.com/helper.php?id=1100000003807362398http://factordb.com/cert.php?id=1100000003807362398http://factordb.com/certchain.php?fid=1100000003807362398&action=all&fr=0&to=100http://factordb.com/index.php?query=%283*36%5E%28n%2B2%29-1585%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635254990591B333×36592+11594924http://factordb.com/index.php?id=1100000002634138415&open=primehttp://factordb.com/index.php?showid=1100000002634138415&base=36http://factordb.com/index.php?showid=1100000002634138415http://factordb.com/helper.php?id=1100000002634138415http://factordb.com/cert.php?id=1100000002634138415http://factordb.com/certchain.php?fid=1100000002634138415&action=all&fr=0&to=100http://factordb.com/index.php?query=333*36%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635255J675T(19×36676+331)/356761052http://factordb.com/index.php?id=1100000002332534943&open=primehttp://factordb.com/index.php?showid=1100000002332534943&base=36http://factordb.com/index.php?showid=1100000002332534943http://factordb.com/helper.php?id=1100000002332534943http://factordb.com/cert.php?id=1100000002332534943http://factordb.com/certchain.php?fid=1100000002332534943&action=all&fr=0&to=100http://factordb.com/index.php?query=%2819*36%5E%28n%2B1%29%2B331%29%2F35&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
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3635282VL07258J1137×367259+19726111301http://factordb.com/index.php?id=1100000003807362489&open=primehttp://factordb.com/index.php?showid=1100000003807362489&base=36http://factordb.com/index.php?showid=1100000003807362489http://factordb.com/helper.php?id=1100000003807362489http://factordb.com/cert.php?id=1100000003807362489http://factordb.com/certchain.php?fid=1100000003807362489&action=all&fr=0&to=100http://factordb.com/index.php?query=1137*36%5E%28n%2B1%29%2B19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
3635283J10117LJ(19×3610119+2501)/351011915748http://factordb.com/index.php?id=1100000003807362491&open=primehttp://factordb.com/index.php?showid=1100000003807362491&base=36http://factordb.com/index.php?showid=1100000003807362491http://factordb.com/helper.php?id=1100000003807362491http://factordb.com/cert.php?id=1100000003807362491http://factordb.com/certchain.php?fid=1100000003807362491&action=all&fr=0&to=100http://factordb.com/index.php?query=%2819*36%5E%28n%2B2%29%2B2501%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show

Condensed table for bases 2 ≤ b ≤ 36: (the bases b = 11, 13, 16, 17, 19, 21~23, 25~36 data assumes the primality of the probable primes) (This data assumes that a number > 1025000 which has passed the Miller–Rabin primality tests (https://t5k.org/prove/prove2_3.html, https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test, https://en.wikipedia.org/wiki/Strong_pseudoprime, https://t5k.org/glossary/xpage/MillersTest.html, https://t5k.org/glossary/xpage/StrongPRP.html, https://www.rieselprime.de/ziki/Miller-Rabin_pseudoprimality_test, https://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html, https://mathworld.wolfram.com/StrongPseudoprime.html, http://www.numericana.com/answer/pseudo.htm#rabin, http://www.numericana.com/answer/pseudo.htm#strong, http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm, http://ntheory.org/data/spsps.txt, https://faculty.lynchburg.edu/~nicely/misc/mpzspsp.html, https://miller-rabin.appspot.com/, http://www.pi-e.de/Miller-Rabin-Pseudoprimzahlen.htm (in German), http://factordb.com/prooffailed.php, https://sites.google.com/view/strong-pseudoprime, https://sites.google.com/view/bases-strong-pseudoprime, https://oeis.org/A001262, https://oeis.org/A020229, https://oeis.org/A020230, https://oeis.org/A020231, https://oeis.org/A020232, https://oeis.org/A020233, https://oeis.org/A020234, https://oeis.org/A020235, https://oeis.org/A020236, https://oeis.org/A020237, https://oeis.org/A020238, https://oeis.org/A020239, https://oeis.org/A020240, https://oeis.org/A020241, https://oeis.org/A020242, https://oeis.org/A020243, https://oeis.org/A020244, https://oeis.org/A020245, https://oeis.org/A020246, https://oeis.org/A020247, https://oeis.org/A020248, https://oeis.org/A020249, https://oeis.org/A020250, https://oeis.org/A020251, https://oeis.org/A020252, https://oeis.org/A020253, https://oeis.org/A020254, https://oeis.org/A020255, https://oeis.org/A020256, https://oeis.org/A020257, https://oeis.org/A020258, https://oeis.org/A020259, https://oeis.org/A020260, https://oeis.org/A020261, https://oeis.org/A020262, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773, https://oeis.org/A014233, https://oeis.org/A006945, https://oeis.org/A089825, https://oeis.org/A181782, https://oeis.org/A071294, https://oeis.org/A141768, https://oeis.org/A195328, https://oeis.org/A329759, https://oeis.org/A298756) to all prime bases p < 64 and has passed the Baillie–PSW primality test (https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test, https://mathworld.wolfram.com/Baillie-PSWPrimalityTest.html, http://pseudoprime.com/dopo.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_318.pdf)) and has trial factored (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) to 1016 is in fact prime, since in some cases (e.g. b = 11) a candidate for minimal prime base b is too large to be proven prime rigorously, this candidate for minimal prime base 11 has 65263 decimal digits, while the top record ordinary prime (https://t5k.org/glossary/xpage/OrdinaryPrime.html) (i.e. neither N−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1) nor N+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) can be ≥ 1/4 factored (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm)) has 86453 decimal digits (the entry of this prime in top definitely primes is https://t5k.org/primes/page.php?id=136044), see https://t5k.org/top20/page.php?id=27 and https://t5k.org/primes/search.php?Comment=ECPP&OnList=all&Number=1000000&Style=HTML and http://factordb.com/certoverview.php?digits=300&perpage=1000&skip=0&descending=on)

For every solved base b, we give the number of minimal primes (or probable prime, which is a minimal prime assuming its primality) and the top 10 minimal primes (or probable prime, which is a minimal prime assuming its primality) in the table below, and the minimal primes (or probable prime, which is a minimal prime assuming its primality) are exactly the elements in the "kernel b" file. For every unsolved base b, we give the current greatest lower bound (https://en.wikipedia.org/wiki/Greatest_lower_bound, https://mathworld.wolfram.com/GreatestLowerBound.html) and the current least upper bound (https://en.wikipedia.org/wiki/Least_upper_bound, https://mathworld.wolfram.com/LeastUpperBound.html) for the number of minimal primes (or probable prime, which is a minimal prime assuming its primality) and the top 10 known minimal primes (or probable prime, which is a minimal prime assuming its primality) and the number of families of the form x{y}z (where x and z are strings (may be empty) of digits in base b, y is a digit in base b) which have no known prime (or strong probable prime) members > b, nor can be ruled out as only containing composites (only count the numbers > b) (by covering congruence, algebraic factorization, or combine of them) in the table below, and the greatest lower bound (https://en.wikipedia.org/wiki/Greatest_lower_bound, https://mathworld.wolfram.com/GreatestLowerBound.html) for the number of minimal primes (or probable prime, which is a minimal prime assuming its primality) is exactly the number of the elements in the "kernel b" file (also, the currently known minimal primes (or probable prime, which is a minimal prime assuming its primality) are exactly the elements in the "kernel b" file), and the left families (i.e. families which have no known prime (or strong probable prime) members > b, nor can be ruled out as only containing composites (only count the numbers > b) (by covering congruence, algebraic factorization, or combine of them)) (all of them are linear families, i.e. of the form x{y}z (where x and z are strings (may be empty) of digits in base b, y is a digit in base b)) are exactly the families in the "left b" file, since such linear families can contain at most one minimal prime (in fact, they must contain exactly one minimal prime if they are not covered by another left family (e.g. the base 19 left family 5{H}05 is covered by another base 19 left family 5{H}5, thus the base 19 left family 5{H}05 may contain no minimal primes) if we assume the heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, https://web.archive.org/web/20230911032453/https://www.utm.edu/staff/caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf), https://arxiv.org/pdf/2103.04483.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_113.pdf)) that all unsolved families have a prime, this is reasonable, references: https://t5k.org/mersenne/heuristic.html, https://t5k.org/notes/faq/NextMersenne.html, https://t5k.org/glossary/xpage/Repunit.html, https://web.archive.org/web/20100628035147/http://www.math.niu.edu/~rusin/known-math/98/exp_primes, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, https://mathoverflow.net/questions/268918/density-of-primes-in-sequences-of-the-form-anb, https://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Mathematics/2023_September_25#Are_there_infinitely_many_primes_of_the_form_1000%E2%80%A60007,_333%E2%80%A63331,_7111%E2%80%A6111,_or_3444%E2%80%A64447_in_base_10?, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf), https://www.mersenneforum.org/showthread.php?t=12327, https://oeis.org/A234285 (the comment by Farideh Firoozbakht, although this comment is not true, there is no prime for s = 509203 and s = −78557, s = 509203 has a covering set of {3, 5, 7, 13, 17, 241}, and s = −78557 has a covering set of {3, 5, 7, 13, 19, 37, 73}), https://www.mersenneforum.org/showpost.php?p=564786&postcount=3, https://www.mersenneforum.org/showpost.php?p=461665&postcount=7, https://www.mersenneforum.org/showpost.php?p=354505&postcount=5, https://www.mersenneforum.org/showpost.php?p=344985&postcount=293, https://www.mersenneforum.org/showpost.php?p=625978&postcount=1027, https://www.primegrid.com/forum_thread.php?id=5093&nowrap=true#66471, https://www.primegrid.com/forum_thread.php?id=4935&nowrap=true#63813, also the graphs https://t5k.org/gifs/lg_lg_Mn.gif (for the family {1} in base b = 2) and https://t5k.org/gifs/repunits.gif (for the family {1} in base b = 10) and https://www.mersenneforum.org/attachment.php?attachmentid=4010&d=1642088235 (for the family 2{0}1 in base b = 3)), the least upper bound (https://en.wikipedia.org/wiki/Least_upper_bound, https://mathworld.wolfram.com/LeastUpperBound.html) for the number of minimal primes (or probable prime, which is a minimal prime assuming its primality) is given by the sum of the greatest lower bound for the number of minimal primes (or probable prime, which is a minimal prime assuming its primality) (i.e. the number of elements in the "kernel b" file) and the number of left families (i.e. the number of families in the "left b" file), also, we give the searching limit of length for the linear families, if there are any more minimal primes (or probable prime, which is a minimal prime assuming its primality) base b they must have > this many digits in base b.

Four coincidences (https://en.wikipedia.org/wiki/Mathematical_coincidence, https://mathworld.wolfram.com/Coincidence.html):

  • The length of the largest minimal prime in bases b = 18 and b = 20 are both 6271 (in theory, b = 20 should be about 1.65 times as b = 18).
  • The length of the 10401st minimal prime in base b = 17 is exactly 10401.
  • Bases b = 20 and b = 23 have two same type of families which have the minimal prime with the same length: (b−1)×bn+1 (both have length 15: J00000000000001 and M00000000000001, respectively) and bn−(b−1) (both have length 17: JJJJJJJJJJJJJJJJ1 and MMMMMMMMMMMMMMMM1, respectively).
  • A minimal prime in base b = 34 has length exactly 100000 (it is GFGC999965).

(for the factorization of the numbers in these families and the N−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) and N+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) of these (probable) primes, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=snfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#smallpolynomial, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=gnfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) may be used, they have SNFS polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt and https://alfredreichlg.de/10w7/10w7.txt and https://web.archive.org/web/20020320010222/http://proth.cjb.net/ and http://web.archive.org/web/20111104173105/http://www.immortaltheory.com/NumberTheory/2nm3_db.txt and https://www.asahi-net.or.jp/~KC2H-MSM/mathland/aba/index.htm and https://www.alpertron.com.ar/BRILLIANT.HTM and https://www.alpertron.com.ar/BRILLIANT3.HTM and https://www.alpertron.com.ar/BRILLIANT4.HTM and https://www.alpertron.com.ar/BRILLIANT2.HTM and https://oeis.org/wiki/Factors_of_33*2%5En%2B1 and https://oeis.org/wiki/Factors_of_33*2%5En-1 and https://web.archive.org/web/20111018190410/http://www.sr5.psp-project.de/s5stats.html (section "Top ten factors:") and https://web.archive.org/web/20111018190339/http://www.sr5.psp-project.de/r5stats.html (section "Top ten factors:") and https://brnikat.com/nums/cullen_woodall/cw.html and https://oeis.org/wiki/OEIS_sequences_needing_factors#Near_powers.2C_factorials.2C_and_primorials (sections "near-powers with b = 2" and "near-powers with b = 3" and "near-powers with b = 5" and "near-powers with b = 6" and "near-powers with b = 7" and "near-powers with b = 10" and "near-powers with b > 10") and https://www.mersenneforum.org/showthread.php?t=9554 and https://www.mersenneforum.org/showthread.php?t=9167 and https://www.mersenneforum.org/showpost.php?p=644144&postcount=5, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm and https://www.mersenneforum.org/showthread.php?t=12962)

(link to the factorization of the first 200 numbers in corresponding families of these minimal primes (start with the smallest number in these families which is > b), to show the small prime factors (and their period) (i.e. show whether these numbers are rough numbers) and the algebraic factors (and their period) and let you feel the Nash weight (or difficulty) of these families (high or low), all small prime factors (< 1012, by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172)) and all algebraic factors (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/53335.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/83336.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=algebraic&maxrows=10000, https://sites.google.com/view/algebraic-factors-of-xn-kyn, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://www.mersenneforum.org/showpost.php?p=96560&postcount=99, https://www.mersenneforum.org/showpost.php?p=96651&postcount=101, https://www.mersenneforum.org/showthread.php?t=21916, https://www.mersenneforum.org/showpost.php?p=196598&postcount=492, https://www.mersenneforum.org/showpost.php?p=203083&postcount=149, https://www.mersenneforum.org/showpost.php?p=206065&postcount=192, https://www.mersenneforum.org/showpost.php?p=208044&postcount=260, https://www.mersenneforum.org/showpost.php?p=210533&postcount=336, https://www.mersenneforum.org/showpost.php?p=452132&postcount=66, https://www.mersenneforum.org/showpost.php?p=451337&postcount=32, https://www.mersenneforum.org/showpost.php?p=208852&postcount=227, https://www.mersenneforum.org/showpost.php?p=232904&postcount=604, https://www.mersenneforum.org/showpost.php?p=383690&postcount=1, https://www.mersenneforum.org/showpost.php?p=207886&postcount=253, https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php, https://www.alpertron.com.ar/POLFACT.HTM, https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) of the N−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) and N+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) of these (probable) primes and the first 200 numbers (start with the smallest n making the number > b (if n = 0 already makes the number > b, then start with n = 0)) in corresponding families of these (probable) primes were added to factordb)

bnumber of minimal primes (or probable prime, which is a minimal prime assuming its primality) base bbase-b form of the top 10 known minimal primes (or probable prime, which is a minimal prime assuming its primality) base b (write "dn" if there are 5 or more (n) consecutive same digits d)length of the top 10 known minimal primes (or probable prime, which is a minimal prime assuming its primality) base blength of the top 10 known minimal primes (or probable prime, which is a minimal prime assuming its primality) base b in decimalalgebraic ((a×bn+c)/gcd(a+c,b−1)) form of the top 10 known minimal primes (or probable prime, which is a minimal prime assuming its primality) base bfactordb entry of the top 10 known minimal primes (or probable prime, which is a minimal prime assuming its primality)the top 10 known minimal primes (or probable prime, which is a minimal prime assuming its primality) written in base b (use lower case letters instead of upper case letters)the top 10 known minimal primes (or probable prime, which is a minimal prime assuming its primality) written in decimalfactorization of the first 200 numbers in corresponding family (n is the number of digits in the "{}", start with the smallest n making the number > b (if n = 0 already makes the number > b, then start with n = 0) of the top 10 known minimal primes)number of left families in base b (all of these left families are linear families)searching limit of length for the left families in base b (if there are different searching limits for the left families in base b, choose the lowest searching limit)
2111213http://factordb.com/index.php?id=3&open=ecmhttp://factordb.com/index.php?showid=3&base=2http://factordb.com/index.php?showid=30
33111
21
12
3
2
2
2
1
1
13
7
5
http://factordb.com/index.php?id=13&open=ecm
http://factordb.com/index.php?id=7&open=ecm
http://factordb.com/index.php?id=5&open=ecm
http://factordb.com/index.php?showid=13&base=3
http://factordb.com/index.php?showid=7&base=3
http://factordb.com/index.php?showid=5&base=3
http://factordb.com/index.php?showid=13
http://factordb.com/index.php?showid=7
http://factordb.com/index.php?showid=5


0
45221
31
23
13
11
3
2
2
2
2
2
2
2
1
1
41
13
11
7
5
http://factordb.com/index.php?id=41&open=ecm
http://factordb.com/index.php?id=13&open=ecm
http://factordb.com/index.php?id=11&open=ecm
http://factordb.com/index.php?id=7&open=ecm
http://factordb.com/index.php?id=5&open=ecm
http://factordb.com/index.php?showid=41&base=4
http://factordb.com/index.php?showid=13&base=4
http://factordb.com/index.php?showid=11&base=4
http://factordb.com/index.php?showid=7&base=4
http://factordb.com/index.php?showid=5&base=4
http://factordb.com/index.php?showid=41
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http://factordb.com/index.php?showid=11
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0
522109313
300031
44441
33331
33001
30301
14444
10103
3101
414
96
6
5
5
5
5
5
5
4
3
67
4
4
4
4
4
4
3
3
3
595+8
9391
3121
2341
2251
1951
1249
653
401
109
http://factordb.com/index.php?id=1100000000034686071&open=ecm
http://factordb.com/index.php?id=9391&open=ecm
http://factordb.com/index.php?id=3121&open=ecm
http://factordb.com/index.php?id=2341&open=ecm
http://factordb.com/index.php?id=2251&open=ecm
http://factordb.com/index.php?id=1951&open=ecm
http://factordb.com/index.php?id=1249&open=ecm
http://factordb.com/index.php?id=653&open=ecm
http://factordb.com/index.php?id=401&open=ecm
http://factordb.com/index.php?id=109&open=ecm
http://factordb.com/index.php?showid=1100000000034686071&base=5
http://factordb.com/index.php?showid=9391&base=5
http://factordb.com/index.php?showid=3121&base=5
http://factordb.com/index.php?showid=2341&base=5
http://factordb.com/index.php?showid=2251&base=5
http://factordb.com/index.php?showid=1951&base=5
http://factordb.com/index.php?showid=1249&base=5
http://factordb.com/index.php?showid=653&base=5
http://factordb.com/index.php?showid=401&base=5
http://factordb.com/index.php?showid=109&base=5
http://factordb.com/index.php?showid=1100000000034686071
http://factordb.com/index.php?showid=9391
http://factordb.com/index.php?showid=3121
http://factordb.com/index.php?showid=2341
http://factordb.com/index.php?showid=2251
http://factordb.com/index.php?showid=1951
http://factordb.com/index.php?showid=1249
http://factordb.com/index.php?showid=653
http://factordb.com/index.php?showid=401
http://factordb.com/index.php?showid=109
http://factordb.com/index.php?query=5%5E%28n%2B2%29%2B8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show








0
61140041
4441
4401
51
45
35
31
25
21
15
5
4
4
2
2
2
2
2
2
2
4
4
4
2
2
2
2
2
2
2
5209
1033
1009
31
29
23
19
17
13
11
http://factordb.com/index.php?id=5209&open=ecm
http://factordb.com/index.php?id=1033&open=ecm
http://factordb.com/index.php?id=1009&open=ecm
http://factordb.com/index.php?id=31&open=ecm
http://factordb.com/index.php?id=29&open=ecm
http://factordb.com/index.php?id=23&open=ecm
http://factordb.com/index.php?id=19&open=ecm
http://factordb.com/index.php?id=17&open=ecm
http://factordb.com/index.php?id=13&open=ecm
http://factordb.com/index.php?id=11&open=ecm
http://factordb.com/index.php?showid=5209&base=6
http://factordb.com/index.php?showid=1033&base=6
http://factordb.com/index.php?showid=1009&base=6
http://factordb.com/index.php?showid=31&base=6
http://factordb.com/index.php?showid=29&base=6
http://factordb.com/index.php?showid=23&base=6
http://factordb.com/index.php?showid=19&base=6
http://factordb.com/index.php?showid=17&base=6
http://factordb.com/index.php?showid=13&base=6
http://factordb.com/index.php?showid=11&base=6
http://factordb.com/index.php?showid=5209
http://factordb.com/index.php?showid=1033
http://factordb.com/index.php?showid=1009
http://factordb.com/index.php?showid=31
http://factordb.com/index.php?showid=29
http://factordb.com/index.php?showid=23
http://factordb.com/index.php?showid=19
http://factordb.com/index.php?showid=17
http://factordb.com/index.php?showid=13
http://factordb.com/index.php?showid=11









0
7713161
51071
3601
1100021
531101
351101
300053
150001
100121
40054
17
10
8
7
6
6
6
6
6
5
15
9
7
6
5
5
5
5
5
4
(717−5)/2
36×78+1
(78−47)/2
134471
91631
62819
50459
28813
16871
9643
http://factordb.com/index.php?id=116315256993601&open=ecm
http://factordb.com/index.php?id=207532837&open=ecm
http://factordb.com/index.php?id=2882377&open=ecm
http://factordb.com/index.php?id=134471&open=ecm
http://factordb.com/index.php?id=91631&open=ecm
http://factordb.com/index.php?id=62819&open=ecm
http://factordb.com/index.php?id=50459&open=ecm
http://factordb.com/index.php?id=28813&open=ecm
http://factordb.com/index.php?id=16871&open=ecm
http://factordb.com/index.php?id=9643&open=ecm
http://factordb.com/index.php?showid=116315256993601&base=7
http://factordb.com/index.php?showid=207532837&base=7
http://factordb.com/index.php?showid=2882377&base=7
http://factordb.com/index.php?showid=134471&base=7
http://factordb.com/index.php?showid=91631&base=7
http://factordb.com/index.php?showid=62819&base=7
http://factordb.com/index.php?showid=50459&base=7
http://factordb.com/index.php?showid=28813&base=7
http://factordb.com/index.php?showid=16871&base=7
http://factordb.com/index.php?showid=9643&base=7
http://factordb.com/index.php?showid=116315256993601
http://factordb.com/index.php?showid=207532837
http://factordb.com/index.php?showid=2882377
http://factordb.com/index.php?showid=134471
http://factordb.com/index.php?showid=91631
http://factordb.com/index.php?showid=62819
http://factordb.com/index.php?showid=50459
http://factordb.com/index.php?showid=28813
http://factordb.com/index.php?showid=16871
http://factordb.com/index.php?showid=9643
http://factordb.com/index.php?query=%287%5E%28n%2B1%29-5%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=36*7%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%287%5E%28n%2B2%29-47%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show






0
87542207
51325
7121
7777461
7471
481
55025
5550525
5500525
4577
221
15
13
11
9
9
8
7
7
7
200
14
12
10
9
8
8
7
7
7
(4×8221+17)/7
(5×815−173)/7
813−7
(28669×87−25)/7
(53×88−25)/7
(4×89−25)/7
(5×88−2413)/7
1495381
1474901
(4×87+185)/7
http://factordb.com/index.php?id=1100000000416605822&open=ecm
http://factordb.com/index.php?id=25131694349141&open=ecm
http://factordb.com/index.php?id=549755813881&open=ecm
http://factordb.com/index.php?id=8589035809&open=ecm
http://factordb.com/index.php?id=127027489&open=ecm
http://factordb.com/index.php?id=76695841&open=ecm
http://factordb.com/index.php?id=11983381&open=ecm
http://factordb.com/index.php?id=1495381&open=ecm
http://factordb.com/index.php?id=1474901&open=ecm
http://factordb.com/index.php?id=1198399&open=ecm
http://factordb.com/index.php?showid=1100000000416605822&base=8
http://factordb.com/index.php?showid=25131694349141&base=8
http://factordb.com/index.php?showid=549755813881&base=8
http://factordb.com/index.php?showid=8589035809&base=8
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http://factordb.com/index.php?showid=76695841&base=8
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http://factordb.com/index.php?showid=1100000000416605822
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http://factordb.com/index.php?showid=549755813881
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http://factordb.com/index.php?showid=127027489
http://factordb.com/index.php?showid=76695841
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http://factordb.com/index.php?query=%284*8%5E%28n%2B1%29%2B17%29%2F7&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%285*8%5E%28n%2B2%29-173%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=8%5E%28n%2B1%29-7&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2828669*8%5E%28n%2B1%29-25%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2853*8%5E%28n%2B1%29-25%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%284*8%5E%28n%2B1%29-25%29%2F7&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%285*8%5E%28n%2B3%29-2413%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show


http://factordb.com/index.php?query=%284*8%5E%28n%2B2%29%2B185%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
0
915130115811
2768607
763292
56136
102557
302051
819335
7271507
511361
1011507
1161
689
331
38
28
23
22
19
16
15
1108
657
316
37
26
22
21
19
16
14
3×91160+10
(23×9688−511)/8
(31×9330−19)/4
(409×936−1)/8
927+52
3×922+46
922−454
(527×917−511)/8
(41×915+359)/8
914+412
http://factordb.com/index.php?id=1100000002376318423&open=prime
http://factordb.com/index.php?id=1100000002495467486&open=prime
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http://factordb.com/index.php?id=1100000001554010824&open=ecm
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http://factordb.com/index.php?id=1100000002495736583&open=ecm
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http://factordb.com/index.php?showid=1100000002376318423&base=9
http://factordb.com/index.php?showid=1100000002495467486&base=9
http://factordb.com/index.php?showid=1100000002359003642&base=9
http://factordb.com/index.php?showid=1100000001554010824&base=9
http://factordb.com/index.php?showid=1100000002512830927&base=9
http://factordb.com/index.php?showid=1100000000032261811&base=9
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http://factordb.com/index.php?query=3*9%5E%28n%2B2%29%2B10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2823*9%5E%28n%2B2%29-511%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2831*9%5E%28n%2B1%29-19%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28409*9%5En-1%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=9%5E%28n%2B2%29%2B52&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=3*9%5E%28n%2B2%29%2B46&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=9%5E%28n%2B3%29-454&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28527*9%5E%28n%2B2%29-511%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2841*9%5E%28n%2B2%29%2B359%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=9%5E%28n%2B3%29%2B412&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
0
1077502827
5111
80551
66600049
66000049
60549
22051
5200007
946669
666649
31
12
8
8
8
8
8
7
6
6
31
12
8
8
8
8
8
7
6
6
5×1030+27
(5×1012−41)/9
(725×106−41)/9
66600049
66000049
6×107+49
22×106+1
5200007
946669
666649
http://factordb.com/index.php?id=1100000000204142046&open=ecm
http://factordb.com/index.php?id=555555555551&open=ecm
http://factordb.com/index.php?id=80555551&open=ecm
http://factordb.com/index.php?id=66600049&open=ecm
http://factordb.com/index.php?id=66000049&open=ecm
http://factordb.com/index.php?id=60000049&open=ecm
http://factordb.com/index.php?id=22000001&open=ecm
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http://factordb.com/index.php?showid=1100000000204142046&base=10
http://factordb.com/index.php?showid=555555555551&base=10
http://factordb.com/index.php?showid=80555551&base=10
http://factordb.com/index.php?showid=66600049&base=10
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http://factordb.com/index.php?showid=5200007&base=10
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http://factordb.com/index.php?showid=1100000000204142046
http://factordb.com/index.php?showid=555555555551
http://factordb.com/index.php?showid=80555551
http://factordb.com/index.php?showid=66600049
http://factordb.com/index.php?showid=66000049
http://factordb.com/index.php?showid=60000049
http://factordb.com/index.php?showid=22000001
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http://factordb.com/index.php?showid=946669
http://factordb.com/index.php?showid=666649
http://factordb.com/index.php?query=5*10%5E%28n%2B2%29%2B27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%285*10%5E%28n%2B1%29-41%29%2F9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28725*10%5E%28n%2B1%29-41%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show


http://factordb.com/index.php?query=6*10%5E%28n%2B2%29%2B49&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=22*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show


0
1110685762668
5571011
775944
A71358
8522005
507206
51612A
5012657
1012551
326122
62669
1013
761
715
223
208
163
129
128
124
65263
1055
793
745
233
217
170
134
133
129
(57×1162668−7)/10
(607×111011−7)/10
(7×11761−367)/10
11715−58
(17×11222−111)/2
(557×11206−7)/10
(11163−57)/2
5×11128+62
11127+56
(178×11122−3)/5
http://factordb.com/index.php?id=1100000003573679860&open=prime
http://factordb.com/index.php?id=1100000002361376522&open=prime
http://factordb.com/index.php?id=1100000002505568840&open=prime
http://factordb.com/index.php?id=1100000003576826487&open=prime
http://factordb.com/index.php?id=1100000003576826769&open=ecm
http://factordb.com/index.php?id=1100000002518512744&open=ecm
http://factordb.com/index.php?id=1100000002391585327&open=ecm
http://factordb.com/index.php?id=1100000002632393378&open=ecm
http://factordb.com/index.php?id=1100000002391531300&open=ecm
http://factordb.com/index.php?id=1100000003576826781&open=ecm
http://factordb.com/index.php?showid=1100000003573679860&base=11
http://factordb.com/index.php?showid=1100000002361376522&base=11
http://factordb.com/index.php?showid=1100000002505568840&base=11
http://factordb.com/index.php?showid=1100000003576826487&base=11
http://factordb.com/index.php?showid=1100000003576826769&base=11
http://factordb.com/index.php?showid=1100000002518512744&base=11
http://factordb.com/index.php?showid=1100000002391585327&base=11
http://factordb.com/index.php?showid=1100000002632393378&base=11
http://factordb.com/index.php?showid=1100000002391531300&base=11
http://factordb.com/index.php?showid=1100000003576826781&base=11
http://factordb.com/index.php?showid=1100000003573679860
http://factordb.com/index.php?showid=1100000002361376522
http://factordb.com/index.php?showid=1100000002505568840
http://factordb.com/index.php?showid=1100000003576826487
http://factordb.com/index.php?showid=1100000003576826769
http://factordb.com/index.php?showid=1100000002518512744
http://factordb.com/index.php?showid=1100000002391585327
http://factordb.com/index.php?showid=1100000002632393378
http://factordb.com/index.php?showid=1100000002391531300
http://factordb.com/index.php?showid=1100000003576826781
http://factordb.com/index.php?query=%2857*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28607*11%5En-7%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%287*11%5E%28n%2B2%29-367%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=11%5E%28n%2B2%29-58&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2817*11%5E%28n%2B2%29-111%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28557*11%5En-7%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2811%5E%28n%2B2%29-57%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=5*11%5E%28n%2B2%29%2B62&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=11%5E%28n%2B2%29%2B56&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28178*11%5En-3%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
0
12106403977
B0279B
B699B
AA051
B00099B
AAA0001
BBBAA1
A00065
44AAA1
BBBB1
42
30
9
8
7
7
6
6
6
5
45
33
10
9
8
8
7
7
7
6
4×1241+91
11×1229+119
129−313
130×126+1
32847239
32555521
2985817
2488397
1097113
248821
http://factordb.com/index.php?id=1100000002375054575&open=ecm
http://factordb.com/index.php?id=1100000002354113100&open=ecm
http://factordb.com/index.php?id=5159780039&open=ecm
http://factordb.com/index.php?id=388177921&open=ecm
http://factordb.com/index.php?id=32847239&open=ecm
http://factordb.com/index.php?id=32555521&open=ecm
http://factordb.com/index.php?id=2985817&open=ecm
http://factordb.com/index.php?id=2488397&open=ecm
http://factordb.com/index.php?id=1097113&open=ecm
http://factordb.com/index.php?id=248821&open=ecm
http://factordb.com/index.php?showid=1100000002375054575&base=12
http://factordb.com/index.php?showid=1100000002354113100&base=12
http://factordb.com/index.php?showid=5159780039&base=12
http://factordb.com/index.php?showid=388177921&base=12
http://factordb.com/index.php?showid=32847239&base=12
http://factordb.com/index.php?showid=32555521&base=12
http://factordb.com/index.php?showid=2985817&base=12
http://factordb.com/index.php?showid=2488397&base=12
http://factordb.com/index.php?showid=1097113&base=12
http://factordb.com/index.php?showid=248821&base=12
http://factordb.com/index.php?showid=1100000002375054575
http://factordb.com/index.php?showid=1100000002354113100
http://factordb.com/index.php?showid=5159780039
http://factordb.com/index.php?showid=388177921
http://factordb.com/index.php?showid=32847239
http://factordb.com/index.php?showid=32555521
http://factordb.com/index.php?showid=2985817
http://factordb.com/index.php?showid=2488397
http://factordb.com/index.php?showid=1097113
http://factordb.com/index.php?showid=248821
http://factordb.com/index.php?query=4*12%5E%28n%2B2%29%2B91&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=11*12%5E%28n%2B2%29%2B119&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=12%5E%28n%2B3%29-313&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=130*12%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show





0
133197A3592197A
95197420
8032017111
C523755C
C1063192
B06540BBA
39062661
1770270317
72022972
93015511
592199
197421
32021
23757
10633
6544
6269
2708
2300
1554
659677
219916
35670
26464
11845
7290
6983
3016
2562
1731
(41×13592198+27)/4
(113×13197420−5)/12
8×1332020+183
(149×1323756+79)/12
1310633−50
11×136543+2012
48×136267+1
267×132705+20
93×132298+2
120×131552+1
http://factordb.com/index.php?id=1100000005489162806&open=prime
http://factordb.com/index.php?id=1100000003943359311&open=prime
http://factordb.com/index.php?id=1100000000490878060&open=prime
http://factordb.com/index.php?id=1100000003590647776&open=prime
http://factordb.com/index.php?id=1100000003590493750&open=prime
http://factordb.com/index.php?id=1100000002616382906&open=prime
http://factordb.com/index.php?id=1100000000765961441&open=prime
http://factordb.com/index.php?id=1100000003590430825&open=prime
http://factordb.com/index.php?id=1100000002632396910&open=prime
http://factordb.com/index.php?id=1100000000765961452&open=prime
http://factordb.com/index.php?showid=1100000005489162806&base=13
http://factordb.com/index.php?showid=1100000003943359311&base=13
http://factordb.com/index.php?showid=1100000000490878060&base=13
http://factordb.com/index.php?showid=1100000003590647776&base=13
http://factordb.com/index.php?showid=1100000003590493750&base=13
http://factordb.com/index.php?showid=1100000002616382906&base=13
http://factordb.com/index.php?showid=1100000000765961441&base=13
http://factordb.com/index.php?showid=1100000003590430825&base=13
http://factordb.com/index.php?showid=1100000002632396910&base=13
http://factordb.com/index.php?showid=1100000000765961452&base=13
http://factordb.com/index.php?showid=1100000005489162806
http://factordb.com/index.php?showid=1100000003943359311
http://factordb.com/index.php?showid=1100000000490878060
http://factordb.com/index.php?showid=1100000003590647776
http://factordb.com/index.php?showid=1100000003590493750
http://factordb.com/index.php?showid=1100000002616382906
http://factordb.com/index.php?showid=1100000000765961441
http://factordb.com/index.php?showid=1100000003590430825
http://factordb.com/index.php?showid=1100000002632396910
http://factordb.com/index.php?showid=1100000000765961452
http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=8*13%5E%28n%2B3%29%2B183&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28149*13%5E%28n%2B1%29%2B79%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=13%5E%28n%2B2%29-50&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=11*13%5E%28n%2B3%29%2B2012&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=48*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=267*13%5E%28n%2B2%29%2B20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=93*13%5E%28n%2B1%29%2B2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=120*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
0
146504D19698
34D708
8D14185
886B
408349
8C793
1879B
6B772B
46309
A593
19699
710
144
87
86
81
81
80
65
60
22578
814
165
100
99
93
92
92
74
69
5×1419698−1
47×14708−1
9×14143−79
(8×1487+31)/13
4×1485+65
(116×1480−129)/13
(21×1480+31)/13
(89×1479−1649)/13
(4×1465−667)/13
(10×1460−101)/13
http://factordb.com/index.php?id=1100000000884560233&open=prime
http://factordb.com/index.php?id=1100000001540144903&open=prime
http://factordb.com/index.php?id=1100000003575856650&open=ecm
http://factordb.com/index.php?id=1100000002321014379&open=ecm
http://factordb.com/index.php?id=1100000000823937973&open=ecm
http://factordb.com/index.php?id=1100000002631073246&open=ecm
http://factordb.com/index.php?id=1100000002384401372&open=ecm
http://factordb.com/index.php?id=1100000002631077787&open=ecm
http://factordb.com/index.php?id=1100000000840126683&open=ecm
http://factordb.com/index.php?id=1100000002321038522&open=ecm
http://factordb.com/index.php?showid=1100000000884560233&base=14
http://factordb.com/index.php?showid=1100000001540144903&base=14
http://factordb.com/index.php?showid=1100000003575856650&base=14
http://factordb.com/index.php?showid=1100000002321014379&base=14
http://factordb.com/index.php?showid=1100000000823937973&base=14
http://factordb.com/index.php?showid=1100000002631073246&base=14
http://factordb.com/index.php?showid=1100000002384401372&base=14
http://factordb.com/index.php?showid=1100000002631077787&base=14
http://factordb.com/index.php?showid=1100000000840126683&base=14
http://factordb.com/index.php?showid=1100000002321038522&base=14
http://factordb.com/index.php?showid=1100000000884560233
http://factordb.com/index.php?showid=1100000001540144903
http://factordb.com/index.php?showid=1100000003575856650
http://factordb.com/index.php?showid=1100000002321014379
http://factordb.com/index.php?showid=1100000000823937973
http://factordb.com/index.php?showid=1100000002631073246
http://factordb.com/index.php?showid=1100000002384401372
http://factordb.com/index.php?showid=1100000002631077787
http://factordb.com/index.php?showid=1100000000840126683
http://factordb.com/index.php?showid=1100000002321038522
http://factordb.com/index.php?query=5*14%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=47*14%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=9*14%5E%28n%2B2%29-79&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%288*14%5E%28n%2B1%29%2B31%29%2F13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=4*14%5E%28n%2B2%29%2B65&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28116*14%5E%28n%2B1%29-129%29%2F13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2821*14%5E%28n%2B1%29%2B31%29%2F13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2889*14%5E%28n%2B2%29-1649%29%2F13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%284*14%5E%28n%2B2%29-667%29%2F13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2810*14%5E%28n%2B1%29-101%29%2F13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
0
151284715597
E145397
9610408
773CE
759CCE
503317
EB31
6330261
705024B
B70241
157
148
107
75
62
36
32
30
28
27
185
175
126
88
73
42
38
35
33
32
(15157+59)/2
15148−2558
(66×15106−619)/7
(1575+163)/2
(1562+2413)/2
5×1535+22
(207×1531−11)/14
1398×1527+1
1580×1525+11
172×1525+1
http://factordb.com/index.php?id=1100000002454891840&open=ecm
http://factordb.com/index.php?id=1100000002454900849&open=ecm
http://factordb.com/index.php?id=1100000000823937997&open=ecm
http://factordb.com/index.php?id=1100000003588407143&open=ecm
http://factordb.com/index.php?id=1100000003588407386&open=ecm
http://factordb.com/index.php?id=1100000002632398579&open=ecm
http://factordb.com/index.php?id=1100000002321033312&open=ecm
http://factordb.com/index.php?id=1100000002391199877&open=ecm
http://factordb.com/index.php?id=1100000003588407806&open=ecm
http://factordb.com/index.php?id=1100000000851967288&open=ecm
http://factordb.com/index.php?showid=1100000002454891840&base=15
http://factordb.com/index.php?showid=1100000002454900849&base=15
http://factordb.com/index.php?showid=1100000000823937997&base=15
http://factordb.com/index.php?showid=1100000003588407143&base=15
http://factordb.com/index.php?showid=1100000003588407386&base=15
http://factordb.com/index.php?showid=1100000002632398579&base=15
http://factordb.com/index.php?showid=1100000002321033312&base=15
http://factordb.com/index.php?showid=1100000002391199877&base=15
http://factordb.com/index.php?showid=1100000003588407806&base=15
http://factordb.com/index.php?showid=1100000000851967288&base=15
http://factordb.com/index.php?showid=1100000002454891840
http://factordb.com/index.php?showid=1100000002454900849
http://factordb.com/index.php?showid=1100000000823937997
http://factordb.com/index.php?showid=1100000003588407143
http://factordb.com/index.php?showid=1100000003588407386
http://factordb.com/index.php?showid=1100000002632398579
http://factordb.com/index.php?showid=1100000002321033312
http://factordb.com/index.php?showid=1100000002391199877
http://factordb.com/index.php?showid=1100000003588407806
http://factordb.com/index.php?showid=1100000000851967288
http://factordb.com/index.php?query=%2815%5E%28n%2B2%29%2B59%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=15%5E%28n%2B3%29-2558&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2866*15%5E%28n%2B2%29-619%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2815%5E%28n%2B2%29%2B163%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2815%5E%28n%2B3%29%2B2413%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=5*15%5E%28n%2B2%29%2B22&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28207*15%5En-11%29%2F14&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=1398*15%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=1580*15%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=172*15%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
0
1623473116137AF
472785DD
DB32234
D0B17804
5BC3700D
90354291
300F1960AF
201713321
F81517F
FAF106245
116139
72787
32235
17806
3703
3545
1965
1717
1519
1066
139845
87644
38815
21441
4459
4269
2366
2067
1830
1284
(16116139+619)/5
(4×1672787+2291)/15
(206×1632234−11)/15
(3131×1617804−11)/15
(459×163701+1)/5
9×163544+145
769×161962−81
2×161716+801
(233×161518+97)/15
251×161064−187
http://factordb.com/index.php?id=1100000003851731988&open=prime
http://factordb.com/index.php?id=1100000003615909841&open=prime
http://factordb.com/index.php?id=1100000002383583629&open=prime
http://factordb.com/index.php?id=1100000003589278511&open=prime
http://factordb.com/index.php?id=1100000000993764322&open=prime
http://factordb.com/index.php?id=1100000000633424191&open=prime
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http://factordb.com/index.php?id=1100000003588387610&open=prime
http://factordb.com/index.php?showid=1100000003851731988&base=16
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http://factordb.com/index.php?showid=1100000000633424191&base=16
http://factordb.com/index.php?showid=1100000003588368750&base=16
http://factordb.com/index.php?showid=1100000003588386735&base=16
http://factordb.com/index.php?showid=1100000000633744824&base=16
http://factordb.com/index.php?showid=1100000003588387610&base=16
http://factordb.com/index.php?showid=1100000003851731988
http://factordb.com/index.php?showid=1100000003615909841
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http://factordb.com/index.php?showid=1100000003588387610
http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%284*16%5E%28n%2B2%29%2B2291%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28206*16%5En-11%29%2F15&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%283131*16%5En-11%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28459*16%5E%28n%2B1%29%2B1%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=9*16%5E%28n%2B2%29%2B145&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=769*16%5E%28n%2B2%29-81&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=2*16%5E%28n%2B3%29%2B801&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28233*16%5E%28n%2B1%29%2B97%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=251*16%5E%28n%2B2%29-187&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
0
1710409~10427B671032E
570513101
E9B44732
D0GD37096
G732072F
15024325D
34716074
B3013077D
9D0103985
1090191F
67105
51313
44734
37099
32074
24328
16076
13080
10401
9022
82570
63138
55043
45649
39466
29934
19780
16095
12798
11100
(11×1767105−2411)/16
92×1751311+1
(3963×1744732−11)/16
(60381×1737096−13)/16
(263×1732073+121)/16
22×1724326+13
(887×1716074−7)/16
190×1713078+13
166×1710399+5
179021+32
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http://factordb.com/index.php?id=1100000000765961389&open=prime
http://factordb.com/index.php?id=1100000003883765450&open=prime
http://factordb.com/index.php?id=1100000003848346668&open=prime
http://factordb.com/index.php?id=1100000003838755382&open=prime
http://factordb.com/index.php?id=1100000003815568647&open=prime
http://factordb.com/index.php?id=1100000003802992457&open=prime
http://factordb.com/index.php?id=1100000003782940761&open=prime
http://factordb.com/index.php?id=1100000003782940760&open=prime
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http://factordb.com/index.php?showid=1100000003993647842&base=17
http://factordb.com/index.php?showid=1100000000765961389&base=17
http://factordb.com/index.php?showid=1100000003883765450&base=17
http://factordb.com/index.php?showid=1100000003848346668&base=17
http://factordb.com/index.php?showid=1100000003838755382&base=17
http://factordb.com/index.php?showid=1100000003815568647&base=17
http://factordb.com/index.php?showid=1100000003802992457&base=17
http://factordb.com/index.php?showid=1100000003782940761&base=17
http://factordb.com/index.php?showid=1100000003782940760&base=17
http://factordb.com/index.php?showid=1100000001100010542&base=17
http://factordb.com/index.php?showid=1100000003993647842
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http://factordb.com/index.php?showid=1100000003838755382
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http://factordb.com/index.php?showid=1100000003782940761
http://factordb.com/index.php?showid=1100000003782940760
http://factordb.com/index.php?showid=1100000001100010542
http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29-2411%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=92*17%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%283963*17%5En-11%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2860381*17%5En-13%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28263*17%5E%28n%2B1%29%2B121%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=22*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28887*17%5En-7%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=190*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=166*17%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=17%5E%28n%2B2%29%2B32&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
18100000
18549C06268C5
H766FH
80298B
C0116F5
HD93
GG0301
CF305
B196B
CCF145
714G7
6271
768
300
119
94
33
32
21
17
16
7872
965
377
150
118
42
41
27
22
20
12×186270+221
18768−37
8×18299+11
12×18118+275
(302×1893−13)/17
304×1831+1
(219×1831−185)/17
(11×1821−1541)/17
(3891×1815−185)/17
(7×1816+2747)/17
http://factordb.com/index.php?id=1100000003590442437&open=prime
http://factordb.com/index.php?id=1100000003590430490&open=prime
http://factordb.com/index.php?id=1100000002355574745&open=prime
http://factordb.com/index.php?id=1100000002632837015&open=ecm
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http://factordb.com/index.php?id=1100000003590430465&open=ecm
http://factordb.com/index.php?showid=1100000003590442437&base=18
http://factordb.com/index.php?showid=1100000003590430490&base=18
http://factordb.com/index.php?showid=1100000002355574745&base=18
http://factordb.com/index.php?showid=1100000002632837015&base=18
http://factordb.com/index.php?showid=1100000002321052894&base=18
http://factordb.com/index.php?showid=1100000000819230161&base=18
http://factordb.com/index.php?showid=1100000002631240657&base=18
http://factordb.com/index.php?showid=1100000003590430474&base=18
http://factordb.com/index.php?showid=1100000003590430470&base=18
http://factordb.com/index.php?showid=1100000003590430465&base=18
http://factordb.com/index.php?showid=1100000003590442437
http://factordb.com/index.php?showid=1100000003590430490
http://factordb.com/index.php?showid=1100000002355574745
http://factordb.com/index.php?showid=1100000002632837015
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http://factordb.com/index.php?showid=1100000002631240657
http://factordb.com/index.php?showid=1100000003590430474
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http://factordb.com/index.php?showid=1100000003590430465
http://factordb.com/index.php?query=12*18%5E%28n%2B2%29%2B221&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=18%5E%28n%2B2%29-37&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=8*18%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=12*18%5E%28n%2B2%29%2B275&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28302*18%5En-13%29%2F17&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=304*18%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28219*18%5E%28n%2B1%29-185%29%2F17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2811*18%5E%28n%2B2%29-1541%29%2F17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%283891*18%5E%28n%2B1%29-185%29%2F17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%287*18%5E%28n%2B2%29%2B2747%29%2F17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
0
1931412~31435H862916
D90730469
4F0498476
2482247
2458867A
9042994G
DB36272
333531088
B26588FG
10227907717
86292
73049
49850
48225
45888
42996
36273
31091
26590
22795
110347
93412
63746
61667
58679
54981
46385
39757
34002
29148
(17×1986292−215)/18
256×1973047+9
91×1949848+6
(1948225+44)/9
(1945888+926)/9
9×1942995+16
(245×1936272−11)/18
(20579×1931088−5)/18
(11×1926590+1447)/18
1922794+50566
http://factordb.com/index.php?id=1100000004163040839&open=prime
http://factordb.com/index.php?id=1100000003998413751&open=prime
http://factordb.com/index.php?id=1100000000808118332&open=prime
http://factordb.com/index.php?id=1100000003949188041&open=prime
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http://factordb.com/index.php?id=1100000003968087931&open=prime
http://factordb.com/index.php?id=1100000003949187975&open=prime
http://factordb.com/index.php?showid=1100000004163040839&base=19
http://factordb.com/index.php?showid=1100000003998413751&base=19
http://factordb.com/index.php?showid=1100000000808118332&base=19
http://factordb.com/index.php?showid=1100000003949188041&base=19
http://factordb.com/index.php?showid=1100000003949189035&base=19
http://factordb.com/index.php?showid=1100000000808118328&base=19
http://factordb.com/index.php?showid=1100000003968090004&base=19
http://factordb.com/index.php?showid=1100000003949189435&base=19
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http://factordb.com/index.php?showid=1100000003949187975&base=19
http://factordb.com/index.php?showid=1100000004163040839
http://factordb.com/index.php?showid=1100000003998413751
http://factordb.com/index.php?showid=1100000000808118332
http://factordb.com/index.php?showid=1100000003949188041
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http://factordb.com/index.php?showid=1100000000808118328
http://factordb.com/index.php?showid=1100000003968090004
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http://factordb.com/index.php?showid=1100000003968087931
http://factordb.com/index.php?showid=1100000003949187975
http://factordb.com/index.php?query=%2817*19%5E%28n%2B1%29-215%29%2F18&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=256*19%5E%28n%2B1%29%2B9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=91*19%5E%28n%2B1%29%2B6&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2819%5E%28n%2B1%29%2B44%29%2F9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2819%5E%28n%2B2%29%2B926%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=9*19%5E%28n%2B1%29%2B16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28245*19%5En-11%29%2F18&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2820579*19%5En-5%29%2F18&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2811*19%5E%28n%2B2%29%2B1447%29%2F18&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=19%5E%28n%2B4%29%2B50566&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
23100000
203314G06269D
CD2449
501163AJ
J65505J
JCJ629
E566C7
3A5273
G44799
EC04297
40387404B
6271
2450
1166
658
631
568
529
449
432
392
8159
3188
1517
857
821
739
688
585
562
510
16×206270+13
(241×202449−13)/19
5×201165+219
20658−7881
393×20629−1
(14×20568−907)/19
(67×20528−143)/19
(16×20449−2809)/19
292×20430+7
4×20391+32091
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http://factordb.com/index.php?id=1100000002325393915&open=prime
http://factordb.com/index.php?id=1100000003590502412&open=prime
http://factordb.com/index.php?id=1100000003590502490&open=prime
http://factordb.com/index.php?id=1100000001559454258&open=prime
http://factordb.com/index.php?id=1100000003590502516&open=prime
http://factordb.com/index.php?id=1100000003590502531&open=prime
http://factordb.com/index.php?id=1100000000840126753&open=prime
http://factordb.com/index.php?id=1100000002633348702&open=prime
http://factordb.com/index.php?id=1100000003590502563&open=prime
http://factordb.com/index.php?showid=1100000003590539457&base=20
http://factordb.com/index.php?showid=1100000002325393915&base=20
http://factordb.com/index.php?showid=1100000003590502412&base=20
http://factordb.com/index.php?showid=1100000003590502490&base=20
http://factordb.com/index.php?showid=1100000001559454258&base=20
http://factordb.com/index.php?showid=1100000003590502516&base=20
http://factordb.com/index.php?showid=1100000003590502531&base=20
http://factordb.com/index.php?showid=1100000000840126753&base=20
http://factordb.com/index.php?showid=1100000002633348702&base=20
http://factordb.com/index.php?showid=1100000003590502563&base=20
http://factordb.com/index.php?showid=1100000003590539457
http://factordb.com/index.php?showid=1100000002325393915
http://factordb.com/index.php?showid=1100000003590502412
http://factordb.com/index.php?showid=1100000003590502490
http://factordb.com/index.php?showid=1100000001559454258
http://factordb.com/index.php?showid=1100000003590502516
http://factordb.com/index.php?showid=1100000003590502531
http://factordb.com/index.php?showid=1100000000840126753
http://factordb.com/index.php?showid=1100000002633348702
http://factordb.com/index.php?showid=1100000003590502563
http://factordb.com/index.php?query=16*20%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28241*20%5En-13%29%2F19&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=5*20%5E%28n%2B2%29%2B219&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=20%5E%28n%2B3%29-7881&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=393*20%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2814*20%5E%28n%2B2%29-907%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2867*20%5E%28n%2B1%29-143%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2816*20%5E%28n%2B2%29-2809%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=292*20%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=4*20%5E%28n%2B4%29%2B32091&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
0
2113382~1339440473339G
B9045019E5
HD37414
BD35027B
9903323999H
530606FEK
4329236B
J233046J
9211260D
5D0198481
47336
45023
37415
35029
33244
30609
29238
23306
21128
19851
62588
59531
49471
46316
43956
40472
38659
30816
27936
26247
4×2147335+205
240×2145021+299
(353×2137414−13)/20
(233×2135028−53)/20
198×2133242+4175
(2130609+18455)/4
(83×2129237+157)/20
(19×2123306−5479)/20
(9×2121128−3709)/20
118×2119849+1
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http://factordb.com/index.php?id=1100000003996110311&open=prime
http://factordb.com/index.php?id=1100000003996110479&open=prime
http://factordb.com/index.php?id=1100000003996110718&open=prime
http://factordb.com/index.php?id=1100000003996110944&open=prime
http://factordb.com/index.php?id=1100000003996111130&open=prime
http://factordb.com/index.php?id=1100000003996112263&open=prime
http://factordb.com/index.php?id=1100000003996112521&open=prime
http://factordb.com/index.php?id=1100000003996112710&open=prime
http://factordb.com/index.php?id=1100000000777265872&open=prime
http://factordb.com/index.php?showid=1100000000808118331&base=21
http://factordb.com/index.php?showid=1100000003996110311&base=21
http://factordb.com/index.php?showid=1100000003996110479&base=21
http://factordb.com/index.php?showid=1100000003996110718&base=21
http://factordb.com/index.php?showid=1100000003996110944&base=21
http://factordb.com/index.php?showid=1100000003996111130&base=21
http://factordb.com/index.php?showid=1100000003996112263&base=21
http://factordb.com/index.php?showid=1100000003996112521&base=21
http://factordb.com/index.php?showid=1100000003996112710&base=21
http://factordb.com/index.php?showid=1100000000777265872&base=21
http://factordb.com/index.php?showid=1100000000808118331
http://factordb.com/index.php?showid=1100000003996110311
http://factordb.com/index.php?showid=1100000003996110479
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http://factordb.com/index.php?showid=1100000003996112521
http://factordb.com/index.php?showid=1100000003996112710
http://factordb.com/index.php?showid=1100000000777265872
http://factordb.com/index.php?query=4*21%5E%28n%2B2%29%2B205&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=240*21%5E%28n%2B2%29%2B299&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28353*21%5En-13%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28233*21%5E%28n%2B1%29-53%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=198*21%5E%28n%2B3%29%2B4175&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2821%5E%28n%2B3%29%2B18455%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2883*21%5E%28n%2B1%29%2B157%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2819*21%5E%28n%2B2%29-5479%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%289*21%5E%28n%2B2%29-3709%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=118*21%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
12100000
228003BK220015
738152L
L2385KE7
7959K7
J0767IGGJ
K0760EC1
I626AF
E60496L
L483G3
L0454B63
22003
3817
2388
961
772
764
628
499
485
458
29538
5124
3206
1290
1037
1026
843
670
652
615
(251×2222002−335)/21
(223817−289)/3
222388−653
(22961+857)/3
19×22771+199779
20×22763+7041
(6×22628−1259)/7
314×22497+21
22485−129
21×22457+5459
http://factordb.com/index.php?id=1100000003594696838&open=prime
http://factordb.com/index.php?id=1100000003591359839&open=prime
http://factordb.com/index.php?id=1100000003591360774&open=prime
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http://factordb.com/index.php?showid=1100000003594696838&base=22
http://factordb.com/index.php?showid=1100000003591359839&base=22
http://factordb.com/index.php?showid=1100000003591360774&base=22
http://factordb.com/index.php?showid=1100000003591361817&base=22
http://factordb.com/index.php?showid=1100000003591362567&base=22
http://factordb.com/index.php?showid=1100000000632724415&base=22
http://factordb.com/index.php?showid=1100000000632724334&base=22
http://factordb.com/index.php?showid=1100000000632703239&base=22
http://factordb.com/index.php?showid=1100000003591364730&base=22
http://factordb.com/index.php?showid=1100000003591365331&base=22
http://factordb.com/index.php?showid=1100000003594696838
http://factordb.com/index.php?showid=1100000003591359839
http://factordb.com/index.php?showid=1100000003591360774
http://factordb.com/index.php?showid=1100000003591361817
http://factordb.com/index.php?showid=1100000003591362567
http://factordb.com/index.php?showid=1100000000632724415
http://factordb.com/index.php?showid=1100000000632724334
http://factordb.com/index.php?showid=1100000000632703239
http://factordb.com/index.php?showid=1100000003591364730
http://factordb.com/index.php?showid=1100000003591365331
http://factordb.com/index.php?query=%28251*22%5E%28n%2B1%29-335%29%2F21&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2822%5E%28n%2B2%29-289%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=22%5E%28n%2B3%29-653&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2822%5E%28n%2B2%29%2B857%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=19*22%5E%28n%2B4%29%2B199779&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=20*22%5E%28n%2B3%29%2B7041&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%286*22%5E%28n%2B2%29-1259%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=314*22%5E%28n%2B1%29%2B21&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=22%5E%28n%2B2%29-129&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=21*22%5E%28n%2B3%29%2B5459&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
0
2365178~65265B093046FB
L86444D
AJ810654
207356098
J68217G4
D7066770B
5F623406
A7M761532
B30611365
EJ52169
93049
86445
81067
73563
68219
66773
62342
61535
61139
52170
126708
117715
110391
100172
92896
90927
84893
83794
83255
71042
11×2393048+356
(21×2386445−197)/22
(239×2381066−349)/22
2×2373562+215
(19×2368219−1867)/22
306×2366771+11
(125×2362341−213)/22
(120413×2361532−7)/22
256×2361137+5
(327×2352169−19)/22
http://factordb.com/index.php?id=1100000004691540361&open=prime
http://factordb.com/index.php?id=1100000004691546739&open=prime
http://factordb.com/index.php?id=1100000004691548070&open=prime
http://factordb.com/index.php?id=1100000004691548569&open=prime
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http://factordb.com/index.php?id=1100000004691551005&open=prime
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http://factordb.com/index.php?showid=1100000004691540361&base=23
http://factordb.com/index.php?showid=1100000004691546739&base=23
http://factordb.com/index.php?showid=1100000004691548070&base=23
http://factordb.com/index.php?showid=1100000004691548569&base=23
http://factordb.com/index.php?showid=1100000004691549462&base=23
http://factordb.com/index.php?showid=1100000004691549803&base=23
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http://factordb.com/index.php?showid=1100000004691556967&base=23
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http://factordb.com/index.php?showid=1100000004691540361
http://factordb.com/index.php?showid=1100000004691546739
http://factordb.com/index.php?showid=1100000004691548070
http://factordb.com/index.php?showid=1100000004691548569
http://factordb.com/index.php?showid=1100000004691549462
http://factordb.com/index.php?showid=1100000004691549803
http://factordb.com/index.php?showid=1100000004691551005
http://factordb.com/index.php?showid=1100000004691556967
http://factordb.com/index.php?showid=1100000004691557254
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http://factordb.com/index.php?query=256*23%5E%28n%2B1%29%2B5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28327*23%5En-19%29%2F22&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
87100000
243409N00N8129LN
88N5951
A029518ID
D2698LD
N2644LLN
BC0331B
203137
C7298
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I0241I5
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334
315
299
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244
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201×245951−1
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276×24332+11
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0
25133639~133724E98396FOO
1J710962721
NB0855985NH
D7081581JJ7
F080054HL
J01075943E7
K677715I
LO66377KC
KJD0633991
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81586
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111915
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92796
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85130
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27676×2596273+1
586×2585601+3717
332×2581584+12357
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22×2566379−113
12988×2563400+1
7×2560896+207526
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http://factordb.com/index.php?query=12988*25%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=7*25%5E%28n%2B4%29%2B207526&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
85100000
2625255~25259M0611862BB
J044303KCB
6K233005
LD0209757
720279OL
5193916F
9GDK15920P
M8772P
K04364I5
J4222P
61190
44307
23302
20978
20281
19393
15924
8773
4367
4223
86583
62694
32972
29684
28697
27440
22532
12414
6180
5976
22×2661189+1649
19×2644306+13843
(34×2623301−79)/5
559×2620976+7
(7×2620281+11393)/25
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(32569×2615921+21)/5
(22×268773+53)/25
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(19×264223+131)/25
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http://factordb.com/index.php?query=20*26%5E%28n%2B2%29%2B473&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
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4100000
27102852~102896CA0F88883A
GNN078795N
O44L660167
NJ064369H
ME496409G
PH0478901
QF47165AF5
J040791PD
51039164I07
NGN036329N
88887
78799
66020
64372
49643
47893
47169
40794
39169
36333
127230
112790
94499
92140
71058
68553
67516
58391
56065
52006
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12308×2778796+23
(457829×2766017−385)/26
640×2764370+17
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692×2747891+1
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17222×2736330+23
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44100000
2825528~25529O4O945359
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29355205~355367H049949IDL
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16250000
302619OT34205
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54882J
C010221
M0547SS7
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50527
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814
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0
31569323~5694002IIF91805
B088309APO9
J0T77516
J09077128NNN
D69861QO
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3163864−17574
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http://factordb.com/index.php?showid=1100000006965865343
http://factordb.com/index.php?query=%284997*31%5En-1%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=11*31%5E%28n%2B4%29%2B322688&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2817699*31%5En-29%29%2F30&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=18268*31%5E%28n%2B3%29%2B22839&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2813*31%5E%28n%2B2%29%2B12407%29%2F30&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=9348*31%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2819*31%5E%28n%2B1%29-199%29%2F30&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=23*31%5E%28n%2B3%29%2B2970&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=400205*31%5E%28n%2B2%29%2B912&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=31%5E%28n%2B3%29-17574&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
77100000
32168882~169002V9958363
689074AF
877700H
Q77401EQQQ3
8772493
JM76028L
E72919IL
B067680CB
GK66076F
KN65022
99585
89076
77701
77406
77250
76030
72921
67683
66078
65023
149891
134073
116952
116508
116273
114437
109757
101873
99458
97870
3299585−829
(6×3289076+4241)/31
(8×3277701+271)/31
(26×3277406−390071011)/31
(8×3277250−163)/31
(611×3276029−53)/31
(14×3272921+4171)/31
11×3267682+395
(516×3266077−175)/31
(643×3265022−23)/31
http://factordb.com/index.php?id=1100000005514892191&open=prime
http://factordb.com/index.php?id=1100000005514897129&open=prime
http://factordb.com/index.php?id=1100000005514901700&open=prime
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http://factordb.com/index.php?id=1100000005514918574&open=prime
http://factordb.com/index.php?id=1100000005514922523&open=prime
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http://factordb.com/index.php?id=1100000004591654467&open=prime
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http://factordb.com/index.php?id=1100000004591654952&open=prime
http://factordb.com/index.php?showid=1100000005514892191&base=32
http://factordb.com/index.php?showid=1100000005514897129&base=32
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http://factordb.com/index.php?showid=1100000005514918574&base=32
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http://factordb.com/index.php?showid=1100000004591654373&base=32
http://factordb.com/index.php?showid=1100000004591654467&base=32
http://factordb.com/index.php?showid=1100000004591654632&base=32
http://factordb.com/index.php?showid=1100000004591654952&base=32
http://factordb.com/index.php?showid=1100000005514892191
http://factordb.com/index.php?showid=1100000005514897129
http://factordb.com/index.php?showid=1100000005514901700
http://factordb.com/index.php?showid=1100000005514915338
http://factordb.com/index.php?showid=1100000005514918574
http://factordb.com/index.php?showid=1100000005514922523
http://factordb.com/index.php?showid=1100000004591654373
http://factordb.com/index.php?showid=1100000004591654467
http://factordb.com/index.php?showid=1100000004591654632
http://factordb.com/index.php?showid=1100000004591654952
http://factordb.com/index.php?query=32%5E%28n%2B2%29-829&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%286*32%5E%28n%2B2%29%2B4241%29%2F31&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%288*32%5E%28n%2B1%29%2B271%29%2F31&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2826*32%5E%28n%2B5%29-390071011%29%2F31&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%288*32%5E%28n%2B1%29-163%29%2F31&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28611*32%5E%28n%2B1%29-53%29%2F31&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2814*32%5E%28n%2B2%29%2B4171%29%2F31&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=11*32%5E%28n%2B2%29%2B395&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28516*32%5E%28n%2B1%29-175%29%2F31&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28643*32%5En-23%29%2F31&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
120100000
33280012~280093DP95093M5
HJ094295J
9093597Q
9F093157N
789449333H
K3807516K
D801079UD
VFU72204FK
J687152BJ
DF068367J
95096
94298
93599
93160
89453
80754
80110
72208
68718
68370
144405
143193
142131
141465
135835
122626
121648
109649
104350
103821
(441×3395095−3833)/32
580×3394296+19
9×3393598+26
312×3393158+23
(7×3389453−4743239)/32
(643×3380753+3709)/32
(13×3380110−121453)/32
(16623×3372206−8095)/16
(19×3368718−600883)/32
444×3368368+19
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http://factordb.com/index.php?id=1100000005652362811&open=prime
http://factordb.com/index.php?id=1100000005652375073&open=prime
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http://factordb.com/index.php?id=1100000005652430746&open=prime
http://factordb.com/index.php?id=1100000005652446200&open=prime
http://factordb.com/index.php?id=1100000005652461592&open=prime
http://factordb.com/index.php?id=1100000004614764298&open=prime
http://factordb.com/index.php?id=1100000004614770536&open=prime
http://factordb.com/index.php?id=1100000004614784274&open=prime
http://factordb.com/index.php?showid=1100000005652348775&base=33
http://factordb.com/index.php?showid=1100000005652362811&base=33
http://factordb.com/index.php?showid=1100000005652375073&base=33
http://factordb.com/index.php?showid=1100000005652389776&base=33
http://factordb.com/index.php?showid=1100000005652430746&base=33
http://factordb.com/index.php?showid=1100000005652446200&base=33
http://factordb.com/index.php?showid=1100000005652461592&base=33
http://factordb.com/index.php?showid=1100000004614764298&base=33
http://factordb.com/index.php?showid=1100000004614770536&base=33
http://factordb.com/index.php?showid=1100000004614784274&base=33
http://factordb.com/index.php?showid=1100000005652348775
http://factordb.com/index.php?showid=1100000005652362811
http://factordb.com/index.php?showid=1100000005652375073
http://factordb.com/index.php?showid=1100000005652389776
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http://factordb.com/index.php?showid=1100000005652446200
http://factordb.com/index.php?showid=1100000005652461592
http://factordb.com/index.php?showid=1100000004614764298
http://factordb.com/index.php?showid=1100000004614770536
http://factordb.com/index.php?showid=1100000004614784274
http://factordb.com/index.php?query=%28441*33%5E%28n%2B2%29-3833%29%2F32&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=580*33%5E%28n%2B1%29%2B19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=9*33%5E%28n%2B1%29%2B26&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=312*33%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%287*33%5E%28n%2B4%29-4743239%29%2F32&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28643*33%5E%28n%2B2%29%2B3709%29%2F32&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2813*33%5E%28n%2B3%29-121453%29%2F32&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2816623*33%5E%28n%2B2%29-8095%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2819*33%5E%28n%2B3%29-600883%29%2F32&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=444*33%5E%28n%2B1%29%2B19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
81100000
34184785~184832GFGC999965
9097950FJ
NM085218KX
F83189H2HP
P79441444P
6770278X
XQIQ72241D
T66530IF
466152B
2EEC660397
100000
97953
85222
83193
79445
77029
72245
66532
66153
66043
153148
150013
130516
127408
121669
117968
110642
101893
101312
101143
(209246×3499997−81)/11
9×3497952+529
804×3485220+713
(5×3483193+700233)/11
(25×3479445−28062367)/33
(2×3477029+1043)/11
(1288676×3472242−455)/33
(29×3466532−12833)/33
(4×3466153+227)/33
(30826×3466040−59)/11
http://factordb.com/index.php?id=1100000004702891268&open=prime
http://factordb.com/index.php?id=1100000004702894713&open=prime
http://factordb.com/index.php?id=1100000004702900996&open=prime
http://factordb.com/index.php?id=1100000004702910376&open=prime
http://factordb.com/index.php?id=1100000004702913746&open=prime
http://factordb.com/index.php?id=1100000004702918600&open=prime
http://factordb.com/index.php?id=1100000004399656529&open=prime
http://factordb.com/index.php?id=1100000004399657696&open=prime
http://factordb.com/index.php?id=1100000004399658651&open=prime
http://factordb.com/index.php?id=1100000004399659716&open=prime
http://factordb.com/index.php?showid=1100000004702891268&base=34
http://factordb.com/index.php?showid=1100000004702894713&base=34
http://factordb.com/index.php?showid=1100000004702900996&base=34
http://factordb.com/index.php?showid=1100000004702910376&base=34
http://factordb.com/index.php?showid=1100000004702913746&base=34
http://factordb.com/index.php?showid=1100000004702918600&base=34
http://factordb.com/index.php?showid=1100000004399656529&base=34
http://factordb.com/index.php?showid=1100000004399657696&base=34
http://factordb.com/index.php?showid=1100000004399658651&base=34
http://factordb.com/index.php?showid=1100000004399659716&base=34
http://factordb.com/index.php?showid=1100000004702891268
http://factordb.com/index.php?showid=1100000004702894713
http://factordb.com/index.php?showid=1100000004702900996
http://factordb.com/index.php?showid=1100000004702910376
http://factordb.com/index.php?showid=1100000004702913746
http://factordb.com/index.php?showid=1100000004702918600
http://factordb.com/index.php?showid=1100000004399656529
http://factordb.com/index.php?showid=1100000004399657696
http://factordb.com/index.php?showid=1100000004399658651
http://factordb.com/index.php?showid=1100000004399659716
http://factordb.com/index.php?query=%28209246*34%5E%28n%2B1%29-81%29%2F11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=9*34%5E%28n%2B2%29%2B529&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=804*34%5E%28n%2B2%29%2B713&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%285*34%5E%28n%2B4%29%2B700233%29%2F11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2825*34%5E%28n%2B4%29-28062367%29%2F33&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%282*34%5E%28n%2B2%29%2B1043%29%2F11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%281288676*34%5E%28n%2B1%29-455%29%2F33&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2829*34%5E%28n%2B2%29-12833%29%2F33&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%284*34%5E%28n%2B1%29%2B227%29%2F33&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2830826*34%5E%28n%2B1%29-59%29%2F11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
47100000
35≥714196(239 linear left families and 134 non-linear left families, totally 373 left families)
3635286~35290P81993SZ
S0750078H
7K26567Z
J10117LJ
VL07258J
EO06177V
FZ57773P
T0946181
RY4562H
OZ3932AZ
81995
75010
26569
10119
7261
6180
5780
4621
4564
3935
127609
116739
41349
15748
11301
9618
8996
7192
7103
6124
(5×3681995+821)/7
28×3675009+305
(53×3626568+101)/7
(19×3610119+2501)/35
1137×367259+19
528×366178+31
16×365779−1163
(36549×364619−289)/35
(979×364563−629)/35
25×363934−901
http://factordb.com/index.php?id=1100000002394962083&open=prime
http://factordb.com/index.php?id=1100000004020085177&open=prime
http://factordb.com/index.php?id=1100000003896952461&open=prime
http://factordb.com/index.php?id=1100000003807362491&open=prime
http://factordb.com/index.php?id=1100000003807362489&open=prime
http://factordb.com/index.php?id=1100000003807362488&open=prime
http://factordb.com/index.php?id=1100000003807362487&open=prime
http://factordb.com/index.php?id=1100000003807362486&open=prime
http://factordb.com/index.php?id=1100000003807362485&open=prime
http://factordb.com/index.php?id=1100000000840634476&open=prime
http://factordb.com/index.php?showid=1100000002394962083&base=36
http://factordb.com/index.php?showid=1100000004020085177&base=36
http://factordb.com/index.php?showid=1100000003896952461&base=36
http://factordb.com/index.php?showid=1100000003807362491&base=36
http://factordb.com/index.php?showid=1100000003807362489&base=36
http://factordb.com/index.php?showid=1100000003807362488&base=36
http://factordb.com/index.php?showid=1100000003807362487&base=36
http://factordb.com/index.php?showid=1100000003807362486&base=36
http://factordb.com/index.php?showid=1100000003807362485&base=36
http://factordb.com/index.php?showid=1100000000840634476&base=36
http://factordb.com/index.php?showid=1100000002394962083
http://factordb.com/index.php?showid=1100000004020085177
http://factordb.com/index.php?showid=1100000003896952461
http://factordb.com/index.php?showid=1100000003807362491
http://factordb.com/index.php?showid=1100000003807362489
http://factordb.com/index.php?showid=1100000003807362488
http://factordb.com/index.php?showid=1100000003807362487
http://factordb.com/index.php?showid=1100000003807362486
http://factordb.com/index.php?showid=1100000003807362485
http://factordb.com/index.php?showid=1100000000840634476
http://factordb.com/index.php?query=%285*36%5E%28n%2B2%29%2B821%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2819*36%5E%28n%2B2%29%2B2501%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=1137*36%5E%28n%2B1%29%2B19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=528*36%5E%28n%2B1%29%2B31&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=16*36%5E%28n%2B2%29-1163&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%2836549*36%5E%28n%2B1%29-289%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=%28979*36%5E%28n%2B1%29-629%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
http://factordb.com/index.php?query=25*36%5E%28n%2B2%29-901&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
4100000

These "unsolved" families in fact have larger (probable) primes (found by other projects), but since the length of these (probable) primes are larger than the search limit of the corresponding bases b, and they are not in the kernel files (nor counted in the condensed table above) since they may not be the next minimal primes in base b, and the indices of these minimal primes in base b are unknown: (since the family {L}G in base b = 27 covers two other unsolved families {L}0G and N9{L}G in base b = 27, thus the family {L}0G only needs to search to length 101106 and the family N9{L}G only needs to search to length 101107)

(for the factorization of the numbers in these families and the N−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) and N+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) of these (probable) primes, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=snfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#smallpolynomial, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=gnfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) may be used, they have SNFS polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt and https://alfredreichlg.de/10w7/10w7.txt and https://web.archive.org/web/20020320010222/http://proth.cjb.net/ and http://web.archive.org/web/20111104173105/http://www.immortaltheory.com/NumberTheory/2nm3_db.txt and https://www.asahi-net.or.jp/~KC2H-MSM/mathland/aba/index.htm and https://www.alpertron.com.ar/BRILLIANT.HTM and https://www.alpertron.com.ar/BRILLIANT3.HTM and https://www.alpertron.com.ar/BRILLIANT4.HTM and https://www.alpertron.com.ar/BRILLIANT2.HTM and https://oeis.org/wiki/Factors_of_33*2%5En%2B1 and https://oeis.org/wiki/Factors_of_33*2%5En-1 and https://web.archive.org/web/20111018190410/http://www.sr5.psp-project.de/s5stats.html (section "Top ten factors:") and https://web.archive.org/web/20111018190339/http://www.sr5.psp-project.de/r5stats.html (section "Top ten factors:") and https://brnikat.com/nums/cullen_woodall/cw.html and https://oeis.org/wiki/OEIS_sequences_needing_factors#Near_powers.2C_factorials.2C_and_primorials (sections "near-powers with b = 2" and "near-powers with b = 3" and "near-powers with b = 5" and "near-powers with b = 6" and "near-powers with b = 7" and "near-powers with b = 10" and "near-powers with b > 10") and https://www.mersenneforum.org/showthread.php?t=9554 and https://www.mersenneforum.org/showthread.php?t=9167 and https://www.mersenneforum.org/showpost.php?p=644144&postcount=5, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm and https://www.mersenneforum.org/showthread.php?t=12962)

(link to the factorization of the first 200 numbers in these "unsolved" families (start with the smallest number in these families which is > b), to show the small prime factors (and their period) (i.e. show whether these numbers are rough numbers) and the algebraic factors (and their period) and let you feel the Nash weight (or difficulty) of these families (high or low), all small prime factors (< 1012, by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172)) and all algebraic factors (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/53335.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/83336.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=algebraic&maxrows=10000, https://sites.google.com/view/algebraic-factors-of-xn-kyn, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://www.mersenneforum.org/showpost.php?p=96560&postcount=99, https://www.mersenneforum.org/showpost.php?p=96651&postcount=101, https://www.mersenneforum.org/showthread.php?t=21916, https://www.mersenneforum.org/showpost.php?p=196598&postcount=492, https://www.mersenneforum.org/showpost.php?p=203083&postcount=149, https://www.mersenneforum.org/showpost.php?p=206065&postcount=192, https://www.mersenneforum.org/showpost.php?p=208044&postcount=260, https://www.mersenneforum.org/showpost.php?p=210533&postcount=336, https://www.mersenneforum.org/showpost.php?p=452132&postcount=66, https://www.mersenneforum.org/showpost.php?p=451337&postcount=32, https://www.mersenneforum.org/showpost.php?p=208852&postcount=227, https://www.mersenneforum.org/showpost.php?p=232904&postcount=604, https://www.mersenneforum.org/showpost.php?p=383690&postcount=1, https://www.mersenneforum.org/showpost.php?p=207886&postcount=253, https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php, https://www.alpertron.com.ar/POLFACT.HTM, https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) of the N−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) and N+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) of these (probable) primes and the first 200 numbers (start with the smallest n making the number > b (if n = 0 already makes the number > b, then start with n = 0)) in corresponding families of these (probable) primes were added to factordb)

b
(2 ≤ b ≤ 36)
familyalgebraic formbase-b form of the smallest (probable) prime in this family (write "dn" if there are 5 or more (n) consecutive same digits d)length of the smallest (probable) prime in this familylength of the smallest (probable) prime in this family in decimalalgebraic ((a×bn+c)/gcd(a+c,b−1)) form of the smallest (probable) prime in this familyfactordb entry of the smallest (probable) prime in this familythe smallest (probable) prime in this family written in base b (use lower case letters instead of upper case letters)the smallest (probable) prime in this family written in decimalfactorization of the first 200 numbers in corresponding family (n is the number of digits in the "{}", start with the smallest n making the number > b (if n = 0 already makes the number > b, then start with n = 0))the entry of the smallest (probable) prime in this family in the top definitely primes page or the top probable primes pagethe process which the smallest (probable) prime in this family is found inreference of searching the (probable) primes in this family
174{9}(73×17n−9)/16
(n ≥ 1)
49111333111334136991(73×17111333−9)/16http://factordb.com/index.php?id=1100000000808118219&open=primehttp://factordb.com/index.php?showid=1100000000808118219&base=17http://factordb.com/index.php?showid=1100000000808118219http://factordb.com/index.php?query=%2873*17%5En-9%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttp://www.primenumbers.net/prptop/searchform.php?form=%2873*17%5E111333-9%29%2F16&action=Searchsolving the original minimal prime problem (i.e. prime > b is not required) in base b = 17https://github.com/curtisbright/mepn-data/blob/master/data/minimal.17.txt
1797{0}1160×17n+1+1
(n ≥ 0)
9701660471166050204316160×17166048+1http://factordb.com/index.php?id=1100000000765961411&open=primehttp://factordb.com/index.php?showid=1100000000765961411&base=17http://factordb.com/index.php?showid=1100000000765961411http://factordb.com/index.php?query=160*17%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttps://t5k.org/primes/page.php?id=85139solving the Sierpinski conjecture in base b = 17http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S17
17F7{0}1262×17n+1+1
(n ≥ 0)
F701867671186770229811262×17186768+1http://factordb.com/index.php?id=1100000000765961429&open=primehttp://factordb.com/index.php?showid=1100000000765961429&base=17http://factordb.com/index.php?showid=1100000000765961429http://factordb.com/index.php?query=262*17%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttps://t5k.org/primes/page.php?id=85256solving the Sierpinski conjecture in base b = 17http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S17
191E7{0}1634×19n+1+1
(n ≥ 0)
1E701228961122899157158634×19122897+1http://factordb.com/index.php?id=1100000001582289581&open=primehttp://factordb.com/index.php?showid=1100000001582289581&base=19http://factordb.com/index.php?showid=1100000001582289581http://factordb.com/index.php?query=634*19%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show(too small for the top primes page)solving the Sierpinski conjecture in base b = 19http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S19
19FG{6}(904×19n−1)/3
(n ≥ 0)
FG6110984110986141924(904×19110984−1)/3http://factordb.com/index.php?id=1100000000808118212&open=primehttp://factordb.com/index.php?showid=1100000000808118212&base=19http://factordb.com/index.php?showid=1100000000808118212http://factordb.com/index.php?query=%28904*19%5En-1%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttp://www.primenumbers.net/prptop/searchform.php?form=%28904*19%5E110984-1%29%2F3&action=Searchsolving the original minimal prime problem (i.e. prime > b is not required) in base b = 19https://github.com/curtisbright/mepn-data/blob/master/data/minimal.19.txt
21C{F}0K(51×21n+2−1243)/4
(n ≥ 0)
CF4791470K479150633542(51×21479149−1243)/4http://factordb.com/index.php?id=1100000000805209046&open=primehttp://factordb.com/index.php?showid=1100000000805209046&base=21http://factordb.com/index.php?showid=1100000000805209046http://factordb.com/index.php?query=%2851*21%5E%28n%2B2%29-1243%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttp://www.primenumbers.net/prptop/searchform.php?form=%2851*21%5E479149-1243%29%2F4&action=Searchsolving the original minimal prime problem (i.e. prime > b is not required) in base b = 21https://github.com/curtisbright/mepn-data/blob/master/data/minimal.21.txt
238{0}18×23n+1+1
(n ≥ 0)
8011921411192161623408×23119215+1http://factordb.com/index.php?id=1100000000634720609&open=primehttp://factordb.com/index.php?showid=1100000000634720609&base=23http://factordb.com/index.php?showid=1100000000634720609http://factordb.com/index.php?query=8*23%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttps://t5k.org/primes/page.php?id=85951solving the Sierpinski conjecture in base b = 23http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S23
239{E}(106×23n−7)/11
(n ≥ 1)
9E8008738008741090573(106×23800873−7)/11http://factordb.com/index.php?id=1100000000782858648&open=primehttp://factordb.com/index.php?showid=1100000000782858648&base=23http://factordb.com/index.php?showid=1100000000782858648http://factordb.com/index.php?query=%28106*23%5En-7%29%2F11&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttp://www.primenumbers.net/prptop/searchform.php?form=%28106*23%5E800873-7%29%2F11&action=Searchsolving the original minimal prime problem (i.e. prime > b is not required) in base b = 23https://github.com/curtisbright/mepn-data/blob/master/data/minimal.23.txt
2571JD{0}1110488×25n+1+1
(n ≥ 0)
71JD04585491458554641031110488×25458550+1http://factordb.com/index.php?id=1100000002341496334&open=primehttp://factordb.com/index.php?showid=1100000002341496334&base=25http://factordb.com/index.php?showid=1100000002341496334http://factordb.com/index.php?query=110488*25%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttps://t5k.org/primes/page.php?id=111834solving the Sierpinski conjecture in base b = 5
(110488×5r+1 can be prime only if r is even, thus can be converted to 110488×25r/2+1)
https://www.primegrid.com/forum_thread.php?id=5087&nowrap=true#66386
259{6}M(37×25n+1+63)/4
(n ≥ 0)
96136965M136967191472(37×25136966+63)/4http://factordb.com/index.php?id=1100000000808118185&open=primehttp://factordb.com/index.php?showid=1100000000808118185&base=25http://factordb.com/index.php?showid=1100000000808118185http://factordb.com/index.php?query=%2837*25%5E%28n%2B1%29%2B63%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttp://www.primenumbers.net/prptop/searchform.php?form=%2837*25%5E136966%2B63%29%2F4&action=Searchsolving the original minimal prime problem (i.e. prime > b is not required) in base b = 25https://github.com/curtisbright/mepn-data/blob/master/data/minimal.25.txt
25DKJ{0}18644×25n+1+1
(n ≥ 0)
DKJ024680812468123450298644×25246809+1http://factordb.com/index.php?id=1100000003983678207&open=primehttp://factordb.com/index.php?showid=1100000003983678207&base=25http://factordb.com/index.php?showid=1100000003983678207http://factordb.com/index.php?query=8644*25%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttps://t5k.org/primes/page.php?id=94113solving the Sierpinski conjecture in base b = 5
(8644×5r+1 can be prime only if r is even, thus can be converted to 8644×25r/2+1)
https://www.mersenneforum.org/showpost.php?p=224809&postcount=24
25M{F}0F6(181×25n+3−75077)/8
(n ≥ 0)
MF1099880F6109992153763(181×25109991−75077)/8http://factordb.com/index.php?id=1100000000808118206&open=primehttp://factordb.com/index.php?showid=1100000000808118206&base=25http://factordb.com/index.php?showid=1100000000808118206http://factordb.com/index.php?query=%28181*25%5E%28n%2B3%29-75077%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttp://www.primenumbers.net/prptop/searchform.php?form=%28181*25%5E109991-75077%29%2F8&action=Searchsolving the original minimal prime problem (i.e. prime > b is not required) in base b = 25https://github.com/curtisbright/mepn-data/blob/master/data/minimal.25.txt
27A{0}PM10×27n+2+697
(n ≥ 0)
A0109003PM10900615602710×27109005+697http://factordb.com/index.php?id=1100000000808118203&open=primehttp://factordb.com/index.php?showid=1100000000808118203&base=27http://factordb.com/index.php?showid=1100000000808118203http://factordb.com/index.php?query=10*27%5E%28n%2B2%29%2B697&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttp://www.primenumbers.net/prptop/searchform.php?form=10*27%5E109005%2B697&action=Searchsolving the original minimal prime problem (i.e. prime > b is not required) in base b = 27https://github.com/curtisbright/mepn-data/blob/master/data/minimal.27.txt
27{L}G(21×27n+1−151)/26
(n ≥ 1)
L101105G101106144720(21×27101106−151)/26http://factordb.com/index.php?id=1100000000808118209&open=primehttp://factordb.com/index.php?showid=1100000000808118209&base=27http://factordb.com/index.php?showid=1100000000808118209http://factordb.com/index.php?query=%2821*27%5E%28n%2B1%29-151%29%2F26&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttp://www.primenumbers.net/prptop/searchform.php?form=%2821*27%5E101106-151%29%2F26&action=Searchsolving the original minimal prime problem (i.e. prime > b is not required) in base b = 27https://github.com/curtisbright/mepn-data/blob/master/data/minimal.27.txt
2947D{0}13580×29n+1+1
(n ≥ 0)
47D013425311342571963373580×29134254+1http://factordb.com/index.php?id=1100000005145366749&open=primehttp://factordb.com/index.php?showid=1100000005145366749&base=29http://factordb.com/index.php?showid=1100000005145366749http://factordb.com/index.php?query=3580*29%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show(too small for the top primes page)solving the Sierpinski conjecture in base b = 841
(3580×29n+1+1 can be prime only if n+1 is even, thus can be converted to 3580×841(n+1)/2+1)
http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S841
29E9{C}(2908×29n−3)/7
(n ≥ 0)
E9C615016150389942(2908×2961501−3)/7http://factordb.com/index.php?id=1100000000808118274&open=primehttp://factordb.com/index.php?showid=1100000000808118274&base=29http://factordb.com/index.php?showid=1100000000808118274http://factordb.com/index.php?query=%282908*29%5En-3%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttp://www.primenumbers.net/prptop/searchform.php?form=%282908*29%5E61501-3%29%2F7&action=Searchsolving the original minimal prime problem (i.e. prime > b is not required) in base b = 29https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt
29{F}IK(15×29n+2+2561)/28
(n ≥ 0)
F57249IK5725183724(15×2957251+2561)/28http://factordb.com/index.php?id=1100000000819229872&open=primehttp://factordb.com/index.php?showid=1100000000819229872&base=29http://factordb.com/index.php?showid=1100000000819229872http://factordb.com/index.php?query=%2815*29%5E%28n%2B2%29%2B2561%29%2F28&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttp://www.primenumbers.net/prptop/searchform.php?form=%2815*29%5E57251%2B2561%29%2F28&action=Searchsolving the original minimal prime problem (i.e. prime > b is not required) in base b = 29https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt
29{I}FK(9×29n+2−1199)/14
(n ≥ 0)
I110725FK110727161927(9×29110727−1199)/14http://factordb.com/index.php?id=1100000000808118200&open=primehttp://factordb.com/index.php?showid=1100000000808118200&base=29http://factordb.com/index.php?showid=1100000000808118200http://factordb.com/index.php?query=%289*29%5E%28n%2B2%29-1199%29%2F14&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttp://www.primenumbers.net/prptop/searchform.php?form=%289*29%5E110727-1199%29%2F14&action=Searchsolving the original minimal prime problem (i.e. prime > b is not required) in base b = 29https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt
29{L}CE(3×29n+2−1075)/4
(n ≥ 0)
L72087CE72089105423(3×2972089−1075)/4http://factordb.com/index.php?id=1100000000808118245&open=primehttp://factordb.com/index.php?showid=1100000000808118245&base=29http://factordb.com/index.php?showid=1100000000808118245http://factordb.com/index.php?query=%283*29%5E%28n%2B2%29-1075%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttp://www.primenumbers.net/prptop/searchform.php?form=%283*29%5E72089-1075%29%2F4&action=Searchsolving the original minimal prime problem (i.e. prime > b is not required) in base b = 29https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt
29LO{G}(4435×29n−4)/7
(n ≥ 0)
LOG660816608396640(4435×2966081−4)/7http://factordb.com/index.php?id=1100000000808118256&open=primehttp://factordb.com/index.php?showid=1100000000808118256&base=29http://factordb.com/index.php?showid=1100000000808118256http://factordb.com/index.php?query=%284435*29%5En-4%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttp://www.primenumbers.net/prptop/searchform.php?form=%284435*29%5E66081-4%29%2F7&action=Searchsolving the original minimal prime problem (i.e. prime > b is not required) in base b = 29https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt
29LP{C}9(4441×29n+1−24)/7
(n ≥ 0)
LPC75151975154109905(4441×2975152−24)/7http://factordb.com/index.php?id=1100000000808118236&open=primehttp://factordb.com/index.php?showid=1100000000808118236&base=29http://factordb.com/index.php?showid=1100000000808118236http://factordb.com/index.php?query=%284441*29%5E%28n%2B1%29-24%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttp://www.primenumbers.net/prptop/searchform.php?form=%284441*29%5E75152-24%29%2F7&action=Searchsolving the original minimal prime problem (i.e. prime > b is not required) in base b = 29https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt
29O{0}FPL24×29n+3+13361
(n ≥ 0)
O0174236FPL17424025480924×29174239+13361http://factordb.com/index.php?id=1100000000808118178&open=primehttp://factordb.com/index.php?showid=1100000000808118178&base=29http://factordb.com/index.php?showid=1100000000808118178http://factordb.com/index.php?query=24*29%5E%28n%2B3%29%2B13361&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttp://www.primenumbers.net/prptop/searchform.php?form=24*29%5E174239%2B13361&action=Searchsolving the original minimal prime problem (i.e. prime > b is not required) in base b = 29https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt
29P{0}OPR25×29n+3+20936
(n ≥ 0)
P059144OPR591488649825×2959147+20936http://factordb.com/index.php?id=1100000000808118282&open=primehttp://factordb.com/index.php?showid=1100000000808118282&base=29http://factordb.com/index.php?showid=1100000000808118282http://factordb.com/index.php?query=25*29%5E%28n%2B3%29%2B20936&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttp://www.primenumbers.net/prptop/searchform.php?form=25*29%5E59147%2B20936&action=Searchsolving the original minimal prime problem (i.e. prime > b is not required) in base b = 29https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt
29PC{L}(2951×29n−3)/4
(n ≥ 0)
PCL123418123420180490(2951×29123418−3)/4http://factordb.com/index.php?id=1100000000808118196&open=primehttp://factordb.com/index.php?showid=1100000000808118196&base=29http://factordb.com/index.php?showid=1100000000808118196http://factordb.com/index.php?query=%282951*29%5En-3%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttp://www.primenumbers.net/prptop/searchform.php?form=%282951*29%5E123418-3%29%2F4&action=Searchsolving the original minimal prime problem (i.e. prime > b is not required) in base b = 29https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt
29P{L}C(103×29n+1−39)/4
(n ≥ 0)
PL54178C5418079233(103×2954179−39)/4http://factordb.com/index.php?id=1100000000819229869&open=primehttp://factordb.com/index.php?showid=1100000000819229869&base=29http://factordb.com/index.php?showid=1100000000819229869http://factordb.com/index.php?query=%28103*29%5E%28n%2B1%29-39%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttp://www.primenumbers.net/prptop/searchform.php?form=%28103*29%5E54179-39%29%2F4&action=Searchsolving the original minimal prime problem (i.e. prime > b is not required) in base b = 29https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt
29PO{L}0O(2999×29n+2−2427)/4
(n ≥ 0)
POL645540O6455894410(2999×2964556−2427)/4http://factordb.com/index.php?id=1100000000808118259&open=primehttp://factordb.com/index.php?showid=1100000000808118259&base=29http://factordb.com/index.php?showid=1100000000808118259http://factordb.com/index.php?query=%282999*29%5E%28n%2B2%29-2427%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttp://www.primenumbers.net/prptop/searchform.php?form=%282999*29%5E64556-2427%29%2F4&action=Searchsolving the original minimal prime problem (i.e. prime > b is not required) in base b = 29https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt
29Q9R{0}922154×29n+1+9
(n ≥ 0)
Q9R0520499520537612322154×2952050+9http://factordb.com/index.php?id=1100000000819229865&open=primehttp://factordb.com/index.php?showid=1100000000819229865&base=29http://factordb.com/index.php?showid=1100000000819229865http://factordb.com/index.php?query=22154*29%5E%28n%2B1%29%2B9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttp://www.primenumbers.net/prptop/searchform.php?form=22154*29%5E52050%2B9&action=Searchsolving the original minimal prime problem (i.e. prime > b is not required) in base b = 29https://github.com/curtisbright/mepn-data/blob/master/data/minimal.29.txt
32NU{0}1766×32n+1+1
(n ≥ 0)
NU06618631661866996208766×32661864+1http://factordb.com/index.php?id=1100000003813355148&open=primehttp://factordb.com/index.php?showid=1100000003813355148&base=32http://factordb.com/index.php?showid=1100000003813355148http://factordb.com/index.php?query=766*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttps://t5k.org/primes/page.php?id=134216finding the Proth primes k×2r+1 for k = 383http://www.prothsearch.com/riesel1a.html
3309321 is the smallest exponent r == 1 mod 5 (≥ 6) for k = 383
(since 766×32n+1+1 = 383×25×(n+1)+1+1, thus we need an exponent r == 1 mod 5 for 383×2r+1, i.e. the Proth number for k = 383, and since n ≥ 0, 5×(n+1)+1 must be ≥ 5×1+1 = 6)
33N7{0}1766×33n+1+1
(n ≥ 0)
N706104111610414926923766×33610412+1http://factordb.com/index.php?id=1100000000838755581&open=primehttp://factordb.com/index.php?showid=1100000000838755581&base=33http://factordb.com/index.php?showid=1100000000838755581http://factordb.com/index.php?query=766*33%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showhttps://t5k.org/primes/page.php?id=121575solving the Sierpinski conjecture in base b = 33http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S33
351B{0}146×35n+1+1
(n ≥ 0)
1B0560611560648656646×3556062+1http://factordb.com/index.php?id=1100000000885460611&open=primehttp://factordb.com/index.php?showid=1100000000885460611&base=35http://factordb.com/index.php?showid=1100000000885460611http://factordb.com/index.php?query=46*35%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show(too small for the top primes page)solving the Sierpinski conjecture in base b = 35http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S35

These unsolved families in fact have larger search limit of lengths (searched by other projects) than the search limit of the corresponding bases b:

(for the factorization of the numbers in these families and the N−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) and N+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) of these (probable) primes, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=snfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#smallpolynomial, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=gnfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/) may be used, they have SNFS polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt and https://alfredreichlg.de/10w7/10w7.txt and https://web.archive.org/web/20020320010222/http://proth.cjb.net/ and http://web.archive.org/web/20111104173105/http://www.immortaltheory.com/NumberTheory/2nm3_db.txt and https://www.asahi-net.or.jp/~KC2H-MSM/mathland/aba/index.htm and https://www.alpertron.com.ar/BRILLIANT.HTM and https://www.alpertron.com.ar/BRILLIANT3.HTM and https://www.alpertron.com.ar/BRILLIANT4.HTM and https://www.alpertron.com.ar/BRILLIANT2.HTM and https://oeis.org/wiki/Factors_of_33*2%5En%2B1 and https://oeis.org/wiki/Factors_of_33*2%5En-1 and https://web.archive.org/web/20111018190410/http://www.sr5.psp-project.de/s5stats.html (section "Top ten factors:") and https://web.archive.org/web/20111018190339/http://www.sr5.psp-project.de/r5stats.html (section "Top ten factors:") and https://brnikat.com/nums/cullen_woodall/cw.html and https://oeis.org/wiki/OEIS_sequences_needing_factors#Near_powers.2C_factorials.2C_and_primorials (sections "near-powers with b = 2" and "near-powers with b = 3" and "near-powers with b = 5" and "near-powers with b = 6" and "near-powers with b = 7" and "near-powers with b = 10" and "near-powers with b > 10") and https://www.mersenneforum.org/showthread.php?t=9554 and https://www.mersenneforum.org/showthread.php?t=9167 and https://www.mersenneforum.org/showpost.php?p=644144&postcount=5, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm and https://www.mersenneforum.org/showthread.php?t=12962)

(link to the factorization of the first 200 numbers in these unsolved families (start with the smallest number in these families which is > b), to show the small prime factors (and their period) (i.e. show whether these numbers are rough numbers) and the algebraic factors (and their period) and let you feel the Nash weight (or difficulty) of these families (high or low), all small prime factors (< 1012, by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172)) and all algebraic factors (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/53335.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/83336.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=algebraic&maxrows=10000, https://sites.google.com/view/algebraic-factors-of-xn-kyn, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://www.mersenneforum.org/showpost.php?p=96560&postcount=99, https://www.mersenneforum.org/showpost.php?p=96651&postcount=101, https://www.mersenneforum.org/showthread.php?t=21916, https://www.mersenneforum.org/showpost.php?p=196598&postcount=492, https://www.mersenneforum.org/showpost.php?p=203083&postcount=149, https://www.mersenneforum.org/showpost.php?p=206065&postcount=192, https://www.mersenneforum.org/showpost.php?p=208044&postcount=260, https://www.mersenneforum.org/showpost.php?p=210533&postcount=336, https://www.mersenneforum.org/showpost.php?p=452132&postcount=66, https://www.mersenneforum.org/showpost.php?p=451337&postcount=32, https://www.mersenneforum.org/showpost.php?p=208852&postcount=227, https://www.mersenneforum.org/showpost.php?p=232904&postcount=604, https://www.mersenneforum.org/showpost.php?p=383690&postcount=1, https://www.mersenneforum.org/showpost.php?p=207886&postcount=253, https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php, https://www.alpertron.com.ar/POLFACT.HTM, https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) of the N−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) and N+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) of these (probable) primes and the first 200 numbers (start with the smallest n making the number > b (if n = 0 already makes the number > b, then start with n = 0)) in corresponding families of these (probable) primes were added to factordb)

b
(2 ≤ b ≤ 36)
familyalgebraic formtrue search limit of length for this familyfactorization of the first 200 numbers in this family (n is the number of digits in the "{}", start with the smallest n making the number > b (if n = 0 already makes the number > b, then start with n = 0))the process of searching the (probable) primes in this familyreference of searching the (probable) primes in this family
17F1{9}(4105×17n−9)/16
(n ≥ 0)
1000000http://factordb.com/index.php?query=%284105*17%5En-9%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showsolving the original minimal prime problem (i.e. prime > b is not required) in base b = 17https://github.com/curtisbright/mepn-data/blob/master/data/sieve.17.txt
19EE1{6}(15964×19n−1)/3
(n ≥ 0)
707350http://factordb.com/index.php?query=%2815964*19%5En-1%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showsolving the original minimal prime problem (i.e. prime > b is not required) in base b = 19https://github.com/curtisbright/mepn-data/blob/master/data/sieve.19.txt
21G{0}FK16×21n+2+335
(n ≥ 0)
506722http://factordb.com/index.php?query=16*21%5E%28n%2B2%29%2B335&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showsolving the original minimal prime problem (i.e. prime > b is not required) in base b = 21https://github.com/curtisbright/mepn-data/blob/master/data/sieve.21.txt
23H3{0}1394×23n+1+1
(n ≥ 0)
800000http://factordb.com/index.php?query=394*23%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showsolving the Sierpinski conjecture in base b = 529
(394×23n+1+1 can be prime only if n+1 is even, thus can be converted to 394×529(n+1)/2+1)
http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S529
23JH{0}1454×23n+1+1
(n ≥ 0)
800000http://factordb.com/index.php?query=454*23%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showsolving the Sierpinski conjecture in base b = 529
(454×23n+1+1 can be prime only if n+1 is even, thus can be converted to 454×529(n+1)/2+1)
http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm#S529
25D71J{0}1207544×25n+1+1
(n ≥ 0)
600000http://factordb.com/index.php?query=207544*25%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showsolving the Sierpinski conjecture in base b = 25http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base25-reserve.htm
25EF{O}366×25n−1
(n ≥ 0)
600000http://factordb.com/index.php?query=366*25%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showsolving the Riesel conjecture in base b = 25http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base25-reserve.htm
31{F}G(31n+1+1)/2
(n ≥ 1)
16777215http://factordb.com/index.php?query=%2831%5E%28n%2B1%29%2B1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showfinding the generalized half Fermat primes (b2r+1)/2 in base b = 31
((bn+1+1)/2 can be prime only if n+1 is power of 2, thus can be converted to (b2r+1)/2)
for n+1 ≤ 219 = 524288 see http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt, and the numbers with n+1 = 220 = 1048576, 221 = 2097152, 222 = 4194304, 223 = 8388608 are divisible by 37339791361, 5138022401, 998244353, 2121143418881, respectively (these four prime factors are found by trial division), but the number with n+1 = 224 = 16777216 has no prime factor < 248 (also use trial division to show this), thus the smallest prime candidate is n+1 = 224 = 16777216, and its length is 16777216
324{0}14×32n+1+1
(n ≥ 0)
1717986918http://factordb.com/index.php?query=4*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showfinding the Fermat primes 22r+1
(4×32n+1+1 = 2n+7+1, and 2n+7+1 can be prime only if 5×n+7 is a power of 2, thus can be converted to 22r+1)
http://www.prothsearch.com/fermat.html, 2r == 2 mod 5 if and only if r == 1 mod 4, and the smallest r == 1 mod 4 (and r > 2) such that 22r+1 may be prime is r = 33, and thus the smallest possible prime is n+1 = (233−2)/5 = 1717986918, and its length is 1717986919 (since 4×32n+1+1 = 25×(n+1)+2+1, thus we need an exponent s == 2 mod 5 for 2s+1 = 22r+1 (if 2s+1 is prime, then s is a power of 2, thus we can let s = 2r), and 2r == 2 mod 5 if and only if r == 1 mod 4, and since n ≥ 0, 5×(n+1)+2 must be ≥ 5×1+2 = 7, thus r = log2(5×(n+1)+2) must be > 2)
32G{0}116×32n+1+1
(n ≥ 0)
3435973836http://factordb.com/index.php?query=16*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showfinding the Fermat primes 22r+1
(16×32n+1+1 = 2n+9+1, and 2n+9+1 can be prime only if 5×n+9 is a power of 2, thus can be converted to 22r+1)
http://www.prothsearch.com/fermat.html, 2r == 4 mod 5 if and only if r == 2 mod 4, and the smallest r == 2 mod 4 (and r > 3) such that 22r+1 may be prime is r = 34, and thus the smallest possible prime is n+1 = (234−4)/5 = 3435973836, and its length is 3435973837 (since 16×32n+1+1 = 25×(n+1)+4+1, thus we need an exponent s == 4 mod 5 for 2s+1 = 22r+1 (if 2s+1 is prime, then s is a power of 2, thus we can let s = 2r), and 2r == 4 mod 5 if and only if r == 2 mod 4, and since n ≥ 0, 5×(n+1)+4 must be ≥ 5×1+4 = 9, thus r = log2(5×(n+1)+4) must be > 3)
32NG{0}1752×32n+1+1
(n ≥ 0)
1800000http://factordb.com/index.php?query=752*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showfinding the Proth primes k×2r+1 for k = 47http://www.prothsearch.com/riesel1.html
k = 47 is searched to exponent 9000000 with no exponent == 4 mod 5 (≥ 9) has been found
(since 752×32n+1+1 = 47×25×(n+1)+4+1, thus we need an exponent r == 4 mod 5 for 47×2r+1, i.e. the Proth number for k = 47, and since n ≥ 0, 5×(n+1)+4 must be ≥ 5×1+4 = 9)
32UG{0}1976×32n+1+1
(n ≥ 0)
1000000http://factordb.com/index.php?query=976*32%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showfinding the Proth primes k×2r+1 for k = 61http://www.prothsearch.com/riesel1.html
k = 61 is searched to exponent 5000000 with no exponent == 4 mod 5 (≥ 9) has been found
(since 976×32n+1+1 = 61×25×(n+1)+4+1, thus we need an exponent r == 4 mod 5 for 61×2r+1, i.e. the Proth number for k = 61, and since n ≥ 0, 5×(n+1)+4 must be ≥ 5×1+4 = 9)
32S{V}29×32n−1
(n ≥ 1)
2000000http://factordb.com/index.php?query=29*32%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Showsolving the Riesel conjecture in base b = 1024
(29×32n−1 can be prime only if n is even, thus can be converted to 29×1024n/2−1)
http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm#R1024
(note: the composite number SV1745576 = 29×321745576−1 was submitted to top definitely primes in 2016 (it is a false prime and definitely composite) and had been deleted from top definitely primes, see https://web.archive.org/web/20211102171448/http://primes.utm.edu/primes/page.php?id=122375&deleted=1 and https://www.mersenneforum.org/showpost.php?p=445097&postcount=539 and https://www.mersenneforum.org/showpost.php?p=445118&postcount=543 and https://www.mersenneforum.org/showpost.php?p=445137&postcount=559 and https://www.mersenneforum.org/showpost.php?p=445153&postcount=561)

Also, the minimal primes in the families in the table "This problem covers finding the smallest prime in these families in the same base b (or proving that such prime does not exist), since the smallest prime in these families (if exists) must be a minimal prime in base b" in this "README" file for bases 2 ≤ b ≤ 36 and their lengths (in base b) and their indices in the set of the minimal primes base b are (according to this "README" file, the smallest prime in these families (if exists) must be a minimal prime in base b):

b
(2 ≤ b ≤ 36)
(bn−1)/(b−1)
(n ≥ 2)
bn+1
(n ≥ 1)
(bn+1)/2
(odd b, n ≥ 2)
bn+1
(b ≥ 3, n ≥ 1)
bn−1
(b ≥ 3, n ≥ 1)
bn+2
(b ≥ 3, n ≥ 1)
bn−2
(b ≥ 3, n ≥ 2)
bn+1
(b ≥ 4, n ≥ 1)
bn−1
(b ≥ 4, n ≥ 1)
bn+3
(b ≥ 4, n ≥ 1)
bn−3
(b ≥ 4, n ≥ 2)
bn+1
(b ≥ 5, n ≥ 1)
bn−1
(b ≥ 5, n ≥ 1)
bn+4
(b ≥ 5, n ≥ 1)
bn−4
(b ≥ 5, n ≥ 2)
bn+1
(b ≥ 6, n ≥ 1)
bn−1
(b ≥ 6, n ≥ 1)
bn+1
(b ≥ 7, n ≥ 1)
bn−1
(b ≥ 7, n ≥ 1)
bn+1
(b ≥ 8, n ≥ 1)
bn−1
(b ≥ 8, n ≥ 1)
bn+1
(b ≥ 9, n ≥ 1)
bn−1
(b ≥ 9, n ≥ 1)
bn+1
(b ≥ 10, n ≥ 1)
bn−1
(b ≥ 10, n ≥ 1)
10×bn+1
(b ≥ 11, n ≥ 1)
10×bn−1
(b ≥ 11, n ≥ 1)
11×bn+1
(b ≥ 12, n ≥ 1)
11×bn−1
(b ≥ 12, n ≥ 1)
12×bn+1
(b ≥ 13, n ≥ 1)
12×bn−1
(b ≥ 13, n ≥ 1)
(b−1)×bn+1
(n ≥ 1)
(b−1)×bn−1
(n ≥ 1)
bn+(b−1)
(n ≥ 1)
bn−(b−1)
(n ≥ 2)
corresponding form
({x} or x{y} or {x}y or x{0}y)
{x}x{0}y{x}yx{0}yx{y}x{0}y{x}yx{0}yx{y}x{0}y{x}yx{0}yx{y}x{0}y{x}yx{0}yx{y}x{0}yx{y}x{0}yx{y}x{0}yx{y}x{0}yx{y}x{0}yx{y}x{0}yx{y}x{0}yx{y}x{0}yx{y}x{0}y{x}y
the value of x11(b−1)/2211b−1321b−1431b−1546576879810911101211b−1b−21b−1
the value of y1(b+1)/21b−12b−21b−13b−31b−14b−41b−11b−11b−11b−11b−11b−11b−11b−11b−1b−11
211 = 3 (length 2, index 1)11 = 3 (length 2, index 1)11 = 3 (length 2, index 1)11 = 3 (length 2, index 1)11 = 3 (length 2, index 1)11 = 3 (length 2, index 1)
3111 = 13 (length 3, index 3)always divisible by 212 = 5 (length 2, index 1)21 = 7 (length 2, index 2)12 = 5 (length 2, index 1)12 = 5 (length 2, index 1)21 = 7 (length 2, index 2)21 = 7 (length 2, index 2)12 = 5 (length 2, index 1)12 = 5 (length 2, index 1)21 = 7 (length 2, index 2)
411 = 5 (length 2, index 1)11 = 5 (length 2, index 1)always divisible by 313 = 7 (length 2, index 2)always divisible by 2always divisible by 231 = 13 (length 2, index 4)23 = 11 (length 2, index 3)13 = 7 (length 2, index 2)31 = 13 (length 2, index 4)31 = 13 (length 2, index 4)23 = 11 (length 2, index 3)13 = 7 (length 2, index 2)31 = 13 (length 2, index 4)
5111 = 31 (length 3, index 8)always divisible by 223 = 13 (length 2, index 3)21 = 11 (length 2, index 2)14444 = 1249 (length 5, index 16)12 = 7 (length 2, index 1)43 = 23 (length 2, index 6)always divisible by 2
(in fact, always divisible by 4)
always divisible by 2always divisible by 2
(in fact, always divisible by 4)
always divisible by 2401 = 101 (length 3, index 12)34 = 19 (length 2, index 5)104 = 29 (length 3, index 7)44441 = 3121 (length 5, index 20)401 = 101 (length 3, index 12)34 = 19 (length 2, index 5)104 = 29 (length 3, index 7)44441 = 3121 (length 5, index 20)
611 = 7 (length 2, index 1)11 = 7 (length 2, index 1)21 = 13 (length 2, index 3)15 = 11 (length 2, index 2)always divisible by 2always divisible by 231 = 19 (length 2, index 5)25 = 17 (length 2, index 4)always divisible by 3always divisible by 3always divisible by 535 = 23 (length 2, index 6)always divisible by 2
(in fact, always divisible by 4 for n ≥ 2)
always divisible by 2
(in fact, always divisible by 4)
51 = 31 (length 2, index 8)45 = 29 (length 2, index 7)51 = 31 (length 2, index 8)45 = 29 (length 2, index 7)15 = 11 (length 2, index 2)51 = 31 (length 2, index 8)
711111 = 2801 (length 5, index 53)always divisible by 23334 = 1201 (length 4, index 41)always divisible by 316 = 13 (length 2, index 2)always divisible by 365 = 47 (length 2, index 11)always divisible by 2always divisible by 2always divisible by 2always divisible by 241 = 29 (length 2, index 6)always divisible by 314 = 11 (length 2, index 1)always divisible by 3always divisible by 2
(in fact, always divisible by 6)
always divisible by 2always divisible by 2
(in fact, always divisible by 6)
always divisible by 261 = 43 (length 2, index 10)56 = 41 (length 2, index 9)61 = 43 (length 2, index 10)56 = 41 (length 2, index 9)
8111 = 73 (length 3, index 16)sum-of-two-cubes factorization
8n+1 = (2n+1) × (4n−2n+1)
21 = 17 (length 2, index 3)177 = 127 (length 3, index 21)always divisible by 2always divisible by 2301 = 193 (length 3, index 24)27 = 23 (length 2, index 5)13 = 11 (length 2, index 1)75 = 61 (length 2, index 14)401 = 257 (length 3, index 27)37 = 31 (length 2, index 7)always divisible by 2
(in fact, always divisible by 4)
always divisible by 2
(in fact, always divisible by 4)
51 = 41 (length 2, index 9)47777 = 20479 (length 5, index 58)always divisible by 757 = 47 (length 2, index 11)701 = 449 (length 3, index 39)6777 = 3583 (length 4, index 55)701 = 449 (length 3, index 39)6777 = 3583 (length 4, index 55)107 = 71 (length 3, index 15)7121 = 813−7 (length 13, index 73)
9difference-of-two-squares factorization
(9n−1)/8 = (3n−1) × (3n+1) / 8
always divisible by 245 = 41 (length 2, index 9)21 = 19 (length 2, index 4)18 = 17 (length 2, index 3)12 = 11 (length 2, index 1)87 = 79 (length 2, index 18)always divisible by 2
(in fact, always divisible by 4)
always divisible by 2always divisible by 2
(in fact, always divisible by 12)
always divisible by 2
(in fact, always divisible by 6)
41 = 37 (length 2, index 8)difference-of-two-squares factorization
4×9n−1 = (2×3n−1) × (2×3n+1)
14 = 13 (length 2, index 2)difference-of-two-squares factorization
9n−4 = (3n−2) × (3n+2)
always divisible by 2always divisible by 2
(in fact, always divisible by 4)
601 = 487 (length 3, index 44)58 = 53 (length 2, index 12)always divisible by 2
(in fact, always divisible by 8)
always divisible by 281 = 73 (length 2, index 17)78 = 71 (length 2, index 16)81 = 73 (length 2, index 17)78 = 71 (length 2, index 16)18 = 17 (length 2, index 3)81 = 73 (length 2, index 17)
1011 = 11 (length 2, index 1)11 = 11 (length 2, index 1)always divisible by 319 = 19 (length 2, index 4)always divisible by 2
(in fact, always divisible by 6)
always divisible by 231 = 31 (length 2, index 7)29 = 29 (length 2, index 6)13 = 13 (length 2, index 2)97 = 97 (length 2, index 21)41 = 41 (length 2, index 9)always divisible by 3always divisible by 2
(in fact, always divisible by 4 for n ≥ 2)
always divisible by 2
(in fact, always divisible by 12)
always divisible by 3499 = 499 (length 3, index 30)61 = 61 (length 2, index 14)59 = 59 (length 2, index 13)71 = 71 (length 2, index 16)always divisible by 3always divisible by 3
(in fact, always divisible by 9)
79 = 79 (length 2, index 18)9001 = 9001 (length 4, index 56)89 = 89 (length 2, index 20)9001 = 9001 (length 4, index 56)89 = 89 (length 2, index 20)19 = 19 (length 2, index 4)991 = 991 (length 3, index 44)
11117 = (1117−1)/10 (length 17, index 975)always divisible by 256 = 61 (length 2, index 13)21 = 23 (length 2, index 4)1AA = 241 (length 3, index 37)12 = 13 (length 2, index 1)AAA9 = 14639 (length 4, index 227)always divisible by 2always divisible by 2always divisible by 2always divisible by 2always divisible by 53A = 43 (length 2, index 9)always divisible by 5AA7 = 1327 (length 3, index 112)always divisible by 2
(in fact, always divisible by some element of {22,2×3})
divisible by 2×3 if n is even, divisible by 22 if n is odd
always divisible by 2
(in fact, always divisible by some element of {22,2×3})
divisible by 22 if n is even, divisible by 2×3 if n is odd
61 = 67 (length 2, index 14)always divisible by 5always divisible by 2
(in fact, always divisible by some element of {22,2×3})
divisible by 22 if n is even, divisible by 2×3 if n is odd
always divisible by 2
(in fact, always divisible by some element of {22,2×3})
divisible by 2×3 if n is even, divisible by 22 if n is odd
81 = 89 (length 2, index 19)7AA = 967 (length 3, index 88)always divisible by 2
(in fact, always divisible by 10)
always divisible by 2A091 = 10×1110+1 (length 11, index 905)9A = 109 (length 2, index 24)A091 = 10×1110+1 (length 11, index 905)9A = 109 (length 2, index 24)10A = 131 (length 3, index 26)AA1 = 1321 (length 3, index 111)
1211 = 13 (length 2, index 1)11 = 13 (length 2, index 1)2001 = 3456 (length 4, index 58)1B = 23 (length 2, index 4)always divisible by 2always divisible by 231 = 37 (length 2, index 7)2BB = 431 (length 3, index 34)always divisible by 3always divisible by 3401 = 577 (length 3, index 35)3B = 47 (length 2, index 10)always divisible by 2
(in fact, always divisible by 4)
always divisible by 2
(in fact, always divisible by 4)
51 = 61 (length 2, index 13)4B = 59 (length 2, index 12)61 = 73 (length 2, index 16)5B = 71 (length 2, index 15)701 = 1009 (length 3, index 43)6B = 83 (length 2, index 18)81 = 97 (length 2, index 20)7BB = 1151 (length 3, index 51)91 = 109 (length 2, index 24)8B = 107 (length 2, index 23)always divisible by 119BB = 1439 (length 3, index 54)B001 = 19009 (length 4, index 84)AB = 131 (length 2, index 27)B001 = 19009 (length 4, index 84)AB = 131 (length 2, index 27)1B = 23 (length 2, index 4)BBBB1 = 248821 (length 5, index 97)
1311111 = 30941 (length 5, index 494)always divisible by 26667 = 14281 (length 4, index 305)always divisible by 31CC = 337 (length 3, index 48)always divisible by 3CB = 167 (length 2, index 33)always divisible by 2
(in fact, always divisible by 4)
always divisible by 2always divisible by 2
(in fact, always divisible by 4)
always divisible by 241 = 53 (length 2, index 10)always divisible by 314 = 17 (length 2, index 1)always divisible by 3always divisible by 2
(in fact, always divisible by 6)
always divisible by 2
(in fact, always divisible by 4)
61 = 79 (length 2, index 16)5CC = 1013 (length 3, index 101)always divisible by 2
(in fact, always divisible by 4)
always divisible by 2
(in fact, always divisible by 6)
always divisible by 37C = 103 (length 2, index 21)always divisible by 2always divisible by 2
(in fact, always divisible by 4)
A1 = 131 (length 2, index 26)always divisible by 3always divisible by 2
(in fact, always divisible by 12)
always divisible by 2C1 = 157 (length 2, index 31)BCC = 2027 (length 3, index 176)C1 = 157 (length 2, index 31)BCC = 2027 (length 3, index 176)10C = 181 (length 3, index 34)CB = 167 (length 2, index 33)
14111 = 211 (length 3, index 40)101 = 197 (length 3, index 39)21 = 29 (length 2, index 4)1DDDD = 76831 (length 5, index 233)always divisible by 2always divisible by 231 = 43 (length 2, index 8)2D = 41 (length 2, index 7)13 = 17 (length 2, index 1)DB = 193 (length 2, index 38)always divisible by some element of {3,5}
divisible by 3 if n is odd, divisible by 5 if n is even
always divisible by some element of {3,5}
divisible by 3 if n is even, divisible by 5 if n is odd
always divisible by 2
(in fact, always divisible by 4 for n ≥ 2)
(in fact, also always divisible by some element of {22×3,22×5} for n ≥ 2)
divisible by 22×3 if n ≥ 2 is odd, divisible by 22×5 if n ≥ 2 is even
always divisible by 2
(in fact, always divisible by 4)
(in fact, also always divisible by some element of {22×3,22×5})
divisible by 22×3 if n is even, divisible by 22×5 if n is odd
51 = 71 (length 2, index 14)4D19698 = 5×1419698−1 (length 19699, index 650)6051 = 6×146+1 (length 7, index 468)5D = 83 (length 2, index 17)701 = 1373 (length 3, index 78)6D = 97 (length 2, index 19)81 = 113 (length 2, index 24)7DD = 1567 (length 3, index 83)91 = 127 (length 2, index 25)combine of factor 5 and difference-of-two-squares factorization
divisible by 5 if n is odd, 9×14n−1 = (3×14n/2−1) × (3×14n/2+1) if n is even
A051 = 10×146+1 (length 7, index 488)9D = 139 (length 2, index 28)always divisible by some element of {3,5}
divisible by 3 if n is even, divisible by 5 if n is odd
always divisible by some element of {3,5}
divisible by 3 if n is odd, divisible by 5 if n is even
always divisible by 13BD = 167 (length 2, index 33)D01 = 2549 (length 3, index 120)CD = 181 (length 2, index 36)1015D = 1416+13 (length 17, index 606)DD1 = 2731 (length 3, index 131)
15111 = 241 (length 3, index 43)always divisible by 278 = 113 (length 2, index 24)21 = 31 (length 2, index 5)1E = 29 (length 2, index 4)12 = 17 (length 2, index 1)ED = 223 (length 2, index 42)always divisible by 2always divisible by 2always divisible by 2
(in fact, always divisible by 6)
always divisible by 2
(in fact, always divisible by 6)
41 = 61 (length 2, index 12)3E = 59 (length 2, index 11)14 = 19 (length 2, index 2)EEB = 3371 (length 3, index 175)always divisible by 2always divisible by 2always divisible by 75E = 89 (length 2, index 18)always divisible by 2always divisible by 2801 = 1801 (length 3, index 118)always divisible by 7always divisible by 2always divisible by 2A1 = 151 (length 2, index 30)9E = 149 (length 2, index 29)always divisible by 2always divisible by 2C1 = 181 (length 2, index 36)BE = 179 (length 2, index 35)E1 = 211 (length 2, index 41)DE14 = 14×1514−1 (length 15, index 1252)1E = 29 (length 2, index 4)E1 = 211 (length 2, index 41)
1611 = 17 (length 2, index 1)11 = 17 (length 2, index 1)always divisible by 31F = 31 (length 2, index 5)always divisible by 2always divisible by 2301 = 769 (length 3, index 69)2F = 47 (length 2, index 9)13 = 19 (length 2, index 2)FFD = 4093 (length 3, index 197)always divisible by 5
(in fact, also Aurifeuillean factorization of x4+4×y4)
4×16n+1 = (2×4n−2×2n+1) × (2×4n+2×2n+1)
always divisible by 3
(in fact, also difference-of-two-squares factorization
4×16n−1 = (2×4n−1) × (2×4n+1)
always divisible by 2
(in fact, always divisible by 20)
(in fact, also Aurifeuillean factorization of x4+4×y4)
16n+4 = (4n−2×2n+2) × (4n+2×2n+2)
always divisible by 2
(in fact, always divisible by 12)
(in fact, also difference-of-two-squares factorization
16n−4 = (4n−2) × (4n+2)
always divisible by 34F = 79 (length 2, index 16)61 = 97 (length 2, index 19)always divisible by 571 = 113 (length 2, index 24)always divisible by 3always divisible by 37F = 127 (length 2, index 25)always divisible by 5difference-of-two-squares factorization
9×16n−1 = (3×4n−1) × (3×4n+1)
A001 = 40961 (length 4, index 364)always divisible by 3always divisible by 3always divisible by 5C1 = 193 (length 2, index 38)BF = 191 (length 2, index 37)F1 = 241 (length 2, index 47)EF = 239 (length 2, index 46)1F = 31 (length 2, index 5)F1 = 241 (length 2, index 47)
17111 = 307 (length 3, index 56)always divisible by 28889 = 41761 (length 4, index 811)20461 = 2×1747+1 (length 48, index 10094)1GG = 577 (length 3, index 86)12 = 19 (length 2, index 1)GGGGGF = 24137567 (length 6, index 4999)always divisible by 2
(in fact, always divisible by 4)
always divisible by 2always divisible by 2
(in fact, always divisible by 4)
always divisible by 24000001 = 96550277 (length 7, index 5138)3G = 67 (length 2, index 12)104 = 293 (length 3, index 55)GGD = 4909 (length 3, index 378)always divisible by 2always divisible by 2
(in fact, always divisible by 4)
61 = 103 (length 2, index 20)5G = 101 (length 2, index 19)always divisible by 2
(in fact, always divisible by 8)
always divisible by 281 = 137 (length 2, index 26)7GG = 2311 (length 3, index 209)always divisible by 2always divisible by 2
(in fact, always divisible by 8)
A013551 = 10×171356+1 (length 1357, index 10370)9G117 = 10×17117−1 (length 118, index 10239)always divisible by 2
(in fact, always divisible by 4)
always divisible by 2C01 = 3469 (length 3, index 292)BGG = 3467 (length 3, index 291)G0001 = 1336337 (length 5, index 3039)FG = 271 (length 2, index 51)1000G = 83537 (length 5, index 1348)G10F = 1711−16 (length 11, index 8834)
1811 = 19 (length 2, index 1)11 = 19 (length 2, index 1)21 = 37 (length 2, index 5)1HH = 647 (length 3, index 66)always divisible by 2always divisible by 23001 = 17497 (length 4, index 195)2H = 53 (length 2, index 9)always divisible by 3always divisible by 341 = 73 (length 2, index 14)3H = 71 (length 2, index 13)always divisible by 2
(in fact, always divisible by 4 for n ≥ 2)
always divisible by 2
(in fact, always divisible by 4)
501 = 1621 (length 3, index 83)4H = 89 (length 2, index 17)61 = 109 (length 2, index 22)5H = 107 (length 2, index 21)71 = 127 (length 2, index 24)6HH = 2267 (length 3, index 99)801 = 2593 (length 3, index 105)7HH = 2591 (length 3, index 104)91 = 163 (length 2, index 31)8H11 = 9×1811−1 (length 12, index 532)A1 = 181 (length 2, index 35)9H = 179 (length 2, index 34)B1 = 199 (length 2, index 39)AH = 197 (length 2, index 38)C01 = 3889 (length 3, index 135)BH8 = 12×188−1 (length 9, index 517)H1 = 307 (length 2, index 56)GHH = 5507 (length 3, index 178)100H = 5849 (length 4, index 185)H1 = 307 (length 2, index 56)
19119 = (1919−1)/18 (length 19, index 29382)always divisible by 29A = 181 (length 2, index 34)always divisible by 31I = 37 (length 2, index 4)always divisible by 3IH = 359 (length 2, index 64)always divisible by 2always divisible by 2always divisible by 2always divisible by 24001 = 27437 (length 4, index 748)always divisible by 3
(in fact, also combine of factor 5 and difference-of-two-squares factorization)
divisible by 5 if n is odd, 4×19n−1 = (2×19n/2−1) × (2×19n/2+1) if n is even
14 = 23 (length 2, index 1)always divisible by 3
(in fact, also combine of factor 5 and difference-of-two-squares factorization)
divisible by 5 if n is odd, 19n−4 = (19n/2−2) × (19n/2+2) if n is even
always divisible by 2
(in fact, always divisible by 6)
always divisible by 260131 = 6×1914+1 (length 15, index 28464)5I = 113 (length 2, index 22)always divisible by 2always divisible by 2
(in fact, always divisible by 6)
always divisible by 3
(in fact, always divisible by 9)
7I = 151 (length 2, index 28)always divisible by 2always divisible by 2
(in fact, also combine of factor 5 and difference-of-two-squares factorization)
divisible by 5 if n is odd, 9×19n−1 = (3×19n/2−1) × (3×19n/2+1) if n is even
A1 = 191 (length 2, index 35)always divisible by 3
(in fact, always divisible by 9)
always divisible by 2
(in fact, always divisible by 6)
always divisible by 2C1 = 229 (length 2, index 42)BI = 227 (length 2, index 41)I0281 = 18×1929+1 (length 30, index 30322)HI6 = 18×196−1 (length 7, index 17286)1I = 37 (length 2, index 4)II1 = 6841 (length 3, index 496)
20111 = 421 (length 3, index 73)101 = 401 (length 3, index 71)21 = 41 (length 2, index 5)1J10 = 2×2010−1 (length 11, index 3015)always divisible by 2always divisible by 231 = 61 (length 2, index 10)2J = 59 (length 2, index 9)13 = 23 (length 2, index 1)JH = 397 (length 2, index 70)401 = 1601 (length 3, index 120)3J = 79 (length 2, index 14)always divisible by 2
(in fact, always divisible by 4)
always divisible by 2
(in fact, always divisible by 4)
51 = 101 (length 2, index 18)4JJ = 1999 (length 3, index 145)60141 = 6×2015+1 (length 16, index 3165)5JJ = 2399 (length 3, index 157)701 = 2801 (length 3, index 162)6J = 139 (length 2, index 26)always divisible by some element of {3,7}
divisible by 3 if n is even, divisible by 7 if n is odd
always divisible by some element of {3,7}
divisible by 3 if n is odd, divisible by 7 if n is even
91 = 181 (length 2, index 34)8J = 179 (length 2, index 33)A01 = 4001 (length 3, index 200)9J = 199 (length 2, index 38)B001 = 88001 (length 4, index 572)AJ8 = 11×208−1 (length 9, index 2844)C1 = 241 (length 2, index 45)BJ = 239 (length 2, index 44)J0131 = 19×2014+1 (length 15, index 3160)IJ = 379 (length 2, index 67)10J = 419 (length 3, index 72)J161 = 2017−19 (length 17, index 3185)
21111 = 463 (length 3, index 78)always divisible by 2AAAB = 97241 (length 4, index 941)21 = 43 (length 2, index 6)1K = 41 (length 2, index 5)12 = 23 (length 2, index 1)KJ = 439 (length 2, index 77)always divisible by 2
(in fact, always divisible by 4)
always divisible by 2always divisible by 2
(in fact, always divisible by 12)
always divisible by 2
(in fact, always divisible by 6)
always divisible by 53K = 83 (length 2, index 15)always divisible by 5KKH = 9257 (length 3, index 501)always divisible by 2always divisible by 2
(in fact, always divisible by 4)
61 = 127 (length 2, index 23)always divisible by 5always divisible by 2
(in fact, always divisible by 4)
always divisible by 2801 = 3529 (length 3, index 239)7K = 167 (length 2, index 31)always divisible by 2
(in fact, always divisible by 10)
always divisible by 2
(in fact, always divisible by 4)
A1 = 211 (length 2, index 39)9KK = 4409 (length 3, index 288)always divisible by 2
(in fact, always divisible by 4)
always divisible by 2
(in fact, always divisible by 10)
C091 = 12×2110+1 (length 11, index 11383)BK = 251 (length 2, index 46)K1 = 421 (length 2, index 74)JK = 419 (length 2, index 73)1K = 41 (length 2, index 5)K1 = 421 (length 2, index 74)
2211 = 23 (length 2, index 1)11 = 23 (length 2, index 1)always divisible by 31L = 43 (length 2, index 6)always divisible by 2
(in fact, always divisible by 6)
always divisible by 231 = 67 (length 2, index 11)2LL = 1451 (length 3, index 127)103 = 487 (length 3, index 85)L5J = 226−3 (length 6, index 4633)41 = 89 (length 2, index 16)always divisible by 3always divisible by 2
(in fact, always divisible by 4 for n ≥ 2)
always divisible by 2
(in fact, always divisible by 12)
always divisible by 34L = 109 (length 2, index 21)always divisible by 75L = 131 (length 2, index 24)701 = 3389 (length 3, index 184)always divisible by 3always divisible by 3always divisible by 791 = 199 (length 2, index 38)8L = 197 (length 2, index 37)A0141 = 10×2215+1 (length 16, index 7653)always divisible by 3always divisible by 3AL = 241 (length 2, index 45)C0001 = 2811073 (length 5, index 2186)BL = 263 (length 2, index 48)L1 = 463 (length 2, index 82)KL = 461 (length 2, index 81)1L = 43 (length 2, index 6)L1 = 463 (length 2, index 82)

Also, if we write all minimal primes base b, then we will write these numbers of each digits: (the bases b = 11, 13, 16, 17, 19, 21~23, 25~36 data assumes the primality of the probable primes) (This data assumes that a number > 1025000 which has passed the Miller–Rabin primality tests (https://t5k.org/prove/prove2_3.html, https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test, https://en.wikipedia.org/wiki/Strong_pseudoprime, https://t5k.org/glossary/xpage/MillersTest.html, https://t5k.org/glossary/xpage/StrongPRP.html, https://www.rieselprime.de/ziki/Miller-Rabin_pseudoprimality_test, https://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html, https://mathworld.wolfram.com/StrongPseudoprime.html, http://www.numericana.com/answer/pseudo.htm#rabin, http://www.numericana.com/answer/pseudo.htm#strong, http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm, http://ntheory.org/data/spsps.txt, https://faculty.lynchburg.edu/~nicely/misc/mpzspsp.html, https://miller-rabin.appspot.com/, http://www.pi-e.de/Miller-Rabin-Pseudoprimzahlen.htm (in German), http://factordb.com/prooffailed.php, https://sites.google.com/view/strong-pseudoprime, https://sites.google.com/view/bases-strong-pseudoprime, https://oeis.org/A001262, https://oeis.org/A020229, https://oeis.org/A020230, https://oeis.org/A020231, https://oeis.org/A020232, https://oeis.org/A020233, https://oeis.org/A020234, https://oeis.org/A020235, https://oeis.org/A020236, https://oeis.org/A020237, https://oeis.org/A020238, https://oeis.org/A020239, https://oeis.org/A020240, https://oeis.org/A020241, https://oeis.org/A020242, https://oeis.org/A020243, https://oeis.org/A020244, https://oeis.org/A020245, https://oeis.org/A020246, https://oeis.org/A020247, https://oeis.org/A020248, https://oeis.org/A020249, https://oeis.org/A020250, https://oeis.org/A020251, https://oeis.org/A020252, https://oeis.org/A020253, https://oeis.org/A020254, https://oeis.org/A020255, https://oeis.org/A020256, https://oeis.org/A020257, https://oeis.org/A020258, https://oeis.org/A020259, https://oeis.org/A020260, https://oeis.org/A020261, https://oeis.org/A020262, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773, https://oeis.org/A014233, https://oeis.org/A006945, https://oeis.org/A089825, https://oeis.org/A181782, https://oeis.org/A071294, https://oeis.org/A141768, https://oeis.org/A195328, https://oeis.org/A329759, https://oeis.org/A298756) to all prime bases p < 64 and has passed the Baillie–PSW primality test (https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test, https://mathworld.wolfram.com/Baillie-PSWPrimalityTest.html, http://pseudoprime.com/dopo.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_318.pdf)) and has trial factored (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) to 1016 is in fact prime, since in some cases (e.g. b = 11) a candidate for minimal prime base b is too large to be proven prime rigorously, this candidate for minimal prime base 11 has 65263 decimal digits, while the top record ordinary prime (https://t5k.org/glossary/xpage/OrdinaryPrime.html) (i.e. neither N−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1) nor N+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) can be ≥ 1/4 factored (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm)) has 86453 decimal digits (the entry of this prime in top definitely primes is https://t5k.org/primes/page.php?id=136044), see https://t5k.org/top20/page.php?id=27 and https://t5k.org/primes/search.php?Comment=ECPP&OnList=all&Number=1000000&Style=HTML and http://factordb.com/certoverview.php?digits=300&perpage=1000&skip=0&descending=on)

(some digits of the unsolved bases b (i.e. the bases b = 17, 19, 21, 23, 25 ≤ b ≤ 29, 31 ≤ b ≤ 36) only have the greatest lower bound (https://en.wikipedia.org/wiki/Greatest_lower_bound, https://mathworld.wolfram.com/GreatestLowerBound.html) of the numbers of the digits, and some (but not all) of these digits (such digits are exactly the digits which do not appear in the y of any left family x{y}z (where x and z are strings (may be empty) of digits in base b, y is a digit in base b)) also have the least upper bound (https://en.wikipedia.org/wiki/Least_upper_bound, https://mathworld.wolfram.com/LeastUpperBound.html) of the numbers of the digits, and these digits also have the exact numbers of the digits (such digits are exactly the digits which do not appear in the y of any left family x{y}z (where x and z are strings (may be empty) of digits in base b, y is a digit in base b) neither appear in the x or z of any left family which is covered by another left family (e.g. the base 19 left family 5{H}05 is covered by another base 19 left family 5{H}5, thus the base 19 left family 5{H}05 may contain no minimal primes)) or a larger greatest lower bound (https://en.wikipedia.org/wiki/Greatest_lower_bound, https://mathworld.wolfram.com/GreatestLowerBound.html) of the numbers of the digits (such digits are exactly the digits which do not appear in the y of any left family x{y}z (where x and z are strings (may be empty) of digits in base b, y is a digit in base b) but appear in the x or z of some left families which are covered by other left families (e.g. the base 19 left family 5{H}05 is covered by another base 19 left family 5{H}5, thus the base 19 left family 5{H}05 may contain no minimal primes)) when assume the heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, https://web.archive.org/web/20230911032453/https://www.utm.edu/staff/caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf), https://arxiv.org/pdf/2103.04483.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_113.pdf)) that all left families have a prime (these exact numbers are the numbers in the brackets in the table below), this is reasonable, references: https://t5k.org/mersenne/heuristic.html, https://t5k.org/notes/faq/NextMersenne.html, https://t5k.org/glossary/xpage/Repunit.html, https://web.archive.org/web/20100628035147/http://www.math.niu.edu/~rusin/known-math/98/exp_primes, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, https://mathoverflow.net/questions/268918/density-of-primes-in-sequences-of-the-form-anb, https://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Mathematics/2023_September_25#Are_there_infinitely_many_primes_of_the_form_1000%E2%80%A60007,_333%E2%80%A63331,_7111%E2%80%A6111,_or_3444%E2%80%A64447_in_base_10?, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf), https://www.mersenneforum.org/showthread.php?t=12327, https://oeis.org/A234285 (the comment by Farideh Firoozbakht, although this comment is not true, there is no prime for s = 509203 and s = −78557, s = 509203 has a covering set of {3, 5, 7, 13, 17, 241}, and s = −78557 has a covering set of {3, 5, 7, 13, 19, 37, 73}), https://www.mersenneforum.org/showpost.php?p=564786&postcount=3, https://www.mersenneforum.org/showpost.php?p=461665&postcount=7, https://www.mersenneforum.org/showpost.php?p=354505&postcount=5, https://www.mersenneforum.org/showpost.php?p=344985&postcount=293, https://www.mersenneforum.org/showpost.php?p=625978&postcount=1027, https://www.primegrid.com/forum_thread.php?id=5093&nowrap=true#66471, https://www.primegrid.com/forum_thread.php?id=4935&nowrap=true#63813, also the graphs https://t5k.org/gifs/lg_lg_Mn.gif (for the family {1} in base b = 2) and https://t5k.org/gifs/repunits.gif (for the family {1} in base b = 10) and https://www.mersenneforum.org/attachment.php?attachmentid=4010&d=1642088235 (for the family 2{0}1 in base b = 3), also, for the digits which appear in the y of some left families x{y}z (where x and z are strings (may be empty) of digits in base b, y is a digit in base b), they have no least upper bound (https://en.wikipedia.org/wiki/Least_upper_bound, https://mathworld.wolfram.com/LeastUpperBound.html) of the numbers of the digits, but have a larger greatest lower bound (https://en.wikipedia.org/wiki/Greatest_lower_bound, https://mathworld.wolfram.com/GreatestLowerBound.html) of the numbers of the digits, this greatest lower bound (when assume the heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, https://web.archive.org/web/20230911032453/https://www.utm.edu/staff/caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf), https://arxiv.org/pdf/2103.04483.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_113.pdf)) that all left families have a prime) is calculated by the searching limit of length for the left families (e.g. if the searching limit of length for a left family is 100000, this means that the smallest prime in this family has length ≥100001))

b
(2 ≤ b ≤ 36)
total number of digitsthe number of digits 0the number of digits 1the number of digits 2the number of digits 3the number of digits 4the number of digits 5the number of digits 6the number of digits 7the number of digits 8the number of digits 9the number of digits Athe number of digits Bthe number of digits Cthe number of digits Dthe number of digits Ethe number of digits Fthe number of digits Gthe number of digits Hthe number of digits Ithe number of digits Jthe number of digits Kthe number of digits Lthe number of digits Mthe number of digits Nthe number of digits Othe number of digits Pthe number of digits Qthe number of digits Rthe number of digits Sthe number of digits Tthe number of digits Uthe number of digits Vthe number of digits Wthe number of digits Xthe number of digits Ythe number of digits Z
2202
37052
4110533
51691052442214
629392285
72884373216724519
852322491324283482559
930041350174321082410735779458
103106933279194521312234
117541426665232272507221514251669173575921395
1243310542284252314434383869
139214876652512194785940741001225813759442385356751339706712261
1425404847289451864882933186273826149827419620909
1582861644373101493165492386983375698229816165689677
16278146128313693881166807461065762583127031520174151194577313486916185
17≥582405
(≥2382423)
≥196456
(≥896443)
2395~2401
(2401)
14793256~3257
(3257)
10059~10061
(10061)
≥37679
(≥137680)
7235≥66029
(≥266032)
1604~1605
(1605)
≥11607
(≥211612)
11164~11166
(11166)
≥138279
(≥438280)
3352≥65689
(≥265690)
10456~10459
(10459)
≥13027
(≥113032)
2639
18101067089128409439155148212869513715415129371107207900

Links for top (probable) primes: (also pages for the largest known prime: https://en.wikipedia.org/wiki/Largest_known_prime_number, https://en.wikipedia.org/wiki/List_of_largest_known_primes_and_probable_primes, http://www.numericana.com/answer/primes.htm#history, and related pages: https://web.archive.org/web/20210416051711/https://en.wikipedia.org/wiki/Titanic_prime, https://web.archive.org/web/20210219091535/https://en.wikipedia.org/wiki/Gigantic_prime, https://en.wikipedia.org/wiki/Megaprime, https://t5k.org/glossary/xpage/TitanicPrime.html, https://t5k.org/glossary/xpage/GiganticPrime.html, https://t5k.org/glossary/xpage/Megaprime.html, https://www.rieselprime.de/ziki/Titanic_prime, https://www.rieselprime.de/ziki/Gigantic_prime, https://www.rieselprime.de/ziki/Megaprime, https://www.rieselprime.de/ziki/Gigaprime, https://mathworld.wolfram.com/TitanicPrime.html, https://mathworld.wolfram.com/GiganticPrime.html, titanic primes are the primes with ≥ 1000 decimal digits, gigantic primes are the primes with ≥ 10000 decimal digits, megaprimes are the primes with ≥ 1000000 decimal digits, bevaprimes (also called gigaprimes) are the primes with ≥ 1000000000 decimal digits)

and the search result page for the top (probable) primes of special forms: (note: a large prime of the form (a×bn+c)/d with small a, b, c, d and large n can be easily proven prime if and only if c = ±1 and d = 1)

(Note: The top definitely primes page converts the perfect power (https://oeis.org/A001597, https://en.wikipedia.org/wiki/Perfect_power, https://mathworld.wolfram.com/PerfectPower.html, https://www.numbersaplenty.com/set/perfect_power/) bases (i.e. b = mr with r > 1) to their "ground bases" (https://oeis.org/A052410) (i.e. b = m), i.e. the bases are normalized, e.g. it converts the prime 2805222×252805222+1 to 2805222×55610444+1 (i.e. converts base 25 = 52 to base 5) (see https://t5k.org/primes/page.php?id=129893 for the entry of this prime in the top definitely primes page), and it converts the prime 2622×121810960−1 to 2622×111621920−1 (i.e. converts base 121 = 112 to base 11) (see https://t5k.org/primes/page.php?id=119929 for the entry of this prime in the top definitely primes page), thus do not search the perfect power bases in the top definitely primes page (otherwise, you will find nothing!), see https://www.mersenneforum.org/showpost.php?p=121374&postcount=1 and https://www.mersenneforum.org/showpost.php?p=656659&postcount=1, however, unlike the top definitely primes page, the top probable primes page does not convert, e.g. it has the probable primes (161025393+1)/17 (see http://www.primenumbers.net/prptop/searchform.php?form=%2816%5E1025393%2B1%29%2F17&action=Search for the entry of this probable prime in the top probable primes page) and (9860029+1)/10 (see http://www.primenumbers.net/prptop/searchform.php?form=%289%5E860029%2B1%29%2F10&action=Search for the entry of this probable prime in the top probable primes page), and does not convert base 16 = 24 to base 2 and and does not convert base 9 = 32 to base 3, thus of course you can search the perfect power bases in the top probable primes page (unlike the top definitely primes page))

Definitely primes (i.e. c = ±1 and d = 1):

Probable primes (i.e. c ≠ ±1 or/and d ≠ 1):

Index pages of searching the primes of the form (a×bn+c)/gcd(a+c,b−1):

  1. https://www.primegrid.com/
  2. http://www.noprimeleftbehind.net/stats/
  3. http://www.noprimeleftbehind.net/crus/
  4. https://web.archive.org/web/20230706153536/http://www.noprimeleftbehind.net/gary/
  5. http://www.prothsearch.com/
  6. https://web.archive.org/web/20210817181915/http://www.15k.org/
  7. https://web.archive.org/web/20091027064415/http://www.geocities.com/primes_r_us/riesel/index.html
  8. https://web.archive.org/web/20231223043356/https://www.mersenneforum.org/321search/index.html
  9. https://web.archive.org/web/20110601231527/http://www.bodang.com/12121/
  10. https://www.rieselprime.de/default.htm
  11. https://www.rieselprime.de/Related/ (subpages not available in this page, the subpages are https://www.rieselprime.de/Related/RieselTwinSG.htm and https://www.rieselprime.de/Related/LiskovetsGallot.htm)
  12. http://www.fermatquotient.com/
  13. https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html
  14. http://www.primenumbers.net/
  15. https://pzktupel.de/ktuplets.php
  16. http://harvey563.tripod.com/
  17. http://guenter.loeh.name/
  18. http://www.fermatsearch.org/
  19. http://jeppesn.dk/
  20. https://web.archive.org/web/20231002190634/http://yves.gallot.pagesperso-orange.fr/primes/index.html
  21. http://boincvm.proxyma.ru:30080/test4vm/index.php
  22. http://prpnet.primegrid.com/
  23. https://stdkmd.net/ (sections https://stdkmd.net/nrr/prime/ and https://stdkmd.net/nrr/cert/ and https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#primenumbers and https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#probableprimenumbers)
  24. https://kurtbeschorner.de/ (section "Rprime project; base 10")
  25. http://www.elektrosoft.it/matematica/repunit/repunit.htm
  26. https://www.mersenne.org/
  27. https://www.mersenne.ca/
  28. https://web.archive.org/web/20211011120227/http://mprime.s3-website.us-west-1.amazonaws.com/
  29. http://kenta.blogspot.com/2012/11/ezgxggdm-dual-sierpinski-problem.html
  30. https://oeis.org/A076336/a076336c.html (special situation: in theory this reference should be http://web.archive.org/web/20080908010544/http://sierpinski.insider.com/dual, but this page is excluded from the Wayback Machine, thus linked to the cached copy page)
  31. http://www.doublemersennes.org/
  32. http://www.hoegge.dk/mersenne/NMC.html

Index pages of factoring the numbers of the form (a×bn+c)/gcd(a+c,b−1):

  1. https://homes.cerias.purdue.edu/~ssw/cun/index.html
  2. https://maths-people.anu.edu.au/~brent/factors.html
  3. http://myfactors.mooo.com/
  4. https://web.archive.org/web/20190315214330/http://cage.ugent.be/~jdemeyer/cunningham/
  5. https://www.mersenne.org/
  6. https://www.mersenne.ca/
  7. https://mers.sourceforge.io/mersenne.html
  8. http://www.fermatsearch.org/
  9. http://www.prothsearch.com/ (section "Fermat numbers")
  10. https://web.archive.org/web/20211011120227/http://mprime.s3-website.us-west-1.amazonaws.com/
  11. http://www.doublemersennes.org/
  12. https://web.archive.org/web/20231126215849/http://bearnol.is-a-geek.com/Mersenneplustwo/Mersenneplustwo.html (warning: currently it is still unknown whether the number 22127−1−1 is prime, in fact, many subpages in https://web.archive.org/web/20151025022357/http://www.bearnol.pwp.blueyonder.co.uk/Math/ (including https://web.archive.org/web/20151025022406/http://www.bearnol.pwp.blueyonder.co.uk/Math/mmprime.html) are only hoaxes (although some subpages are not), e.g. Riemann hypothesis (https://en.wikipedia.org/wiki/Riemann_hypothesis, https://t5k.org/glossary/xpage/RiemannHypothesis.html, https://mathworld.wolfram.com/RiemannHypothesis.html, http://www.numericana.com/answer/open.htm#rh) and abc conjecture (https://en.wikipedia.org/wiki/Abc_conjecture, https://mathworld.wolfram.com/abcConjecture.html, http://www.numericana.com/answer/open.htm#abc) and Collatz conjecture (https://en.wikipedia.org/wiki/Collatz_conjecture, https://mathworld.wolfram.com/CollatzProblem.html, http://www.numericana.com/answer/open.htm#collatz, http://www.ericr.nl/wondrous/index.html, http://www.ericr.nl/wondrous/delrecs.html, http://www.ericr.nl/wondrous/pathrecs.html, http://www.ericr.nl/wondrous/comprecs.html, http://www.ericr.nl/wondrous/glidrecs.html, http://www.ericr.nl/wondrous/residues.html, http://www.ericr.nl/wondrous/classrec.html, http://www.ericr.nl/wondrous/strengths.html, http://www.ericr.nl/wondrous/progress.html, http://www.ericr.nl/wondrous/techpage.html, http://www.ericr.nl/wondrous/showsteps.html, https://pcbarina.fit.vutbr.cz/, https://pcbarina.fit.vutbr.cz/path-records.htm, http://sweet.ua.pt/tos/3x_plus_1.html, http://www.rechenkraft.net/yoyo/y_status_col.php, https://oeis.org/A006370, https://oeis.org/A070165, https://oeis.org/A006577, https://oeis.org/A006877, https://oeis.org/A006878, https://oeis.org/A025586, https://oeis.org/A006884, https://oeis.org/A006885) and Beal's conjecture (https://en.wikipedia.org/wiki/Beal_conjecture, https://t5k.org/glossary/xpage/BealsConjecture.html, https://mathworld.wolfram.com/BealsConjecture.html) and Goldbach's conjecture (https://en.wikipedia.org/wiki/Goldbach%27s_conjecture, https://t5k.org/glossary/xpage/GoldbachConjecture.html, https://mathworld.wolfram.com/GoldbachConjecture.html, http://www.numericana.com/answer/open.htm#goldbach, https://oeis.org/A045917, https://oeis.org/A035026, https://oeis.org/A002375, https://oeis.org/A002372) and twin prime conjecture (https://en.wikipedia.org/wiki/Twin_prime_conjecture, https://t5k.org/glossary/xpage/TwinPrimeConjecture.html, https://mathworld.wolfram.com/TwinPrimeConjecture.html, http://www.numericana.com/answer/open.htm#twin, https://oeis.org/A001097, https://oeis.org/A077800, https://oeis.org/A001359, https://oeis.org/A006512) are currently still unsolved problems, but this page has false proofs of Riemann hypothesis and Collatz conjecture and Beal's conjecture and twin prime conjecture, and has false disproofs of abc conjecture and Goldbach's conjecture, also, the Catalan-Mersenne number conjecture (https://en.wikipedia.org/wiki/Catalan%E2%80%93Mersenne_number_conjecture, https://mathworld.wolfram.com/Catalan-MersenneNumber.html, https://t5k.org/mersenne/index.html#c, https://oeis.org/A007013) is currently still an unsolved problem, it is still unknown whether the number 22127−1−1 is prime, and the New Mersenne conjecture (https://en.wikipedia.org/wiki/New_Mersenne_conjecture, https://t5k.org/glossary/xpage/NewMersenneConjecture.html, https://mathworld.wolfram.com/NewMersennePrimeConjecture.html, https://t5k.org/mersenne/NewMersenneConjecture.html, http://www.hoegge.dk/mersenne/NMC.html, http://www.primenumbers.net/rl/nmc/, https://web.archive.org/web/20010809112857/http://orca.st.usm.edu/~cwcurry/NMC.html, https://web.archive.org/web/20211128180850/http://mprime.s3-website.us-west-1.amazonaws.com/new_mersenne_conjecture.html, https://oeis.org/A107360, https://oeis.org/A122834, https://oeis.org/A120334) is currently also still an unsolved problem, however, it is known that the Catalan-Mersenne number conjecture and the New Mersenne conjecture is not both true, i.e. at least one of them is false, since the prime 886407410000361345663448535540258622490179142922169401 divides the number (22127−1+1)/3 (this prime factor is found by trial division, all prime factors of (22127−1+1)/3 and all prime factors of 22127−1−1 are == 1 mod 2×(2127−1)), see https://www.mersenneforum.org/showpost.php?p=549829&postcount=1 and https://t5k.org/curios/cpage/40578.html)
  13. https://web.archive.org/web/20130530210800/http://www.euronet.nl/users/bota/medium-p.htm
  14. https://stdkmd.net/
  15. https://kurtbeschorner.de/
  16. https://repunit-koide.jimdofree.com/
  17. https://gmplib.org/~tege/repunit.html
  18. https://web.archive.org/web/20070708171301/http://www.ludwigsgymnasium.de/unterr/mathe/prim/zehnp.htm
  19. http://chesswanks.com/pxp/repfactors.html
  20. https://alfredreichlg.de/
  21. https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html
  22. http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/index.htm
  23. http://mklasson.com/factors/index.php
  24. https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt
  25. https://web.archive.org/web/20020320010222/http://proth.cjb.net/
  26. https://brnikat.com/nums/index.html
  27. https://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/index.htm
  28. https://www.lirmm.fr/~ochem/opn/
  29. https://web.archive.org/web/20120426061657/http://oddperfect.org/
  30. https://web.archive.org/web/20190211112446/http://home.earthlink.net/~elevensmooth/
  31. https://2721.hddkillers.com/
  32. https://web.archive.org/web/20081006071311/http://www-staff.maths.uts.edu.au/~rons/fact/fact.htm
  33. https://homes.cerias.purdue.edu/~ssw/bell/index.html
  34. http://www.loria.fr/~zimmerma/ecmnet/ (finding the factors of the numbers)
  35. https://www.rechenkraft.net/yoyo/ (finding the factors of the numbers)
  36. http://escatter11.fullerton.edu/nfs/ (finding the factors of the numbers)

OEIS sequences for the exponents n for the primes in given families in given base b (only list those for families {x}, x{y}, {x}y, x{0}y (where x and y are any digits in base b), since the smallest prime in these families must be minimal primes in the same base b (except the cases that the repeating digit (i.e. y in x{y}, or x in {x}y) is 1), also only count the primes > b):

base b = 2:

base b = 3:

base b = 4:

base b = 5:

base b = 6:

base b = 7:

base b = 8:

base b = 9:

base b = 10:

base b = 11:

base b = 12:

base b = 13:

base b = 14:

base b = 15:

base b = 16:

base b = 17:

base b = 18:

base b = 19:

base b = 20:

base b = 21:

base b = 22:

base b = 23:

base b = 24:

base b = 25:

base b = 26:

base b = 27:

base b = 28:

base b = 29:

base b = 30:

base b = 31:

base b = 32:

(none of these sequences is currently in OEIS)

base b = 33:

base b = 34:

base b = 35:

base b = 36:

References of minimal primes (https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics), https://t5k.org/glossary/xpage/MinimalPrime.html) (the original definition, i.e. prime > base (b) is not required):

  1. http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_11.pdf) (base b = 10)
  2. https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_17.pdf) (bases 2 ≤ b ≤ 30)
  3. https://cs.uwaterloo.ca/~shallit/Papers/br10.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_18.pdf) (bases 2 ≤ b ≤ 30)
  4. https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_19.pdf) (bases 2 ≤ b ≤ 30)
  5. https://doi.org/10.1080/10586458.2015.1064048 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_20.pdf) (bases 2 ≤ b ≤ 30)
  6. https://scholar.colorado.edu/downloads/hh63sw661 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_16.pdf) (bases 2 ≤ b ≤ 10) (warning: the datas for bases b = 8 and b = 10 have errors, the data for base b = 8 misses the prime 6101 and the data for base b = 10 misses the primes 9001 and 9049 and instead wrongly includes the primes 90001, 90469, and 9000049, and the correct values of Sm for bases b = 8 and b = 10 are 15 and 26 (instead of 14 and 27), respectively, also, the "number of minimal primes base b" and the "length of the largest minimal prime base b" are not the same sizes of b but the same sizes of eγ×(b−1)×eulerphi(b), this article has this error is because it only search bases 2 ≤ b ≤ 10, and for the data of 2 ≤ b ≤ 10 for the original minimal problem, you may think that they are the same sizes of b (however, if you extend the data to 2 ≤ b ≤ 16, then you will know that they are not the same sizes of b), since bases b = 7 and b = 9 have very large differences of the "number of minimal primes base b" between the original minimal problem and this new minimal prime problem (b = 7: 9 v.s. 71, b = 9: 12 v.s. 151), and bases b = 5 and b = 8 and b = 9 have very large differences of the "length of the largest minimal prime base b" between the original minimal problem and this new minimal prime problem (b = 5: 5 v.s. 96, b = 8: 9 v.s. 221, b = 9: 4 v.s. 1161))
  7. https://github.com/curtisbright/mepn-data (bases 2 ≤ b ≤ 30)
  8. https://github.com/curtisbright/mepn (bases 2 ≤ b ≤ 30)
  9. https://github.com/RaymondDevillers/primes (bases 28 ≤ b ≤ 50)
  10. http://recursed.blogspot.com/2006/12/prime-game.html (base b = 10)
  11. https://inzitan.blogspot.com/2007/07/prime-game.html (in Spain) (base b = 10)
  12. http://www.pourlascience.fr/ewb_pages/a/article-nombres-premiers-inevitables-et-pyramidaux-24978.php (in French) (base b = 10)
  13. http://villemin.gerard.free.fr/aNombre/TYPMULTI/PremInev.htm (base b = 10)
  14. https://schoolbag.info/mathematics/numbers/66.html (base b = 10)
  15. https://www.microsiervos.com/archivo/ciencia/2-3-5-7-11.html (in Spain) (base b = 10)
  16. https://math.stackexchange.com/questions/58292/how-to-check-if-an-integer-has-a-prime-number-in-it (base b = 10)
  17. https://www.metafilter.com/62794/3-is-an-odd-prime-5-is-an-odd-prime-7-is-an-odd-prime-9-is-a-very-odd-prime (base b = 10)
  18. https://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1165031124 (base b = 10)
  19. https://www.cristal.univ-lille.fr/profil/jdelahay/pls:2002:094.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_100.pdf) (bases 2 ≤ b ≤ 10) (warning: the data for base b = 8 has error, the data misses the primes 444641 and 444444441)
  20. https://logs.esolangs.org/freenode-esoteric/2011-02-04.html (bases 2 ≤ b ≤ 10) (warning: the data for base b = 8 has error, the data misses the prime 111 and instead wrongly includes the primes 1101, 101111, 600111, 1000011, 1000111, 4411111, 64111111, 601111111, 41111111111111111, and possibly 6111111111111111111111 if the author of this article continues to search, in fact, the data for base b = 8 misses not only 111 but also 1011)
  21. http://www.bitman.name/math/article/730 (in Italian) (bases 2 ≤ b ≤ 20)
  22. http://www.bitman.name/math/table/497 (in Italian) (bases 2 ≤ b ≤ 16)
  23. http://www.bitman.name/math/table/498 (in Italian) (base b = 17)
  24. http://www.bitman.name/math/table/499 (in Italian) (base b = 18)
  25. http://www.bitman.name/math/table/500 (in Italian) (base b = 19)
  26. http://www.bitman.name/math/table/501 (in Italian) (base b = 20)
  27. https://www.primepuzzles.net/puzzles/puzz_178.htm (base b = 10)
  28. https://oeis.org/A326609 (largest minimal prime in base b)
  29. https://oeis.org/A330049 (length of the largest minimal prime in base b)
  30. https://oeis.org/A330048 (number of minimal primes in base b)
  31. https://oeis.org/A071062 (base b = 10)
  32. https://oeis.org/A110600 (base b = 12)

Other researches for the digits of the primes:

Left-truncatable primes (https://en.wikipedia.org/wiki/Truncatable_prime, https://t5k.org/glossary/xpage/LeftTruncatablePrime.html, https://mathworld.wolfram.com/TruncatablePrime.html, https://www.numbersaplenty.com/set/truncatable_prime/), i.e. every nonempty suffix is prime:

  1. http://primerecords.dk/left-truncatable.txt (base b = 10)
  2. http://chesswanks.com/num/LTPs/ (bases 3 ≤ b ≤ 120)
  3. https://rosettacode.org/wiki/Find_largest_left_truncatable_prime_in_a_given_base (bases 3 ≤ b ≤ 17)
  4. https://www.ams.org/journals/mcom/1977-31-137/S0025-5718-1977-0427213-2/S0025-5718-1977-0427213-2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_28.pdf) (bases 3 ≤ b ≤ 11)
  5. http://www.primerecords.dk/left-truncatable.htm (base b = 10)
  6. http://rosettacode.org/wiki/Truncatable_primes (base b = 10)
  7. https://www.primepuzzles.net/puzzles/puzz_002.htm (base b = 10)
  8. https://web.archive.org/web/20041204160717/http://www.wschnei.de/digit-related-numbers/circular-primes.html (base b = 10)
  9. http://www.bitman.name/math/article/1155 (in Italian) (bases 2 ≤ b ≤ 20)
  10. http://www.bitman.name/math/table/524 (in Italian) (bases 2 ≤ b ≤ 20)
  11. https://oeis.org/A103443 (largest left-truncatable prime in base b)
  12. https://oeis.org/A103463 (length of the largest left-truncatable prime in base b)
  13. https://oeis.org/A076623 (number of left-truncatable primes in base b)
  14. https://oeis.org/A024785 (base b = 10)
  15. https://oeis.org/A129940 (base b = 4)
  16. https://oeis.org/A024779 (base b = 4, written in base b)
  17. https://oeis.org/A129941 (base b = 5)
  18. https://oeis.org/A024780 (base b = 5, written in base b)
  19. https://oeis.org/A129942 (base b = 6)
  20. https://oeis.org/A024781 (base b = 6, written in base b)
  21. https://oeis.org/A129943 (base b = 7)
  22. https://oeis.org/A024782 (base b = 7, written in base b)
  23. https://oeis.org/A129944 (base b = 8)
  24. https://oeis.org/A024783 (base b = 8, written in base b)
  25. https://oeis.org/A129945 (base b = 9)
  26. https://oeis.org/A024784 (base b = 9, written in base b)

Right-truncatable primes (https://en.wikipedia.org/wiki/Truncatable_prime, https://t5k.org/glossary/xpage/RightTruncatablePrime.html, https://mathworld.wolfram.com/TruncatablePrime.html, https://www.numbersaplenty.com/set/truncatable_prime/), i.e. every nonempty prefix is prime:

  1. http://primerecords.dk/right-truncatable.txt (base b = 10)
  2. http://fatphil.org/maths/rtp/rtp.html (bases 3 ≤ b ≤ 90)
  3. https://www.ams.org/journals/mcom/1977-31-137/S0025-5718-1977-0427213-2/S0025-5718-1977-0427213-2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_28.pdf) (bases 3 ≤ b ≤ 15)
  4. http://rosettacode.org/wiki/Truncatable_primes (base b = 10)
  5. https://www.primepuzzles.net/puzzles/puzz_002.htm (base b = 10)
  6. https://web.archive.org/web/20041204160717/http://www.wschnei.de/digit-related-numbers/circular-primes.html (base b = 10)
  7. http://www.bitman.name/math/article/1155 (in Italian) (bases 2 ≤ b ≤ 20)
  8. http://www.bitman.name/math/table/525 (in Italian) (bases 2 ≤ b ≤ 20)
  9. https://oeis.org/A023107 (largest right-truncatable prime in base b)
  10. https://oeis.org/A103483 (length of the largest right-truncatable prime in base b)
  11. https://oeis.org/A076586 (number of right-truncatable primes in base b)
  12. https://oeis.org/A024770 (base b = 10)
  13. https://oeis.org/A129669 (base b = 3)
  14. https://oeis.org/A024763 (base b = 3, written in base b)
  15. https://oeis.org/A129670 (base b = 4)
  16. https://oeis.org/A024764 (base b = 4, written in base b)
  17. https://oeis.org/A129671 (base b = 5)
  18. https://oeis.org/A024765 (base b = 5, written in base b)
  19. https://oeis.org/A129672 (base b = 6)
  20. https://oeis.org/A024766 (base b = 6, written in base b)
  21. https://oeis.org/A129673 (base b = 7)
  22. https://oeis.org/A024767 (base b = 7, written in base b)
  23. https://oeis.org/A129692 (base b = 8)
  24. https://oeis.org/A024768 (base b = 8, written in base b)
  25. https://oeis.org/A129693 (base b = 9)
  26. https://oeis.org/A024769 (base b = 9, written in base b)
  27. https://oeis.org/A237600 (base b = 16)

Other researches for the minimal elements of other subsets of positive integers written in the positional numeral system with radix b, as digit strings with subsequence ordering:

Primes == 1 mod 4:

  1. https://www.primepuzzles.net/puzzles/puzz_178.htm (base b = 10)
  2. https://github.com/curtisbright/mepn-data/blob/master/data/primes1mod4/minimal.10.txt (base b = 10)
  3. https://www.primepuzzles.net/puzzles/Minimal%20Primes%204k+1,%204k-1,%20pu%20178.doc (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/doc_1.doc) (base b = 10) (warning: the data has many errors, the data wrongly including many primes which are not minimal primes, including the largest "minimal 4×k+1 prime" in the list: 9630493, this prime is not a minimal 4×k+1 prime since 9949 is also a prime == 1 mod 4, and 9949 is a subsequence of 9630493, there are 146 (instead of 173) minimal 4×k+1 primes and 113 (instead of 138) minimal 4×k−1 primes, and the largest minimal 4×k+1 prime is 87733 = (8×1079−503)/9 instead of 9630493 = 10633−507)
  4. https://oeis.org/A111055 (base b = 10)
  5. https://oeis.org/A111057 (base b = 12)

Primes == 3 mod 4:

  1. https://www.primepuzzles.net/puzzles/puzz_178.htm (base b = 10)
  2. https://github.com/curtisbright/mepn-data/blob/master/data/primes3mod4/minimal.10.txt (base b = 10)
  3. https://www.primepuzzles.net/puzzles/Minimal%20Primes%204k+1,%204k-1,%20pu%20178.doc (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/doc_1.doc) (base b = 10) (warning: the data has many errors, the data wrongly including many primes which are not minimal primes, including the largest "minimal 4×k+1 prime" in the list: 9630493, this prime is not a minimal 4×k+1 prime since 9949 is also a prime == 1 mod 4, and 9949 is a subsequence of 9630493, there are 146 (instead of 173) minimal 4×k+1 primes and 113 (instead of 138) minimal 4×k−1 primes, and the largest minimal 4×k+1 prime is 87733 = (8×1079−503)/9 instead of 9630493 = 10633−507)
  4. https://oeis.org/A111056 (base b = 10) (warning: the b-file does not include the prime 21915199 since the limit of the numbers in the b-files in OEIS is 101000−1)

Palindromic primes:

  1. https://www.primepuzzles.net/puzzles/puzz_178.htm (base b = 10)
  2. https://oeis.org/A114835 (base b = 10) (warning: the b-file does not include the probable prime 9943401999 since the limit of the numbers in the b-files in OEIS is 101000−1)

Composites:

  1. https://github.com/curtisbright/mepn-data/tree/master/data/composites (bases 2 ≤ b ≤ 30)
  2. http://www.bitman.name/math/table/504 (bases 2 ≤ b ≤ 20)
  3. https://oeis.org/A071070 (base b = 10)

Squares:

  1. http://recursed.blogspot.com/2006/12/prime-game.html (base b = 10)
  2. https://oeis.org/A130448 (base b = 10)

Powers of 2:

  1. https://oeis.org/A071071/a071071.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_13.pdf) (base b = 10)
  2. https://oeis.org/A071071 (base b = 10)

Multiples of 3:

  1. https://oeis.org/A071073 (base b = 10)

Multiples of 4:

  1. https://oeis.org/A071072 (base b = 10)

Other sets:

  1. https://arxiv.org/pdf/1607.01548.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_14.pdf) (sums of three squares, quadratic residues mod 6, quadratic residues mod 7, range of Euler's totient function, range of "Euler's totient function + 3", range of Dedekind psi function, perfect numbers) (base b = 10)
  2. https://nntdm.net/papers/nntdm-25/NNTDM-25-1-036-047.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_15.pdf) (range of "Euler's totient function + n", for 0 ≤ n ≤ 5) (base b = 10)

The prime curios for them:

The sets of the minimal elements of these sets under the subsequence ordering are:

set (in decimal, i.e. base b = 10)the set of the minimal elements under the subsequence orderingnumber of such elementslength of the longest such element
primes == 1 mod 45, 13, 17, 29, 37, 41, 61, 73, 89, 97, 101, 109, 149, 181, 233, 277, 281, 349, 409, 433, 449, 677, 701, 709, 769, 821, 877, 881, 1669, 2221, 3001, 3121, 3169, 3221, 3301, 3833, 4969, 4993, 6469, 6833, 6949, 7121, 7477, 7949, 9001, 9049, 9221, 9649, 9833, 9901, 9949, 11969, 19121, 20021, 20201, 21121, 23021, 23201, 43669, 44777, 47777, 60493, 60649, 66749, 80833, 90121, 91121, 91921, 91969, 94693, 111121, 112121, 119921, 199921, 220301, 466369, 470077, 666493, 666649, 772721, 777221, 777781, 779981, 799921, 800333, 803333, 806033, 833033, 833633, 860333, 863633, 901169, 946369, 946669, 999169, 1111169, 1999969, 4007077, 4044077, 4400477, 4666693, 8000033, 8000633, 8006633, 8600633, 8660033, 8830033, 8863333, 8866633, 22000001, 40400077, 44040077, 60000049, 66000049, 66600049, 79999981, 80666633, 83333333, 86606633, 86666633, 88600033, 88883033, 88886033, 400000477, 400444477, 444000077, 444044477, 836666333, 866663333, 888803633, 888806333, 888880633, 888886333, 8888800033, 8888888033, 88888883333, 440444444477, 7777777777921, 8888888888333, 40000000000777, 44444444400077, 40444444444444477, 44444444444444477, 88888888888888633, 999999999999999121, 8773314679
primes == 3 mod 43, 7, 11, 19, 59, 251, 491, 499, 691, 991, 2099, 2699, 2999, 4051, 4451, 4651, 5051, 5651, 5851, 6299, 6451, 6551, 6899, 8291, 8699, 8951, 8999, 9551, 9851, 22091, 22291, 66851, 80051, 80651, 84551, 85451, 86851, 88651, 92899, 98299, 98899, 200891, 208891, 228299, 282299, 545551, 608851, 686051, 822299, 828899, 848851, 866051, 880091, 885551, 888091, 888451, 902299, 909299, 909899, 2000291, 2888299, 2888891, 8000099, 8000891, 8000899, 8028299, 8808299, 8808551, 8880551, 8888851, 9000451, 9000899, 9908099, 9980099, 9990899, 9998099, 9999299, 60000851, 60008651, 60086651, 60866651, 68666651, 80088299, 80555551, 80888299, 88808099, 88808899, 88880899, 90000299, 90080099, 222222899, 800888899, 808802899, 808880099, 808888099, 888800299, 888822899, 992222299, 2222288899, 8808888899, 8888800099, 8888888299, 8888888891, 48555555551, 555555555551, 999999999899, 88888888888099, 2228888888888899, 9222222222222299, 2288888888888888888888899, 84399, 864751, 2191519911319153
palindromic primes2, 3, 5, 7, 11, 919, 94049, 94649, 94849, 94949, 96469, 98689, 9809089, 9888889, 9889889, 9908099, 9980899, 9989899, 900808009, 906686609, 906989609, 908000809, 908444809, 908808809, 909848909, 960898069, 968999869, 988000889, 989040989, 996686699, 996989699, 999686999, 90689098609, 90899999809, 90999899909, 96099899069, 96600800669, 96609890669, 98000000089, 98844444889, 9009004009009, 9099094909909, 9600098900069, 9668000008669, 9699998999969, 9844444444489, 9899900099989, 9900004000099, 9900994990099, 900006898600009, 900904444409009, 966666989666669, 966668909866669, 966699989996669, 999090040090999, 999904444409999, 90000006860000009, 90000008480000009, 90000089998000009, 90999444444499909, 96000060806000069, 99900944444900999, 99990009490009999, 99999884448899999, 9000090994990900009, 9000094444444900009, 9666666080806666669, 9666666668666666669, 9909999994999999099, 9999444444444449999, 9999909994999099999, 9999990994990999999, 900000000080000000009, 900999994444499999009, 90000000009490000000009, 90909444444444444490909, 98999999444444499999989, 9904444444444444444444099, 999999999844444448999999999, 90944444444444444444444444909, 99999999999944444999999999999, 99999999999999499999999999999, 9999999999990004000999999999999, 900000000999999949999999000000009, 989999999999998444899999999999989, 9000000999999999994999999999990000009, ..., 9943401999, ... (this set is not known to be complete)≥ 87≥ 34023
composites4, 6, 8, 9, 10, 12, 15, 20, 21, 22, 25, 27, 30, 32, 33, 35, 50, 51, 52, 55, 57, 70, 72, 75, 77, 111, 117, 171, 371, 711, 713, 731323
squares1, 4, 9, 25, 36, 576, 676, 7056, 80656, 665856, 2027776, 2802276, 22282727076, 77770707876, 78807087076, 7888885568656, 8782782707776, 72822772707876, 555006880085056, 782280288087076, 827702888070276, 888288787822276, 2282820800707876, 7880082008070276, 80077778877070276, 88778000807227876, 782828878078078276, 872727072820287876, 2707700770820007076, 7078287780880770276, 7808287827720727876, 8008002202002207876, 27282772777702807876, 70880800720008787876, 72887222220777087876, 80028077888770207876, 80880700827207270276, 87078270070088278276, 88002002000028027076, 2882278278888228807876, 8770777780888228887076, 77700027222828822007876, 702087807788807888287876, 788708087882007280808827876, 880070008077808877000002276, 888000227087070707880827076, 888077027227228277087787076, 888588886555505085888555556, 7770000800780088788282227776, 7782727788888878708800870276, 5000060065066660656065066555556, 8070008800822880080708800087876, 80787870808888808272077777227076, 800008088070820870870077778827876, 822822722220080888878078820887876, ... (this set is currently not known, and might be extremely difficult to found)≥ 55≥ 33
powers of 21, 2, 4, 8, 65536 (this set is conjectured to be complete by Jeffrey Shallit, but not proven, this conjecture is true if all powers of 2 except 65536 contain at least one of 1, 2, 4, 8, only powers of 16 can be exceptions)≥ 5≥ 5
multiples of 30, 3, 6, 9, 12, 15, 18, 21, 24, 27, 42, 45, 48, 51, 54, 57, 72, 75, 78, 81, 84, 87, 111, 114, 117, 141, 144, 147, 171, 174, 177, 222, 225, 228, 252, 255, 258, 282, 285, 288, 411, 414, 417, 441, 444, 447, 471, 474, 477, 522, 525, 528, 552, 555, 558, 582, 585, 588, 711, 714, 717, 741, 744, 747, 771, 774, 777, 822, 825, 828, 852, 855, 858, 882, 885, 888763
multiples of 40, 4, 8, 12, 16, 32, 36, 52, 56, 72, 76, 92, 96132
range of Euler's totient function1, 2, 4, 6, 8, 30, 70, 500, 900, 990, 5590, 9550, 5555555555501312
range of Dedekind psi function1, 3, 4, 6, 8, 20, 72, 90, 222, 252, 500, 522, 552, 570, 592, 750, 770, 992, 7000, 56902070
range of "Euler's totient function + 1"2, 3, 5, 7, 9, 11, 41, 61, 8192
range of "Euler's totient function + 2"3, 4, 6, 8, 10, 12, 20, 22, 50, 72, 90, 770, 992, 5592, 9552, 555555555552 (this set is conjectured to be complete, but not proven, this conjecture is true if and only if there are no totients of the form 6{9}8, and such totients are conjectured not exist, since such totients are == 2 mod 12, thus must be of the form (p−1)×pn with p prime and n odd)16~1712 or > 5000
range of "Euler's totient function + 3"4, 5, 7, 9, 11, 13, 21, 23, 31, 33, 61, 63, 81, 83142
range of "Euler's totient function + 4"5, 6, 8, 10, 12, 14, 20, 22, 24, 32, 34, 40, 44, 70, 74, 92, 300, 472, 772, 900, 904, 994 (this set is conjectured to be complete, but not proven, this conjecture is true if and only if there are no totients of the form {3,9}26 or {3,9}86, and such totients are conjectured not exist, since such totients are == 2 mod 12, thus must be of the form (p−1)×pn with p prime and n odd)≥ 223 or > 5000
range of "Euler's totient function + 5"6, 7, 9, 11, 13, 15, 21, 23, 25, 33, 35, 41, 45, 51, 53, 83, 85, 301, 443, 505, 801, 881, 555555555555 (this set is conjectured to be complete, but not proven, this conjecture is true if and only if there are no totients of the form 3{9}8, and such totients are conjectured not exist, since such totients are == 2 mod 12, thus must be of the form (p−1)×pn with p prime and n odd)23~2412 or > 5000
perfect numbers6, 28 (this set is conjectured to be complete, but not proven, this conjecture is true if there are no odd perfect numbers, since all even perfect numbers end with either 6 or 28, and odd perfect numbers are conjectured not exist, this is a famous open problem)≥ 22 or > 1500

(for the primality certificate for the largest element in the minimal set of "primes == 3 mod 4" (21915199), see http://factordb.com/cert.php?id=1100000000301493137 and https://stdkmd.net/nrr/cert/2/2w99_19153.zip and http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/Certif/shallit.certif.gz, note that this prime is also an element of the minimal set of the primes > 100 in base b = 100, since this prime is written 2:229575:99 in base b = 100 (use the character ":" to saparate the digits for bases b > 36 (and just use decimal to write the digits), just like https://baseconvert.com/ and https://baseconvert.com/high-precision), and both families 2:{22} in base b = 100 and {22}:99 in base b = 100 contain no primes, however, the element 9943401999 in the minimal set of "palindromic primes" is only a probable prime, i.e. not a definitely prime)

(for the references of the famous open problem for the existence of odd perfect numbers, see https://web.archive.org/web/20120426061657/http://oddperfect.org/ and https://www.lirmm.fr/~ochem/opn/ and https://mathworld.wolfram.com/OddPerfectNumber.html and https://maths-people.anu.edu.au/~brent/pub/pub116.html and https://maths-people.anu.edu.au/~brent/pub/pub100.html and https://maths-people.anu.edu.au/~brent/pub/pub106.html and https://maths-people.anu.edu.au/~brent/pd/rpb116a.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_398.pdf) and https://maths-people.anu.edu.au/~brent/pd/rpb116.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_399.pdf) and https://maths-people.anu.edu.au/~brent/pd/rpb116p.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_400.pdf) and https://maths-people.anu.edu.au/~brent/pd/rpb100a.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_401.pdf) and https://maths-people.anu.edu.au/~brent/pd/rpb100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_402.pdf) and https://maths-people.anu.edu.au/~brent/pd/rpb100s.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_403.pdf) and https://www.lirmm.fr/~ochem/opn/opn.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_404.pdf) and https://www.lirmm.fr/~ochem/opn/opnf.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_437.pdf) and https://math.colgate.edu/~integers/x79/x79.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_438.pdf) and https://www.lirmm.fr/~ochem/opn/opn_slide.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_439.pdf) and https://www.ams.org/journals/mcom/2007-76-260/S0025-5718-07-02033-9/S0025-5718-07-02033-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_440.pdf) and https://maths-people.anu.edu.au/~brent/pd/rpb106i.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_405.pdf), currently it is know that there is no odd perfect number ≤ 101500)

Besides, the sets of the minimal elements of other classical sets under the subsequence ordering are:

set (in decimal, i.e. base b = 10)the set of the minimal elements under the subsequence orderingnumber of such elementslength of the longest such element
cubes1, 8, 27, 64, 343, 729, 3375, 4096, 35937, 39304, 46656, 50653, 79507, 97336, 300763, 405224, 456533, 474552, 493039, 636056, 704969, 3307949, 4330747, 5545233, 5639752, 5735339, 6539203, 9663597, 23393656, 23639903, 29503629, 37933056, 40353607, 45499293, 50243409, 54439939, 57066625, 57960603, 70444997, 70957944, 73560059, 76765625, 95443993, 202262003, 236029032, 350402625, 377933067, 379503424, 445943744, 454756609, 537367797, 549353259, 563559976, 567663552, 773620632, 907039232, ... (this set is currently not known, and might be extremely difficult to found)≥ 56≥ 9
4th powers1, 256, 625, 4096, 20736, 65536, 456976, 4477456, 8503056, 9834496, 59969536, 78074896, 84934656, 303595776, 362673936, 688747536, ... (this set is currently not known, and might be extremely difficult to found)≥ 16≥ 9
perfect powers1, 4, 8, 9, 25, 27, 32, 36, 576, 676, 3375, 7056, 7776, 50653, 20226200, ... (this set is currently not known, and might be extremely difficult to found)≥ 15≥ 8
powerful numbers1, 4, 8, 9, 25, 27, 32, 36, 72, 200, 500, 576, 675, 676, 3375, 7056, 7776, 50653, 60552, 357075, 570375, 665523, 735075, 766656, 7555707, ... (this set is currently not known, and might be extremely difficult to found)≥ 25≥ 7
prime powers2, 3, 4, 5, 7, 8, 9, 11, 16, 61102
powers of 31, 3, 9, 27 (this set is conjectured to be complete, but not proven, this conjecture is true if all powers of 3 except 27 contain at least one of 1, 3, 9, only (4×m+3)th powers of 3 can be exceptions)≥ 4≥ 2
squarefree numbers1, 2, 3, 5, 6, 7, 89, 94, 409, 449, 498, 499, 998133
triangular numbers1, 3, 6, 28, 45, 55, 78, 820, 990, 2775, 7750, 9870, 24090, 25200, 40470, 49770, 57970, 70500, 97020, 292995, 299925, 422740, 442270, 588070, 702705, 749700, 870540, 2474200, 4744740, 5727420, 7279020, 7799275, 8588440, 20740020, 27524490, 27792240, 40002040, 52270200, 54920440, ... (this set is currently not known, and might be extremely difficult to found)≥ 39≥ 8
Pronic numbers2, 6, 30, 90, 110, 870, 5550, 5700, 7140, 8010, 15500, 15750, 48180, 50400, 50850, 57840, 144780, 147840, 475410, 504810, 757770, 845480, 884540, 1477440, 4517750, 4754580, 7185080, 7450170, 10077450, 10500840, 14588580, 17770440, 17854850, 40710780, 41480040, 41544470, 45501770, 47755010, 54471780, ... (this set is currently not known, and might be extremely difficult to found)≥ 39≥ 8
semiprimes4, 6, 9, 10, 15, 21, 22, 25, 33, 35, 38, 51, 55, 57, 58, 77, 82, 85, 87, 111, 118, 123, 178, 183, 203, 237, 278, 301, 302, 327, 371, 502, 703, 713, 718, 723, 731, 753, 781, 803, 813, 818, 831, 1137, 1317, 3007, 3117, 8801, 8881, 28883, 50003, 80081, 888883, 7000001, 8000011557
Mersenne primes3, 7, 8191 (this set is conjectured to be complete, but not proven, this conjecture is true if all Mersenne primes other than 8191 contain either 3 or 7, only Mersenne primes with exponents == 1 mod 4 can be exceptions)≥ 3≥ 4
Fermat primes3, 5, 17 (this set is conjectured to be complete, but not proven, this conjecture is true if all Fermat primes other than 17 contain either 3 or 5, of course this conjecture is true if 65537 is the largest Fermat prime)≥ 3≥ 2
factorial numbers1, 2, 6, 5040 (this set is conjectured to be complete, but not proven, this conjecture is true if all factorial numbers other than 5040 contain at least one of 1, 2, 6)≥ 4≥ 4
primorial numbers1, 2, 6, 30 (this set is conjectured to be complete, but not proven, this conjecture is true if all primorial numbers other than 30 contain at least one of 1, 2, 6)≥ 4≥ 2
Fibonacci numbers1, 2, 3, 5, 8 (this set is conjectured to be complete, but not proven, this conjecture is true if all Fibonacci numbers contain at least one of 1, 2, 3, 5, 8)≥ 5≥ 1
Lucas numbers1, 3, 4, 7, 29 (this set is conjectured to be complete, but not proven, this conjecture is true if all Lucas numbers other than 29 contain at least one of 1, 3, 4, 7)≥ 5≥ 2

Tools about this research: (in fact, you can also use Wolfram Alpha (https://www.wolframalpha.com/) or online Magma calculator (http://magma.maths.usyd.edu.au/calc/) or Pari/GP (https://pari.math.u-bordeaux.fr/) or Wolfram Mathematica (https://www.wolfram.com/mathematica/) or Maple (https://www.maplesoft.com/)) (many of these tools have a limit of 253, this is because the limit of the significant figures (https://en.wikipedia.org/wiki/Significant_figures, https://mathworld.wolfram.com/SignificantDigits.html) of the double-precision floating-point format (https://en.wikipedia.org/wiki/Double-precision_floating-point_format, https://mathworld.wolfram.com/Floating-PointNumber.html) is 53 bits (https://en.wikipedia.org/wiki/Bit, https://mathworld.wolfram.com/Bit.html))

Prime checkers:

  1. https://t5k.org/curios/includes/primetest.php (limit: 253−1)
  2. https://www.numberempire.com/primenumbers.php (choose "Check") (also give the index for the primes ≤ 109−1) (limit: 101000−1)
  3. http://www.numbertheory.org/php/lucas.html (no limit)
  4. https://www.alpertron.com.ar/ECM.HTM (click "Prime") (no limit)
  5. http://www.javascripter.net/faq/numberisprime.htm (just type "isPrime(n)") (no limit)
  6. http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm (just type "isPrimeMR18(n)") (no limit)
  7. http://www.javascripter.net/math/calculators/100digitbigintcalculator.htm (just type x and click "prime?") (no limit)
  8. http://www.prime-numbers.org/ (use the box "Key in a number to check it's a prime number or not" and click "Check") (limit: 1011−1)
  9. https://www.walter-fendt.de/html5/men/primenumbers_en.htm (limit: 1012)
  10. http://www.math.com/students/calculators/source/prime-number.htm (limit: 253)
  11. https://www.dcode.fr/primality-test (no limit)
  12. https://www.calculatorsoup.com/calculators/math/prime-number-calculator.php (limit: 1013−1)
  13. https://onlinemathtools.com/test-prime-number (choose "Test All Numbers") (no limit)
  14. https://www.bigprimes.net/primalitytest (use the first box) (limit: 253−4)
  15. https://www.archimedes-lab.org/primOmatic.html (limit: 1012−1)
  16. http://www.sonic.net/~undoc/java/PrimeCalc.html (click "y=IsPrime(x)") (no limit)
  17. https://web.archive.org/web/20021011204430/http://www.miroe.de/proth/pr_prim.htm (limit: 1010−34)
  18. https://www.bigprimes.net/cruncher/ (limit: 106−1)
  19. https://math.tools/numbers/is-prime/ (limit: 231−1)
  20. http://www.primzahlen.de/primzahltests/testverfahren.htm (in German) (use the box "Miller-Rabin-Test" and click "Test") (limit: 49106×1010)
  21. http://www.proftnj.com/calcprem.htm (in French) (use the box "Rechercher si un nombre est premier" and click "Rechercher") (limit: 253)
  22. http://www.positiveintegers.org/ (just enter the number) (limit: 1000000)
  23. https://numdic.com/ (just enter the number) (limit: 700000)
  24. https://numbermatics.com/ (just enter the number) (limit: 1026+2)
  25. https://metanumbers.com/ (just enter the number) (limit: 263−1)
  26. https://int.darkbyte.ru/ (just enter the number) (limit: 231−2)
  27. https://www.numbersaplenty.com/ (just enter the number) (limit: 1015−1)
  28. http://factordb.com/ (online factor database, "P" means definitely prime, "PRP" means unproven probable prime, "FF" means composite and fully factored, "CF" means composite and has known proper factor but not fully factored, "C" means composite and has no known proper factor, "U" means number with unknown status, "Unit" means the number "1", "Zero" means the number "0", see http://factordb.com/status.html, also, this online factor database uses colors to show the status of the numbers, black means definitely prime ("P"), blue means composite ("C" or "CF" or "FF"), brown means unproven probable prime ("PRP"), red means number with unknown status ("U"), also, this online factor database automatically stores all numbers < 1018 and automatically checks the probable-primality of the numbers with unknown status < 102000 (see http://factordb.com/show_config.php) and automatically reserves to the probable-primality tests of the numbers with unknown status < 1049999 and automatically trial factors all numbers with the primes p < 5000000 (see http://factordb.com/show_config.php) and automatically algebraic factors all numbers (see https://www.mersenneforum.org/showpost.php?p=640196&postcount=1) and automatically proves the primality of the probable primes < 10299 (see https://web.archive.org/web/20150812084455/http://factordb.com/status.php) and automatically factors the composites < 1069 (see https://web.archive.org/web/20150812084455/http://factordb.com/status.php)) (limit: 1010000000−1 (see http://factordb.com/index.php?query=999999999%5E999999999%2B1) or 104000000−1 for Fibonacci and Lucas numbers (see http://factordb.com/index.php?query=I999999999 and http://factordb.com/index.php?query=L999999999) or 102000000−1 for factorial and primorial (see http://factordb.com/index.php?query=999999999%21%2B1 and http://factordb.com/index.php?query=999999999%23%2B1))
  29. https://578d0722p8.goho.co/index.html (more types of numbers in the online factor database) (limit: 1010000000−1 (see http://factordb.com/index.php?query=999999999%5E999999999%2B1) or 104000000−1 for Fibonacci and Lucas numbers (see http://factordb.com/index.php?query=I999999999 and http://factordb.com/index.php?query=L999999999) or 102000000−1 for factorial and primorial (see http://factordb.com/index.php?query=999999999%21%2B1 and http://factordb.com/index.php?query=999999999%23%2B1))
  30. https://t5k.org/nthprime/ (calculate the nth prime (use the box "Nth prime"), calculate the index of a prime (use the box "Pi function")) (limit: 3×1013)
  31. http://www.numbertheory.org/php/nprime.html (calculate the next (probable) prime above N, in fact, links 2, 7, 11, 12, 13 can also calculate the next prime above N, besides, links 2, 7 can also calculate the previous prime below N, for link 2 choose "Find next" to calculate the next prime above N, or choose "Find previous" to calculate the previous prime below N, for link 7 just type x and click "next p" to calculate the next prime above N, or just type x and click "prev p" to calculate the previous prime below N, for link 11 use the second box to calculate the next prime above N, for link 13 click "y=NextPrime(x)" to calculate the next prime above N) (no limit)
  32. https://www.dcode.fr/prime-numbers-search (calculate the nth prime (choose "Find the Nth Prime Number"), calculate the index of a prime (choose "Find π(n) the number of primes less than or equal to N"), calculate the next (probable) prime above N (choose "Find Next Prime after N"), calculate the previous (probable) prime below N (choose "Find Previous Prime before N"), in fact, links 2, 7, 11, 12, 13 can also calculate the next prime above N, besides, links 2, 7 can also calculate the previous prime below N, for link 2 choose "Find next" to calculate the next prime above N, or choose "Find previous" to calculate the previous prime below N, for link 7 just type x and click "next p" to calculate the next prime above N, or just type x and click "prev p" to calculate the previous prime below N, for link 11 use the second box to calculate the next prime above N, for link 13 click "y=NextPrime(x)" to calculate the next prime above N) (no limit)
  33. http://factordb.com/nextprime.php (calculate the next (probable) prime above N, in fact, links 2, 7, 11, 12, 13 can also calculate the next prime above N, besides, links 2, 7 can also calculate the previous prime below N, for link 2 choose "Find next" to calculate the next prime above N, or choose "Find previous" to calculate the previous prime below N, for link 7 just type x and click "next p" to calculate the next prime above N, or just type x and click "prev p" to calculate the previous prime below N, for link 11 use the second box to calculate the next prime above N, for link 13 click "y=NextPrime(x)" to calculate the next prime above N) (limit: 102000−1)

Integer factorizers:

  1. https://www.numberempire.com/numberfactorizer.php (limit: 1070−1)
  2. https://www.alpertron.com.ar/ECM.HTM (click "Factor") (no limit)
  3. http://www.numbertheory.org/php/factor.html (limit: 1025−1)
  4. http://www.numbertheory.org/php/lprimefactor.html (only find the least prime factor) (limit: 1025−1)
  5. https://www.mersenne.ca/factor.php (use the box "Get prime factorization for small (<=50 decimal digits) numbers") (limit: 1050−1)
  6. http://www.javascripter.net/math/calculators/primefactorscalculator.htm (limit: 1020−1)
  7. http://www.javascripter.net/faq/numberisprime.htm (only find the least prime factor, just type "leastFactor(n)") (no limit)
  8. http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm (only find the least prime factor, just type "leastFactor(n)") (no limit)
  9. https://www.walter-fendt.de/html5/men/primenumbers_en.htm (limit: 1012)
  10. https://web.archive.org/web/20230122202627/https://betaprojects.com/calculators/prime_factors.html (limit: 253−1)
  11. https://www.emathhelp.net/calculators/pre-algebra/prime-factorization-calculator/ (limit: 109)
  12. https://primefan.tripod.com/Factorer.html (limit: 253)
  13. https://www.dcode.fr/prime-factors-decomposition (no limit)
  14. https://www.calculatorsoup.com/calculators/math/prime-factors.php (also all (prime or composite or unit) factors of N) (limit: 1013−1)
  15. https://www.calculator.net/prime-factorization-calculator.html (limit: 1014−1)
  16. http://www.se16.info/js/factor.htm (limit: 253)
  17. https://web.archive.org/web/20230918044810/http://math.fau.edu/Richman/mla/factor-f.htm (limit: 253)
  18. http://www.rsok.com/~jrm/factor.html (no limit)
  19. http://www.brennen.net/primes/FactorApplet.html (need run with Java)
  20. https://web.archive.org/web/20161004191531/http://britton.disted.camosun.bc.ca/jbprimefactor.htm (limit: 106)
  21. http://wims.unice.fr/~wims/en_tool~algebra~factor.en.html (no limit)
  22. http://www.analyzemath.com/Calculators_3/prime_factors.html (limit: 253)
  23. https://www.archimedes-lab.org/primOmatic.html (limit: 1012−1)
  24. https://math.tools/calculator/prime-factors/ (limit: 231−1)
  25. https://www.bigprimes.net/cruncher/ (all (prime or composite or unit) factors of N) (limit: 1017−1)
  26. http://www.proftnj.com/calcprem.htm (in French) (use the box "Factoriser un nombre" and click "Factoriser") (limit: 253)
  27. http://www.positiveintegers.org/ (just enter the number) (also all (prime or composite or unit) factors of N) (limit: 1000000)
  28. https://numdic.com/ (just enter the number) (also all (prime or composite or unit) factors of N) (limit: 700000)
  29. https://numbermatics.com/ (just enter the number) (also all (prime or composite or unit) factors of N) (limit: 1026+2)
  30. https://metanumbers.com/ (just enter the number) (also all (prime or composite or unit) factors of N) (limit: 263−1)
  31. https://int.darkbyte.ru/ (just enter the number) (also all (prime or composite or unit) factors of N) (limit: 231−2)
  32. https://www.numbersaplenty.com/ (just enter the number) (also all (prime or composite or unit) factors of N) (limit: 1015−1)
  33. http://factordb.com/ (online factor database, "P" means definitely prime, "PRP" means unproven probable prime, "FF" means composite and fully factored, "CF" means composite and has known proper factor but not fully factored, "C" means composite and has no known proper factor, "U" means number with unknown status, "Unit" means the number "1", "Zero" means the number "0", see http://factordb.com/status.html, also, this online factor database uses colors to show the status of the numbers, black means definitely prime ("P"), blue means composite ("C" or "CF" or "FF"), brown means unproven probable prime ("PRP"), red means number with unknown status ("U"), also, this online factor database automatically stores all numbers < 1018 and automatically checks the probable-primality of the numbers with unknown status < 102000 (see http://factordb.com/show_config.php) and automatically reserves to the probable-primality tests of the numbers with unknown status < 1049999 and automatically trial factors all numbers with the primes p < 5000000 (see http://factordb.com/show_config.php) and automatically algebraic factors all numbers (see https://www.mersenneforum.org/showpost.php?p=640196&postcount=1) and automatically proves the primality of the probable primes < 10299 (see https://web.archive.org/web/20150812084455/http://factordb.com/status.php) and automatically factors the composites < 1069 (see https://web.archive.org/web/20150812084455/http://factordb.com/status.php)) (limit: 1010000000−1 (see http://factordb.com/index.php?query=999999999%5E999999999%2B1) or 104000000−1 for Fibonacci and Lucas numbers (see http://factordb.com/index.php?query=I999999999 and http://factordb.com/index.php?query=L999999999) or 102000000−1 for factorial and primorial (see http://factordb.com/index.php?query=999999999%21%2B1 and http://factordb.com/index.php?query=999999999%23%2B1))
  34. https://578d0722p8.goho.co/index.html (more types of numbers in the online factor database) (limit: 1010000000−1 (see http://factordb.com/index.php?query=999999999%5E999999999%2B1) or 104000000−1 for Fibonacci and Lucas numbers (see http://factordb.com/index.php?query=I999999999 and http://factordb.com/index.php?query=L999999999) or 102000000−1 for factorial and primorial (see http://factordb.com/index.php?query=999999999%21%2B1 and http://factordb.com/index.php?query=999999999%23%2B1))
  35. http://myfactorcollection.mooo.com:8090/dbio.html (online factor database for numbers of the form bn±1)
  36. http://myfactorcollection.mooo.com:8090/interactive.html (online factor database for numbers of the form bn±1) (only list the greatest prime factor (for fully factored numbers) or the composite factor with no known proper factor (for non-fully factored numbers), and only list the number of decimal digits of this factor if this factor is > 106, but you can click the lattices to see other prime factors) (the lattices saparated to two lattices means the number has Aurifeuillean factorization, and for such lattices, the left lattice is for the Aurifeuillean L part, and the right lattice is for the Aurifeuillean M part)
  37. https://web.archive.org/web/20120722020628/http://homes.cerias.purdue.edu/~ssw/cun/prime.php (online factor database for numbers of the form bn±1 for 2 ≤ b ≤ 12)
  38. https://web.archive.org/web/20120330032919/http://homes.cerias.purdue.edu/~ssw/cun/clientold.html (online factor database for numbers of the form bn±1 for 2 ≤ b ≤ 12)

Prime generators:

  1. http://www.numbertheory.org/php/prime_generator.html
  2. http://www.se16.info/js/prime.htm
  3. http://www.rsok.com/~jrm/printprimes.html
  4. http://www.sonic.net/~undoc/java/PrimeCalc.html (click "ShowPrimes(x..y)")
  5. http://www.primzahlen.de/primzahltests/testverfahren.htm (in German) (use the box "Primzahlgenerator" and click "erzeugen")

Base converters:

  1. https://baseconvert.com/ (any base b) (no limit)
  2. https://baseconvert.com/high-precision (any base b) (no limit)
  3. https://baseconvert.com/ieee-754-floating-point (for IEEE 754 (https://en.wikipedia.org/wiki/IEEE_754), only bases b = 2, 10, 16) (no limit)
  4. https://www.calculand.com/unit-converter/zahlen.php?og=Base+2-36&ug=1 (bases 2 ≤ b ≤ 36) (use lower case letters instead of upper case letters) (limit: 12 digits in the input base b)
  5. https://www.calculand.com/unit-converter/zahlen.php?og=Base62&ug=1 (covert base b = 62 to any base 2 ≤ b ≤ 200) (no limit)
  6. https://www.calculand.com/unit-converter/zahlen.php?og=Base64&ug=1 (covert base b = 64 to any base 2 ≤ b ≤ 200) (no limit)
  7. https://www.calculand.com/unit-converter/zahlen.php?og=Base85&ug=1 (covert base b = 85 to any base 2 ≤ b ≤ 200) (no limit)
  8. https://www.calculand.com/unit-converter/zahlen.php?og=System+calculand&ug=1 (bases 2 ≤ b ≤ 200) (limit: 253)
  9. http://www.unitconversion.org/unit_converter/numbers.html (bases 2 ≤ b ≤ 36) (limit: 253)
  10. http://www.unitconversion.org/unit_converter/numbers-ex.html (bases 2 ≤ b ≤ 36) (limit: 253)
  11. https://web.archive.org/web/20161126114837/http://bmanolov.free.fr/numbaseconv.php (bases 2 ≤ b ≤ 36) (limit: 253)
  12. https://math.tools/calculator/base/ (bases 2 ≤ b ≤ 64) (no limit)
  13. http://extraconversion.com/base-number (bases 2 ≤ b ≤ 36) (use lower case letters instead of upper case letters) (limit: 263−1)
  14. http://www.math.com/students/converters/source/base.htm (any base b) (limit: 252)
  15. https://www.dcode.fr/base-n-convert (bases 2 ≤ b ≤ 62) (no limit)
  16. https://www.convertworld.com/en/numerals/ (bases b = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 20) (limit: 253)
  17. https://www.cut-the-knot.org/binary.shtml (bases b = 2, 3, 5, 8, 10, 12, 16, 36) (limit: 253)
  18. https://www.cut-the-knot.org/Curriculum/Algorithms/BaseConversion.shtml (bases b = 2, 8, 10, 16) (limit: 253)
  19. http://www.tonymarston.net/php-mysql/converter.php (bases b = 2, 8, 10, 16, 36) (no limit)
  20. https://web.archive.org/web/20230918043634/http://math.fau.edu/Richman/mla/convert.htm (any base b) (limit: 253)
  21. https://web.archive.org/web/20190629223750/http://thedevtoolkit.com/tools/base_conversion (bases 2 ≤ b ≤ 36) (use lower case letters instead of upper case letters) (no limit)
  22. https://web.archive.org/web/20170204004954/http://ultrastudio.org/en/MechengburakalkanApplet-1.7.zip (bases 2 ≤ b ≤ 36) (use lower case letters instead of upper case letters) (limit: 253)
  23. https://www.bigprimes.net/cruncher/ (bases b = 2, 3, 4, 5, 8, 10, 16) (no limit)
  24. http://www.mathsisfun.com/numbers/convert-base.php (bases b = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 20, 26) (use 0−9, A−H, J−K instead of 0−9, A−J for b = 20, and use Z, A−Y instead of 0−9, A−P for b = 26) (no limit)
  25. http://www.kwuntung.net/hkunit/base/base.php (in Chinese) (bases 2 ≤ b ≤ 36) (use lower case letters instead of upper case letters) (limit: 253)
  26. https://linesegment.web.fc2.com/application/math/numbers/RadixConversion.html (in Japanese) (bases 2 ≤ b ≤ 36 and b = 62) (limit: 253)
  27. http://www.positiveintegers.org/ (just enter the number) (bases 2 ≤ b ≤ 36) (limit: 1000000)
  28. https://numdic.com/ (just enter the number) (bases b = 2, 3, 4, 5, 6, 7, 8, 10, 16, 24, 36) (use lower case letters instead of upper case letters) (limit: 700000)
  29. https://numbermatics.com/ (just enter the number) (bases b = 2, 16, 36) (limit: 1026+2)
  30. https://metanumbers.com/ (just enter the number) (bases b = 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 36) (use lower case letters instead of upper case letters) (limit: 263−1)
  31. https://int.darkbyte.ru/ (just enter the number) (bases 2 ≤ b ≤ 36) (use lower case letters instead of upper case letters) (limit: 231−2)
  32. https://www.numbersaplenty.com/ (just enter the number) (bases 2 ≤ b ≤ 16) (use lower case letters instead of upper case letters) (limit: 1015−1)
  33. http://factordb.com/index.php?showid=1000000000000000127 (you can change the "showid" to the ID for your number) (bases 2 ≤ b ≤ 36) (use lower case letters instead of upper case letters) (limit: 1010000000−1 (see http://factordb.com/index.php?query=999999999%5E999999999%2B1) or 104000000−1 for Fibonacci and Lucas numbers (see http://factordb.com/index.php?query=I999999999 and http://factordb.com/index.php?query=L999999999) or 102000000−1 for factorial and primorial (see http://factordb.com/index.php?query=999999999%21%2B1 and http://factordb.com/index.php?query=999999999%23%2B1))

Expression generators:

  1. https://stdkmd.net/nrr/exprgen.htm (only support base 10 forms)
  2. https://www.numberempire.com/simplifyexpression.php (e.g. for the form 5{7} in base 11, type "5*11^n+7*(11^n-1)/10")

(In fact, you can use Wolfram Alpha (https://www.wolframalpha.com/, https://en.wikipedia.org/wiki/WolframAlpha) and online Magma calculator (http://magma.maths.usyd.edu.au/calc/) for prime checker, integer factorizer, base converter, expression generator (e.g. https://www.wolframalpha.com/input?i=Is+11111111111111111+prime%3F, https://www.wolframalpha.com/input?i=PrimeQ%5B11111111111111111%5D (for prime checker, change 11111111111111111 to your number) and https://www.wolframalpha.com/input?i=Factorization+of+11111111111111111, https://www.wolframalpha.com/input?i=Factorization%5B11111111111111111%5D (for integer factorizer, change 11111111111111111 to your number) and https://www.wolframalpha.com/input?i=11111111111111111+in+base+36, https://www.wolframalpha.com/input?i=BaseForm%5B11111111111111111%2C36%5D (for base converter, use lower case letters instead of upper case letters, change 11111111111111111 to your number and change 36 to your base b) and https://www.wolframalpha.com/input?i=5*11%5En%2B7*%2811%5En-1%29%2F10, https://www.wolframalpha.com/input?i=Simplify%5B5*11%5En%2B7*%2811%5En-1%29%2F10%5D (for expression generator, change 5*11^n+7*(11^n-1)/10 to your form, see the "Alternate form" box, for more information for these tools in the wolfram language (https://en.wikipedia.org/wiki/Wolfram_Language) see https://reference.wolfram.com/language/ref/PrimeQ.html and https://reference.wolfram.com/language/ref/FactorInteger.html and https://reference.wolfram.com/language/ref/BaseForm.html and https://reference.wolfram.com/language/ref/Simplify.html)), besides, many mathematical softwares (https://en.wikipedia.org/wiki/Mathematical_software) also already have prime checkers, integer factorizers, base converters, expression generators, such as these (you can download these softwares by clicking the links):)

  1. Maple (https://www.maplesoft.com/, https://en.wikipedia.org/wiki/Maple_(software))
  2. Wolfram Mathematica (https://www.wolfram.com/mathematica/, https://en.wikipedia.org/wiki/Wolfram_Mathematica)
  3. Pari/GP (https://pari.math.u-bordeaux.fr/, https://en.wikipedia.org/wiki/PARI/GP)
  4. Python (https://www.python.org/, https://en.wikipedia.org/wiki/Python_(programming_language))
  5. GMP (https://gmplib.org/, https://en.wikipedia.org/wiki/GNU_Multiple_Precision_Arithmetic_Library)
  6. Magma (http://magma.maths.usyd.edu.au/, https://en.wikipedia.org/wiki/Magma_(computer_algebra_system))
  7. SageMath (https://www.sagemath.org/, https://en.wikipedia.org/wiki/SageMath)

Lists of small primes: (see https://t5k.org/notes/faq/LongestList.html, long lists just waste storage, and if placed on the Internet, they just waste bandwidth, since small primes are too easy to find, they can be found far faster than they can be read from a hard disk, so no one bothers to keep long lists (say past 1012), also see https://t5k.org/notes/faq/x_digit_primes.html, although we can quickly determine whether a number < 10300 is prime or not, but there is no list of all primes < 10300, since the space which we need is larger than the number of elementary particles (https://en.wikipedia.org/wiki/Elementary_particle) (i.e. quarks (https://en.wikipedia.org/wiki/Quark) and electrons (https://en.wikipedia.org/wiki/Electron)) in the universe (https://en.wikipedia.org/wiki/Universe), also see https://www.mersenneforum.org/showthread.php?t=24417)

  1. https://t5k.org/lists/small/1000.txt (limit: first 1000 primes, up to 7919)
  2. https://t5k.org/lists/small/10000.txt (limit: first 10000 primes, up to 104729)
  3. https://t5k.org/lists/small/100000.txt (limit: first 100008 primes, up to 1299827)
  4. https://t5k.org/lists/small/millions/ (limit: first 50000000 primes, up to 982451653)
  5. https://oeis.org/A000040/b000040.txt (limit: first 10000 primes, up to 104729)
  6. https://oeis.org/A000040/a000040.txt (limit: first 100000 primes, up to 1299709)
  7. https://oeis.org/A000040/b000040_1.txt (limit: first 500000 primes, up to 7368787)
  8. https://oeis.org/A000040/a000040_1B.7z (limit: first 1000000000 primes, up to 22801763489)
  9. http://www.prime-numbers.org/ (limit: first 4118044813 primes, up to 99999999977)
  10. https://web.archive.org/web/20091027064420/http://geocities.com/primes_r_us/small/index.html (limit: first 5800000 primes, up to 100711433)
  11. http://prime-numbers.org/sample.zip (limit: first 1000 primes, up to 7919)
  12. https://metanumbers.com/prime-numbers (limit: first 1229 primes, up to 9973)
  13. https://www.numberempire.com/primenumberstable.php (page limit: first 25 primes, up to 97, total limit: first 1229 primes, up to 9973)
  14. https://www.calculatorsoup.com/calculators/math/prime-numbers.php (limit: first 1009 primes, up to 8011)
  15. https://www2.cs.arizona.edu/icon/oddsends/primes.htm (limit: first 50000 primes, up to 611953)
  16. https://www.numbersaplenty.com/set/prime_number/more.php (limit: first 600 primes, up to 4409)
  17. https://web.archive.org/web/20050412073754/http://www.mindspring.com/~benbradley/basep.hpp (limit: first 6542 primes, up to 65521)
  18. https://cdn1.byjus.com/wp-content/uploads/2021/10/Prime-Numbers-from-1-to-1000.png (limit: first 168 primes, up to 997)
  19. http://noe-education.org/D11102.php (in French) (limit: first 5133 primes, up to 49999)
  20. https://math.tools/numbers/prime/ (limit: first 150 primes, up to 863)
  21. https://math.tools/list/prime-numbers (limit: first 3000 primes, up to 27449) (warning: this site uses index n−1 for the nth prime)
  22. https://web.archive.org/web/20060513054350/http://www.walter-fendt.de/m14i/primes_i.htm (in Italian) (limit: first 37607912018 primes, up to 999999999989)
  23. https://primefan.tripod.com/500Primes1.html (limit: first 501 primes, up to 3581) (warning: this site incorrectly includes 1 as a prime and misses the primes 3229 and 3329, thus this site actually lists the first 500−1+2 = 501 primes)
  24. https://www.gutenberg.org/files/65/65.txt (limit: first 101341 primes, up to 1318699)
  25. https://web.archive.org/web/20231024091219/http://www.primos.mat.br/indexen.html (page limit: first 168 primes, up to 997, total limit: first 37607912018 primes, up to 999999999989)
  26. https://www.walter-fendt.de/html5/men/primenumbers_en.htm (limit: first 37607912018 primes, up to 999999999989)
  27. https://www.rsok.com/~jrm/first100primes.html (limit: first 100 primes, up to 541)
  28. http://www.rsok.com/~jrm/printprimes.html (limit: first 98222287 primes, up to 1999999973)
  29. http://mrteverett.com/numbers/primes/factest.txt (limit: first 14172 primes, up to 153757) (warning: this site does not list the primes 2 and 3 and start with the prime 5, thus this site actually only has 14172−2 = 14170 primes)
  30. http://www.primzahlen.de/primzahltests/2-100003.htm (in German) (limit: first 9593 primes, up to 100003)
  31. https://jocelyn.quizz.chat/np/cache/index.html (in French) (limit: first 1000000 primes, up to 15485863)
  32. http://www.sosmath.com/tables/prime/prime.html (limit: first 1000 primes, up to 7919)
  33. https://www.bigprimes.net/archive/prime (page limit: first 100 primes, up to 541, total limit: first 1000099 primes, up to 15487457)
  34. https://web.archive.org/web/20201130071856/http://www.mathematical.com/primelist1to100kk.html (limit: first 664579 primes, up to 9999991)
  35. https://web.archive.org/web/20191118082053/http://www.tsm-resources.com/alists/prim.html (limit: first 2000 primes, up to 17389)
  36. https://web.archive.org/web/20090917191047/http://planetmath.org/encyclopedia/FirstThousandPositivePrimeNumbers.html (limit: first 1000 primes, up to 7919)
  37. https://openlibrary.org/books/OL16553580M (limit: first 664999 primes, up to 10006721)
  38. https://sites.google.com/view/the-first-25000-primes (limit: first 25000 primes, up to 287117)
  39. https://faculty.lynchburg.edu/~nicely/gaps/gaplist.html (the longest list ever calculated, with all primes < 264 (but unlikely other lists here, the primes are not all stored))
  40. https://en.wikipedia.org/wiki/List_of_prime_numbers#The_first_1000_prime_numbers (limit: first 1000 primes, up to 7919)

Also http://5.199.134.130/primes.tar.xz and http://5.199.134.130/primes.tar.xz.SHA256SUM and http://5.199.134.130/primes.tar.xz.par2 and http://5.199.134.130/primes.tar.xz.vol00+10.par2 and the .xz files in http://5.199.134.130/primes/ ("n.xz" for the definitely primes with n decimal digits for 19 ≤ n ≤ 9999, http://5.199.134.130/primes/above9999.xz for all definitely primes with ≥ 10000 decimal digits, http://5.199.134.130/primes/PRP.xz fpr all probable primes) and https://download.mersenne.ca/factordb_2024-03-21_primes.tar.xz.torrent for all primes and probable primes > 1018 in factordb (i.e. the numbers in either http://factordb.com/listtype.php?t=4 or http://factordb.com/listtype.php?t=1) (pixz is recommended for unpacking. "pixz -d < primes.tar.xz | tar -xv"), also http://factordb.com/dlprp.php?n=1000000 for 10000 random chosen small unproven probable primes in factordb (i.e. the numbers in http://factordb.com/listtype.php?t=1), also http://factordb.com/dlc.php?n=1000000 for 5000 random chosen small composites with no known proper (prime or composite) factors in factordb (i.e. the numbers in http://factordb.com/listtype.php?t=3), also http://factordb.com/primobatch.php?digits=300&files=10&parts=1&start=Generate+zip and http://factordb.com/primobatch.php?digits=300&files=32000&parts=32&start=Generate+zip for the input files for Primo in factordb, also see http://factordb.com/distribution.php?size=20&start=0&lim=500000 and http://factordb.com/types_pie.php?large for the distribution of numbers by digits and status (http://factordb.com/status.html, http://factordb.com/distribution.php) in factordb, also http://5.199.134.130/certificates.tar.xz and http://5.199.134.130/certificates.tar.xz.SHA256SUM and http://5.199.134.130/certificates.tar.xz.par2 and http://5.199.134.130/certificates.tar.xz.vol00+10.par2 and the .xz files in http://5.199.134.130/certificates/ ("n.xz" for the primality certificates of the definitely primes with n decimal digits for 300 ≤ n ≤ 5679 and selected 5680 ≤ n ≤ 86453) for all primality certificates in factordb, also http://factordb.com/certtop.php for the primality certificates by user in factordb, also http://factordb.com/certtypetop.php for the primality certificates by software in factordb.

Lists of factorizations of small integers:

  1. http://primefan.tripod.com/500factored.html (page limit: 500, total limit: 2500)
  2. http://www.sosmath.com/tables/factor/factor.html (page limit: 200, total limit: 1000)
  3. https://sites.google.com/view/prime-factorization-of-integer (limit: 25000)
  4. https://web.archive.org/web/20060210182347/http://bearnol.is-a-geek.com/Panfur%20Project/ (limit: 20000000, however, currently the data is only available up to 1500000 in the archive pages, the data for numbers > 1500000 has no available archive pages) (warning: this site does not factor the composite numbers 15, 51, 85, 91, 255, 435, 451, 561, 595, 679, 703, 771, 1105, 1261, 1285, 1351, 1387, ...)
  5. http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/ (limit: 10000)
  6. http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/?infinity (no limit)
  7. https://oeis.org/A027746/b027746.txt (flattened rows, the Nth row ends with the term with index "the term N of https://oeis.org/A022559") (limit: 2048)
  8. https://oeis.org/A112798/b112798.txt (replace the nth prime by n, flattened rows, the Nth row ends with the term with index "the term N of https://oeis.org/A022559") (limit: 3275)
  9. https://oeis.org/A027750/a027750.txt (all (prime or composite or unit) factors of N) (limit: 10000)
  10. https://oeis.org/A027750/b027750.txt (all (prime or composite or unit) factors of N, flattened rows, the Nth row ends with the term with index "the term N of https://oeis.org/A006218") (limit: 1000)
  11. https://oeis.org/A054841/b054841.txt (the nth rightmost decimal digit represents the exponent of the nth prime, but only for the exponents ≤ 9, thus the number will be incorrect if N is divisible by a 10th power (> 1), thus, the number is incorrect only for 9 values of N, namely N = 1024, 2048, 3072, 4096, 5120, 6144, 7168, 8192, 9216, since the limit of N is 10000) (limit: 10000)
  12. http://factorzone.tripod.com/factors.htm (all (prime or composite or unit) factors of N) (limit: 1100)
  13. http://functions.wolfram.com/NumberTheoryFunctions/Divisors/03/02 (all (prime or composite or unit) factors of N) (limit: 50)
  14. https://www.asahi-net.or.jp/~KC2H-MSM/cn/old/p01n002.txt (limit: 1001)
  15. https://en.wikipedia.org/wiki/Table_of_prime_factors (limit: 1000)
  16. https://en.wikipedia.org/wiki/Table_of_divisors (all (prime or composite or unit) factors of N) (limit: 1000)
  17. http://factordb.com/index.php?query=n&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (from factordb) (the limit in the list is 199, but you can change the number to get the list of the factorizations of larger integers (at most 1000000000000000000000000000198, since the start value of n in factordb must be ≤ 999999999999999999999999999999))
  18. http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=20001&FExp=1&TExp=1&c0=-&EN=&LM= (from an online factor database for numbers of the form bn±1) (the limit in the list is 20000, but you can change the number to get the list of the factorizations of larger integers (at most 1099999, since the base b in http://myfactors.mooo.com/ must be ≤ 1100000))
  19. http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=20001&FExp=1&TExp=1&c0=%2B&EN=&LM= (from an online factor database for numbers of the form bn±1) (the limit in the list is 20002, but you can change the number to get the list of the factorizations of larger integers (at most 1100001, since the base b in http://myfactors.mooo.com/ must be ≤ 1100000))

Lists of small integers in various bases:

  1. https://sites.google.com/view/integer-in-various-base (bases 2 ≤ b ≤ 36) (limit: 12500)
  2. https://math.tools/table/bases-table/ (bases 2 ≤ b ≤ 35) (use lower case letters instead of upper case letters) (limit: 256)
  3. https://math.tools/table/base/ (bases 2 ≤ b ≤ 64) (limit: 64, together with 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000)
  4. https://linesegment.web.fc2.com/application/math/numbers/RadixConversion.html (in Japanese) (bases 2 ≤ b ≤ 36 and b = 62) (limit: 1024 for b = 2 and b = 4, 2187 for b = 3, 3125 for b = 5, b4 for 6 ≤ b ≤ 8, 729 for b = 9, 3993 for b = 11, 3456 for b = 12, 4394 for b = 13, b3 for 14 ≤ b ≤ 17, b2 for 18 ≤ b ≤ 36, 3843 for b = 62)
  5. https://en.wikipedia.org/wiki/Table_of_bases (bases 2 ≤ b ≤ 36) (limit: 1296)

Also, programs related to this research: (some of these programs can also be downloaded in http://www.fermatsearch.org/download.php or https://www.mersenne.org/download/freeware.php or https://download.mersenne.ca/) (some of these programs need to use GMP (https://gmplib.org/))

Primality (or probable primality) testing (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/index.html) programs (https://www.rieselprime.de/ziki/Primality_testing_program):

  1. LLR (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64)
  2. PFGW (https://sourceforge.net/projects/openpfgw/, https://t5k.org/bios/page.php?id=175, https://www.rieselprime.de/ziki/PFGW, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/pfgw_win_4.0.3)
  3. Primo (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309)
  4. Proth.exe (https://t5k.org/programs/gallot/, https://t5k.org/bios/page.php?id=411, https://www.rieselprime.de/ziki/Proth.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/proth)
  5. CHG (https://www.mersenneforum.org/attachment.php?attachmentid=21133&d=1571237465, https://t5k.org/bios/page.php?id=797, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/CHG)

Sieving (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.primegrid.com/stats_psp_sieve.php, https://www.primegrid.com/stats_pps_sieve.php, https://www.primegrid.com/stats_sr5_sieve.php, https://www.primegrid.com/stats_cw_sieve.php, https://www.primegrid.com/sieving/rsp/) programs (https://www.rieselprime.de/ziki/Sieving_program):

  1. SRSieve (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, https://www.mersenneforum.org/showpost.php?p=631129&postcount=1, http://www.rieselprime.de/dl/CRUS_pack.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve_1.8.2, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve-other-programs, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2cl.exe)
  2. MTSieve (https://sourceforge.net/projects/mtsieve/, https://www.mersenneforum.org/rogue/mtsieve.html, https://download.mersenne.ca/mtsieve, https://t5k.org/bios/page.php?id=449, https://www.rieselprime.de/ziki/Mtsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/mtsieve_2.4.8)
  3. NewPGen (https://t5k.org/programs/NewPGen/, https://t5k.org/bios/page.php?id=105, https://www.rieselprime.de/ziki/NewPGen, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/newpgen, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/newpgenlinux)

Integer factoring (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) programs (https://www.rieselprime.de/ziki/Factoring_program): (for more integer factoring programs see https://www.mersenneforum.org/showthread.php?t=3255)

  1. GMP-ECM (https://web.archive.org/web/20210803045418/https://gforge.inria.fr/projects/ecm, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/ecm704dev-svn2990-win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/ecm704dev-svn2990-linux64, https://www.rieselprime.de/ziki/GMP-ECM)
  2. MSieve (https://sourceforge.net/projects/msieve, https://download.mersenne.ca/msieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/msieve153_win64)
  3. GGNFS (http://sourceforge.net/projects/ggnfs, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/GGNFS)
  4. CADO-NFS (https://web.archive.org/web/20210506173015/http://cado-nfs.gforge.inria.fr/index.html, https://www.rieselprime.de/ziki/CADO-NFS, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/cado-nfs-2.3.0)
  5. YAFU (http://bbuhrow.googlepages.com/home, https://github.com/bbuhrow/yafu)
  6. YTools (https://github.com/bbuhrow/ytools)
  7. YSieve (https://github.com/bbuhrow/ysieve)

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