(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 for broken links (using the wayback machine (https://web.archive.org/, https://en.wikipedia.org/wiki/Wayback_Machine)), 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_cw_sieve.php) 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 ".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 the only one exception: the page "OEIS sequences needing factors" (https://oeis.org/wiki/OEIS_sequences_needing_factors) in the OEIS 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)), 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://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://www.kwuntung.net/hkunit/base/base.php (in Chinese), https://linesegment.web.fc2.com/application/math/numbers/RadixConversion.html (in Japanese), also see https://t5k.org/notes/words.html for the English words which are prime numbers when viewed as a number base 36, 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 = m^r 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 = m^r 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 2805222×25^2805222+1 to 2805222×5^5610444+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 2622×11^1621920−1 (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://mersenneforum.org/showpost.php?p=121374&postcount=1 and https://mersenneforum.org/showpost.php?p=643173&postcount=103, 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://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, 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/), 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 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), 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 y ∈ L such that x is a subsequence of y}
• sup(L) = {x ∈ Σ* : there exists y ∈ L 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)) Σ_b = {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):

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 = 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)

• b = 2: binary (https://en.wikipedia.org/wiki/Binary_number, https://www.rieselprime.de/ziki/Binary, https://mathworld.wolfram.com/Binary.html, https://oeis.org/A007088)
• b = 3: ternary (https://en.wikipedia.org/wiki/Ternary_numeral_system, https://mathworld.wolfram.com/Ternary.html, https://oeis.org/A007089)
• b = 4: quaternary (https://en.wikipedia.org/wiki/Quaternary_numeral_system, https://mathworld.wolfram.com/Quaternary.html, https://oeis.org/A007090)
• b = 5: quinary (https://en.wikipedia.org/wiki/Quinary, https://oeis.org/A007091)
• b = 6: senary (https://en.wikipedia.org/wiki/Senary, https://oeis.org/A007092)
• b = 7: septenary (https://web.archive.org/web/20141218113439/https://en.wikipedia.org/wiki/Septenary, https://fr.wikipedia.org/wiki/Syst%C3%A8me_sept%C3%A9naire (in Franch), https://ja.wikipedia.org/wiki/%E4%B8%83%E9%80%B2%E6%B3%95 (in Japanese), https://oeis.org/A007093)
• b = 8: octal (https://en.wikipedia.org/wiki/Octal, https://mathworld.wolfram.com/Octal.html, https://oeis.org/A007094)
• b = 9: nonary (https://web.archive.org/web/20141218113440/https://en.wikipedia.org/wiki/Nonary, https://fr.wikipedia.org/wiki/Syst%C3%A8me_nonaire (in Franch), https://ja.wikipedia.org/wiki/%E4%B9%9D%E9%80%B2%E6%B3%95 (in Japanese), https://oeis.org/A007095)
• b = 10: decimal (the standard system) (https://en.wikipedia.org/wiki/Decimal, https://www.rieselprime.de/ziki/Decimal, https://mathworld.wolfram.com/Decimal.html, https://oeis.org/A001477)
• b = 11: undecimal (https://web.archive.org/web/20141218113441/https://en.wikipedia.org/wiki/Base_11, https://fr.wikipedia.org/wiki/Syst%C3%A8me_und%C3%A9cimal (in Franch), https://ja.wikipedia.org/wiki/%E5%8D%81%E4%B8%80%E9%80%B2%E6%B3%95 (in Japanese), https://oeis.org/A055649)
• b = 12: duodecimal (https://en.wikipedia.org/wiki/Duodecimal, https://mathworld.wolfram.com/Duodecimal.html, https://oeis.org/A049872)
• b = 13: tridecimal (https://web.archive.org/web/20141218113443/https://en.wikipedia.org/wiki/Base_13, https://fr.wikipedia.org/wiki/Syst%C3%A8me_trid%C3%A9cimal (in Franch), https://ja.wikipedia.org/wiki/%E5%8D%81%E4%B8%89%E9%80%B2%E6%B3%95 (in Japanese), https://oeis.org/A055648)
• b = 14: tetradecimal (https://web.archive.org/web/20141218113444/https://en.wikipedia.org/wiki/Tetradecimal, https://oeis.org/A055647)
• b = 15: pentadecimal (https://web.archive.org/web/20141218113445/https://en.wikipedia.org/wiki/Pentadecimal, https://ja.wikipedia.org/wiki/%E5%8D%81%E4%BA%94%E9%80%B2%E6%B3%95 (in Japanese), https://oeis.org/A055646)
• b = 16: hexadecimal (https://en.wikipedia.org/wiki/Hexadecimal, https://mathworld.wolfram.com/Hexadecimal.html, https://oeis.org/A055645)
• b = 20: vigesimal (https://en.wikipedia.org/wiki/Vigesimal, https://mathworld.wolfram.com/Vigesimal.html, https://oeis.org/A055644)
• b = 24: quadravigesimal (https://ja.wikipedia.org/wiki/%E4%BA%8C%E5%8D%81%E5%9B%9B%E9%80%B2%E6%B3%95 (in Japanese))
• b = 36: hexatrigesimal (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 Franch), https://ja.wikipedia.org/wiki/%E4%B8%89%E5%8D%81%E5%85%AD%E9%80%B2%E6%B3%95 (in Japanese))

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(x)×f(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) 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).

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 Z_n) 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) 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://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/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(2^22...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+j−q)² − ((g×k+2×g+k+1)×(h+j)+h−z)² − (2×n+p+q+z−e)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×(u²−a))²−1)×(n+4×d×y)²+1−(x+c×u)²)² - (n+l+v−y)² − ((a²−1)×l²+1−m²)² − (a×i+k+1−l−i)² − (p+l×(a−n−1)+b×(2×a×n+2×a−n²−2×n−2)−m)² − (q+y×(a−p−1)+s×(2×a×p+2×a−p²−2×p−2)−x)² − (z+p×l×(a−p)+t×(2×a×p−p²−1)−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)

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 S_m 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 e^{γ×(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://mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://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://mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://mersenneforum.org/showthread.php?t=10910, https://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://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)) with k-values < b, 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://mersenneforum.org/showpost.php?p=144991&postcount=1, https://mersenneforum.org/showthread.php?t=10761, https://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 bⁿ−k (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://mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://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://mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://mersenneforum.org/showthread.php?t=10910, https://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://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 b^r" covers "finding the smallest prime of the form k×bⁿ+1 and k×bⁿ−1 and bⁿ+k and bⁿ−k (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://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://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((b−eulerphi(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), 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 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), while neither is true 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://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://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://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")

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%A60007.2C_333%E2%80%A63331.2C_7111%E2%80%A6111.2C_or_3444%E2%80%A64447_in_base_10.3F 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://mersenneforum.org/showpost.php?p=564786&postcount=3 and https://mersenneforum.org/showpost.php?p=461665&postcount=7 and https://mersenneforum.org/showpost.php?p=344985&postcount=293 and https://mersenneforum.org/showpost.php?p=625978&postcount=1027)

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) e^{γ×(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, e^{γ×(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 e^{γ×(b−1)×eulerphi(b)} has exponential growth for b, and "largest minimal prime base b" is roughly b^{eγ×(b−1)×eulerphi(b)}, which has double exponential growth (https://en.wikipedia.org/wiki/Double_exponential_function) for b. (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)

(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) e^{γ×(b−1)×eulerphi(b)}, and e^{γ×(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.

(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 (m ≠ n) 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://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://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://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/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_7.3F. (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, 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.3F for the proof)

The lengths of the minimal primes in base b 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 13 family A3_nA, this family has been searched to n = 500000 with no prime or probable prime found, we can use ">500000" for the n of the smallest prime in the base 13 family A3_nA (while for the n of the smallest prime in the base 13 family 95_n, it is 197420), ">500000" includes infinity (since infinity is > 500000) but does not includes 0 or −1, it is still possible that there is no prime in the base 13 family A3_nA, 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, http://www.utm.edu/~caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.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. 2^82589933−1, 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), 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. 2^82589933−1, 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 19, 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 = 23 (or 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 = 23 (or 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) e^{γ×(b−1)×eulerphi(b)}, and e^{γ×(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)

(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/primesize.txt, https://stdkmd.net/nrr/prime/primesize.zip, https://stdkmd.net/nrr/prime/primecount.htm, https://stdkmd.net/nrr/prime/primecount.txt, https://stdkmd.net/nrr/prime/primedifficulty.htm, 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)

• x{0}y (see https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf) for the case b = 2, this is exactly the case x×bⁿ+y, the case y = 1 is exactly x×bⁿ+1 (exactly the generalized Sierpinski problem to base b with k = x), also see https://math.stackexchange.com/questions/805465/is-my-proof-correct-regarding-the-non-primality-of-2-cdot-17a-1, and the case x = 1 is exactly bⁿ+y (exactly the generalized dual Sierpinski problem to base b with k = y), see https://mathoverflow.net/questions/268918/density-of-primes-in-sequences-of-the-form-anb and https://math.stackexchange.com/questions/597234/least-prime-of-the-form-38n31 and https://math.stackexchange.com/questions/760966/is-324455n-ever-prime)
• x{y} (unless y = 1) (see https://stdkmd.net/nrr/abbbb.htm for the case b = 10)
• {x}y (unless x = 1) (see https://stdkmd.net/nrr/aaaab.htm for the case b = 10)
• x{0}yz (unless there is a prime of the form x{0}y or x{0}z) (this is exactly the case x×bⁿ+yz, the case x = 1 is exactly bⁿ+yz (exactly the generalized dual Sierpinski problem to base b with k = yz))
• xy{0}z (unless there is a prime of the form x{0}z or y{0}z) (this is exactly the case xy×bⁿ+z, the case z = 1 is exactly xy×bⁿ+1 (exactly the generalized Sierpinski problem to base b with k = xy))
• xy{x} (unless either x = 1 or there is a prime of the form y{x} (or both)) (see https://stdkmd.net/nrr/abaaa.htm for the case b = 10)
• {x}yx (unless either x = 1 or there is a prime of the form {x}y (or both)) (see https://stdkmd.net/nrr/aaaba.htm for the case b = 10)

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/records.htm#nrprime, https://stdkmd.net/nrr/records.htm#nrprp, https://stdkmd.net/nrr/records.htm#nrpprime, 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):

• Prove that all elements in S, when read as base b representation, are primes > b.
• Prove that all proper subsequence of all elements in S, when read as base b representation, which are > b, are composite (https://en.wikipedia.org/wiki/Composite_number, https://t5k.org/glossary/xpage/Composite.html, https://www.rieselprime.de/ziki/Composite_number, https://mathworld.wolfram.com/CompositeNumber.html, https://oeis.org/A002808).
• Prove that all primes > b, when written in base b, contain at least one element in S as subsequence (equivalently, prove that all strings not containing any element in S as subsequence, when read as base b representation, which are > b, are composite).

("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), P ⟶ Q 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. "X ◁ Y" 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) 11 ◁ p, 31 ◁ p, 41 ◁ p, 61 ◁ p, or 71 ◁ p. 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 251 ◁ p. If 8 ◁ y, then 281 ◁ p. If 9 ◁ y, then 29 ◁ p. Hence we may assume all digits of y are 0 or 2.

If 22 ◁ y, then 2221 ◁ p. 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 20201 ◁ p. 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 521 ◁ p. If 9 ◁ y, then 59 ◁ p. Hence we may assume all digits of y are 0, 5, or 8.

If 05 ◁ y, then 5051 ◁ p. If 08 ◁ y, then 5081 ◁ p. If 50 ◁ y, then 5501 ◁ p. If 58 ◁ y, then 5581 ◁ p. If 80 ◁ y, then 5801 ◁ p. If 85 ◁ y, then 5851 ◁ p. 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 821 ◁ p. If 8 ◁ y, then 881 ◁ p. If 9 ◁ y, then 89 ◁ p. Hence we may assume all digits of y are 0 or 5.

If 50 ◁ y, then 8501 ◁ p. 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 991 ◁ p. Hence we may assume all digits of y are 0, 2, 5, or 8.

If 00 ◁ y, then 9001 ◁ p. If 22 ◁ y, then 9221 ◁ p. If 55 ◁ y, then 9551 ◁ p. 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 9851 ◁ p. 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) 13 ◁ p, 23 ◁ p, 43 ◁ p, 53 ◁ p, 73 ◁ p, or 83 ◁ p. 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) 17 ◁ p, 37 ◁ p, 47 ◁ p, 67 ◁ p, or 97 ◁ p. 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 227 ◁ p. If 5 ◁ y, then 257 ◁ p. If 7 ◁ y, then 277 ◁ p. Hence we may assume all digits of y are 0 or 8.

If 08 ◁ y, then 2087 ◁ p. 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 × 40_n1 = 280_n7, so p cannot be prime.

Case 3.2: p begins with 5.

In this case we can write p = 5y7. If 5 ◁ y, then 557 ◁ p. If 7 ◁ y, then 577 ◁ p. If 8 ◁ y, then 587 ◁ p. 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 50207 ◁ p. 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 727 ◁ p. If 5 ◁ y, then 757 ◁ p. If 8 ◁ y, then 787 ◁ p. 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 827 ◁ p. If 5 ◁ y, then 857 ◁ p. If 7 ◁ y, then 877 ◁ p. If 8 ◁ y, then 887 ◁ p. 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) 19 ◁ p, 29 ◁ p, 59 ◁ p, 79 ◁ p, or 89 ◁ p. Hence we may assume all digits of x are 0, 3, 4, 6, or 9.

If 44 ◁ x, then 449 ◁ p. 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 349 ◁ p.

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 409 ◁ p. If 3 ◁ y, then 43 ◁ p. If 9 ◁ y, then 499 ◁ p. 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 × 6_n7 = 46_n9, 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 6469 ◁ p. If 9 ◁ z, then 499 ◁ p. Hence we may assume z is empty.

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

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

If 666 ◁ y, then 666649 ◁ p. If 00000 ◁ y, then 60000049 ◁ p. 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 9049 ◁ p. If 3 ◁ y, then 349 ◁ p. If 6 ◁ y, then 9649 ◁ p. If 9 ◁ y, then 9949 ◁ p. 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 S_m 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 e^{γ×(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://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 https://web.archive.org/web/20170515153924/http://bitc.bme.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)) 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/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php)) 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_cw_sieve.php) 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://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/A005936, https://oeis.org/A005938, 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/A063994, https://oeis.org/A194946, https://oeis.org/A195327, https://oeis.org/A002997, https://oeis.org/A191311, https://oeis.org/A090086, https://oeis.org/A007535) (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://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://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://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://mersenneforum.org/showpost.php?p=602219&postcount=35, https://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://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/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php). 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/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php).

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 b^k×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://mersenneforum.org/showpost.php?p=96560&postcount=99, https://mersenneforum.org/showpost.php?p=96651&postcount=101, https://mersenneforum.org/showthread.php?t=21916, https://mersenneforum.org/showpost.php?p=196598&postcount=492, https://mersenneforum.org/showpost.php?p=203083&postcount=149, https://mersenneforum.org/showpost.php?p=206065&postcount=192, https://mersenneforum.org/showpost.php?p=208044&postcount=260, https://mersenneforum.org/showpost.php?p=210533&postcount=336, https://mersenneforum.org/showpost.php?p=452132&postcount=66, https://mersenneforum.org/showpost.php?p=451337&postcount=32, https://mersenneforum.org/showpost.php?p=208852&postcount=227, https://mersenneforum.org/showpost.php?p=232904&postcount=604, https://mersenneforum.org/showpost.php?p=383690&postcount=1, https://mersenneforum.org/showpost.php?p=207886&postcount=253, https://mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php (e.g. for the family 3{8} in base 9, type "4*9^n-1", and it will tell you that this form can be factored to (2×3ⁿ−1) × (2×3ⁿ+1)), https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/). In fact, if n = 2×m is even, then b^2×m−1 = (b^m−1) × (b^m+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://stdkmd.net/nrr/records.htm#probableprimenumbers, https://stdkmd.net/nrr/repunit/prpfactors.htm, 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))