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README.md
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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 (2<sup>2<sup>7</sup>−1</sup>−1 = 2<sup>127</sup>−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 (2<sup>2<sup>4</sup></sup>+1 = 2<sup>16</sup>+1) → 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 10<sup>60</sup>) → 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_notation) 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_notation) with radix (https://en.wikipedia.org/wiki/Radix, https://t5k.org/glossary/xpage/Radix.html, https://www.rieselprime.de/ziki/Base, https://mathworld.wolfram.com/Radix.html) *b*, as digit (https://en.wikipedia.org/wiki/Numerical_digit, https://www.rieselprime.de/ziki/Digit, https://mathworld.wolfram.com/Digit.html) strings (https://en.wikipedia.org/wiki/String_(computer_science), https://mathworld.wolfram.com/String.html) under the subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html) ordering (https://en.wikipedia.org/wiki/Partially_ordered_set, https://mathworld.wolfram.com/PartialOrder.html, https://mathworld.wolfram.com/PartiallyOrderedSet.html), for 2 ≤ *b* ≤ 36 (I stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the symbols 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 Arabic numerals (https://en.wikipedia.org/wiki/Arabic_numerals, https://mathworld.wolfram.com/ArabicNumeral.html) and the 26 Latin letters (https://en.wikipedia.org/wiki/Latin_alphabet, https://en.wikipedia.org/wiki/ISO_basic_Latin_alphabet), also the 36 characters in Morse code (https://en.wikipedia.org/wiki/Morse_code) (see https://upload.wikimedia.org/wikipedia/commons/b/b5/International_Morse_Code.svg), also the 36 characters in NATO phonetic alphabet (https://en.wikipedia.org/wiki/NATO_phonetic_alphabet, http://www.nato.int/cps/en/natohq/declassified_136216.htm, https://web.archive.org/web/20190626195301/https://www.icao.int/secretariat/PostalHistory/annex_10_aeronautical_telecommunications.htm) (see https://upload.wikimedia.org/wikipedia/commons/e/e0/FAA_Phonetic_and_Morse_Chart2.svg), also the 36 characters in international maritime signal flags (https://en.wikipedia.org/wiki/International_maritime_signal_flags, http://www.quadibloc.com/other/flaint.htm)), i.e. bases 2 ≤ *b* ≤ 36 are case-insensitive (https://en.wikipedia.org/wiki/Case-insensitive) alphanumeric (https://en.wikipedia.org/wiki/Alphanumericals) numeral system using ASCII (https://en.wikipedia.org/wiki/ASCII) characters (https://en.wikipedia.org/wiki/Character_(computing)), while bases *b* > 36 are not, references: http://www.tonymarston.net/php-mysql/converter.html, https://www.dcode.fr/base-36-cipher, http://www.urticator.net/essay/5/567.html, http://www.urticator.net/essay/6/624.html, https://docs.python.org/3/library/functions.html#int, https://numpy.org/doc/stable/reference/generated/numpy.base_repr.html, https://reference.wolfram.com/language/ref/BaseForm.html, https://support.microsoft.com/en-us/office/base-function-2ef61411-aee9-4f29-a811-1c42456c6342, https://www.cut-the-knot.org/recurrence/word_primes.shtml, https://oeis.org/A072922, https://oeis.org/A073421, https://oeis.org/A002488 (the Alonso del Arte comment in Jul 01 2012), https://en.wikipedia.org/wiki/Base36, https://web.archive.org/web/20150320103231/https://en.wikipedia.org/wiki/Base_36, https://fr.wikipedia.org/wiki/Syst%C3%A8me_%C3%A0_base_36 (in French), https://zh.wikipedia.org/wiki/%E4%B8%89%E5%8D%81%E5%85%AD%E8%BF%9B%E5%88%B6 (in Chinese), https://ja.wikipedia.org/wiki/%E4%B8%89%E5%8D%81%E5%85%AD%E9%80%B2%E6%B3%95 (in Japanese), https://baseconvert.com/, https://baseconvert.com/high-precision, https://www.calculand.com/unit-converter/zahlen.php?og=Base+2-36&ug=1, http://www.unitconversion.org/unit_converter/numbers.html, http://www.unitconversion.org/unit_converter/numbers-ex.html, http://extraconversion.com/base-number, https://web.archive.org/web/20190629223750/http://thedevtoolkit.com/tools/base_conversion, http://www.kwuntung.net/hkunit/base/base.php (in Chinese), https://linesegment.web.fc2.com/application/math/numbers/RadixConversion.html (in Japanese), also https://jpbenney.blogspot.com/2021/08/pentatrigesimal.html and https://jpbenney.blogspot.com/2021/08/pentatrigesimal-periods.html and https://jpbenney.blogspot.com/2021/08/can-you-recognise-this-list.html (although they use base *b* = 35 instead of base *b* = 36, since they do not use the Latin letter O as a digit), also see https://t5k.org/notes/words.html for the English words which are prime numbers when viewed as a number base 36 (also see https://oeis.org/A038842), also, the pangrams (https://en.wikipedia.org/wiki/Pangram, https://web.archive.org/web/20141012231620/http://en.wikipedia.org/wiki/List_of_pangrams, http://clagnut.com/blog/2380/, http://rinkworks.com/words/pangrams.shtml, https://web.archive.org/web/20160612205649/http://rec-puzzles.org/index.php/Pangram, http://www.cl.cam.ac.uk/~mgk25/ucs/examples/quickbrown.txt, https://web.archive.org/web/20141009121409/http://www.fatrazie.com/EWpangram.html, http://www.fun-with-words.com/pang_example.html, https://web.archive.org/web/20180702041918/http://users.tinyonline.co.uk/gswithenbank/pangrams.htm, https://web.archive.org/web/20131001184201/http://www.p22.com/products/pangramcontest.html, http://dailypangram.tumblr.com/, https://wordsmith.org/pangram/) together with the digits 0−9 will be pandigital numbers (https://en.wikipedia.org/wiki/Pandigital_number, https://mathworld.wolfram.com/PandigitalNumber.html, https://oeis.org/A171102, https://oeis.org/A050288, https://oeis.org/A049363, https://oeis.org/A185122, https://arxiv.org/pdf/2403.20304.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_441.pdf)) in base *b* = 36, e.g. the famous pangram "The quick brown fox jumps over the lazy dog" (https://en.wikipedia.org/wiki/The_quick_brown_fox_jumps_over_the_lazy_dog) together with the digits 0−9 is a pandigital number in base *b* = 36, see http://factordb.com/index.php?showid=1100000002639212386&base=36 (use lower case letters instead of upper case letters) (although this number is not prime (it is divisible by 2), see http://factordb.com/index.php?id=1100000002639212386&open=ecm for its prime factorization), and the shorter pangram "Waltz, bad nymph, for quick jigs vex" together with the digits 0−9 is a pandigital number in base *b* = 36, see http://factordb.com/index.php?showid=1100000007048545780&base=36 (use lower case letters instead of upper case letters) (although this number is not prime (it is divisible by 3), see http://factordb.com/index.php?id=1100000007048545780&open=ecm for its prime factorization), also, the palindromes (https://en.wikipedia.org/wiki/Palindrome) will be palindromic numbers (https://en.wikipedia.org/wiki/Palindromic_number, https://en.wikipedia.org/wiki/Palindromic_prime, https://t5k.org/glossary/xpage/PalindromicPrime.html, https://mathworld.wolfram.com/PalindromicNumber.html, https://mathworld.wolfram.com/PalindromicPrime.html, https://www.numbersaplenty.com/set/palindromic_number/, https://t5k.org/top20/page.php?id=53, https://t5k.org/primes/search.php?Comment=Palindrome&OnList=all&Number=1000000&Style=HTML, https://stdkmd.net/nrr/abbba.htm, https://stdkmd.net/nrr/aabaa.htm, https://stdkmd.net/nrr/prime/prime_nrpl.htm, https://stdkmd.net/nrr/prime/prime_nrpl.txt, https://stdkmd.net/nrr/prime/prime_n2pl.htm, https://stdkmd.net/nrr/prime/prime_n2pl.txt, https://stdkmd.net/nrr/prime/prime_n3pl.htm, https://stdkmd.net/nrr/prime/prime_n3pl.txt, https://stdkmd.net/nrr/prime/prime_pd.htm, https://stdkmd.net/nrr/prime/prime_pd.txt, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#pdprime, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#nrpprime, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#pdprp, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#nrpprp, https://www.asahi-net.or.jp/~KC2H-MSM/mathland/aba/index.htm, http://factordb.com/tables.php?open=3, http://factordb.com/tables.php?open=5, http://factordb.com/tables.php?open=6, https://oeis.org/A002113, https://oeis.org/A002385) in base *b* = 36, e.g. the longest known palindrome word "saippuakivikauppias" is a palindromic number in base *b* = 36, see http://factordb.com/index.php?showid=1100000003910734942&base=36 (use lower case letters instead of upper case letters) (although this number is not prime (it is divisible by 2), see http://factordb.com/index.php?id=1100000003910734942&open=ecm for its prime factorization), and Demetri Martin's Palindrome in https://web.archive.org/web/20150923005512/http://classes.yale.edu/fractals/panorama/Literature/Martin/MartinPalindrome.html is a palindromic number in base *b* = 36, see http://factordb.com/index.php?showid=1100000004709774731&base=36 (use lower case letters instead of upper case letters) (although this number is not prime (it is divisible by 23), see http://factordb.com/index.php?id=1100000004709774731&open=ecm for its prime factorization), **(you can try to convert these numbers to base *b* = 36: 133, 391, 417, 853, 1030, 15238, 35665, 36825, 599609, 620303, 630308, 739172, 957182, 1329077, 1334693, 19353617, 43427410, 816958261, 1421722899, 23508730562, you will get something interesting! A similar example is the fraction 1480479987/1679615 in base *b* = 36, see https://www.reddit.com/r/funny/comments/1dhog4/wolfram_alpha_gives_up/)** also, the digits in bases 2 ≤ *b* ≤ 36 can use either sixteen-segment display (https://en.wikipedia.org/wiki/Sixteen-segment_display) or fourteen-segment display (https://en.wikipedia.org/wiki/Fourteen-segment_display) to show (and thus we may define "dihedral prime" (https://oeis.org/A134996, https://en.wikipedia.org/wiki/Dihedral_prime, https://t5k.org/glossary/xpage/DihedralPrime.html, https://mathworld.wolfram.com/DihedralPrime.html) in these bases *b*, using either sixteen-segment display (https://en.wikipedia.org/wiki/Sixteen-segment_display) or fourteen-segment display (https://en.wikipedia.org/wiki/Fourteen-segment_display)), see https://upload.wikimedia.org/wikipedia/commons/5/5b/Sixteen-segment_display_0-9_A-Z.gif and https://upload.wikimedia.org/wikipedia/commons/1/17/Arabic_number_on_a_14_segment_display.gif and https://upload.wikimedia.org/wikipedia/commons/2/2b/Latin_alphabet_on_a_14_segment_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 = 5<sup>2</sup>, 27 = 3<sup>3</sup>, 32 = 2<sup>5</sup>, 36 = 6<sup>2</sup> (*n* is divisible by all numbers less than or equal to the square root (https://en.wikipedia.org/wiki/Square_root, https://www.rieselprime.de/ziki/Square_root, https://mathworld.wolfram.com/SquareRoot.html) of *n* if and only if *n* is a divisor of 24, and for *n* = 3, 8, 24 (i.e. the unitary divisors (https://en.wikipedia.org/wiki/Unitary_divisor, https://mathworld.wolfram.com/UnitaryDivisor.html) > 1 of 24, or the divisors *d* of 24 such that *d*+1 is square), the smallest non-divisor of *n* is exactly the square root (https://en.wikipedia.org/wiki/Square_root, https://www.rieselprime.de/ziki/Square_root, https://mathworld.wolfram.com/SquareRoot.html) of *n*+1, also, 24 is the largest number *n* such that the smallest composite number coprime to *n* is exactly *n*+1, and all such *n* are exactly 3, 8, 24 (i.e. the unitary divisors (https://en.wikipedia.org/wiki/Unitary_divisor, https://mathworld.wolfram.com/UnitaryDivisor.html) > 1 of 24, or the divisors *d* of 24 such that *d*+1 is square), also, 25 = 5<sup>2</sup> is exactly 24×(24+1)/24, 27 = 3<sup>3</sup> is exactly 24×(8+1)/8, 32 = 2<sup>5</sup> is exactly 24×(3+1)/3, i.e. they are 24×(*d*+1)/*d* for these divisors *d* of 24, and the numbers *d*+1 are all squares), and the main problem in this project in perfect power bases *b* are more interesting since a large minimal prime in base *b* = *m*<sup>*r*</sup> 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*<sup>*r*</sup> 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<sup>2805222</sup>+1 to 2805222×5<sup>5610444</sup>+1 (i.e. converts base 25 = 5<sup>2</sup> 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 2622×121<sup>810960</sup>−1 to 2622×11<sup>1621920</sup>−1 (i.e. converts base 121 = 11<sup>2</sup> to base 11) (see https://t5k.org/primes/page.php?id=119929 for the entry of this prime in the top definitely primes page), see https://www.mersenneforum.org/showpost.php?p=121374&postcount=1 and https://www.mersenneforum.org/showpost.php?p=656659&postcount=1 and https://www.mersenneforum.org/showpost.php?p=643173&postcount=9, and 36 is exactly 24 + (half of 24), besides, the number 36 also has other number theory properties, e.g. it is the smallest perfect power (https://oeis.org/A001597, https://en.wikipedia.org/wiki/Perfect_power, https://mathworld.wolfram.com/PerfectPower.html, https://www.numbersaplenty.com/set/perfect_power/) which is not prime power (https://oeis.org/A246655, https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html), it is the smallest square triangular number (https://en.wikipedia.org/wiki/Square_triangular_number, https://mathworld.wolfram.com/SquareTriangularNumber.html, https://oeis.org/A001110) (i.e. a number which is both square (https://en.wikipedia.org/wiki/Square_number, https://www.rieselprime.de/ziki/Square_number, https://mathworld.wolfram.com/SquareNumber.html, https://www.numbersaplenty.com/set/square_number/, https://oeis.org/A000290) and triangular (https://en.wikipedia.org/wiki/Triangular_number, https://mathworld.wolfram.com/TriangularNumber.html, https://www.numbersaplenty.com/set/triangular_number/, https://oeis.org/A000217)) greater than 1, it is the smallest number greater than 1 which is neither prime power (https://oeis.org/A246655, https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html) nor squarefree (https://en.wikipedia.org/wiki/Square-free_integer, https://mathworld.wolfram.com/Squarefree.html, https://oeis.org/A005117), it is a highly composite number (https://en.wikipedia.org/wiki/Highly_composite_number, https://mathworld.wolfram.com/HighlyCompositeNumber.html, https://www.numbersaplenty.com/set/highly_composite_number/, https://oeis.org/A002182) (highly composite numbers are more suitable for the stopping base *b* since it is more convenient for the fractions (https://en.wikipedia.org/wiki/Fraction, https://mathworld.wolfram.com/Fraction.html) with small denominators, since all fractions with denominators dividing (https://en.wikipedia.org/wiki/Divides, https://t5k.org/glossary/xpage/Divides.html, https://t5k.org/glossary/xpage/Divisor.html, https://www.rieselprime.de/ziki/Factor, https://mathworld.wolfram.com/Divides.html, https://mathworld.wolfram.com/Divisor.html, https://mathworld.wolfram.com/Divisible.html, http://www.numericana.com/answer/primes.htm#divisor) the base *b* have only one digit after the radix point (https://en.wikipedia.org/wiki/Radix_point), this is the generalization of repeating decimal (https://en.wikipedia.org/wiki/Repeating_decimal, https://t5k.org/glossary/xpage/PeriodOfADecimal.html, https://mathworld.wolfram.com/RepeatingDecimal.html) to other bases *b*, also base *b*+*k* will have more same congruence for small modulos (https://en.wikipedia.org/wiki/Integers_modulo_n, https://mathworld.wolfram.com/Mod.html), thus have more same properties in the main problem in this project since they are congruent mod many small numbers (thus no need to repeat them, and stop at such base *b* is better), and for the case for base *b* = 36, it is divisible by 2 and 3, and the number one less than 36 is the product of the next two primes (5 and 7), it can approximate many fractions well for its size), it is a highly abundant number (https://en.wikipedia.org/wiki/Highly_abundant_number, https://oeis.org/A002093), it is a superabundant number (https://en.wikipedia.org/wiki/Superabundant_number, https://mathworld.wolfram.com/SuperabundantNumber.html, https://www.numbersaplenty.com/set/superabundant_number/, https://oeis.org/A004394), all even perfect numbers (https://en.wikipedia.org/wiki/Perfect_number, https://en.wikipedia.org/wiki/List_of_Mersenne_primes_and_perfect_numbers, https://t5k.org/glossary/xpage/PerfectNumber.html, https://www.rieselprime.de/ziki/Perfect_number, https://mathworld.wolfram.com/PerfectNumber.html, https://mathworld.wolfram.com/EvenPerfectNumber.html, https://www.numbersaplenty.com/set/perfect_number/, https://www.numericana.com/answer/numbers.htm#perfect, https://t5k.org/notes/proofs/EvenPerfect.html, https://t5k.org/notes/proofs/Theorem3.html, https://oeis.org/A000396) except 6 end with the digit S in base *b* = 36 (the first twelve perfect numbers (of course all are even, since currently there are no known odd perfect numbers) written in base *b* = 36 are 6, S, DS, 69S, JZ3LS, 3Y26PDS, 1R4ZG4XS, HIO94MA0BJLS, 49GANX6QRHCA3LSAR3NSUI9S, 2C0BVYDSTGQL620GUGC18745PAZSVNZIPDS, 21O3YNZFCCMBFGSF6MZB3QSTL75SELZYD2D9DTAPDS, SPN7SJDRBBR9EKYVZLYMKAIYODC0J4OYKTWFTY3W9V1MZQSXS), and if there exists an odd perfect number (https://web.archive.org/web/20120426061657/http://oddperfect.org/, https://www.lirmm.fr/~ochem/opn/, https://en.wikipedia.org/wiki/Perfect_number, https://en.wikipedia.org/wiki/List_of_Mersenne_primes_and_perfect_numbers, https://t5k.org/glossary/xpage/PerfectNumber.html, https://www.rieselprime.de/ziki/Perfect_number, https://mathworld.wolfram.com/PerfectNumber.html, https://mathworld.wolfram.com/OddPerfectNumber.html, https://maths-people.anu.edu.au/~brent/pub/pub116.html, https://maths-people.anu.edu.au/~brent/pub/pub100.html, https://maths-people.anu.edu.au/~brent/pub/pub106.html, https://maths-people.anu.edu.au/~brent/pd/rpb116a.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_398.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb116.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_399.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb116p.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_400.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb100a.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_401.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_402.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb100s.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_403.pdf), https://www.lirmm.fr/~ochem/opn/opn.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_404.pdf), https://www.lirmm.fr/~ochem/opn/opnf.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_437.pdf), https://math.colgate.edu/~integers/x79/x79.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_438.pdf), https://www.lirmm.fr/~ochem/opn/opn_slide.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_439.pdf), https://www.ams.org/journals/mcom/2007-76-260/S0025-5718-07-02033-9/S0025-5718-07-02033-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_440.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb106i.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_405.pdf)), then it must be end with one of the digits {1,9,D,P} in base *b* = 36, but whether there exists an odd perfect number is a famous open problem (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://t5k.org/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, http://www.numericana.com/answer/open.htm, https://t5k.org/notes/conjectures/), and currently it is only know that there is no odd perfect number ≤ 10<sup>1500</sup>, also all records for highest point of trajectory before reaching 1 in the sequence of the Collatz conjecture (https://en.wikipedia.org/wiki/Collatz_conjecture, https://mathworld.wolfram.com/CollatzProblem.html, http://www.numericana.com/answer/open.htm#collatz, http://www.ericr.nl/wondrous/index.html, http://www.ericr.nl/wondrous/delrecs.html, http://www.ericr.nl/wondrous/pathrecs.html, http://www.ericr.nl/wondrous/comprecs.html, http://www.ericr.nl/wondrous/glidrecs.html, http://www.ericr.nl/wondrous/residues.html, http://www.ericr.nl/wondrous/classrec.html, http://www.ericr.nl/wondrous/strengths.html, http://www.ericr.nl/wondrous/progress.html, http://www.ericr.nl/wondrous/techpage.html, http://www.ericr.nl/wondrous/showsteps.html, https://pcbarina.fit.vutbr.cz/, https://pcbarina.fit.vutbr.cz/path-records.htm, http://sweet.ua.pt/tos/3x_plus_1.html, http://www.rechenkraft.net/yoyo/y_status_col.php, https://oeis.org/A006370, https://oeis.org/A070165, https://oeis.org/A006577, https://oeis.org/A006877, https://oeis.org/A006878, https://oeis.org/A025586, https://oeis.org/A006884, https://oeis.org/A006885) except 1 and 2 end with the digit G in base *b* = 36 (the first twenty record numbers written in base *b* = 36 are 1, 2, G, 1G, 4G, 74G, A4G, UDG, W1G, 5DAG, RDAG, 41YQG, 4URDG, G55MG, TUQSG, 1RBMDG, 201Q4G, 9T814G, PZ884G, 151S9DG, also, the famous "sequence of the Collatz conjecture" starting with the number R (27 in decimal) (i.e. the sequence https://oeis.org/A008884, which gives the record number 74G (9232 in decimal)) written in base *b* = 36 is R, 2A, 15, 3G, 1Q, V, 2M, 1B, 3Y, 1Z, 5Y, 2Z, 8Y, 4H, DG, 6Q, 3D, A4, 52, 2J, 7M, 3T, BG, 5Q, 2V, 8M, 4B, CY, 6H, JG, 9Q, 4V, EM, 7B, LY, AZ, WY, GH, 1DG, OQ, CD, 114, IK, 9A, 4N, DY, 6Z, KY, AH, VG, FQ, 7V, NM, BT, ZG, HQ, 8V, QM, DB, 13Y, JZ, 1NY, TZ, 2HY, 18Z, 3QY, 1VH, 5MG, 2T8, 1EM, PB, 23Y, 11Z, 35Y, 1KZ, 4QY, 2DH, 74G, 3K8, 1S4, W2, G1, 1C4, O2, C1, 104, I2, 91, R4, DK, 6S, 3E, 1P, 54, 2K, 1A, N, 1Y, Z, 2Y, 1H, 4G, 28, 14, K, A, 5, G, 8, 4, 2, 1), also 36 is the number of nonzero numbers (which you can place a bet, you cannot place a bet on the zero numbers) in a roulette (https://en.wikipedia.org/wiki/Roulette), also the famous 36 officers problem (https://en.wikipedia.org/wiki/Thirty-six_officers_problem, https://mathworld.wolfram.com/36OfficerProblem.html) (36 is the only number beside 4 such that this problem has no solutions), also 36 is the largest number *n*<sup>2</sup> (*n*<sup>2</sup> is the total number of squares in an *n*×*n* board) such that it is possible to color all *n*<sup>2</sup> squares in an *n*×*n* board with black and white with no squares (with all four vertices on the *n*<sup>2</sup> squares in the *n*×*n* board) having all four vertices all black or all white (the total number of squares with all four vertices on the *n*<sup>2</sup> squares in the *n*×*n* board is *n*<sup>2</sup>×(*n*<sup>2</sup>−1)/12 (https://oeis.org/A002415), e.g. the total number of squares with all four vertices on the 36 squares in the 6×6 board is 105), also 36 is the total number of chemical elements (https://en.wikipedia.org/wiki/Chemical_element) in the first 4 periods (https://en.wikipedia.org/wiki/Period_(periodic_table)) of the periodic table (https://en.wikipedia.org/wiki/Periodic_table, https://iupac.org/what-we-do/periodic-table-of-elements/, https://ptable.com/) (i.e. from Hydrogen (H, *Z* = 1, https://en.wikipedia.org/wiki/Hydrogen) to Krypton (Kr, *Z* = 36, https://en.wikipedia.org/wiki/Krypton)), and 4 is the smallest *n* such that the chemical elements in the first *n* periods of the periodic table contain chemical elements in all 18 groups (https://en.wikipedia.org/wiki/Group_(periodic_table)), i.e. the chemical elements in the first 4 periods are exactly the chemical elements with atomic numbers (https://en.wikipedia.org/wiki/Atomic_number) *Z* ≤ 36 (the atomic number is also related to number theory, e.g. there are 92 "atomic elements" for the look-and-say sequence (https://en.wikipedia.org/wiki/Look-and-say_sequence, https://mathworld.wolfram.com/LookandSaySequence.html, http://www.se16.info/js/looknsay.htm, https://oeis.org/A001155, https://oeis.org/A005150, https://oeis.org/A006751, https://oeis.org/A006715, https://oeis.org/A049064, https://oeis.org/A001387, https://oeis.org/A001388, https://oeis.org/A001389, https://oeis.org/A045918) in bases *b* ≥ 4 (although in base *b* = 2 there are only 13 "atomic elements" and its Conway's constant (the limit of the ratio of two consecutive terms of the look-and-say sequence with any start number except 111, including the limit of the ratio of two consecutive terms in the sequences https://oeis.org/A001609 and https://oeis.org/A049194) is the root of *x*<sup>3</sup>−*x*<sup>2</sup>−1 (i.e. the number https://oeis.org/A092526), see https://www.math.uni-bielefeld.de/~sillke/SEQUENCES/series001 and https://www.nathanieljohnston.com/2010/11/the-binary-look-and-say-sequence/ and https://arxiv.org/pdf/2004.06414.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_442.pdf), and in base *b* = 3 there are only 27 "atomic elements" and its Conway's constant (the limit of the ratio of two consecutive terms of the look-and-say sequence with any start number except 22, including the limit of the ratio of two consecutive terms in the sequence https://oeis.org/A046639) is the root of *x*<sup>3</sup>−*x*−1 (i.e. the number https://oeis.org/A060006), see https://www.nathanieljohnston.com/2011/01/further-variants-of-the-look-and-say-sequence/) (note that the base *b* "Look and Say" number *LS*(*n*,*b*) is small when *n* is a large minimal prime in base *b*, e.g. for the largest minimal prime in decimal (base *b* = 10), i.e. 5000000000000000000000000000027, its "Look and Say" number *LS*(5000000000000000000000000000027,10) is only 152801217, since it can be read as "one 5, twenty-eight 0's, one 2, one 7"), and John Conway named these 92 "atomic elements" after the 92 naturally-occurring chemical elements up to uranium, see https://mathworld.wolfram.com/CosmologicalTheorem.html and http://www.se16.info/js/lands2.htm and https://www.nathanieljohnston.com/2010/10/a-derivation-of-conways-degree-71-look-and-say-polynomial/ and https://web.archive.org/web/20000820121520/http://www.mathsoft.com/asolve/constant/cnwy/cnwy4.gif and https://web.archive.org/web/20000820121758/http://www.mathsoft.com/asolve/constant/cnwy/cnwy5.gif and https://oeis.org/A119566 (note that the lowest polynomial with Conway's constant (the limit of the ratio of two consecutive terms of the look-and-say sequence with any start number except 22, including the limit of the ratio of two consecutive terms in the sequences https://oeis.org/A022471 and https://oeis.org/A005341) (https://oeis.org/A014715) as a root (this polynomial is called Conway's polynomial) has degree 71, thus it has 72 (= double of 36) coefficients (including 0), see https://oeis.org/A137275, and the 71 roots of this polynomial have 36 different complex conjugates (https://en.wikipedia.org/wiki/Complex_conjugate, https://mathworld.wolfram.com/ComplexConjugate.html), since Conway's constant is the only one real root of this polynomial, and a complex number is the complex conjugate of itself if and only if it is a real number), also http://factordb.com/index.php?showid=1100000004405258711&base=36 (use lower case letters instead of upper case letters) is a large number (although this number is not prime, see http://factordb.com/index.php?id=1100000004405258711&open=ecm for its prime factorization), which when written in base *b* = 36, is the concatenation of the atomic numbers (https://en.wikipedia.org/wiki/Atomic_number) and the chemical symbol (https://en.wikipedia.org/wiki/Chemical_symbol) of all currently known chemical elements, i.e. the chemical elements with atomic numbers ≤ 118), also the next number 37 is the smallest irregular prime (https://en.wikipedia.org/wiki/Irregular_prime, https://t5k.org/glossary/xpage/Regular.html, https://mathworld.wolfram.com/IrregularPrime.html, https://t5k.org/top20/page.php?id=26, https://t5k.org/primes/search.php?Comment=^Irregular&OnList=all&Number=1000000&Style=HTML, https://arxiv.org/pdf/0912.2121.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_237.pdf), https://math.dartmouth.edu/~carlp/irreg.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_239.pdf), https://www.ams.org/journals/mcom/1978-32-142/S0025-5718-1978-0491465-4/S0025-5718-1978-0491465-4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_240.pdf), https://www.ams.org/journals/mcom/1974-28-126/S0025-5718-1974-0347727-0/S0025-5718-1974-0347727-0.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_294.pdf), https://www.ams.org/journals/mcom/1975-29-129/S0025-5718-1975-0376606-9/S0025-5718-1975-0376606-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_295.pdf), https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1197511-5/S0025-5718-1993-1197511-5.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_300.pdf), https://oeis.org/A000928) (thus, all prime number ≤ 36 are regular, and 36 is the largest number satisfying this property), also the previous number 35 is the smallest number *n* > 1 such that *gcd*(*n*, *b*<sup>*n*</sup>−*b*) = 1 for some *b* (see https://oeis.org/A121707 and https://oeis.org/A321487 and https://oeis.org/A267999 and https://oeis.org/A306097) and the smallest semiprime (https://en.wikipedia.org/wiki/Semiprime, https://t5k.org/glossary/xpage/Semiprime.html, https://mathworld.wolfram.com/Semiprime.html, https://www.numbersaplenty.com/set/semiprime/, https://oeis.org/A001358) which is not in the range of the primary pretenders (i.e. is the smallest *n* such that *b*<sup>*n*</sup> == *b* mod *n*, for some *b*) (see https://oeis.org/A000790 and https://oeis.org/A108574) and likely to be the largest semiprime (https://en.wikipedia.org/wiki/Semiprime, https://t5k.org/glossary/xpage/Semiprime.html, https://mathworld.wolfram.com/Semiprime.html, https://www.numbersaplenty.com/set/semiprime/, https://oeis.org/A001358) *n* such that *b*<sup>*n*+1</sup> == 1 mod *n* for every *b* coprime to *n* (see https://oeis.org/A208728) and the smallest non-perfect power composite number coprime to 6 (i.e. not divisible by either 2 or 3) and the smallest quasi-Carmichael number (see https://oeis.org/A257750) and the smallest composite squarefree (https://en.wikipedia.org/wiki/Square-free_integer, https://mathworld.wolfram.com/Squarefree.html, https://oeis.org/A005117) number *k* such that *k*<sup>2</sup>−1 is divisible by *p*−1 and *p*+1, where *p* are all the prime factors of *k* (see https://oeis.org/A306685) (all of these properties mean that the number 35+1 = 36 has many divisors), but these are not the main reason, the main reason is base *b* = 36 is the maximum base for which it is possible to write the numbers with the symbols 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 Arabic numerals (https://en.wikipedia.org/wiki/Arabic_numerals, https://mathworld.wolfram.com/ArabicNumeral.html) and the 26 Latin letters (https://en.wikipedia.org/wiki/Latin_alphabet, https://en.wikipedia.org/wiki/ISO_basic_Latin_alphabet)), i.e. bases 2 ≤ *b* ≤ 36 are case-insensitive (https://en.wikipedia.org/wiki/Case-insensitive) alphanumeric (https://en.wikipedia.org/wiki/Alphanumericals) numeral system using ASCII (https://en.wikipedia.org/wiki/ASCII) characters (https://en.wikipedia.org/wiki/Character_(computing)), while bases *b* > 36 are not)**
A string (https://en.wikipedia.org/wiki/String_(computer_science), https://mathworld.wolfram.com/String.html) *x* is a subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html) of another string *y*, if *x* can be obtained from *y* by deleting zero or more of the characters (https://en.wikipedia.org/wiki/Character_(computing)) (in this project, digits (https://en.wikipedia.org/wiki/Numerical_digit, https://www.rieselprime.de/ziki/Digit, https://mathworld.wolfram.com/Digit.html)) in *y*. For example, 514 is a subsequence of 352148, "STRING" is a subsequence of "MEISTERSINGER". In contrast, 758 is not a subsequence of 378259, "ABC" is not a subsequence of "CBACACBA", since the characters (in this project, digits) must be in the same order. The empty string (https://en.wikipedia.org/wiki/Empty_string) *𝜆* is a subsequence of every string. There are 2<sup>*n*</sup> subsequences of a string with length *n*, e.g. the subsequences of 123456 are (totally 2<sup>6</sup> = 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<sub>*b*</sub> = {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):
|*b*|the base *b* representations of the prime numbers (https://en.wikipedia.org/wiki/Prime_number, https://t5k.org/glossary/xpage/Prime.html, https://www.rieselprime.de/ziki/Prime, https://mathworld.wolfram.com/PrimeNumber.html, https://www.numbersaplenty.com/set/prime_number/, http://www.numericana.com/answer/primes.htm#definition, http://irvinemclean.com/maths/pfaq2.htm, https://oeis.org/A000040, https://t5k.org/lists/small/1000.txt, https://t5k.org/lists/small/10000.txt, https://t5k.org/lists/small/100000.txt, https://t5k.org/lists/small/millions/) > *b* (only list the first 1000 elements in the sets)|
|---|---|
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1000110011, 1000111001, 1000111011, 1001000001, 1001001011, 1001010001, 1001010111, 1001011001, 1001011111, 1001100101, 1001101001, 1001101011, 1001110111, 1010000001, 1010000011, 1010000111, 1010001101, 1010010011, 1010010101, 1010100001, 1010100101, 1010101011, 1010110011, 1010111101, 1011000101, 1011001111, 1011010111, 1011011101, 1011100011, 1011100111, 1011101111, 1011110101, 1011111001, 1100000001, 1100000101, 1100010011, 1100011101, 1100101001, 1100101011, 1100110101, 1100110111, 1100111011, 1100111101, 1101000111, 1101010101, 1101011001, 1101011011, 1101011111, 1101101101, 1101110001, 1101110011, 1101110111, 1110001011, 1110001111, 1110010111, 1110100001, 1110101001, 1110101101, 1110110011, 1110111001, 1111000111, 1111001011, 1111010001, 1111010111, 1111011111, 1111100101, 1111110001, 1111110101, 1111111011, 1111111101, 10000000111, 10000001001, 10000001111, 10000011001, 10000011011, 10000100101, 10000100111, 10000101101, 10000111111, 10001000011, 10001000101, 10001001001, 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111101111111, 111110010101, 111110100001, 111110100011, 111110100111, 111110101101, 111110110011, 111110110101, 111110111011, 111111010001, 111111010011, 111111011001, 111111101001, 111111101111, 111111111011, 111111111101, 1000000000011, 1000000001111, 1000000011111, 1000000100001, 1000000100101, 1000000101011, 1000000111001, 1000000111101, 1000000111111, 1000001010001, 1000001101001, 1000001110011, 1000001111001, 1000001111011, 1000010000101, 1000010000111, 1000010010001, 1000010010011, 1000010011101, 1000010100011, 1000010100101, 1000010101111, 1000010110001, 1000010111011, 1000011000001, 1000011001001, 1000011100111, 1000011110001, 1000011110011, 1000011111101, 1000100000101, 1000100001011, 1000100010101, 1000100100111, 1000100101101, 1000100111001, 1000101000101, 1000101000111, 1000101011001, 1000101011111, 1000101100011, 1000101101001, 1000101101111, 1000110000001, 1000110000011, 1000110001101, 1000110011011, 1000110100001, 1000110100101, 1000110100111, 1000110101011, 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1101010110001, 1101010111001, 1101011001001, 1101011001111, 1101011010101, 1101011010111, 1101011100011, 1101011110011, 1101011111011, 1101011111111, 1101100000101, 1101100100011, 1101100100101, 1101100101111, 1101100110001, 1101100110111, 1101100111011, 1101101000001, 1101101000111, 1101101001111, 1101101010101, 1101101011001, 1101101100101, 1101101101011, 1101101110011, 1101101111111, 1101110000011, 1101110010001, 1101110011101, 1101110100111, 1101110111111, 1101111000101, 1101111010001, 1101111010111, 1101111011001, 1101111101111, 1101111110111, 1110000001001, 1110000010011, 1110000011001, 1110000100111, 1110000101011, 1110000101101, 1110000110011, 1110000111101, 1110001000101, 1110001001011, 1110001001111, 1110001010101, 1110001110011, 1110010000001, 1110010001011, 1110010001101, 1110010011001, 1110010100011, 1110010100101, 1110010110101, 1110010110111, 1110011001001, 1110011100001, 1110011110011, 1110011111001, 1110100001001, 1110100011011, 1110100100001, 1110100100011, 1110100110101, 1110100111001, 1110100111111, 1110101000001, 1110101001011, 1110101010011, 1110101011101, 1110101100011, 1110101101001, 1110101110001, 1110101110101, 1110101111011, 1110101111101, 1110110000111, 1110110001001, 1110110010101, 1110110011001, 1110110011111, 1110110100101, 1110110100111, 1110110110011, 1110110110111, 1110111000101, 1110111010111, 1110111011011, 1110111100001, 1110111110101, 1110111111001, 1111000000001, 1111000000111, 1111000001011, 1111000010011, 1111000010111, 1111000100101, 1111000101011, 1111000101111, 1111000111101, 1111001001001, 1111001001101, 1111001001111, 1111001101101, 1111001110001, 1111010001001, 1111010001111, 1111010010101, 1111010100001, 1111010101101, 1111010111011, 1111011000001, 1111011000101, 1111011000111, 1111011001011, 1111011011101, 1111011100011, 1111011101111, 1111011110111, ...|
|3|12, 21, 102, 111, 122, 201, 212, 1002, 1011, 1101, 1112, 1121, 1202, 1222, 2012, 2021, 2111, 2122, 2201, 2221, 10002, 10022, 10121, 10202, 10211, 10222, 11001, 11012, 11201, 11212, 12002, 12011, 12112, 12121, 12211, 20001, 20012, 20102, 20122, 20201, 21002, 21011, 21022, 21101, 21211, 22021, 22102, 22111, 22122, 22212, 22221, 100022, 100112, 100202, 100222, 101001, 101021, 101102, 101111, 101212, 102101, 102112, 102121, 102202, 110021, 110111, 110212, 110221, 111002, 111022, 111121, 111211, 112001, 112012, 112102, 112201, 112212, 120011, 120112, 120121, 120222, 121001, 121021, 121102, 121122, 121221, 122002, 122011, 122022, 122202, 200001, 200012, 200111, 200122, 200212, 201022, 201101, 202001, 202021, 202122, 202212, 210002, 210011, 210101, 210202, 210222, 211012, 211021, 211111, 211201, 211212, 211221, 212101, 212202, 212211, 212222, 220012, 220102, 220111, 220221, 221002, 221022, 221121, 221222, 222021, 222122, 222221, 1000011, 1000101, 1000112, 1000211, 1001001, 1001012, 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12111212, 12111221, 12112011, 12112222, 12120001, 12120021, 12120212, 12121002, 12121112, 12121121, 12121211, 12122021, 12122212, 12122221, 12200002, 12200022, 12200211, 12200222, 12201001, 12201201, 12202121, 12202222, 12210012, 12210021, 12210122, 12210201, 12211002, 12211011, 12211112, 12211202, 12211211, 12212012, 12212021, 12212122, 12212212, 12220011, 12221021, 12221122, 12221201, 12222002, 12222101, 12222121, 12222222, 20000122, 20000212, 20001022, 20001202, 20001211, 20002111, 20002201, 20002212, 20010002, 20010022, 20010222, 20011001, 20011102, 20011221, 20012011, 20012022, 20012101, 20012112, 20020102, 20020111, 20020221, 20021011, 20021202, 20022001, 20022021, 20022111, 20100011, 20100202, 20100211, 20100222, 20101012, 20101021, 20101111, 20101201, 20102002, 20102022, 20102202, 20110012, 20110212, 20110221, 20111011, 20111022, 20111222, 20112021, 20120011, 20120022, 20120101, 20120112, 20120202, 20120211, 20121021, 20121102, 20121221, 20200001, 20200102, 20200122, 20201002, 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21212122, 21212201, 21212221, 21221001, 21221012, 21221111, 21221212, 21222002, 21222022, 21222121, 21222211, 22000021, 22000102, 22000122, 22000201, 22000221, 22001002, 22001022, 22001101, 22001202, 22001211, 22002102, 22002122, 22010101, 22010112, 22010222, 22011111, 22012112, 22012202, 22020111, 22020122, 22021022, 22021121, 22021211, 22021222, 22022012, 22022201, 22022221, 22100011, 22100112, 22100121, 22100222, 22101102, 22101201, 22102002, 22102011, 22102112, 22102211, 22110021, 22110122, 22111112, 22111121, 22111202, 22112001, 22112021, 22112102, 22112201, 22120101, 22120202, 22120222, 22121012, 22121021, 22121111, 22121212, 22122022, 22122101, 22122202, 22122222, 22200012, 22200102, 22200201, 22200221, 22201022, 22201112, 22201121, 22201211, 22202001, 22202021, 22202122, 22202221, 22210211, 22211001, 22211212, 22211221, 22212121, 22212202, 22220001, 22220102, 22221112, 22221211, 22222111, 22222122, 22222201, 100000002, 100000022, 100000101, 100000121, 100000202, 100001102, 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4058, 4063, 4069, 4076, 4081, 4085, 4087, 4098, 40A3, 40A7, 40A9, 4104, 4124, 412A, 4131, 4135, 4151, 4153, 4157, 4168, 416A, 4175, 4179, 41A2, 41A8, 4203, 4207, 4214, 4223, 4252, 4267, 4269, 4274, 4278, 427A, 4283, 4285, 4294, 42A7, 4302, 4306, 4313, 4322, 4328, 4346, 434A, 4351, 4357, 4384, 4388, 4395, 43A4, 43AA, 4405, 4412, 4418, 4429, 4432, 4438, 443A, 4445, 4449, 4454, 4456, 4465, 4467, 4481, 4487, 44A5, 44A9, 450A, 4522, 4548, 4553, 4571, 4575, 4591, 4599, 45A4, 45A8, 4603, 4616, 4621, 4627, 4636, 4638, 4647, 4658, 4665, 4674, 4676, 4685, 4692, 46A3, 4702, 4724, 4726, 472A, 4737, 4742, 4746, 4753, 476A, 4779, 4784, 478A, 4791, 4797, 47A6, 4807, 4809, 4818, 4823, 4829, 4834, 4841, 4847, 4856, 4861, 4863, 4869, 4874, 487A, 4889, 4896, 4908, 4913, 4933, 4935, 4951, 4955, 4962, 4971, 4999, 49A6, 4A12, 4A16, 4A18, 4A27, 4A32, 4A34, 4A3A, 4A43, 4A5A, 4A67, 4A78, 4A94, 4AA9, 5004, 5006, 5017, 5022, 5031, 5033, 5042, 5044, 504A, 5059, 5071, 5075, 5097, 5099, 5103, 5105, 5114, 5116, 5125, 5143, 5147, 5149, 5152, 515A, 5174, 517A, 5185, 5187, 5198, 5202, 520A, 5213, 5219, 5246, 5248, 5257, 5259, 5264, 5268, 5273, 5279, 5286, 5291, 5295, 52A6, 5301, 5309, 531A, 5323, 5336, 5347, 5356, 5378, 5383, 5394, 539A, 53A1, 5411, 5419, 5435, 5444, 544A, 5462, 5466, 5468, 5473, 5482, 548A, 5495, 5499, 54A4, 5521, 5534, 5543, 5545, 5556, 5565, 5567, 5581, 5583, 559A, 5611, 5628, 5633, 5648, 5664, 566A, 5671, 5688, 5691, 5697, 5699, 56A8, 5705, 5714, 571A, 5725, 5732, 5736, 5741, 5743, 5752, 5754, 5765, 5769, 5774, 577A, 5781, 5792, 5796, 57A9, 5815, 5819, 5824, 5842, 5846, 5853, 5859, 5862, 586A, 5873, 5886, 5891, 5895, 58A8, 5909, 5912, 5914, 5941, 5945, 5967, 5972, 5978, 5989, 599A, 5A02, 5A08, 5A11, 5A13, 5A17, 5A33, 5A39, 5A4A, 5A57, 5A62, 5A66, 5A77, 5A79, ...|
|12|11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 105, 107, 111, 117, 11B, 125, 12B, 131, 13B, 141, 145, 147, 157, 167, 16B, 171, 175, 17B, 181, 18B, 195, 19B, 1A5, 1A7, 1B1, 1B5, 1B7, 205, 217, 21B, 221, 225, 237, 241, 24B, 251, 255, 25B, 267, 271, 277, 27B, 285, 291, 295, 2A1, 2AB, 2B1, 2BB, 301, 307, 30B, 315, 321, 325, 327, 32B, 33B, 347, 34B, 357, 35B, 365, 375, 377, 391, 397, 3A5, 3AB, 3B5, 3B7, 401, 40B, 415, 41B, 421, 427, 431, 435, 437, 447, 455, 457, 45B, 465, 46B, 471, 481, 485, 48B, 497, 4A5, 4B1, 4BB, 507, 511, 517, 51B, 527, 531, 535, 541, 545, 557, 565, 575, 577, 585, 587, 58B, 591, 59B, 5B1, 5B5, 5B7, 5BB, 611, 615, 617, 61B, 637, 63B, 647, 655, 661, 665, 66B, 675, 687, 68B, 695, 69B, 6A7, 6B1, 701, 705, 70B, 711, 71B, 721, 727, 735, 737, 745, 747, 751, 767, 76B, 771, 775, 77B, 785, 791, 797, 7A1, 7BB, 801, 80B, 817, 825, 82B, 835, 841, 851, 855, 85B, 865, 867, 871, 881, 88B, 8A5, 8A7, 8AB, 8B5, 8B7, 901, 905, 907, 90B, 91B, 921, 927, 955, 95B, 965, 971, 987, 995, 9A7, 9AB, 9B1, 9B5, 9BB, A07, A0B, A11, A17, A27, A35, A37, A3B, A41, A45, A4B, A5B, A6B, A77, A87, A91, A95, A9B, AA7, AAB, AB7, ABB, B11, B15, B1B, B21, B25, B2B, B31, B37, B45, B61, B67, B6B, B71, B91, B95, B97, BA5, BB5, BB7, 1005, 1011, 1017, 1021, 1027, 1041, 1047, 104B, 1051, 1061, 106B, 107B, 1087, 109B, 10B1, 10B7, 10BB, 1101, 1105, 1107, 1115, 1125, 112B, 1135, 114B, 1151, 1165, 1167, 1185, 118B, 1197, 11A1, 11A5, 11A7, 11AB, 11B7, 1201, 120B, 1211, 121B, 1231, 123B, 1245, 1255, 1257, 125B, 1261, 126B, 127B, 1281, 1295, 1297, 12A1, 12A5, 12A7, 12B5, 1301, 1317, 1337, 133B, 1345, 1351, 1365, 1367, 136B, 1377, 138B, 1391, 1395, 13A1, 13A7, 13B1, 13B5, 1405, 1407, 1425, 142B, 1431, 1437, 143B, 1445, 1457, 1461, 1465, 1467, 1471, 1475, 147B, 148B, 1495, 149B, 14B1, 14B5, 14BB, 150B, 1517, 1521, 1525, 1547, 1561, 156B, 1577, 157B, 1585, 1587, 1591, 15AB, 15BB, 1601, 1615, 1621, 1625, 1635, 1647, 1655, 1657, 165B, 1667, 1671, 1677, 167B, 1681, 1685, 168B, 1697, 169B, 16A1, 16A7, 16B5, 16B7, 1705, 1711, 1715, 1727, 1735, 1745, 1747, 1751, 1755, 1757, 176B, 1781, 1785, 178B, 1797, 17A1, 17A5, 17BB, 1807, 1815, 181B, 1825, 1831, 183B, 184B, 1861, 1865, 186B, 1875, 1877, 189B, 18A1, 18AB, 18B7, 18BB, 1911, 1915, 1921, 1931, 1937, 1947, 194B, 1955, 1971, 197B, 1981, 1995, 19B7, 19BB, 1A01, 1A11, 1A17, 1A1B, 1A2B, 1A35, 1A41, 1A45, 1A51, 1A6B, 1A71, 1A75, 1A77, 1A87, 1AAB, 1AB1, 1AB7, 1B01, 1B07, 1B0B, 1B15, 1B17, 1B27, 1B2B, 1B3B, 1B41, 1B4B, 1B51, 1B65, 1B67, 1B7B, 1B85, 1BA1, 1BB5, 2001, 2005, 2007, 200B, 2011, 202B, 2037, 2047, 2051, 205B, 2061, 2065, 206B, 2071, 2077, 2085, 2087, 2097, 20A5, 20A7, 20B5, 2107, 2111, 2115, 211B, 2127, 2131, 2137, 214B, 215B, 2161, 2165, 2177, 2181, 2185, 2191, 219B, 21A7, 21B1, 21B7, 2215, 221B, 2221, 222B, 2241, 2245, 224B, 2265, 2267, 2275, 2287, 228B, 2291, 229B, 22B1, 22B5, 2301, 2317, 231B, 2325, 2327, 232B, 2335, 2337, 2347, 234B, 2367, 2385, 2395, 2397, 239B, 23A5, 23AB, 23B1, 23B7, 2415, 2417, 2421, 2435, 243B, 244B, 2451, 2457, 2467, 247B, 2481, 2485, 248B, 24A1, 24A5, 24A7, 2501, 2521, 252B, 2535, 2537, 2545, 2547, 2555, 2557, 2565, 256B, 2571, 257B, 2581, 258B, 2595, 25A1, 2607, 2615, 2617, 2625, 2631, 2637, 2645, 265B, 2665, 2675, 2685, 2687, 26A1, 26A7, 26AB, 26B5, 26BB, 2715, 2717, 2725, 2737, 2741, 2745, 2747, 274B, 276B, 2771, 2781, 2787, 279B, 27A7, 27B1, 27B7, 2811, 2825, 2827, 282B, 2835, 2837, 2841, 2847, 2855, 285B, 286B, 287B, 2895, 2897, 28A1, 28A5, 28BB, 2907, 2927, 292B, 2931, 2935, 293B, 2941, 2951, 2955, 2967, 2991, 299B, 29A5, 29B5, 2A07, 2A11, 2A1B, 2A2B, 2A31, 2A35, 2A3B, 2A47, 2A51, 2A5B, 2A61, 2A65, 2A77, 2A81, 2A87, 2A8B, 2A95, 2A97, 2AA5, 2AA7, 2ABB, 2B0B, 2B17, 2B31, 2B35, 2B3B, 2B4B, 2B51, 2B57, 2B61, 2B67, 2B8B, 2B95, 2BA7, 2BAB, 2BB7, 3005, 3011, 3021, 3037, 303B, 3041, 3045, 3065, 3075, 307B, 3081, 3095, 309B, 30A5, 30B7, 3105, 3117, 311B, 3145, 314B, 3155, 315B, 3167, 3171, 3175, 3177, 3187, 3191, 3195, 3197, 31A1, 31BB, 3205, 3207, 320B, 3225, 3227, 322B, 323B, 3241, 3247, 324B, 3271, 3277, 3281, 3285, 3291, 329B, 3307, 331B, 3321, 3327, 332B, 3331, 3335, 3337, 3345, 3357, 3361, 3365, 3371, 337B, 3385, 33A1, 33A5, 33A7, 33B1, 3417, 341B, 3427, 3435, 343B, 3445, 3451, 3457, 3467, 346B, 3475, 3477, 3481, 3485, 348B, 3491, 349B, 34A1, 34B5, 34BB, 3517, 351B, 352B, 3541, 3565, 356B, 3587, 358B, 35A5, 35B1, 35B7, 35BB, 3605, 3617, 3621, 3627, 3635, 3637, 3645, 3655, 3661, 366B, 3671, 367B, 3687, 3697, 36A5, 3705, 3707, 370B, 3717, 3721, 3725, 3731, 3747, 3755, 375B, 3765, 3767, 3771, 377B, 378B, 3791, 379B, 37A5, 37AB, 37B5, 3801, 3807, 3815, 381B, 3821, 3827, 3831, 3837, 3845, 3851, 3871, 3877, 3895, 3897, 38B1, 38B5, 3901, 390B, 3935, 3941, 3957, 395B, 3961, 396B, 3975, 3977, 3981, 3985, 399B, 39A7, 39B7, 3A11, 3A25, 3A2B, 3A31, 3A41, 3A47, 3A55, 3A57, 3A65, 3A67, 3A71, 3A7B, 3A91, 3A95, 3AB5, 3AB7, 3B0B, 3B11, 3B1B, 3B21, 3B2B, 3B47, 3B4B, 3B51, 3B55, 3B61, 3B75, 3B7B, 3B85, 3B87, 3B97, 3BAB, 3BB7, 3BBB, 4005, 402B, 4031, 403B, 4041, 4047, 404B, 4055, 405B, 4067, 4071, 4075, 4085, 408B, 4097, 40A7, 40AB, 4101, 4111, 411B, 413B, 4145, 4155, 415B, 4161, 417B, 4187, 41A1, 41AB, 41B5, 4207, 420B, 4211, 4217, 4225, 4231, 4237, 423B, 4245, 426B, 4281, 428B, 4291, 42A1, 42AB, 42B1, 4305, 4307, 4321, 4341, 4357, 4361, 4375, 438B, 4395, 4397, 43B1, 43B5, 43BB, 4401, 440B, 4417, 4425, 442B, 4435, 4441, 4445, 444B, 4451, 445B, 4461, 4471, 4475, 447B, 4485, 4487, 4497, 449B, 44B1, 4507, 450B, 4515, 4531, 4535, 4541, 4547, 454B, 4557, 455B, 4571, 4577, 457B, 4591, 45A1, 45A5, 45A7, 4611, 4615, 4635, 463B, 4645, 4655, 4665, 4677, 4681, 4685, 4687, 468B, 46A5, 46AB, 46BB, 4707, 4711, 4715, 4725, 4727, ...|
|13|14, 16, 1A, 23, 25, 2B, 32, 34, 38, 41, 47, 49, 52, 56, 58, 61, 65, 6B, 76, 7A, 7C, 83, 85, 89, 9A, A1, A7, A9, B6, B8, C1, C7, CB, 104, 10A, 10C, 119, 11B, 122, 124, 133, 142, 146, 148, 14C, 155, 157, 164, 16A, 173, 179, 17B, 184, 188, 18A, 197, 1A8, 1AC, 1B1, 1B5, 1C6, 1CC, 209, 20B, 212, 218, 223, 229, 232, 236, 23C, 247, 24B, 256, 263, 265, 272, 274, 27A, 281, 287, 292, 296, 298, 29C, 2AB, 2B6, 2BA, 2C5, 2C9, 302, 311, 313, 328, 331, 33B, 344, 34A, 34C, 355, 362, 368, 371, 373, 379, 382, 386, 388, 397, 3A4, 3A6, 3AA, 3B3, 3B9, 3BB, 3CA, 401, 407, 412, 41C, 427, 434, 43C, 445, 44B, 452, 45A, 463, 467, 472, 476, 487, 494, 4A3, 4A5, 4B2, 4B4, 4B8, 4BA, 4C7, 508, 50C, 511, 515, 526, 52A, 52C, 533, 54A, 551, 559, 566, 571, 575, 57B, 584, 595, 599, 5A2, 5A8, 5B3, 5B9, 5C8, 5CC, 605, 607, 614, 616, 61C, 629, 62B, 638, 63A, 643, 658, 65C, 661, 665, 66B, 674, 67C, 685, 68B, 6A7, 6A9, 6B6, 6C1, 6CB, 704, 70A, 715, 724, 728, 731, 737, 739, 742, 751, 75B, 773, 775, 779, 782, 784, 78A, 791, 793, 797, 7A6, 7A8, 7B1, 809, 812, 818, 823, 838, 845, 856, 85A, 85C, 863, 869, 874, 878, 87A, 883, 892, 89C, 8A1, 8A5, 8A7, 8AB, 8B4, 8C3, 902, 90A, 919, 922, 926, 92C, 937, 93B, 946, 94A, 95B, 962, 968, 96A, 971, 977, 979, 982, 98C, 9A6, 9AC, 9B3, 9B5, A03, A07, A09, A16, A25, A27, A34, A3C, A45, A4B, A54, A69, A72, A76, A78, A87, A94, AA3, AAB, AC1, B02, B08, B0C, B11, B15, B17, B24, B33, B39, B42, B57, B59, B6C, B71, B8A, B93, B9B, BA4, BA8, BAA, BB1, BB9, BC2, BCC, C01, C0B, C1C, C29, C32, C41, C43, C47, C49, C56, C65, C67, C7A, C7C, C85, C89, C8B, C98, CA3, CB8, 1006, 100A, 1013, 101B, 1031, 1033, 1037, 1042, 1055, 1057, 105B, 1066, 106C, 1075, 1079, 1088, 108A, 10A6, 10AC, 10B1, 10B7, 10BB, 10C4, 1105, 110B, 1112, 1114, 111A, 1121, 1127, 1136, 113C, 1145, 1156, 115A, 1163, 1172, 117A, 1183, 1187, 11A7, 11BC, 11C9, 1204, 1208, 1211, 1213, 1219, 1235, 1244, 1246, 1259, 1264, 1268, 1277, 1288, 1295, 1297, 129B, 12A6, 12AC, 12B5, 12B9, 12BB, 12C2, 12C8, 1303, 1307, 1309, 1312, 131C, 1321, 132B, 1336, 133A, 134B, 1358, 1367, 1369, 1372, 1376, 1378, 138B, 139C, 13A3, 13A9, 13B4, 13BA, 13C1, 1406, 1411, 141B, 1424, 142A, 1435, 1442, 1451, 1462, 1466, 146C, 1475, 1477, 1499, 149B, 14A8, 14B3, 14B7, 14C8, 14CC, 1507, 1516, 151C, 152B, 1532, 1538, 1552, 155C, 1561, 1574, 1594, 1598, 159A, 15A9, 15B2, 15B6, 15C5, 15CB, 1606, 160A, 1615, 1631, 1633, 1637, 1639, 1648, 166A, 166C, 1675, 167B, 1684, 1688, 1691, 1693, 16A2, 16A6, 16B5, 16B7, 16C4, 16C6, 1709, 170B, 1721, 1727, 1741, 1754, 175C, 1763, 1765, 1769, 176B, 1787, 1792, 17A1, 17A7, 17B4, 17B6, 17BA, 17C3, 17C5, 17CB, 1808, 180A, 1819, 1826, 1828, 1835, 1846, 184C, 1853, 1859, 1864, 186A, 1873, 1886, 1895, 1897, 189B, 18AC, 18B5, 18B9, 18C4, 1901, 1909, 1912, 1918, 1934, 193A, 193C, 1949, 195A, 1961, 1967, 197C, 1981, 198B, 199C, 19A3, 19A5, 19B2, 19C3, 19C7, 1A02, 1A17, 1A1B, 1A24, 1A26, 1A2A, 1A33, 1A35, 1A44, 1A48, 1A62, 1A7B, 1A8A, 1A8C, 1A93, 1A99, 1AA2, 1AA4, 1AAA, 1AC6, 1AC8, 1B01, 1B14, 1B1A, 1B29, 1B2B, 1B34, 1B43, 1B56, 1B58, 1B5C, 1B65, 1B76, 1B7A, 1B7C, 1B94, 1BB2, 1BBC, 1BC5, 1BC7, 1C04, 1C06, 1C13, 1C15, 1C22, 1C28, 1C2A, 1C37, 1C39, 1C46, 1C4C, 1C57, 1C7B, 1C88, 1C8A, 1C97, 1CA2, 1CA8, 1CB5, 1CCA, 2003, 2012, 2021, 2023, 2038, 2041, 2045, 204B, 2054, 2069, 206B, 2078, 2089, 2092, 2096, 2098, 209C, 20BA, 20BC, 20CB, 2104, 2117, 2122, 2128, 2131, 2146, 2159, 215B, 2162, 2168, 216A, 2173, 2179, 2186, 218C, 219B, 21AA, 21C2, 21C4, 21CA, 2201, 2216, 2221, 223C, 2243, 2245, 2249, 2252, 2254, 2263, 2267, 2278, 229C, 22A9, 22B2, 22C1, 2302, 2308, 2315, 2324, 2326, 232A, 2333, 233B, 2344, 2351, 2353, 2357, 2368, 2371, 2377, 237B, 2384, 2386, 2393, 2395, 23A8, 23B7, 23C2, 2407, 240B, 2414, 2423, 2425, 242B, 2434, 243A, 245C, 2465, 2476, 247A, 2485, 2492, 249A, 24A9, 24C1, 24C5, 24C7, 24CB, 2519, 2528, 2531, 2533, 2546, 254C, 2555, 2566, 2573, 2584, 2588, 25AC, 25B5, 25BB, 25C4, 25CC, 2605, 2609, 260B, 261A, 2623, 2627, 2629, 2632, 264B, 2654, 2656, 265A, 2672, 2674, 2678, 2687, 2689, 2692, 2696, 26B6, 26BC, 26C5, 26C9, 2704, 2711, 2737, 274A, 274C, 2755, 2759, 275B, 2762, 2764, 2771, 2782, 2788, 278C, 2797, 27A4, 27AA, 27C4, 27C8, 27CA, 2803, 2827, 282B, 2836, 2843, 2849, 2852, 285A, 2863, 2872, 2876, 287C, 2881, 2887, 288B, 2894, 2896, 28A3, 28A5, 28B8, 28C1, 2908, 290C, 291B, 292C, 2951, 2957, 2971, 2975, 298A, 2995, 299B, 29A2, 29A8, 29B9, 29C2, 29C8, 2A05, 2A07, 2A14, 2A23, 2A2B, 2A38, 2A3A, 2A47, 2A52, 2A61, 2A6B, 2A89, 2A8B, 2A92, 2A9A, 2AA3, 2AA7, 2AB2, 2AC7, 2B04, 2B0A, 2B13, 2B15, 2B1B, 2B28, 2B37, 2B39, 2B46, 2B4C, 2B55, 2B5B, 2B66, 2B6C, 2B79, 2B82, 2B84, 2B8A, 2B93, 2B99, 2BA6, 2BB1, 2BCC, 2C05, 2C21, 2C23, 2C38, 2C3C, 2C47, 2C54, 2C78, 2C83, 2C98, 2C9C, 2CA1, 2CAB, 2CB4, 2CB6, 2CBC, 2CC3, 3008, 3013, 3022, 3037, 304A, 3053, 3055, 3064, 306A, 3077, 3079, 3086, 3088, 3091, 309B, 30AC, 30B3, 3101, 3103, 3116, 3118, 3125, 3127, 3134, 314B, 3152, 3154, 3158, 3163, 3176, 317C, 3185, 3187, 3196, 31A9, 31B4, 31B8, 31C1, 3215, 3217, 3224, 3226, 322C, 3233, 3239, 3242, 324A, 3253, 3257, 3266, 326C, 3277, 3286, 328A, 329B, 32AA, 32B7, 3305, 330B, 331A, 3323, 3325, 3341, 3349, 3361, 336B, 3374, 3385, 3389, 338B, 3394, 33A1, 33A9, 33B2, 33B6, 33BC, 3413, 3424, 3431, 3433, 3442, 344C, 3451, 3464, 3466, 347B, 3499, 34B1, 34B7, 34CA, 3512, 3518, 351A, 3532, 3536, 353C, 3541, 354B, 3556, 3563, 3569, 3572, 357A, 3581, 3587, 3589, 3596, 3598, 35A7, 35AB, 35B4, 35BA, 35BC, 35CB, 3602, 3613, 3628, 362C, 3635, 364C, 3653, 365B, 3664, 3668, 3673, 3677, 3688, 3691, 3695, 36A6, 36B5, 36B9, 36BB, 3712, 3716, 3734, 373A, 3743, 3752, 3761, 3772, 3778, 377C, 3781, 3785, 379A, 37A3, 37B2, 37BA, 37C3, 37C7, 3806, 3808, 3817, ...|
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695, 6A3, 6A5, 6AB, 6D3, 6D9, 701, 709, 71D, 729, 739, 73D, 741, 745, 74B, 755, 759, 75B, 763, 771, 77B, 77D, 783, 785, 789, 791, 79D, 7AB, 7B5, 7C3, 7C9, 7CD, 7D5, 7DD, 803, 80B, 811, 821, 825, 82B, 82D, 833, 839, 83B, 843, 84D, 865, 86B, 871, 873, 88D, 893, 895, 8A1, 8AD, 8B1, 8BB, 8C5, 8CB, 8D3, 8D9, 90D, 915, 919, 91B, 929, 935, 943, 94B, 95D, 96D, 975, 979, 97B, 981, 983, 98D, 99B, 9A3, 9A9, 9BD, 9C1, 9D3, 9D5, A0D, A15, A1D, A25, A29, A2B, A31, A39, A41, A4B, A4D, A59, A69, A75, A7B, A89, A8B, A91, A93, A9D, AAB, AAD, AC1, AC3, AC9, ACD, AD1, ADB, B05, B19, B35, B39, B41, B49, B5B, B5D, B63, B6B, B7D, B81, B85, B8D, B95, B9B, BA1, BAD, BB1, BC9, BD1, BD3, BD9, BDD, C05, C15, C1B, C21, C23, C29, C2D, C35, C43, C49, C51, C61, C65, C6B, C79, C83, C89, C8D, CAB, CC1, CCB, CD5, CD9, D01, D03, D09, D23, D31, D33, D45, D4D, D53, D61, D71, D7B, D7D, D83, D8B, D93, D99, D9D, DA1, DA5, DAB, DB5, DB9, DBB, DC3, DCD, DD1, DDB, 1005, 1009, 1019, 1025, 1033, 1035, 103B, 1041, 1043, 1055, 1065, 1069, 1071, 1079, 1081, 1085, 1099, 10A3, 10AD, 10B5, 10BB, 10C5, 10D1, 10DD, 110D, 1113, 1119, 1121, 1123, 1143, 1145, 1151, 1159, 115D, 116D, 1173, 117B, 1189, 1191, 119D, 11A3, 11A9, 11C1, 11CB, 11CD, 1201, 121D, 1223, 1225, 1233, 1239, 123D, 124B, 1253, 125B, 1261, 1269, 1283, 1285, 1289, 128B, 1299, 12B9, 12BB, 12C3, 12C9, 12D1, 12D5, 12DB, 12DD, 130B, 1311, 131D, 1321, 132B, 132D, 1341, 1343, 1355, 135B, 1373, 1385, 138D, 1393, 1395, 1399, 139B, 13B5, 13BD, 13CB, 13D3, 13DD, 1401, 1405, 140B, 140D, 1415, 1421, 1423, 1431, 143B, 143D, 1449, 1459, 1461, 1465, 146B, 1475, 147B, 1483, 1495, 14A3, 14A5, 14A9, 14B9, 14C1, 14C5, 14CD, 14D9, 1503, 1509, 1511, 1529, 1531, 1533, 153D, 154D, 1553, 1559, 156D, 1571, 157B, 158B, 1591, 1593, 159D, 15AD, 15B3, 15BB, 15D1, 15D5, 15DB, 15DD, 1603, 1609, 160B, 1619, 161D, 1635, 164D, 165B, 165D, 1663, 1669, 1671, 1673, 1679, 1693, 1695, 169B, 16AD, 16B5, 16C3, 16C5, 16CB, 16D9, 170B, 170D, 1713, 1719, 1729, 172D, 1731, 1745, 1761, 176B, 1773, 1775, 1781, 1783, 178D, 1791, 179B, 17A3, 17A5, 17B1, 17B3, 17BD, 17C5, 17CD, 1811, 181B, 181D, 1829, 1833, 1839, 1845, 1859, 1861, 186D, 187B, 187D, 1893, 1899, 189D, 18A5, 18AB, 18C1, 18C3, 18CD, 18DD, 1905, 1909, 190B, 1911, 192B, 192D, 193B, 1943, 1955, 195D, 1965, 196B, 1981, 1993, 1995, 1999, 19A1, 19A3, 19A9, 19B1, 19BB, 19C3, 19D1, 19DD, 1A13, 1A15, 1A1B, 1A21, 1A35, 1A3D, 1A59, 1A5D, 1A61, 1A65, 1A6B, 1A6D, 1A7B, 1A81, 1A91, 1AB3, 1ABD, 1AC5, 1AD3, 1B03, 1B09, 1B15, 1B23, 1B25, 1B29, 1B31, 1B39, 1B41, 1B4B, 1B4D, 1B53, 1B63, 1B69, 1B71, 1B75, 1B7B, 1B7D, 1B89, 1B8B, 1B9D, 1BAB, 1BB5, 1BC9, 1BCD, 1BD5, 1C03, 1C05, 1C0B, 1C13, 1C19, 1C39, 1C41, 1C51, 1C55, 1C5D, 1C69, 1C73, 1C81, 1C95, 1C99, 1C9B, 1CA1, 1CBB, 1CC9, 1CD1, 1CD3, 1D05, 1D0B, 1D13, 1D23, 1D2D, 1D3D, 1D43, 1D65, 1D6B, 1D73, 1D79, 1D83, 1D89, 1D8D, 1D91, 1D9D, 1DA5, 1DA9, 1DAB, 1DB3, 1DCB, 1DD3, 1DD5, 1DD9, 200D, 2011, 2015, 2023, 2025, 202B, 2031, 204D, 2055, 205B, 2061, 2069, 2075, 2099, 20AB, 20AD, 20B5, 20B9, 20BB, 20C1, 20C3, 20CD, 20DD, 2105, 2109, 2113, 211D, 2125, 213B, 2141, 2143, 2149, 216B, 2171, 2179, 2185, 218B, 2193, 219B, 21A3, 21B1, 21B5, 21BB, 21BD, 21C5, 21C9, 21D1, 21D3, 21DD, 2201, 2213, 2219, 2231, 2235, 2243, 2253, 2273, 2279, 2291, 2295, 22A9, 22B3, 22B9, 22BD, 22C5, 22D5, 22DB, 2303, 230D, 2311, 231B, 2329, 2333, 233D, 2341, 234B, 2355, 2363, 236D, 2389, 238B, 2391, 2399, 23A1, 23A5, 23AD, 23C3, 23CD, 23D5, 23DB, 23DD, 2405, 2411, 241D, 2421, 242B, 2433, 2439, 2441, 2449, 2451, 245B, 2463, 2465, 246B, 2473, 2479, 2485, 248D, 24A9, 24B1, 24C9, 24CB, 2501, 2505, 250D, 2519, 253B, 2545, 2559, 255D, 2561, 256B, 2573, 2575, 257B, 2581, 2595, 259D, 25AB, 25C1, 25D3, 25D9, 25DB, 2609, 2611, 261B, 261D, 2629, 262B, 2633, 263D, 264D, 2653, 266D, 2671, 2683, 2685, 2691, 2693, 269D, 26B5, 26B9, 26BB, 26C1, 26C9, 26DB, 2703, 2709, 270B, 2719, 272B, 2735, 2739, 2741, 2763, 2765, 2771, 2773, 2779, 277D, 2785, 278B, 2795, 279B, 27A1, 27AD, 27B5, 27BD, 27CB, 27D1, 2801, 280D, 2819, 2835, 283B, 2849, 2851, 2853, 286B, 2875, 2889, 2895, 289B, 28AB, 28B1, 28B3, 28B9, 28C5, 28CD, 28D5, 28D9, 2901, 2923, 2933, 293D, 2941, 294D, 2959, 295B, 296D, 2971, 2985, 29A1, 29B5, 29BB, 29CD, 2A03, 2A09, 2A0B, 2A21, 2A25, 2A2B, 2A2D, 2A39, 2A43, 2A4D, 2A55, 2A5B, 2A65, 2A69, 2A71, 2A73, 2A7D, 2A81, 2A8D, 2A93, 2A99, 2AA1, 2AA3, 2AB1, 2AB5, 2AC5, 2AD9, 2ADD, 2B05, 2B1B, 2B21, 2B29, 2B31, 2B35, 2B3D, 2B43, 2B53, 2B59, 2B5D, 2B6D, 2B7B, 2B81, 2B83, 2BA5, 2BA9, 2BC5, 2BCB, 2BD3, 2C01, 2C0D, 2C1D, 2C25, 2C29, 2C2B, 2C31, 2C45, 2C4B, 2C59, 2C63, 2C69, 2C6D, 2C7B, 2C7D, 2C8B, ...|
|15|12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 102, 104, 108, 10E, 111, 11B, 122, 128, 12E, 131, 137, 13B, 13D, 148, 157, 15B, 15D, 162, 171, 177, 182, 184, 188, 18E, 197, 19D, 1A4, 1A8, 1AE, 1B7, 1BB, 1C4, 1CE, 1D1, 1DB, 1DD, 1E4, 1E8, 1EE, 207, 20B, 20D, 212, 21E, 227, 22B, 234, 238, 23E, 24B, 24D, 261, 267, 272, 278, 27E, 281, 287, 292, 298, 29E, 2A1, 2A7, 2AD, 2B2, 2B4, 2C1, 2CB, 2CD, 2D2, 2D8, 2DE, 2E1, 2ED, 302, 308, 311, 31B, 324, 32E, 337, 33D, 344, 348, 351, 357, 35B, 364, 368, 377, 382, 38E, 391, 39B, 39D, 3A2, 3A4, 3AE, 3BD, 3C2, 3C4, 3C8, 3D7, 3DB, 3DD, 3E2, 407, 40B, 414, 41E, 427, 42B, 432, 438, 447, 44B, 452, 458, 461, 467, 474, 478, 47E, 481, 48B, 48D, 494, 49E, 4A1, 4AB, 4AD, 4B4, 4C7, 4CB, 4CD, 4D2, 4D8, 4DE, 4E7, 4ED, 504, 51B, 51D, 528, 531, 53B, 542, 548, 551, 55D, 562, 568, 56E, 571, 577, 584, 58E, 5A2, 5A4, 5A8, 5AE, 5B1, 5B7, 5BB, 5BD, 5C2, 5CE, 5D1, 5D7, 60B, 612, 618, 621, 634, 63E, 64D, 652, 654, 658, 65E, 667, 66B, 66D, 674, 681, 68B, 68D, 692, 694, 698, 69E, 6AB, 6B8, 6C1, 6CD, 6D4, 6D8, 6DE, 6E7, 6EB, 704, 708, 717, 71B, 722, 724, 728, 72E, 731, 737, 742, 757, 75D, 762, 764, 77D, 782, 784, 78E, 79B, 79D, 7A8, 7B1, 7B7, 7BD, 7C4, 7D7, 7DD, 7E2, 7E4, 801, 80B, 818, 821, 832, 841, 847, 84B, 84D, 852, 854, 85E, 86B, 872, 878, 88B, 88D, 89E, 8A1, 8B8, 8BE, 8C7, 8CD, 8D2, 8D4, 8D8, 8E1, 8E7, 902, 904, 90E, 91D, 928, 92E, 93B, 93D, 942, 944, 94E, 95B, 95D, 96E, 971, 977, 97B, 97D, 988, 991, 9A4, 9BD, 9C2, 9C8, 9D1, 9E2, 9E4, 9E8, A01, A12, A14, A18, A21, A27, A2D, A32, A3E, A41, A58, A5E, A61, A67, A6B, A72, A81, A87, A8B, A8D, A94, A98, A9E, AAB, AB2, AB8, AC7, ACB, AD2, ADE, AE7, AED, B02, B1D, B31, B3B, B44, B48, B4E, B51, B57, B6E, B7B, B7D, B8E, B97, B9B, BA8, BB7, BC2, BC4, BC8, BD1, BD7, BDD, BE2, BE4, BE8, BEE, C07, C0B, C0D, C14, C1E, C21, C2B, C34, C38, C47, C52, C5E, C61, C67, C6B, C6D, C7E, C8D, C92, C98, CA1, CA7, CAB, CBE, CC7, CD2, CD8, CDE, CE7, D02, D0E, D1D, D22, D28, D2E, D31, D4E, D51, D5B, D64, D68, D77, D7B, D84, D91, D97, DA4, DA8, DAE, DC4, DCE, DD1, DE2, E0D, E12, E14, E21, E27, E2B, E38, E3E, E47, E4B, E54, E6B, E6D, E72, E74, E81, E9E, EA1, EA7, EAD, EB4, EB8, EBE, EC1, ECD, ED2, EDE, EE1, EEB, EED, 100E, 1011, 1022, 1028, 103D, 104E, 1057, 105B, 105D, 1062, 1064, 107B, 1084, 1091, 1097, 10A2, 10A4, 10A8, 10AE, 10B1, 10B7, 10C2, 10C4, 10D1, 10DB, 10DD, 10E8, 1107, 110D, 1112, 1118, 1121, 1127, 112D, 113E, 114B, 114D, 1152, 1161, 1167, 116B, 1174, 117E, 1187, 118D, 1194, 11AB, 11B2, 11B4, 11BE, 11CD, 11D2, 11D8, 11EB, 11ED, 1208, 1217, 121B, 121D, 1228, 1237, 123B, 1244, 1257, 125B, 1262, 1264, 1268, 126E, 1271, 127D, 1282, 1297, 12AE, 12BB, 12BD, 12C2, 12C8, 12CE, 12D1, 12D7, 12EE, 1301, 1307, 1318, 131E, 132B, 132D, 1334, 1341, 1352, 1354, 1358, 135E, 136D, 1372, 1374, 1387, 13A1, 13AB, 13B2, 13B4, 13BE, 13C1, 13CB, 13CD, 13D8, 13DE, 13E1, 13EB, 13ED, 1408, 140E, 1417, 1437, 1442, 1444, 144E, 1457, 145D, 1468, 147B, 1482, 148E, 149B, 149D, 14B1, 14B7, 14BB, 14C2, 14C8, 14DB, 14DD, 14E8, 1507, 150D, 1512, 1514, 1518, 1532, 1534, 1541, 1547, 1558, 1561, 1567, 156D, 1581, 1592, 1594, 1598, 159E, 15A1, 15A7, 15AD, 15B8, 15BE, 15CB, 15D8, 15EB, 15ED, 1604, 1608, 161B, 1624, 163D, 1642, 1644, 1648, 164E, 1651, 165D, 1662, 1671, 1691, 169B, 16A2, 16AE, 16BD, 16C4, 16CE, 16DB, 16DD, 16E2, 16E8, 1701, 1707, 1712, 1714, 1718, 1727, 172D, 1734, 1738, 173E, 1741, 174B, 174D, 175E, 176B, 1774, 1787, 178B, 1792, 179E, 17A1, 17A7, 17AD, 17B4, 17D2, 17D8, 17E7, 17EB, 1804, 180E, 1817, 1824, 1837, 183B, 183D, 1842, 185B, 1868, 186E, 1871, 1882, 1888, 188E, 189D, 18A8, 18B7, 18BB, 18DB, 18E2, 18E8, 18EE, 1907, 190D, 1912, 1914, 1921, 1927, 192B, 192D, 1934, 194B, 1952, 1954, 1958, 196B, 196D, 1972, 197E, 1981, 1987, 198B, 19A7, 19AD, 19B4, 19B8, 19C1, 19CB, 19ED, 1A0E, 1A11, 1A17, 1A1B, 1A1D, 1A22, 1A24, 1A2E, 1A3D, 1A44, 1A48, 1A51, 1A5B, 1A62, 1A77, 1A7B, 1A7D, 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191G, 1921, 192B, 192D, 1938, 193C, 1941, 1947, 1949, 1954, 1958, 1965, 1976, 197A, 197G, 1992, 1996, 199E, 19A3, 19A7, 19AF, 19B2, 19BG, 19C5, 19C9, 19D6, 19E1, 19E5, 19E7, 19G3, 19G7, 1A0E, 1A13, 1A19, 1A24, 1A2G, 1A3D, 1A42, 1A46, 1A48, 1A4C, 1A5D, 1A62, 1A6E, 1A75, 1A7B, 1A7F, 1A8A, 1A8C, 1A97, 1AB3, ...|
|18|11, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 10D, 115, 117, 11B, 11H, 127, 12D, 131, 135, 13B, 141, 145, 14D, 155, 157, 15H, 161, 167, 16B, 16H, 177, 17B, 17D, 17H, 18B, 191, 195, 19D, 19H, 1A5, 1AH, 1B1, 1C1, 1C7, 1CH, 1D5, 1DB, 1DD, 1E1, 1EB, 1EH, 1F5, 1F7, 1FD, 1G1, 1G5, 1G7, 1H1, 1HB, 1HD, 1HH, 205, 20B, 20D, 217, 21B, 21H, 227, 22H, 237, 23H, 247, 24D, 251, 255, 25D, 261, 265, 26D, 26H, 27D, 285, 28H, 291, 29B, 29D, 29H, 2A1, 2AB, 2B7, 2BB, 2BD, 2BH, 2CD, 2CH, 2D1, 2D5, 2E7, 2EB, 2F1, 2FB, 2G1, 2G5, 2GB, 2GH, 2HD, 2HH, 305, 30B, 311, 317, 321, 325, 32B, 32D, 335, 337, 33D, 345, 347, 34H, 351, 357, 367, 36B, 36D, 36H, 375, 37B, 381, 387, 38D, 39H, 3A1, 3AB, 3B1, 3BB, 3BH, 3C5, 3CD, 3D7, 3DB, 3DH, 3E5, 3E7, 3ED, 3F7, 3FH, 3GH, 3H1, 3H5, 3HB, 3HD, 401, 405, 407, 40B, 415, 417, 41D, 43B, 43H, 445, 44D, 45D, 465, 471, 475, 477, 47B, 47H, 487, 48B, 48D, 491, 49D, 4A5, 4A7, 4AB, 4AD, 4AH, 4B5, 4BH, 4CB, 4D1, 4DD, 4E1, 4E5, 4EB, 4F1, 4F5, 4FD, 4FH, 4GD, 4GH, 4H5, 4H7, 4HB, 4HH, 501, 507, 50H, 521, 527, 52B, 52D, 541, 545, 547, 54H, 55B, 55D, 565, 56D, 571, 577, 57D, 58D, 591, 595, 597, 5A1, 5AB, 5B5, 5BD, 5CB, 5D7, 5DD, 5DH, 5E1, 5E5, 5E7, 5EH, 5FB, 5FH, 5G5, 5H5, 5H7, 605, 607, 61B, 61H, 627, 62D, 62H, 631, 635, 63D, 641, 64B, 64D, 655, 661, 66B, 66H, 67B, 67D, 67H, 681, 68B, 695, 697, 6A5, 6A7, 6AD, 6AH, 6B1, 6BB, 6C1, 6D1, 6E7, 6EB, 6EH, 6F7, 6G5, 6G7, 6GB, 6H1, 6HH, 701, 705, 70D, 711, 717, 71B, 725, 727, 73B, 73H, 741, 747, 74B, 74H, 75D, 761, 765, 767, 76D, 76H, 775, 77H, 785, 78B, 797, 79B, 79H, 7AB, 7B1, 7B7, 7BB, 7D1, 7E1, 7EB, 7F1, 7F5, 7FB, 7FD, 7G1, 7H5, 7HH, 801, 80H, 817, 81B, 825, 831, 83B, 83D, 83H, 847, 84D, 851, 855, 857, 85B, 85H, 867, 86B, 86D, 871, 87B, 87D, 885, 88D, 88H, 89D, 8A5, 8AH, 8B1, 8B7, 8BB, 8BD, 8CB, 8D7, 8DB, 8DH, 8E7, 8ED, 8EH, 8FH, 8G7, 8GH, 8H5, 8HB, 901, 90B, 915, 921, 925, 92B, 92H, 931, 94B, 94D, 955, 95D, 95H, 96D, 96H, 977, 981, 987, 991, 995, 99B, 9AD, 9B5, 9B7, 9C5, 9DD, 9DH, 9E1, 9ED, 9F1, 9F5, 9FH, 9G5, 9GD, 9GH, 9H7, A0B, A0D, A0H, A11, A1D, A35, A37, A3D, A41, A47, A4B, A4H, A51, A5D, A5H, A6B, A6D, A75, A77, A85, A87, A95, A9B, AAD, ABB, AC1, AC5, AC7, ACB, ACD, ADH, AE7, AF1, AF7, AFH, AG1, AG5, AGB, AGD, AH1, AHB, AHD, B07, B0H, B11, B1B, B27, B2D, B2H, B35, B3D, B41, B47, B55, B5H, B61, B65, B71, B77, B7B, B81, B8B, B91, B97, B9D, BAH, BB5, BB7, BBH, BCD, BCH, BD5, BE5, BE7, BEH, BFD, BFH, BG1, BGB, BH7, BHB, C01, C11, C15, C1B, C1D, C1H, C25, C27, C31, C35, C47, C5B, C65, C67, C6B, C6H, C75, C77, C7D, C8H, C91, C97, CA5, CAB, CB5, CB7, CBD, CC7, CD5, CD7, CDB, CDH, CED, CEH, CF1, CG1, CH7, CHH, D05, D07, D0H, D11, D1B, D1D, D25, D2B, D2D, D35, D37, D3H, D45, D4D, D67, D6H, D71, D7B, D81, D87, D8H, D9H, DA5, DAH, DBB, DBD, DCD, DD1, DD5, DDB, DDH, DEH, DF1, DFB, DG7, DGD, DGH, DH1, DH5, E0B, E0D, E17, E1D, E2B, E31, E37, E3D, E4D, E5B, E5D, E5H, E65, E67, E6D, E71, E7B, E7H, E8B, E95, EA5, EA7, EAD, EAH, EBH, EC7, EDD, EDH, EE1, EE5, EEB, EED, EF7, EFB, EG7, F01, F0B, F0H, F1B, F27, F2D, F35, F3H, F41, F45, F4B, F51, F57, F5H, F61, F65, F71, F77, F7D, F7H, F85, F87, F8H, F91, F9H, FAB, FB1, FC1, FC5, FCB, FD5, FD7, FDD, FE1, FE7, FFH, FG5, FH1, FH5, FHD, G05, G0D, G17, G27, G2B, G2D, G2H, G45, G4H, G55, G57, G65, G6B, G6H, G7D, G85, G91, G95, GAH, GB5, GBB, GBH, GC7, GCD, GCH, GD1, GDD, GE1, GE5, GE7, GED, GFH, GG5, GG7, GGB, GHB, GHD, GHH, H0B, H0D, H11, H15, H2D, H31, H37, H3B, H41, H4B, H67, H75, H77, H7D, H7H, H81, H85, H87, H8H, H9D, HA1, HA5, HAD, HB5, HBB, HCD, HCH, HD1, HD7, HF1, HF5, HFD, HG5, HGB, HGH, HH7, HHD, 1007, 100B, 100H, 1011, 1017, 101B, 101H, 1021, 102B, 102D, 103B, 103H, 1051, 1055, 105H, 106D, 1085, 108B, 109D, 109H, 10AH, 10B7, 10BD, 10BH, 10C5, 10D1, 10D7, 10DD, 10E5, 10E7, 10EH, 10FB, 10G1, 10GB, 10GD, 10H5, 10HD, 1107, 110H, 1125, 1127, 112B, 1131, 1137, 113B, 1141, 1151, 115B, 115H, 1165, 1167, 116D, 1175, 117H, 1181, 118B, 118H, 1195, 119B, 11A1, 11A7, 11AH, 11B5, 11B7, 11BD, 11C1, 11C7, 11CH, 11D7, 11ED, 11F1, 11G5, 11G7, 11H7, 11HB, 1201, 120B, 1225, 122D, 123D, 123H, 1241, 124B, 124H, 1251, 1257, 125B, 126B, 1271, 127D, 128D, 129B, 129H, 12A1, 12AD, 12B1, 12BB, 12BD, 12C5, 12C7, 12CD, 12D5, 12E1, 12E5, 12FB, 12FD, 12GB, 12GD, 12H5, 12H7, 12HH, 1311, 1315, 1317, 131B, 1321, 132H, 1335, 133B, 133D, 1347, 1355, 135D, 135H, 1365, 137H, 1381, 138B, 138D, 1391, 1395, 139B, 139H, 13A7, 13AD, 13AH, 13BB, 13BH, 13C7, 13D1, 13D5, 13E1, 13ED, 13F5, 13GB, 13GH, 13HB, 13HH, 1401, 1415, 141D, 142D, 1435, 143B, 1447, 144B, 144D, 1451, 145B, 1461, 1467, 146B, 146H, 148B, 1497, 149H, 14A1, 14AD, 14B5, 14B7, 14C5, 14C7, 14D7, 14ED, 14FD, 14G1, 14GH, 14HH, 1505, 1507, 1517, 151B, 151H, 1521, 152B, 1531, 153B, 153H, 1545, 154D, 154H, 1555, 1557, 155H, 1561, 156D, 156H, 1575, 157B, 157D, 1587, 158B, 1597, 15A7, 15AB, 15AH, 15C1, 15C5, 15CD, 15D1, 15D5, 15DD, 15DH, 15ED, 15F1, 15F5, 15G1, 15GD, 15GH, 15H1, 160D, 160H, 1625, 162B, 162H, 163B, 1645, 1651, 1657, 165B, 165D, 165H, 166H, 1675, 167H, 1687, 168D, 168H, 169B, 169D, 16A7, 16C1, ...|
|19|14, 1A, 1C, 1I, 23, 25, 29, 2F, 32, 34, 3A, 3E, 3G, 43, 47, 4D, 52, 56, 58, 5C, 5E, 5I, 6D, 6H, 74, 76, 7G, 7I, 85, 8B, 8F, 92, 98, 9A, A1, A3, A7, A9, B2, BE, BI, C1, C5, CB, CD, D4, DA, DG, E3, E5, EB, EF, EH, F8, G3, G7, G9, GD, H8, HE, I5, I7, IB, IH, 106, 10C, 10I, 113, 119, 11H, 122, 12A, 131, 133, 13D, 13F, 142, 146, 14C, 151, 155, 157, 15B, 164, 16C, 16G, 175, 179, 17F, 188, 18A, 199, 19F, 1A6, 1AC, 1AI, 1B1, 1B7, 1BH, 1C4, 1CA, 1CC, 1CI, 1D5, 1D9, 1DB, 1E4, 1EE, 1EG, 1F1, 1F7, 1FD, 1FF, 1G8, 1GC, 1GI, 1H7, 1HH, 1I6, 1IG, 205, 20B, 20H, 212, 21A, 21G, 221, 229, 22D, 238, 23I, 24B, 24D, 254, 256, 25A, 25C, 263, 26H, 272, 274, 278, 283, 287, 289, 28D, 29E, 29I, 2A7, 2AH, 2B6, 2BA, 2BG, 2C3, 2CH, 2D2, 2D8, 2DE, 2E3, 2E9, 2F2, 2F6, 2FC, 2FE, 2G5, 2G7, 2GD, 2H4, 2H6, 2HG, 2HI, 2I5, 304, 308, 30A, 30E, 311, 317, 31F, 322, 328, 33B, 33D, 344, 34C, 353, 359, 35F, 364, 36G, 371, 377, 37D, 37F, 382, 38E, 395, 3A4, 3A6, 3AA, 3AG, 3AI, 3B5, 3B9, 3BB, 3BF, 3C8, 3CA, 3CG, 3EC, 3EI, 3F5, 3FD, 3GC, 3H3, 3HH, 3I2, 3I4, 3I8, 3IE, 403, 407, 409, 40F, 418, 41I, 421, 425, 427, 42B, 42H, 43A, 443, 44B, 454, 45A, 45E, 461, 469, 46D, 472, 476, 481, 485, 48B, 48D, 48H, 494, 496, 49C, 4A3, 4B4, 4BA, 4BE, 4BG, 4D2, 4D6, 4D8, 4DI, 4EB, 4ED, 4F4, 4FC, 4FI, 4G5, 4GB, 4HA, 4HG, 4I1, 4I3, 4IF, 506, 50I, 517, 524, 52I, 535, 539, 53B, 53F, 53H, 548, 551, 557, 55D, 56C, 56E, 57B, 57D, 58G, 593, 59B, 59H, 5A2, 5A4, 5A8, 5AG, 5B3, 5BD, 5BF, 5C6, 5D1, 5DB, 5DH, 5EA, 5EC, 5EG, 5EI, 5F9, 5G2, 5G4, 5H1, 5H3, 5H9, 5HD, 5HF, 5I6, 5IE, 60D, 61I, 623, 629, 62H, 63E, 63G, 641, 649, 656, 658, 65C, 661, 667, 66D, 66H, 67A, 67C, 68F, 692, 694, 69A, 69E, 6A1, 6AF, 6B2, 6B6, 6B8, 6BE, 6BI, 6C5, 6CH, 6D4, 6DA, 6E5, 6E9, 6EF, 6F8, 6FG, 6G3, 6G7, 6HE, 6ID, 704, 70C, 70G, 713, 715, 71B, 72E, 737, 739, 746, 74E, 74I, 75B, 766, 76G, 76I, 773, 77B, 77H, 784, 788, 78A, 78E, 791, 799, 79D, 79F, 7A2, 7AC, 7AE, 7B5, 7BD, 7BH, 7CC, 7D3, 7DF, 7DH, 7E4, 7E8, 7EA, 7F7, 7G2, 7G6, 7GC, 7H1, 7H7, 7HB, 7IA, 7II, 809, 80F, 812, 81A, 821, 82D, 838, 83C, 83I, 845, 847, 85G, 85I, 869, 86H, 872, 87G, 881, 889, 892, 898, 8A1, 8A5, 8AB, 8BC, 8C3, 8C5, 8D2, 8E9, 8ED, 8EF, 8F8, 8FE, 8FI, 8GB, 8GH, 8H6, 8HA, 8HI, 902, 904, 908, 90A, 913, 92C, 92E, 931, 937, 93D, 93H, 944, 946, 94I, 953, 95F, 95H, 968, 96A, 977, 979, 986, 98C, 99D, 9AA, 9AI, 9B3, 9B5, 9B9, 9BB, 9CE, 9D3, 9DF, 9E2, 9EC, 9EE, 9EI, 9F5, 9F7, 9FD, 9G4, 9G6, 9GI, 9H9, 9HB, 9I2, 9IG, A03, A07, A0D, A12, A18, A1E, A2B, A34, A36, A3A, A45, A4B, A4F, A54, A5E, A63, A69, A6F, A7I, A85, A87, A8H, A9C, A9G, AA3, AB2, AB4, ABE, AC9, ACD, ACF, AD6, AE1, AE5, AED, AFC, AFG, AG3, AG5, AG9, AGF, AGH, AHA, AHE, AIF, B0I, B1B, B1D, B1H, B24, B2A, B2C, B2I, B42, B44, B4A, B57, B5D, B66, B68, B6E, B77, B84, B86, B8A, B8G, B9B, B9F, B9H, BAG, BC2, BCC, BCI, BD1, BDB, BDD, BE4, BE6, BEG, BF3, BF5, BFF, BFH, BG8, BGE, BH3, BIE, C05, C07, C0H, C16, C1C, C23, C32, C38, C41, C4D, C4F, C5E, C61, C65, C6B, C6H, C7G, C7I, C89, C94, C9A, C9E, C9G, CA1, CB6, CB8, CC1, CC7, CD4, CDC, CDI, CE5, CF4, CG1, CG3, CG7, CGD, CGF, CH2, CH8, CHI, CI5, CIH, D0A, D19, D1B, D1H, D22, D31, D39, D4E, D4I, D51, D55, D5B, D5D, D66, D6A, D75, D8G, D97, D9D, DA6, DB1, DB7, DBH, DCA, DCC, DCG, DD3, DDB, DDH, DE8, DEA, DEE, DF9, DFF, DG2, DG6, DGC, DGE, DH5, DH7, DI4, DIG, E05, E14, E18, E1E, E27, E29, E2F, E32, E38, E4H, E54, E5I, E63, E6B, E72, E7A, E83, E92, E96, E98, E9C, EAH, EBA, EBG, EBI, ECF, ED2, ED8, EE3, EED, EF8, EFC, EH4, EHA, EHG, EI3, EIB, EIH, F02, F04, F0G, F13, F17, F19, F1F, F2I, F35, F37, F3B, F4A, F4C, F4G, F59, F5B, F5H, F62, F79, F7F, F82, F86, F8E, F95, FAI, FBF, FBH, FC4, FC8, FCA, FCE, FCG, FD7, FE2, FE8, FEC, FF1, FFB, FFH, FGI, FH3, FH5, FHB, G03, G07, G0F, G16, G1C, G1I, G27, G2D, G36, G3A, G3G, G3I, G45, G49, G4F, G4H, G58, G5A, G67, G6D, G7E, G7I, G8B, G96, GAF, GB2, GC3, GC7, GD6, GDE, GE1, GE5, GEB, GF6, GFC, GFI, GG9, GGB, GH2, GHE, GI3, GID, GIF, H06, H0E, H17, H1H, H33, H35, H39, H3H, H44, H48, H4G, H5F, H66, H6C, H6I, H71, H77, H7H, H8A, H8C, H93, H99, H9F, HA2, HAA, HAG, HB7, HBD, HBF, HC2, HC8, HCE, HD5, HDD, HEI, HF5, HG8, HGA, HH9, HHD, HI2, HIC, I14, I1C, I2B, I2F, I2H, I38, I3E, I3G, I43, I47, I56, I5E, I67, I76, I83, I89, I8B, I94, I9A, IA1, IA3, IAD, IAF, IB2, IBC, IC7, ICB, IDG, IDI, IEF, IEH, IF8, IFA, IG1, IH2, IH6, IH8, IHC, II1, IIH, 1004, 100A, 100C, 1015, 1022, 102A, 102E, 1031, 104C, 104E, 1055, 1057, 105D, 105H, 1064, 106A, 106I, 1075, 1079, 1082, 1088, 108G, 1099, 109D, 10A8, 10B1, 10BB, 10CG, 10D3, 10DF, 10E2, 10E4, 10F7, 10FF, 10GE, 10H5, 10HB, 10I6, 10IA, 10IC, 10II, 1109, 110H, 1114, 1118, 111E, 1136, 1141, 114B, 114D, 1156, 115G, 115I, 116F, 116H, 117G, 1192, 11A1, 11A7, 11B4, 11C3, 11C9, 11CB, 11DA, 11DE, 11E1, 11E3, 11ED, 11F2, 11FC, 11FI, 11G5, 11GD, 11GH, 11H4, 11H6, 11HG, 11HI, 11IB, 11IF, 1202, 1208, 120A, 1213, 1217, 1222, 1231, 1235, 123B, 124C, 124G, 1255, 125B, 125F, 1264, 1268, 1273, 1279, 127D, 1288, 1291, 1295, 1297, 12AI, 12B3, 12C8, 12CE, 12D1, 12DD, 12E6, 12F1, 12F7, 12FB, 12FD, 12FH, 12GG, 12H3, 12HF, 12I4, 12IA, 12IE, 1307, 1309, 1312, 132D, 133A, ...|
|20|13, 19, 1B, 1H, 21, 23, 27, 2D, 2J, 31, 37, 3B, 3D, 3J, 43, 49, 4H, 51, 53, 57, 59, 5D, 67, 6B, 6H, 6J, 79, 7B, 7H, 83, 87, 8D, 8J, 91, 9B, 9D, 9H, 9J, AB, B3, B7, B9, BD, BJ, C1, CB, CH, D3, D9, DB, DH, E1, E3, ED, F7, FB, FD, FH, GB, GH, H7, H9, HD, HJ, I7, ID, IJ, J3, J9, JH, 101, 109, 10J, 111, 11B, 11D, 11J, 123, 129, 12H, 131, 133, 137, 13J, 147, 14B, 14J, 153, 159, 161, 163, 171, 177, 17H, 183, 189, 18B, 18H, 197, 19D, 19J, 1A1, 1A7, 1AD, 1AH, 1AJ, 1BB, 1C1, 1C3, 1C7, 1CD, 1CJ, 1D1, 1DD, 1DH, 1E3, 1EB, 1F1, 1F9, 1FJ, 1G7, 1GD, 1GJ, 1H3, 1HB, 1HH, 1I1, 1I9, 1ID, 1J7, 1JH, 209, 20B, 211, 213, 217, 219, 21J, 22D, 22H, 22J, 233, 23H, 241, 243, 247, 257, 25B, 25J, 269, 26H, 271, 277, 27D, 287, 28B, 28H, 293, 29B, 29H, 2A9, 2AD, 2AJ, 2B1, 2BB, 2BD, 2BJ, 2C9, 2CB, 2D1, 2D3, 2D9, 2E7, 2EB, 2ED, 2EH, 2F3, 2F9, 2FH, 2G3, 2G9, 2HB, 2HD, 2I3, 2IB, 2J1, 2J7, 2JD, 301, 30D, 30H, 313, 319, 31B, 31H, 329, 32J, 33H, 33J, 343, 349, 34B, 34H, 351, 353, 357, 35J, 361, 367, 381, 387, 38D, 391, 39J, 3A9, 3B3, 3B7, 3B9, 3BD, 3BJ, 3C7, 3CB, 3CD, 3CJ, 3DB, 3E1, 3E3, 3E7, 3E9, 3ED, 3EJ, 3FB, 3G3, 3GB, 3H3, 3H9, 3HD, 3HJ, 3I7, 3IB, 3IJ, 3J3, 3JH, 401, 407, 409, 40D, 40J, 411, 417, 41H, 42H, 433, 437, 439, 44D, 44H, 44J, 459, 461, 463, 46D, 471, 477, 47D, 47J, 48H, 493, 497, 499, 4A1, 4AB, 4B3, 4BB, 4C7, 4D1, 4D7, 4DB, 4DD, 4DH, 4DJ, 4E9, 4F1, 4F7, 4FD, 4GB, 4GD, 4H9, 4HB, 4ID, 4IJ, 4J7, 4JD, 4JH, 4JJ, 503, 50B, 50H, 517, 519, 51J, 52D, 533, 539, 541, 543, 547, 549, 54J, 55B, 55D, 569, 56B, 56H, 571, 573, 57D, 581, 58J, 5A3, 5A7, 5AD, 5B1, 5BH, 5BJ, 5C3, 5CB, 5D7, 5D9, 5DD, 5E1, 5E7, 5ED, 5EH, 5F9, 5FB, 5GD, 5GJ, 5H1, 5H7, 5HB, 5HH, 5IB, 5IH, 5J1, 5J3, 5J9, 5JD, 5JJ, 60B, 60H, 613, 61H, 621, 627, 62J, 637, 63D, 63H, 653, 661, 66B, 66J, 673, 679, 67B, 67H, 68J, 69B, 69D, 6A9, 6AH, 6B1, 6BD, 6C7, 6CH, 6CJ, 6D3, 6DB, 6DH, 6E3, 6E7, 6E9, 6ED, 6EJ, 6F7, 6FB, 6FD, 6FJ, 6G9, 6GB, 6H1, 6H9, 6HD, 6I7, 6IH, 6J9, 6JB, 6JH, 701, 703, 70J, 71D, 71H, 723, 72B, 72H, 731, 73J, 747, 74H, 753, 759, 75H, 767, 76J, 77D, 77H, 783, 789, 78B, 79J, 7A1, 7AB, 7AJ, 7B3, 7BH, 7C1, 7C9, 7D1, 7D7, 7DJ, 7E3, 7E9, 7F9, 7FJ, 7G1, 7GH, 7I3, 7I7, 7I9, 7J1, 7J7, 7JB, 803, 809, 80H, 811, 819, 82B, 82D, 82H, 82J, 83B, 84J, 851, 857, 85D, 85J, 863, 869, 86B, 873, 877, 87J, 881, 88B, 88D, 899, 89B, 8A7, 8AD, 8BD, 8C9, 8CH, 8D1, 8D3, 8D7, 8D9, 8EB, 8EJ, 8FB, 8FH, 8G7, 8G9, 8GD, 8GJ, 8H1, 8H7, 8HH, 8HJ, 8IB, 8J1, 8J3, 8JD, 907, 90D, 90H, 913, 91B, 91H, 923, 92J, 93B, 93D, 93H, 94B, 94H, 951, 959, 95J, 967, 96D, 96J, 981, 987, 989, 98J, 99D, 99H, 9A3, 9B1, 9B3, 9BD, 9C7, 9CB, 9CD, 9D3, 9DH, 9E1, 9E9, 9F7, 9FB, 9FH, 9FJ, 9G3, 9G9, 9GB, 9H3, 9H7, 9I7, 9J9, A01, A03, A07, A0D, A0J, A11, A17, A29, A2B, A2H, A3D, A3J, A4B, A4D, A4J, A5B, A67, A69, A6D, A6J, A7D, A7H, A7J, A8H, AA1, AAB, AAH, AAJ, AB9, ABB, AC1, AC3, ACD, ACJ, AD1, ADB, ADD, AE3, AE9, AEH, AG7, AGH, AGJ, AH9, AHH, AI3, AID, AJB, AJH, B09, B11, B13, B21, B27, B2B, B2H, B33, B41, B43, B4D, B57, B5D, B5H, B5J, B63, B77, B79, B81, B87, B93, B9B, B9H, BA3, BB1, BBH, BBJ, BC3, BC9, BCB, BCH, BD3, BDD, BDJ, BEB, BF3, BG1, BG3, BG9, BGD, BHB, BHJ, BJ3, BJ7, BJ9, BJD, BJJ, C01, C0D, C0H, C1B, C31, C3B, C3H, C49, C53, C59, C5J, C6B, C6D, C6H, C73, C7B, C7H, C87, C89, C8D, C97, C9D, C9J, CA3, CA9, CAB, CB1, CB3, CBJ, CCB, CCJ, CDH, CE1, CE7, CEJ, CF1, CF7, CFD, CFJ, CH7, CHD, CI7, CIB, CIJ, CJ9, CJH, D09, D17, D1B, D1D, D1H, D31, D3D, D3J, D41, D4H, D53, D59, D63, D6D, D77, D7B, D91, D97, D9D, D9J, DA7, DAD, DAH, DAJ, DBB, DBH, DC1, DC3, DC9, DDB, DDH, DDJ, DE3, DF1, DF3, DF7, DFJ, DG1, DG7, DGB, DHH, DI3, DI9, DID, DJ1, DJB, E13, E1J, E21, E27, E2B, E2D, E2H, E2J, E39, E43, E49, E4D, E51, E5B, E5H, E6H, E71, E73, E79, E8J, E93, E9B, EA1, EA7, EAD, EB1, EB7, EBJ, EC3, EC9, ECB, ECH, ED1, ED7, ED9, EDJ, EE1, EEH, EF3, EG3, EG7, EGJ, EHD, EJ1, EJ7, F07, F0B, F19, F1H, F23, F27, F2D, F37, F3D, F3J, F49, F4B, F51, F5D, F61, F6B, F6D, F73, F7B, F83, F8D, F9H, F9J, FA3, FAB, FAH, FB1, FB9, FC7, FCH, FD3, FD9, FDB, FDH, FE7, FEJ, FF1, FFB, FFH, FG3, FG9, FGH, FH3, FHD, FHJ, FI1, FI7, FID, FIJ, FJ9, FJH, G11, G17, G29, G2B, G39, G3D, G41, G4B, G61, G69, G77, G7B, G7D, G83, G89, G8B, G8H, G91, G9J, GA7, GAJ, GBH, GCD, GCJ, GD1, GDD, GDJ, GE9, GEB, GF1, GF3, GF9, GFJ, GGD, GGH, GI1, GI3, GIJ, GJ1, GJB, GJD, H03, H13, H17, H19, H1D, H21, H2H, H33, H39, H3B, H43, H4J, H57, H5B, H5H, H77, H79, H7J, H81, H87, H8B, H8H, H93, H9B, H9H, HA1, HAD, HAJ, HB7, HBJ, HC3, HCH, HD9, HDJ, HF3, HF9, HG1, HG7, HG9, HHB, HHJ, HIH, HJ7, HJD, I07, I0B, I0D, I0J, I19, I1H, I23, I27, I2D, I43, I4H, I57, I59, I61, I6B, I6D, I79, I7B, I89, I9D, IAB, IAH, IBD, ICB, ICH, ICJ, IDH, IE1, IE7, IE9, IEJ, IF7, IFH, IG3, IG9, IGH, IH1, IH7, IH9, IHJ, II1, IID, IIH, IJ3, IJ9, IJB, J03, J07, J11, J1J, J23, J29, J39, J3D, J41, J47, J4B, J4J, J53, J5H, J63, J67, J71, J7D, J7H, J7J, J99, J9D, JAH, JB3, JB9, JC1, JCD, JD7, JDD, JDH, JDJ, JE3, JF1, JF7, JFJ, JG7, JGD, JGH, JH9, JHB, JI3, JJD, 1009, ...|
|21|12, 18, 1A, 1G, 1K, 21, 25, 2B, 2H, 2J, 34, 38, 3A, 3G, 3K, 45, 4D, 4H, 4J, 52, 54, 58, 61, 65, 6B, 6D, 72, 74, 7A, 7G, 7K, 85, 8B, 8D, 92, 94, 98, 9A, A1, AD, AH, AJ, B2, B8, BA, BK, C5, CB, CH, CJ, D4, D8, DA, DK, ED, EH, EJ, F2, FG, G1, GB, GD, GH, H2, HA, HG, I1, I5, IB, IJ, J2, JA, JK, K1, KB, KD, KJ, 102, 108, 10G, 10K, 111, 115, 11H, 124, 128, 12G, 12K, 135, 13H, 13J, 14G, 151, 15B, 15H, 162, 164, 16A, 16K, 175, 17B, 17D, 17J, 184, 188, 18A, 191, 19B, 19D, 19H, 1A2, 1A8, 1AA, 1B1, 1B5, 1BB, 1BJ, 1C8, 1CG, 1D5, 1DD, 1DJ, 1E4, 1E8, 1EG, 1F1, 1F5, 1FD, 1FH, 1GA, 1GK, 1HB, 1HD, 1I2, 1I4, 1I8, 1IA, 1IK, 1JD, 1JH, 1JJ, 1K2, 1KG, 1KK, 201, 205, 214, 218, 21G, 225, 22D, 22H, 232, 238, 241, 245, 24B, 24H, 254, 25A, 261, 265, 26B, 26D, 272, 274, 27A, 27K, 281, 28B, 28D, 28J, 29G, 29K, 2A1, 2A5, 2AB, 2AH, 2B4, 2BA, 2BG, 2CH, 2CJ, 2D8, 2DG, 2E5, 2EB, 2EH, 2F4, 2FG, 2FK, 2G5, 2GB, 2GD, 2GJ, 2HA, 2HK, 2IH, 2IJ, 2J2, 2J8, 2JA, 2JG, 2JK, 2K1, 2K5, 2KH, 2KJ, 304, 31H, 322, 328, 32G, 33D, 342, 34G, 34K, 351, 355, 35B, 35J, 362, 364, 36A, 371, 37B, 37D, 37H, 37J, 382, 388, 38K, 39B, 39J, 3AA, 3AG, 3AK, 3B5, 3BD, 3BH, 3C4, 3C8, 3D1, 3D5, 3DB, 3DD, 3DH, 3E2, 3E4, 3EA, 3EK, 3FJ, 3G4, 3G8, 3GA, 3HD, 3HH, 3HJ, 3I8, 3IK, 3J1, 3JB, 3JJ, 3K4, 3KA, 3KG, 40D, 40J, 412, 414, 41G, 425, 42H, 434, 43K, 44D, 44J, 452, 454, 458, 45A, 45K, 46B, 46H, 472, 47K, 481, 48H, 48J, 49K, 4A5, 4AD, 4AJ, 4B2, 4B4, 4B8, 4BG, 4C1, 4CB, 4CD, 4D2, 4DG, 4E5, 4EB, 4F2, 4F4, 4F8, 4FA, 4FK, 4GB, 4GD, 4H8, 4HA, 4HG, 4HK, 4I1, 4IB, 4IJ, 4JG, 4KJ, 502, 508, 50G, 51B, 51D, 51H, 524, 52K, 531, 535, 53D, 53J, 544, 548, 54K, 551, 562, 568, 56A, 56G, 56K, 575, 57J, 584, 588, 58A, 58G, 58K, 595, 59H, 5A2, 5A8, 5B1, 5B5, 5BB, 5C2, 5CA, 5CG, 5CK, 5E4, 5F1, 5FB, 5FJ, 5G2, 5G8, 5GA, 5GG, 5HH, 5I8, 5IA, 5J5, 5JD, 5JH, 5K8, 601, 60B, 60D, 60H, 614, 61A, 61G, 61K, 621, 625, 62B, 62J, 632, 634, 63A, 63K, 641, 64B, 64J, 652, 65G, 665, 66H, 66J, 674, 678, 67A, 685, 68J, 692, 698, 69G, 6A1, 6A5, 6B2, 6BA, 6BK, 6C5, 6CB, 6CJ, 6D8, 6DK, 6ED, 6EH, 6F2, 6F8, 6FA, 6GH, 6GJ, 6H8, 6HG, 6HK, 6ID, 6IH, 6J4, 6JG, 6K1, 6KD, 6KH, 702, 711, 71B, 71D, 728, 73D, 73H, 73J, 74A, 74G, 74K, 75B, 75H, 764, 768, 76G, 77H, 77J, 782, 784, 78G, 7A2, 7A4, 7AA, 7AG, 7B1, 7B5, 7BB, 7BD, 7C4, 7C8, 7CK, 7D1, 7DB, 7DD, 7E8, 7EA, 7F5, 7FB, 7GA, 7H5, 7HD, 7HH, 7HJ, 7I2, 7I4, 7J5, 7JD, 7K4, 7KA, 7KK, 801, 805, 80B, 80D, 80J, 818, 81A, 821, 82B, 82D, 832, 83G, 841, 845, 84B, 84J, 854, 85A, 865, 86H, 86J, 872, 87G, 881, 885, 88D, 892, 89A, 89G, 8A1, 8B2, 8B8, 8BA, 8BK, 8CD, 8CH, 8D2, 8DK, 8E1, 8EB, 8F4, 8F8, 8FA, 8FK, 8GD, 8GH, 8H4, 8I1, 8I5, 8IB, 8ID, 8IH, 8J2, 8J4, 8JG, 8JK, 8KJ, 90K, 91B, 91D, 91H, 922, 928, 92A, 92G, 93H, 93J, 944, 94K, 955, 95H, 95J, 964, 96G, 97B, 97D, 97H, 982, 98G, 98K, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9C8, 9CA, 9CK, 9D1, 9DB, 9DH, 9DJ, 9E8, 9EA, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9I2, 9IA, 9IG, 9J5, 9K2, 9K8, 9KK, A0B, A0D, A1A, A1G, A1K, A25, A2B, A38, A3A, A3K, A4D, A4J, A52, A54, A58, A6B, A6D, A74, A7A, A85, A8D, A8J, A94, AA1, AAH, AAJ, AB2, AB8, ABA, ABG, AC1, ACB, ACH, AD8, ADK, AEH, AEJ, AF4, AF8, AG5, AGD, AHG, AHK, AI1, AI5, AIB, AID, AJ4, AJ8, AK1, B0A, B0K, B15, B1H, B2A, B2G, B35, B3H, B3J, B42, B48, B4G, B51, B5B, B5D, B5H, B6A, B6G, B71, B75, B7B, B7D, B82, B84, B8K, B9B, B9J, BAG, BAK, BB5, BBH, BBJ, BC4, BCA, BCG, BE2, BE8, BF1, BF5, BFD, BG2, BGA, BH1, BHJ, BI2, BI4, BI8, BJB, BK2, BK8, BKA, C05, C0B, C0H, C1A, C1K, C2D, C2H, C45, C4B, C4H, C52, C5A, C5G, C5K, C61, C6D, C6J, C72, C74, C7A, C8B, C8H, C8J, C92, C9K, CA1, CA5, CAH, CAJ, CB4, CB8, CCD, CCJ, CD4, CD8, CDG, CE5, CFG, CGB, CGD, CGJ, CH2, CH4, CH8, CHA, CHK, CID, CIJ, CJ2, CJA, CJK, CK5, D04, D08, D0A, D0G, D24, D28, D2G, D35, D3B, D3H, D44, D4A, D51, D55, D5B, D5D, D5J, D62, D68, D6A, D6K, D71, D7H, D82, D91, D95, D9H, DAA, DBH, DC2, DD1, DD5, DE2, DEA, DEG, DEK, DF5, DFJ, DG4, DGA, DGK, DH1, DHB, DI2, DIA, DIK, DJ1, DJB, DJJ, DKA, DKK, E12, E14, E18, E1G, E21, E25, E2D, E3A, E3K, E45, E4B, E4D, E4J, E58, E5K, E61, E6B, E6H, E72, E78, E7G, E81, E8B, E8H, E8J, E94, E9A, E9G, EA5, EAD, EBG, EC1, ED2, ED4, EE1, EE5, EED, EF2, EGB, EGJ, EHG, EHK, EI1, EIB, EIH, EIJ, EJ4, EJ8, EK5, EKD, F04, F11, F1H, F22, F24, F2G, F31, F3B, F3D, F42, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA2, FA4, FA8, FAG, FBB, FBH, FC2, FC4, FCG, FDB, FDJ, FE2, FE8, FFH, FFJ, FG8, FGA, FGG, FGK, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G01, G0D, G12, G25, G2B, G32, G38, G3A, G4B, G4J, G5G, G65, G6B, G74, G78, G7A, G7G, G85, G8D, G8J, G92, G98, GAH, GBA, GBK, GC1, GCD, GD2, GD4, GDK, GE1, GEJ, GG1, GGJ, GH4, GHK, GIH, GJ2, GJ4, GK1, GK5, GKB, GKD, H02, H0A, H0K, H15, H1B, H1J, H22, H28, H2A, H2K, H31, H3D, H3H, H42, H48, H4A, H51, H55, H5J, H6G, H6K, H75, H84, H88, H8G, H91, H95, H9D, H9H, HAA, HAG, HAK, HBD, HC4, HC8, HCA, HDJ, HE2, HF5, HFB, HFH, HG8, HGK, HHD, HHJ, HI2, HI4, HI8, HJ5, HJB, HK2, HKA, HKG, HKK, I0B, I0D, I14, I2D, I38, ...|
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|23|16, 18, 1E, 1I, 1K, 21, 27, 2D, 2F, 2L, 32, 34, 3A, 3E, 3K, 45, 49, 4B, 4F, 4H, 4L, 5C, 5G, 5M, 61, 6B, 6D, 6J, 72, 76, 7C, 7I, 7K, 87, 89, 8D, 8F, 94, 9G, 9K, 9M, A3, A9, AB, AL, B4, BA, BG, BI, C1, C5, C7, CH, D8, DC, DE, DI, E9, EF, F2, F4, F8, FE, FM, G5, GB, GF, GL, H6, HA, HI, I5, I7, IH, IJ, J2, J6, JC, JK, K1, K3, K7, KJ, L4, L8, LG, LK, M3, MF, MH, 10C, 10I, 115, 11B, 11H, 11J, 122, 12C, 12I, 131, 133, 139, 13F, 13J, 13L, 14A, 14K, 14M, 153, 159, 15F, 15H, 166, 16A, 16G, 171, 17B, 17J, 186, 18E, 18K, 193, 197, 19F, 19L, 1A2, 1AA, 1AE, 1B5, 1BF, 1C4, 1C6, 1CG, 1CI, 1CM, 1D1, 1DB, 1E2, 1E6, 1E8, 1EC, 1F3, 1F7, 1F9, 1FD, 1GA, 1GE, 1GM, 1H9, 1HH, 1HL, 1I4, 1IA, 1J1, 1J5, 1JB, 1JH, 1K2, 1K8, 1KK, 1L1, 1L7, 1L9, 1LJ, 1LL, 1M4, 1ME, 1MG, 203, 205, 20B, 216, 21A, 21C, 21G, 21M, 225, 22D, 22J, 232, 241, 243, 24D, 24L, 258, 25E, 25K, 265, 26H, 26L, 274, 27A, 27C, 27I, 287, 28H, 29C, 29E, 29I, 2A1, 2A3, 2A9, 2AD, 2AF, 2AJ, 2B8, 2BA, 2BG, 2D4, 2DA, 2DG, 2E1, 2EJ, 2F6, 2FK, 2G1, 2G3, 2G7, 2GD, 2GL, 2H2, 2H4, 2HA, 2HM, 2I9, 2IB, 2IF, 2IH, 2IL, 2J4, 2JG, 2K5, 2KD, 2L2, 2L8, 2LC, 2LI, 2M3, 2M7, 2MF, 2MJ, 30A, 30E, 30K, 30M, 313, 319, 31B, 31H, 324, 331, 337, 33B, 33D, 34E, 34I, 34K, 357, 35J, 35L, 368, 36G, 36M, 375, 37B, 386, 38C, 38G, 38I, 397, 39H, 3A6, 3AE, 3B7, 3BL, 3C4, 3C8, 3CA, 3CE, 3CG, 3D3, 3DF, 3DL, 3E4, 3EM, 3F1, 3FH, 3FJ, 3GI, 3H1, 3H9, 3HF, 3HJ, 3HL, 3I2, 3IA, 3IG, 3J3, 3J5, 3JF, 3K6, 3KG, 3KM, 3LB, 3LD, 3LH, 3LJ, 3M6, 3MI, 3MK, 40D, 40F, 40L, 412, 414, 41E, 41M, 42H, 43I, 43M, 445, 44D, 456, 458, 45C, 45K, 46D, 46F, 46J, 474, 47A, 47G, 47K, 489, 48B, 49A, 49G, 49I, 4A1, 4A5, 4AB, 4B2, 4B8, 4BC, 4BE, 4BK, 4C1, 4C7, 4CJ, 4D2, 4D8, 4DM, 4E3, 4E9, 4EL, 4F6, 4FC, 4FG, 4GJ, 4HE, 4I1, 4I9, 4ID, 4IJ, 4IL, 4J4, 4K3, 4KF, 4KH, 4LA, 4LI, 4LM, 4MB, 502, 50C, 50E, 50I, 513, 519, 51F, 51J, 51L, 522, 528, 52G, 52K, 52M, 535, 53F, 53H, 544, 54C, 54G, 557, 55H, 566, 568, 56E, 56I, 56K, 57D, 584, 588, 58E, 58M, 595, 599, 5A4, 5AC, 5AM, 5B5, 5BB, 5BJ, 5C6, 5CI, 5D9, 5DD, 5DJ, 5E2, 5E4, 5F9, 5FB, 5FL, 5G6, 5GA, 5H1, 5H5, 5HD, 5I2, 5I8, 5IK, 5J1, 5J7, 5K4, 5KE, 5KG, 5L9, 5MC, 5MG, 5MI, 607, 60D, 60H, 616, 61C, 61K, 621, 629, 638, 63A, 63E, 63G, 645, 65A, 65C, 65I, 661, 667, 66B, 66H, 66J, 678, 67C, 681, 683, 68D, 68F, 698, 69A, 6A3, 6A9, 6B6, 6BM, 6C7, 6CB, 6CD, 6CH, 6CJ, 6DI, 6E3, 6EF, 6EL, 6F8, 6FA, 6FE, 6FK, 6FM, 6G5, 6GF, 6GH, 6H6, 6HG, 6HI, 6I5, 6IJ, 6J2, 6J6, 6JC, 6JK, 6K3, 6K9, 6L2, 6LE, 6LG, 6LK, 6MB, 6MH, 6ML, 706, 70G, 711, 717, 71D, 72C, 72I, 72K, 737, 73L, 742, 748, 753, 755, 75F, 766, 76A, 76C, 76M, 77D, 77H, 782, 78K, 791, 797, 799, 79D, 79J, 79L, 7AA, 7AE, 7BB, 7CA, 7CM, 7D1, 7D5, 7DB, 7DH, 7DJ, 7E2, 7F1, 7F3, 7F9, 7G2, 7G8, 7GK, 7GM, 7H5, 7HH, 7IA, 7IC, 7IG, 7IM, 7JD, 7JH, 7JJ, 7KE, 7LF, 7M2, 7M8, 7MA, 7MK, 7MM, 809, 80B, 80L, 814, 816, 81G, 81I, 825, 82B, 82J, 843, 84D, 84F, 852, 85A, 85G, 863, 86L, 874, 87G, 885, 887, 892, 898, 89C, 89I, 8A1, 8AJ, 8AL, 8B8, 8BM, 8C5, 8C9, 8CB, 8CF, 8DG, 8DI, 8E7, 8ED, 8F6, 8FE, 8FK, 8G3, 8GL, 8HE, 8HG, 8HK, 8I3, 8I5, 8IB, 8IH, 8J4, 8JA, 8JM, 8KB, 8L6, 8L8, 8LE, 8LI, 8MD, 8ML, 90M, 913, 915, 919, 91F, 91H, 926, 92A, 931, 948, 94I, 951, 95D, 964, 96A, 96K, 979, 97B, 97F, 97L, 986, 98C, 98M, 991, 995, 99J, 9A2, 9A8, 9AC, 9AI, 9AK, 9B7, 9B9, 9C2, 9CE, 9CM, 9DH, 9DL, 9E4, 9EG, 9EI, 9F1, 9F7, 9FD, 9GI, 9H1, 9HF, 9HJ, 9I4, 9IE, 9IM, 9JB, 9K6, 9KA, 9KC, 9KG, 9LH, 9M6, 9MC, 9ME, A07, A0D, A0J, A1A, A1K, A2B, A2F, A3M, A45, A4B, A4H, A52, A58, A5C, A5E, A63, A69, A6D, A6F, A6L, A7K, A83, A85, A89, A94, A96, A9A, A9M, AA1, AA7, AAB, ABE, ABK, AC3, AC7, ACF, AD2, AEB, AF4, AF6, AFC, AFG, AFI, AFM, AG1, AGB, AH2, AH8, AHC, AHK, AI7, AID, AJA, AJE, AJG, AJM, AL6, ALA, ALI, AM5, AMB, AMH, B02, B08, B0K, B11, B17, B19, B1F, B1J, B22, B24, B2E, B2G, B39, B3F, B4C, B4G, B55, B5J, B71, B77, B84, B88, B93, B9B, B9H, B9L, BA4, BAI, BB1, BB7, BBH, BBJ, BC6, BCI, BD3, BDD, BDF, BE2, BEA, BEM, BF9, BGA, BGC, BGG, BH1, BH7, BHB, BHJ, BIE, BJ1, BJ7, BJD, BJF, BJL, BK8, BKK, BKM, BL9, BLF, BLL, BM4, BMC, BMI, C05, C0B, C0D, C0J, C12, C18, C1I, C23, C34, C3A, C49, C4B, C56, C5A, C5I, C65, C7C, C7K, C8F, C8J, C8L, C98, C9E, C9G, C9M, CA3, CAL, CB6, CBI, CCD, CD6, CDC, CDE, CE3, CE9, CEJ, CEL, CF8, CFA, CFG, CG3, CGH, CGL, CHM, CI1, CIH, CIJ, CJ6, CJ8, CJI, CKF, CKJ, CKL, CL2, CLA, CM3, CM9, CMF, CMH, D06, D0M, D17, D1B, D1H, D31, D33, D3D, D3F, D3L, D42, D48, D4E, D4M, D55, D59, D5L, D64, D6C, D71, D75, D7J, D88, D8I, D9J, DA2, DAE, DAK, DAM, DBL, DC6, DD1, DDB, DDH, DE8, DEC, DEE, DEK, DF7, DFF, DFL, DG2, DG8, DHF, DI6, DIG, DII, DJ7, DJH, DJJ, DKC, DKE, DL9, DMA, E05, E0B, E14, E1M, E25, E27, E32, E36, E3C, E3E, E41, E49, E4J, E52, E58, E5G, E5K, E63, E65, E6F, E6H, E76, E7A, E7G, E7M, E81, E8D, E8H, E98, EA3, EA7, EAD, EBA, EBE, EBM, EC5, EC9, ECH, ECL, EDC, EDI, EDM, EED, EF2, EF6, EF8, EGF, EGJ, EHK, EI3, EI9, EIL, EJA, EK1, EK7, EKB, EKD, EKH, ELC, ELI, EM7, EMF, EML, F02, F0E, F0G, F15, F2C, F35, F37, ...|
|24|15, 17, 1D, 1H, 1J, 1N, 25, 2B, 2D, 2J, 2N, 31, 37, 3B, 3H, 41, 45, 47, 4B, 4D, 4H, 57, 5B, 5H, 5J, 65, 67, 6D, 6J, 6N, 75, 7B, 7D, 7N, 81, 85, 87, 8J, 97, 9B, 9D, 9H, 9N, A1, AB, AH, AN, B5, B7, BD, BH, BJ, C5, CJ, CN, D1, D5, DJ, E1, EB, ED, EH, EN, F7, FD, FJ, FN, G5, GD, GH, H1, HB, HD, HN, I1, I7, IB, IH, J1, J5, J7, JB, JN, K7, KB, KJ, KN, L5, LH, LJ, MD, MJ, N5, NB, NH, NJ, 101, 10B, 10H, 10N, 111, 117, 11D, 11H, 11J, 127, 12H, 12J, 12N, 135, 13B, 13D, 141, 145, 14B, 14J, 155, 15D, 15N, 167, 16D, 16J, 16N, 177, 17D, 17H, 181, 185, 18J, 195, 19H, 19J, 1A5, 1A7, 1AB, 1AD, 1AN, 1BD, 1BH, 1BJ, 1BN, 1CD, 1CH, 1CJ, 1CN, 1DJ, 1DN, 1E7, 1EH, 1F1, 1F5, 1FB, 1FH, 1G7, 1GB, 1GH, 1GN, 1H7, 1HD, 1I1, 1I5, 1IB, 1ID, 1IN, 1J1, 1J7, 1JH, 1JJ, 1K5, 1K7, 1KD, 1L7, 1LB, 1LD, 1LH, 1LN, 1M5, 1MD, 1MJ, 1N1, 1NN, 201, 20B, 20J, 215, 21B, 21H, 221, 22D, 22H, 22N, 235, 237, 23D, 241, 24B, 255, 257, 25B, 25H, 25J, 261, 265, 267, 26B, 26N, 271, 277, 28H, 28N, 295, 29D, 2A7, 2AH, 2B7, 2BB, 2BD, 2BH, 2BN, 2C7, 2CB, 2CD, 2CJ, 2D7, 2DH, 2DJ, 2DN, 2E1, 2E5, 2EB, 2EN, 2FB, 2FJ, 2G7, 2GD, 2GH, 2GN, 2H7, 2HB, 2HJ, 2HN, 2ID, 2IH, 2IN, 2J1, 2J5, 2JB, 2JD, 2JJ, 2K5, 2L1, 2L7, 2LB, 2LD, 2MD, 2MH, 2MJ, 2N5, 2NH, 2NJ, 305, 30D, 30J, 311, 317, 321, 327, 32B, 32D, 331, 33B, 33N, 347, 34N, 35D, 35J, 35N, 361, 365, 367, 36H, 375, 37B, 37H, 38B, 38D, 395, 397, 3A5, 3AB, 3AJ, 3B1, 3B5, 3B7, 3BB, 3BJ, 3C1, 3CB, 3CD, 3CN, 3DD, 3DN, 3E5, 3EH, 3EJ, 3EN, 3F1, 3FB, 3FN, 3G1, 3GH, 3GJ, 3H1, 3H5, 3H7, 3HH, 3I1, 3IJ, 3JJ, 3JN, 3K5, 3KD, 3L5, 3L7, 3LB, 3LJ, 3MB, 3MD, 3MH, 3N1, 3N7, 3ND, 3NH, 405, 407, 415, 41B, 41D, 41J, 41N, 425, 42J, 431, 435, 437, 43D, 43H, 43N, 44B, 44H, 44N, 45D, 45H, 45N, 46B, 46J, 471, 475, 487, 491, 49B, 49J, 49N, 4A5, 4A7, 4AD, 4BB, 4BN, 4C1, 4CH, 4D1, 4D5, 4DH, 4E7, 4EH, 4EJ, 4EN, 4F7, 4FD, 4FJ, 4FN, 4G1, 4G5, 4GB, 4GJ, 4GN, 4H1, 4H7, 4HH, 4HJ, 4I5, 4ID, 4IH, 4J7, 4JH, 4K5, 4K7, 4KD, 4KH, 4KJ, 4LB, 4M1, 4M5, 4MB, 4MJ, 4N1, 4N5, 4NN, 507, 50H, 50N, 515, 51D, 51N, 52B, 531, 535, 53B, 53H, 53J, 54N, 551, 55B, 55J, 55N, 56D, 56H, 571, 57D, 57J, 587, 58B, 58H, 59D, 59N, 5A1, 5AH, 5BJ, 5BN, 5C1, 5CD, 5CJ, 5CN, 5DB, 5DH, 5E1, 5E5, 5ED, 5FB, 5FD, 5FH, 5FJ, 5G7, 5HB, 5HD, 5HJ, 5I1, 5I7, 5IB, 5IH, 5IJ, 5J7, 5JB, 5JN, 5K1, 5KB, 5KD, 5L5, 5L7, 5LN, 5M5, 5N1, 5NH, 601, 605, 607, 60B, 60D, 61B, 61J, 627, 62D, 62N, 631, 635, 63B, 63D, 63J, 645, 647, 64J, 655, 657, 65H, 667, 66D, 66H, 66N, 677, 67D, 67J, 68B, 68N, 691, 695, 69J, 6A1, 6A5, 6AD, 6AN, 6B7, 6BD, 6BJ, 6CH, 6CN, 6D1, 6DB, 6E1, 6E5, 6EB, 6F5, 6F7, 6FH, 6G7, 6GB, 6GD, 6GN, 6HD, 6HH, 6I1, 6IJ, 6IN, 6J5, 6J7, 6JB, 6JH, 6JJ, 6K7, 6KB, 6L7, 6M5, 6MH, 6MJ, 6MN, 6N5, 6NB, 6ND, 6NJ, 70H, 70J, 711, 71H, 71N, 72B, 72D, 72J, 737, 73N, 741, 745, 74B, 751, 755, 757, 761, 771, 77B, 77H, 77J, 785, 787, 78H, 78J, 795, 79B, 79D, 79N, 7A1, 7AB, 7AH, 7B1, 7C7, 7CH, 7CJ, 7D5, 7DD, 7DJ, 7E5, 7EN, 7F5, 7FH, 7G5, 7G7, 7H1, 7H7, 7HB, 7HH, 7HN, 7IH, 7IJ, 7J5, 7JJ, 7K1, 7K5, 7K7, 7KB, 7LB, 7LD, 7M1, 7M7, 7MN, 7N7, 7ND, 7NJ, 80D, 815, 817, 81B, 81H, 81J, 821, 827, 82H, 82N, 83B, 83N, 84H, 84J, 851, 855, 85N, 867, 877, 87B, 87D, 87H, 87N, 881, 88D, 88H, 897, 8AD, 8AN, 8B5, 8BH, 8C7, 8CD, 8CN, 8DB, 8DD, 8DH, 8DN, 8E7, 8ED, 8EN, 8F1, 8F5, 8FJ, 8G1, 8G7, 8GB, 8GH, 8GJ, 8H5, 8H7, 8HN, 8IB, 8IJ, 8JD, 8JH, 8JN, 8KB, 8KD, 8KJ, 8L1, 8L7, 8MB, 8MH, 8N7, 8NB, 8NJ, 905, 90D, 911, 91J, 91N, 921, 925, 935, 93H, 93N, 941, 94H, 94N, 955, 95J, 965, 96J, 96N, 985, 98B, 98H, 98N, 997, 99D, 99H, 99J, 9A7, 9AD, 9AH, 9AJ, 9B1, 9BN, 9C5, 9C7, 9CB, 9D5, 9D7, 9DB, 9DN, 9E1, 9E7, 9EB, 9FD, 9FJ, 9G1, 9G5, 9GD, 9GN, 9I7, 9IN, 9J1, 9J7, 9JB, 9JD, 9JH, 9JJ, 9K5, 9KJ, 9L1, 9L5, 9LD, 9LN, 9M5, 9N1, 9N5, 9N7, 9ND, A0J, A0N, A17, A1H, A1N, A25, A2D, A2J, A37, A3B, A3H, A3J, A41, A45, A4B, A4D, A4N, A51, A5H, A5N, A6J, A6N, A7B, A81, A95, A9B, AA7, AAB, AB5, ABD, ABJ, ABN, AC5, ACJ, AD1, AD7, ADH, ADJ, AE5, AEH, AF1, AFB, AFD, AFN, AG7, AGJ, AH5, AI5, AI7, AIB, AIJ, AJ1, AJ5, AJD, AK7, AKH, AKN, AL5, AL7, ALD, ALN, AMB, AMD, AMN, AN5, ANB, ANH, B01, B07, B0H, B0N, B11, B17, B1D, B1J, B25, B2D, B3D, B3J, B4H, B4J, B5D, B5H, B61, B6B, B7H, B81, B8J, B8N, B91, B9B, B9H, B9J, BA1, BA5, BAN, BB7, BBJ, BCD, BD5, BDB, BDD, BE1, BE7, BEH, BEJ, BF5, BF7, BFD, BFN, BGD, BGH, BHH, BHJ, BIB, BID, BIN, BJ1, BJB, BK7, BKB, BKD, BKH, BL1, BLH, BLN, BM5, BM7, BMJ, BNB, BNJ, BNN, C05, C1B, C1D, C1N, C21, C27, C2B, C2H, C2N, C37, C3D, C3H, C45, C4B, C4J, C57, C5B, C61, C6D, C6N, C7N, C85, C8H, C8N, C91, C9N, CA7, CB1, CBB, CBH, CC7, CCB, CCD, CCJ, CD5, CDD, CDJ, CDN, CE5, CFB, CG1, CGB, CGD, CH1, CHB, CHD, CI5, CI7, CJ1, CK1, CKJ, CL1, CLH, CMB, CMH, CMJ, CND, CNH, CNN, D01, D0B, D0J, D15, D1B, D1H, D21, D25, D2B, D2D, D2N, D31, D3D, D3H, D3N, D45, D47, D4J, D4N, D5D, D67, D6B, D6H, D7D, D7H, D81, D87, D8B, D8J, D8N, D9D, D9J, D9N, DAD, DB1, DB5, DB7, DCD, DCH, DDH, DDN, DE5, DEH, DF5, DFJ, DG1, DG5, DG7, DGB, DH5, DHB, DHN, DI7, DID, DIH, DJ5, DJ7, DJJ, DL1, DLH, DLJ, ...|
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|27|12, 14, 1A, 1E, 1G, 1K, 1Q, 25, 27, 2D, 2H, 2J, 2P, 32, 38, 3G, 3K, 3M, 3Q, 41, 45, 4J, 4N, 52, 54, 5E, 5G, 5M, 61, 65, 6B, 6H, 6J, 72, 74, 78, 7A, 7M, 87, 8B, 8D, 8H, 8N, 8P, 98, 9E, 9K, 9Q, A1, A7, AB, AD, AN, BA, BE, BG, BK, C7, CD, CN, CP, D2, D8, DG, DM, E1, E5, EB, EJ, EN, F4, FE, FG, FQ, G1, G7, GB, GH, GP, H2, H4, H8, HK, I1, I5, ID, IH, IN, J8, JA, K1, K7, KH, KN, L2, L4, LA, LK, LQ, M5, M7, MD, MJ, MN, MP, NA, NK, NM, NQ, O5, OB, OD, OP, P2, P8, PG, PQ, Q7, QH, QP, 104, 10A, 10E, 10M, 111, 115, 11D, 11H, 124, 12E, 12Q, 131, 13B, 13D, 13H, 13J, 142, 14G, 14K, 14M, 14Q, 15D, 15H, 15J, 15N, 16G, 16K, 171, 17B, 17J, 17N, 182, 188, 18M, 18Q, 195, 19B, 19J, 19P, 1AA, 1AE, 1AK, 1AM, 1B5, 1B7, 1BD, 1BN, 1BP, 1C8, 1CA, 1CG, 1D7, 1DB, 1DD, 1DH, 1DN, 1E2, 1EA, 1EG, 1EM, 1FH, 1FJ, 1G2, 1GA, 1GK, 1GQ, 1H5, 1HD, 1HP, 1I2, 1I8, 1IE, 1IG, 1IM, 1J7, 1JH, 1K8, 1KA, 1KE, 1KK, 1KM, 1L1, 1L5, 1L7, 1LB, 1LN, 1LP, 1M4, 1NB, 1NH, 1NN, 1O4, 1OM, 1P5, 1PJ, 1PN, 1PP, 1Q2, 1Q8, 1QG, 1QK, 1QM, 201, 20D, 20N, 20P, 212, 214, 218, 21E, 21Q, 22B, 22J, 234, 23A, 23E, 23K, 241, 245, 24D, 24H, 254, 258, 25E, 25G, 25K, 25Q, 261, 267, 26H, 27A, 27G, 27K, 27M, 28J, 28N, 28P, 298, 29K, 29M, 2A5, 2AD, 2AJ, 2AP, 2B4, 2BM, 2C1, 2C5, 2C7, 2CJ, 2D2, 2DE, 2DM, 2EB, 2EP, 2F4, 2F8, 2FA, 2FE, 2FG, 2FQ, 2GB, 2GH, 2GN, 2HE, 2HG, 2I5, 2I7, 2J2, 2J8, 2JG, 2JM, 2JQ, 2K1, 2K5, 2KD, 2KJ, 2L2, 2L4, 2LE, 2M1, 2MB, 2MH, 2N2, 2N4, 2N8, 2NA, 2NK, 2O5, 2O7, 2ON, 2OP, 2P4, 2P8, 2PA, 2PK, 2Q1, 2QJ, 30G, 30K, 30Q, 317, 31N, 31P, 322, 32A, 32Q, 331, 335, 33D, 33J, 33P, 342, 34E, 34G, 35B, 35H, 35J, 35P, 362, 368, 36M, 371, 375, 377, 37D, 37H, 37N, 388, 38E, 38K, 397, 39B, 39H, 3A2, 3AA, 3AG, 3AK, 3BJ, 3CA, 3CK, 3D1, 3D5, 3DB, 3DD, 3DJ, 3EE, 3EQ, 3F1, 3FH, 3FP, 3G2, 3GE, 3H1, 3HB, 3HD, 3HH, 3HP, 3I4, 3IA, 3IE, 3IG, 3IK, 3IQ, 3J7, 3JB, 3JD, 3JJ, 3K2, 3K4, 3KE, 3KM, 3KQ, 3LD, 3LN, 3M8, 3MA, 3MG, 3MK, 3MM, 3NB, 3NP, 3O2, 3O8, 3OG, 3OM, 3OQ, 3PH, 3PP, 3Q8, 3QE, 3QK, 401, 40B, 40N, 41A, 41E, 41K, 41Q, 421, 432, 434, 43E, 43M, 43Q, 44D, 44H, 44P, 45A, 45G, 461, 465, 46B, 474, 47E, 47G, 485, 494, 498, 49A, 49M, 4A1, 4A5, 4AH, 4AN, 4B4, 4B8, 4BG, 4CB, 4CD, 4CH, 4CJ, 4D4, 4E5, 4E7, 4ED, 4EJ, 4EP, 4F2, 4F8, 4FA, 4FM, 4FQ, 4GB, 4GD, 4GN, 4GP, 4HE, 4HG, 4I5, 4IB, 4J4, 4JK, 4K1, 4K5, 4K7, 4KB, 4KD, 4L8, 4LG, 4M1, 4M7, 4MH, 4MJ, 4MN, 4N2, 4N4, 4NA, 4NK, 4NM, 4O7, 4OH, 4OJ, 4P2, 4PG, 4PM, 4PQ, 4Q5, 4QD, 4QJ, 4QP, 50E, 50Q, 511, 515, 51J, 51P, 522, 52A, 52K, 531, 537, 53D, 548, 54E, 54G, 54Q, 55D, 55H, 55N, 56E, 56G, 56Q, 57D, 57H, 57J, 582, 58G, 58K, 591, 59J, 59N, 5A2, 5A4, 5A8, 5AE, 5AG, 5B1, 5B5, 5BP, 5CK, 5D5, 5D7, 5DB, 5DH, 5DN, 5DP, 5E4, 5EQ, 5F1, 5F7, 5FN, 5G2, 5GE, 5GG, 5GM, 5H7, 5HN, 5HP, 5I2, 5I8, 5IM, 5IQ, 5J1, 5JJ, 5KG, 5KQ, 5L5, 5L7, 5LH, 5LJ, 5M2, 5M4, 5ME, 5MK, 5MM, 5N5, 5N7, 5NH, 5NN, 5O4, 5P7, 5PH, 5PJ, 5Q2, 5QA, 5QG, 5QQ, 60H, 60N, 618, 61K, 61M, 62D, 62J, 62N, 632, 638, 63Q, 641, 64B, 64P, 654, 658, 65A, 65E, 66B, 66D, 66P, 674, 67K, 681, 687, 68D, 694, 69K, 69M, 69Q, 6A5, 6A7, 6AD, 6AJ, 6B2, 6B8, 6BK, 6C5, 6CN, 6CP, 6D4, 6D8, 6DQ, 6E7, 6F4, 6F8, 6FA, 6FE, 6FK, 6FM, 6G7, 6GB, 6GP, 6I1, 6IB, 6IH, 6J2, 6JG, 6JM, 6K5, 6KH, 6KJ, 6KN, 6L2, 6LA, 6LG, 6LQ, 6M1, 6M5, 6MJ, 6MP, 6N4, 6N8, 6NE, 6NG, 6NQ, 6O1, 6OH, 6P2, 6PA, 6Q1, 6Q5, 6QB, 6QN, 6QP, 704, 70A, 70G, 71H, 71N, 72A, 72E, 72M, 735, 73D, 73P, 74G, 74K, 74M, 74Q, 75N, 768, 76E, 76G, 775, 77B, 77H, 784, 78E, 791, 795, 7A8, 7AE, 7AK, 7AQ, 7B7, 7BD, 7BH, 7BJ, 7C4, 7CA, 7CE, 7CG, 7CM, 7DH, 7DN, 7DP, 7E2, 7EK, 7EM, 7EQ, 7FB, 7FD, 7FJ, 7FN, 7GM, 7H1, 7H7, 7HB, 7HJ, 7I2, 7J7, 7JN, 7JP, 7K4, 7K8, 7KA, 7KE, 7KG, 7KQ, 7LD, 7LJ, 7LN, 7M4, 7ME, 7MK, 7ND, 7NH, 7NJ, 7NP, 7P1, 7P5, 7PD, 7PN, 7Q2, 7Q8, 7QG, 7QM, 807, 80B, 80H, 80J, 80P, 812, 818, 81A, 81K, 81M, 82B, 82H, 83A, 83E, 83Q, 84D, 85E, 85K, 86D, 86H, 878, 87G, 87M, 87Q, 885, 88J, 88P, 894, 89E, 89G, 89Q, 8AB, 8AJ, 8B2, 8B4, 8BE, 8BM, 8C7, 8CH, 8DE, 8DG, 8DK, 8E1, 8E7, 8EB, 8EJ, 8FA, 8FK, 8FQ, 8G5, 8G7, 8GD, 8GN, 8H8, 8HA, 8HK, 8HQ, 8I5, 8IB, 8IJ, 8IP, 8J8, 8JE, 8JG, 8JM, 8K1, 8K7, 8KH, 8KP, 8LM, 8M1, 8MN, 8MP, 8NG, 8NK, 8O1, 8OB, 8PE, 8PM, 8QD, 8QH, 8QJ, 902, 908, 90A, 90G, 90K, 91B, 91J, 924, 92M, 93B, 93H, 93J, 944, 94A, 94K, 94M, 955, 957, 95D, 95N, 96A, 96E, 97B, 97D, 982, 984, 98E, 98G, 98Q, 99J, 99N, 99P, 9A2, 9AA, 9AQ, 9B5, 9BB, 9BD, 9BP, 9CE, 9CM, 9CQ, 9D5, 9E8, 9EA, 9EK, 9EM, 9F1, 9F5, 9FB, 9FH, 9FP, 9G4, 9G8, 9GK, 9GQ, 9H7, 9HJ, 9HN, 9IA, 9IM, 9J5, 9K2, 9K8, 9KK, 9KQ, 9L1, 9LN, 9M4, 9MM, 9N5, 9NB, 9NP, 9O2, 9O4, 9OA, 9OK, 9P1, 9P7, 9PB, 9PH, 9QK, A07, A0H, A0J, A14, A1E, A1G, A25, A27, A2P, A3M, A4D, A4J, A58, A5Q, A65, A67, A6P, A72, A78, A7A, A7K, A81, A8B, A8H, A8N, A94, A98, A9E, A9G, A9Q, AA1, AAD, AAH, AAN, AB2, AB4, ABG, ABK, AC7, ACP, AD2, AD8, AE1, AE5, AED, AEJ, AEN, AF4, AF8, AFM, AG1, AG5, AGJ, AH4, AH8, AHA, AID, AIH, AJE, AJK, AJQ, AKB, AKN, ALA, ALG, ALK, ALM, ALQ, AMH, AMN, AN8, ANG, ANM, ANQ, AOB, AOD, AOP, AQ1, AQH, AQJ, ...|
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|30|11, 17, 1B, 1D, 1H, 1N, 1T, 21, 27, 2B, 2D, 2J, 2N, 2T, 37, 3B, 3D, 3H, 3J, 3N, 47, 4B, 4H, 4J, 4T, 51, 57, 5D, 5H, 5N, 5T, 61, 6B, 6D, 6H, 6J, 71, 7D, 7H, 7J, 7N, 7T, 81, 8B, 8H, 8N, 8T, 91, 97, 9B, 9D, 9N, A7, AB, AD, AH, B1, B7, BH, BJ, BN, BT, C7, CD, CJ, CN, CT, D7, DB, DJ, DT, E1, EB, ED, EJ, EN, ET, F7, FB, FD, FH, FT, G7, GB, GJ, GN, GT, HB, HD, I1, I7, IH, IN, IT, J1, J7, JH, JN, JT, K1, K7, KD, KH, KJ, L1, LB, LD, LH, LN, LT, M1, MD, MH, MN, N1, NB, NJ, NT, O7, OD, OJ, ON, P1, P7, PB, PJ, PN, Q7, QH, QT, R1, RB, RD, RH, RJ, RT, SD, SH, SJ, SN, T7, TB, TD, TH, 107, 10B, 10J, 10T, 117, 11B, 11H, 11N, 127, 12B, 12H, 12N, 131, 137, 13J, 13N, 13T, 141, 14B, 14D, 14J, 14T, 151, 15B, 15D, 15J, 167, 16B, 16D, 16H, 16N, 16T, 177, 17D, 17J, 18B, 18D, 18N, 191, 19B, 19H, 19N, 1A1, 1AD, 1AH, 1AN, 1AT, 1B1, 1B7, 1BJ, 1BT, 1CH, 1CJ, 1CN, 1CT, 1D1, 1D7, 1DB, 1DD, 1DH, 1DT, 1E1, 1E7, 1FB, 1FH, 1FN, 1G1, 1GJ, 1GT, 1HD, 1HH, 1HJ, 1HN, 1HT, 1I7, 1IB, 1ID, 1IJ, 1J1, 1JB, 1JD, 1JH, 1JJ, 1JN, 1JT, 1KB, 1KN, 1L1, 1LD, 1LJ, 1LN, 1LT, 1M7, 1MB, 1MJ, 1MN, 1N7, 1NB, 1NH, 1NJ, 1NN, 1NT, 1O1, 1O7, 1OH, 1P7, 1PD, 1PH, 1PJ, 1QD, 1QH, 1QJ, 1QT, 1RB, 1RD, 1RN, 1S1, 1S7, 1SD, 1SJ, 1T7, 1TD, 1TH, 1TJ, 201, 20B, 20N, 211, 21H, 221, 227, 22B, 22D, 22H, 22J, 22T, 23B, 23H, 23N, 24B, 24D, 24T, 251, 25N, 25T, 267, 26D, 26H, 26J, 26N, 271, 277, 27H, 27J, 27T, 28D, 28N, 28T, 29B, 29D, 29H, 29J, 29T, 2AB, 2AD, 2AT, 2B1, 2B7, 2BB, 2BD, 2BN, 2C1, 2CJ, 2DD, 2DH, 2DN, 2E1, 2EH, 2EJ, 2EN, 2F1, 2FH, 2FJ, 2FN, 2G1, 2G7, 2GD, 2GH, 2GT, 2H1, 2HN, 2HT, 2I1, 2I7, 2IB, 2IH, 2J1, 2J7, 2JB, 2JD, 2JJ, 2JN, 2JT, 2KB, 2KH, 2KN, 2L7, 2LB, 2LH, 2LT, 2M7, 2MD, 2MH, 2ND, 2O1, 2OB, 2OJ, 2ON, 2OT, 2P1, 2P7, 2PT, 2QB, 2QD, 2QT, 2R7, 2RB, 2RN, 2S7, 2SH, 2SJ, 2SN, 2T1, 2T7, 2TD, 2TH, 2TJ, 2TN, 2TT, 307, 30B, 30D, 30J, 30T, 311, 31B, 31J, 31N, 327, 32H, 32T, 331, 337, 33B, 33D, 33T, 34D, 34H, 34N, 351, 357, 35B, 35T, 367, 36H, 36N, 36T, 377, 37H, 37T, 38D, 38H, 38N, 38T, 391, 39T, 3A1, 3AB, 3AJ, 3AN, 3B7, 3BB, 3BJ, 3C1, 3C7, 3CJ, 3CN, 3CT, 3DJ, 3DT, 3E1, 3EH, 3FD, 3FH, 3FJ, 3G1, 3G7, 3GB, 3GN, 3GT, 3H7, 3HB, 3HJ, 3IB, 3ID, 3IH, 3IJ, 3J1, 3JT, 3K1, 3K7, 3KD, 3KJ, 3KN, 3KT, 3L1, 3LD, 3LH, 3LT, 3M1, 3MB, 3MD, 3MT, 3N1, 3NH, 3NN, 3OD, 3OT, 3P7, 3PB, 3PD, 3PH, 3PJ, 3QB, 3QJ, 3R1, 3R7, 3RH, 3RJ, 3RN, 3RT, 3S1, 3S7, 3SH, 3SJ, 3T1, 3TB, 3TD, 3TN, 407, 40D, 40H, 40N, 411, 417, 41D, 41T, 42B, 42D, 42H, 431, 437, 43B, 43J, 43T, 447, 44D, 44J, 45B, 45H, 45J, 45T, 46D, 46H, 46N, 47B, 47D, 47N, 487, 48B, 48D, 48N, 497, 49B, 49J, 4A7, 4AB, 4AH, 4AJ, 4AN, 4AT, 4B1, 4BD, 4BH, 4C7, 4CT, 4DB, 4DD, 4DH, 4DN, 4DT, 4E1, 4E7, 4ET, 4F1, 4F7, 4FN, 4FT, 4GB, 4GD, 4GJ, 4H1, 4HH, 4HJ, 4HN, 4HT, 4ID, 4IH, 4IJ, 4J7, 4K1, 4KB, 4KH, 4KJ, 4KT, 4L1, 4LB, 4LD, 4LN, 4LT, 4M1, 4MB, 4MD, 4MN, 4MT, 4N7, 4O7, 4OH, 4OJ, 4OT, 4P7, 4PD, 4PN, 4QB, 4QH, 4QT, 4RB, 4RD, 4S1, 4S7, 4SB, 4SH, 4SN, 4TB, 4TD, 4TN, 507, 50D, 50H, 50J, 50N, 51H, 51J, 521, 527, 52N, 531, 537, 53D, 541, 54H, 54J, 54N, 54T, 551, 557, 55D, 55N, 55T, 56B, 56N, 57B, 57D, 57J, 57N, 58B, 58J, 59D, 59H, 59J, 59N, 59T, 5A1, 5AD, 5AH, 5B1, 5C1, 5CB, 5CH, 5CT, 5DD, 5DJ, 5DT, 5EB, 5ED, 5EH, 5EN, 5F1, 5F7, 5FH, 5FJ, 5FN, 5G7, 5GD, 5GJ, 5GN, 5GT, 5H1, 5HB, 5HD, 5HT, 5IB, 5IJ, 5J7, 5JB, 5JH, 5JT, 5K1, 5K7, 5KD, 5KJ, 5LH, 5LN, 5M7, 5MB, 5MJ, 5MT, 5N7, 5NJ, 5O7, 5OB, 5OD, 5OH, 5PB, 5PN, 5PT, 5Q1, 5QH, 5QN, 5QT, 5RD, 5RN, 5S7, 5SB, 5TB, 5TH, 5TN, 5TT, 607, 60D, 60H, 60J, 611, 617, 61B, 61D, 61J, 62B, 62H, 62J, 62N, 63B, 63D, 63H, 63T, 641, 647, 64B, 657, 65D, 65J, 65N, 661, 66B, 67D, 67T, 681, 687, 68B, 68D, 68H, 68J, 68T, 69D, 69J, 69N, 6A1, 6AB, 6AH, 6B7, 6BB, 6BD, 6BJ, 6CJ, 6CN, 6D1, 6DB, 6DH, 6DN, 6E1, 6E7, 6EJ, 6EN, 6ET, 6F1, 6F7, 6FB, 6FH, 6FJ, 6FT, 6G1, 6GH, 6GN, 6HD, 6HH, 6HT, 6ID, 6JB, 6JH, 6K7, 6KB, 6KT, 6L7, 6LD, 6LH, 6LN, 6M7, 6MD, 6MJ, 6MT, 6N1, 6NB, 6NN, 6O1, 6OB, 6OD, 6ON, 6P1, 6PD, 6PN, 6QH, 6QJ, 6QN, 6R1, 6R7, 6RB, 6RJ, 6S7, 6SH, 6SN, 6ST, 6T1, 6T7, 6TH, 6TT, 701, 70B, 70H, 70N, 70T, 717, 71D, 71N, 71T, 721, 727, 72D, 72J, 72T, 737, 741, 747, 74T, 751, 75J, 75N, 761, 76B, 77B, 77J, 787, 78B, 78D, 78N, 78T, 791, 797, 79B, 79T, 7A7, 7AJ, 7B7, 7BN, 7BT, 7C1, 7CD, 7CJ, 7CT, 7D1, 7DB, 7DD, 7DJ, 7DT, 7ED, 7EH, 7FB, 7FD, 7FT, 7G1, 7GB, 7GD, 7GN, 7HD, 7HH, 7HJ, 7HN, 7I1, 7IH, 7IN, 7IT, 7J1, 7JD, 7JT, 7K7, 7KB, 7KH, 7LH, 7LJ, 7LT, 7M1, 7M7, 7MB, 7MH, 7MN, 7N1, 7N7, 7NB, 7NN, 7NT, 7O7, 7OJ, 7ON, 7P7, 7PJ, 7PT, 7QN, 7QT, 7RB, 7RH, 7RJ, 7SB, 7SJ, 7T7, 7TH, 7TN, 807, 80B, 80D, 80J, 80T, 817, 81D, 81H, 81N, 82N, 837, 83H, 83J, 841, 84B, 84D, 84T, 851, 85J, 86D, 871, 877, 87N, 88B, 88H, 88J, 897, 89B, 89H, 89J, 89T, 8A7, 8AH, 8AN, 8AT, 8B7, 8BB, 8BH, 8BJ, 8BT, 8C1, 8CD, 8CH, 8CN, 8CT, 8D1, 8DD, 8DH, 8E1, 8EJ, 8EN, 8ET, 8FJ, 8FN, 8G1, 8G7, 8GB, 8GJ, 8GN, 8H7, 8HD, 8HH, 8I1, 8ID, 8IH, 8IJ, 8JJ, 8JN, 8KH, 8KN, 8KT, 8LB, 8LN, 8M7, 8MD, 8MH, 8MJ, 8MN, 8NB, 8NH, 8NT, 8O7, 8OD, 8OH, 8OT, 8P1, 8PD, 8QD, 8QT, 8R1, 8R7, ...|
|31|16, 1A, 1C, 1G, 1M, 1S, 1U, 25, 29, 2B, 2H, 2L, 2R, 34, 38, 3A, 3E, 3G, 3K, 43, 47, 4D, 4F, 4P, 4R, 52, 58, 5C, 5I, 5O, 5Q, 65, 67, 6B, 6D, 6P, 76, 7A, 7C, 7G, 7M, 7O, 83, 89, 8F, 8L, 8N, 8T, 92, 94, 9E, 9S, A1, A3, A7, AL, AR, B6, B8, BC, BI, BQ, C1, C7, CB, CH, CP, CT, D6, DG, DI, DS, DU, E5, E9, EF, EN, ER, ET, F2, FE, FM, FQ, G3, G7, GD, GP, GR, HE, HK, HU, I5, IB, ID, IJ, IT, J4, JA, JC, JI, JO, JS, JU, KB, KL, KN, KR, L2, L8, LA, LM, LQ, M1, M9, MJ, MR, N6, NE, NK, NQ, NU, O7, OD, OH, OP, OT, PC, PM, Q3, Q5, QF, QH, QL, QN, R2, RG, RK, RM, RQ, S9, SD, SF, SJ, T8, TC, TK, TU, U7, UB, UH, UN, 106, 10A, 10G, 10M, 10U, 115, 11H, 11L, 11R, 11T, 128, 12A, 12G, 12Q, 12S, 137, 139, 13F, 142, 146, 148, 14C, 14I, 14O, 151, 157, 15D, 164, 166, 16G, 16O, 173, 179, 17F, 17N, 184, 188, 18E, 18K, 18M, 18S, 199, 19J, 1A6, 1A8, 1AC, 1AI, 1AK, 1AQ, 1AU, 1B1, 1B5, 1BH, 1BJ, 1BP, 1CS, 1D3, 1D9, 1DH, 1E4, 1EE, 1ES, 1F1, 1F3, 1F7, 1FD, 1FL, 1FP, 1FR, 1G2, 1GE, 1GO, 1GQ, 1GU, 1H1, 1H5, 1HB, 1HN, 1I4, 1IC, 1IO, 1IU, 1J3, 1J9, 1JH, 1JL, 1JT, 1K2, 1KG, 1KK, 1KQ, 1KS, 1L1, 1L7, 1L9, 1LF, 1LP, 1ME, 1MK, 1MO, 1MQ, 1NJ, 1NN, 1NP, 1O4, 1OG, 1OI, 1OS, 1P5, 1PB, 1PH, 1PN, 1QA, 1QG, 1QK, 1QM, 1R3, 1RD, 1RP, 1S2, 1SI, 1T1, 1T7, 1TB, 1TD, 1TH, 1TJ, 1TT, 1UA, 1UG, 1UM, 209, 20B, 20R, 20T, 21K, 21Q, 223, 229, 22D, 22F, 22J, 22R, 232, 23C, 23E, 23O, 247, 24H, 24N, 254, 256, 25A, 25C, 25M, 263, 265, 26L, 26N, 26T, 272, 274, 27E, 27M, 289, 292, 296, 29C, 29K, 2A5, 2A7, 2AB, 2AJ, 2B4, 2B6, 2BA, 2BI, 2BO, 2BU, 2C3, 2CF, 2CH, 2D8, 2DE, 2DG, 2DM, 2DQ, 2E1, 2EF, 2EL, 2EP, 2ER, 2F2, 2F6, 2FC, 2FO, 2FU, 2G5, 2GJ, 2GN, 2GT, 2HA, 2HI, 2HO, 2HS, 2IN, 2JA, 2JK, 2JS, 2K1, 2K7, 2K9, 2KF, 2L6, 2LI, 2LK, 2M5, 2MD, 2MH, 2MT, 2NC, 2NM, 2NO, 2NS, 2O5, 2OB, 2OH, 2OL, 2ON, 2OR, 2P2, 2PA, 2PE, 2PG, 2PM, 2Q1, 2Q3, 2QD, 2QL, 2QP, 2R8, 2RI, 2RU, 2S1, 2S7, 2SB, 2SD, 2ST, 2TC, 2TG, 2TM, 2TU, 2U5, 2U9, 2UR, 304, 30E, 30K, 30Q, 313, 31D, 31P, 328, 32C, 32I, 32O, 32Q, 33N, 33P, 344, 34C, 34G, 34U, 353, 35B, 35N, 35T, 36A, 36E, 36K, 379, 37J, 37L, 386, 391, 395, 397, 39J, 39P, 39T, 3AA, 3AG, 3AO, 3AS, 3B5, 3BR, 3BT, 3C2, 3C4, 3CG, 3DD, 3DF, 3DL, 3DR, 3E2, 3E6, 3EC, 3EE, 3EQ, 3EU, 3FB, 3FD, 3FN, 3FP, 3GA, 3GC, 3GS, 3H3, 3HN, 3I8, 3IG, 3IK, 3IM, 3IQ, 3IS, 3JJ, 3JR, 3K8, 3KE, 3KO, 3KQ, 3KU, 3L5, 3L7, 3LD, 3LN, 3LP, 3M6, 3MG, 3MI, 3MS, 3NB, 3NH, 3NL, 3NR, 3O4, 3OA, 3OG, 3P1, 3PD, 3PF, 3PJ, 3Q2, 3Q8, 3QC, 3QK, 3QU, 3R7, 3RD, 3RJ, 3SA, 3SG, 3SI, 3SS, 3TB, 3TF, 3TL, 3U8, 3UA, 3UK, 403, 407, 409, 40J, 412, 416, 41E, 421, 425, 42B, 42D, 42H, 42N, 42P, 436, 43A, 43U, 44L, 452, 454, 458, 45E, 45K, 45M, 45S, 46J, 46L, 46R, 47C, 47I, 47U, 481, 487, 48J, 494, 496, 49A, 49G, 49U, 4A3, 4A5, 4AN, 4BG, 4BQ, 4C1, 4C3, 4CD, 4CF, 4CP, 4CR, 4D6, 4DC, 4DE, 4DO, 4DQ, 4E5, 4EB, 4EJ, 4FI, 4FS, 4FU, 4G9, 4GH, 4GN, 4H2, 4HK, 4HQ, 4I7, 4IJ, 4IL, 4J8, 4JE, 4JI, 4JO, 4JU, 4KH, 4KJ, 4KT, 4LC, 4LI, 4LM, 4LO, 4LS, 4ML, 4MN, 4N4, 4NA, 4NQ, 4O3, 4O9, 4OF, 4P2, 4PI, 4PK, 4PO, 4PU, 4Q1, 4Q7, 4QD, 4QN, 4QT, 4RA, 4RM, 4S9, 4SB, 4SH, 4SL, 4T8, 4TG, 4U9, 4UD, 4UF, 4UJ, 4UP, 4UR, 508, 50C, 50Q, 51P, 524, 52A, 52M, 535, 53B, 53L, 542, 544, 548, 54E, 54M, 54S, 557, 559, 55D, 55R, 562, 568, 56C, 56I, 56K, 56U, 571, 57H, 57T, 586, 58O, 58S, 593, 59F, 59H, 59N, 59T, 5A4, 5B1, 5B7, 5BL, 5BP, 5C2, 5CC, 5CK, 5D1, 5DJ, 5DN, 5DP, 5DT, 5EM, 5F3, 5F9, 5FB, 5FR, 5G2, 5G8, 5GM, 5H1, 5HF, 5HJ, 5II, 5IO, 5IU, 5J5, 5JD, 5JJ, 5JN, 5JP, 5K6, 5KC, 5KG, 5KI, 5KO, 5LF, 5LL, 5LN, 5LR, 5ME, 5MG, 5MK, 5N1, 5N3, 5N9, 5ND, 5O8, 5OE, 5OK, 5OO, 5P1, 5PB, 5QC, 5QS, 5QU, 5R5, 5R9, 5RB, 5RF, 5RH, 5RR, 5SA, 5SG, 5SK, 5SS, 5T7, 5TD, 5U2, 5U6, 5U8, 5UE, 60D, 60H, 60P, 614, 61A, 61G, 61O, 61U, 62B, 62F, 62L, 62N, 62T, 632, 638, 63A, 63K, 63M, 647, 64D, 652, 656, 65I, 661, 66T, 674, 67O, 67S, 68F, 68N, 68T, 692, 698, 69M, 69S, 6A3, 6AD, 6AF, 6AP, 6B6, 6BE, 6BO, 6BQ, 6C5, 6CD, 6CP, 6D4, 6DS, 6DU, 6E3, 6EB, 6EH, 6EL, 6ET, 6FG, 6FQ, 6G1, 6G7, 6G9, 6GF, 6GP, 6H6, 6H8, 6HI, 6HO, 6HU, 6I5, 6ID, 6IJ, 6IT, 6J4, 6J6, 6JC, 6JI, 6JO, 6K3, 6KB, 6L4, 6LA, 6M1, 6M3, 6ML, 6MP, 6N2, 6NC, 6OB, 6OJ, 6P6, 6PA, 6PC, 6PM, 6PS, 6PU, 6Q5, 6Q9, 6QR, 6R4, 6RG, 6S3, 6SJ, 6SP, 6SR, 6T8, 6TE, 6TO, 6TQ, 6U5, 6U7, 6UD, 6UN, 706, 70A, 713, 715, 71L, 71N, 722, 724, 72E, 733, 737, 739, 73D, 73L, 746, 74C, 74I, 74K, 751, 75H, 75P, 75T, 764, 773, 775, 77F, 77H, 77N, 77R, 782, 788, 78G, 78M, 78Q, 797, 79D, 79L, 7A2, 7A6, 7AK, 7B1, 7BB, 7C4, 7CA, 7CM, 7CS, 7CU, 7DL, 7DT, 7EG, 7EQ, 7F1, 7FF, 7FJ, 7FL, 7FR, 7G6, 7GE, 7GK, 7GO, 7GU, 7HT, 7IC, 7IM, 7IO, 7J5, 7JF, 7JH, 7K2, 7K4, 7KM, 7LF, 7M2, 7M8, 7MO, 7NB, 7NH, 7NJ, 7O6, 7OA, 7OG, 7OI, 7OS, 7P5, 7PF, 7PL, 7PR, 7Q4, 7Q8, 7QE, 7QG, 7QQ, 7QS, 7R9, 7RD, 7RJ, 7RP, 7RR, 7S8, 7SC, 7SQ, 7TD, 7TH, 7TN, 7UC, 7UG, 7UO, 7UU, 803, 80B, 80F, 80T, 814, 818, 81M, 823, 827, 829, 838, 83C, 845, 84B, 84H, 84T, 85A, 85O, 85U, 863, 865, 869, 86R, 872, 87E, 87M, 87S, 881, 88D, 88F, 88R, 89Q, 8AB, 8AD, 8AJ, 8BA, ...|
|32|15, 19, 1B, 1F, 1L, 1R, 1T, 23, 27, 29, 2F, 2J, 2P, 31, 35, 37, 3B, 3D, 3H, 3V, 43, 49, 4B, 4L, 4N, 4T, 53, 57, 5D, 5J, 5L, 5V, 61, 65, 67, 6J, 6V, 73, 75, 79, 7F, 7H, 7R, 81, 87, 8D, 8F, 8L, 8P, 8R, 95, 9J, 9N, 9P, 9T, AB, AH, AR, AT, B1, B7, BF, BL, BR, BV, C5, CD, CH, CP, D3, D5, DF, DH, DN, DR, E1, E9, ED, EF, EJ, EV, F7, FB, FJ, FN, FT, G9, GB, GT, H3, HD, HJ, HP, HR, I1, IB, IH, IN, IP, IV, J5, J9, JB, JN, K1, K3, K7, KD, KJ, KL, L1, L5, LB, LJ, LT, M5, MF, MN, MT, N3, N7, NF, NL, NP, O1, O5, OJ, OT, P9, PB, PL, PN, PR, PT, Q7, QL, QP, QR, QV, RD, RH, RJ, RN, SB, SF, SN, T1, T9, TD, TJ, TP, U7, UB, UH, UN, UV, V5, VH, VL, VR, VT, 107, 109, 10F, 10P, 10R, 115, 117, 11D, 11V, 123, 125, 129, 12F, 12L, 12T, 133, 139, 13V, 141, 14B, 14J, 14T, 153, 159, 15H, 15T, 161, 167, 16D, 16F, 16L, 171, 17B, 17T, 17V, 183, 189, 18B, 18H, 18L, 18N, 18R, 197, 199, 19F, 1AH, 1AN, 1AT, 1B5, 1BN, 1C1, 1CF, 1CJ, 1CL, 1CP, 1CV, 1D7, 1DB, 1DD, 1DJ, 1DV, 1E9, 1EB, 1EF, 1EH, 1EL, 1ER, 1F7, 1FJ, 1FR, 1G7, 1GD, 1GH, 1GN, 1GV, 1H3, 1HB, 1HF, 1HT, 1I1, 1I7, 1I9, 1ID, 1IJ, 1IL, 1IR, 1J5, 1JP, 1JV, 1K3, 1K5, 1KT, 1L1, 1L3, 1LD, 1LP, 1LR, 1M5, 1MD, 1MJ, 1MP, 1MV, 1NH, 1NN, 1NR, 1NT, 1O9, 1OJ, 1OV, 1P7, 1PN, 1Q5, 1QB, 1QF, 1QH, 1QL, 1QN, 1R1, 1RD, 1RJ, 1RP, 1SB, 1SD, 1ST, 1SV, 1TL, 1TR, 1U3, 1U9, 1UD, 1UF, 1UJ, 1UR, 1V1, 1VB, 1VD, 1VN, 205, 20F, 20L, 211, 213, 217, 219, 21J, 21V, 221, 22H, 22J, 22P, 22T, 22V, 239, 23H, 243, 24R, 24V, 255, 25D, 25T, 25V, 263, 26B, 26R, 26T, 271, 279, 27F, 27L, 27P, 285, 287, 28T, 293, 295, 29B, 29F, 29L, 2A3, 2A9, 2AD, 2AF, 2AL, 2AP, 2AV, 2BB, 2BH, 2BN, 2C5, 2C9, 2CF, 2CR, 2D3, 2D9, 2DD, 2E7, 2EP, 2F3, 2FB, 2FF, 2FL, 2FN, 2FT, 2GJ, 2GV, 2H1, 2HH, 2HP, 2HT, 2I9, 2IN, 2J1, 2J3, 2J7, 2JF, 2JL, 2JR, 2JV, 2K1, 2K5, 2KB, 2KJ, 2KN, 2KP, 2KV, 2L9, 2LB, 2LL, 2LT, 2M1, 2MF, 2MP, 2N5, 2N7, 2ND, 2NH, 2NJ, 2O3, 2OH, 2OL, 2OR, 2P3, 2P9, 2PD, 2PV, 2Q7, 2QH, 2QN, 2QT, 2R5, 2RF, 2RR, 2S9, 2SD, 2SJ, 2SP, 2SR, 2TN, 2TP, 2U3, 2UB, 2UF, 2UT, 2V1, 2V9, 2VL, 2VR, 307, 30B, 30H, 315, 31F, 31H, 321, 32R, 32V, 331, 33D, 33J, 33N, 343, 349, 34H, 34L, 34T, 35J, 35L, 35P, 35R, 367, 373, 375, 37B, 37H, 37N, 37R, 381, 383, 38F, 38J, 38V, 391, 39B, 39D, 39T, 39V, 3AF, 3AL, 3B9, 3BP, 3C1, 3C5, 3C7, 3CB, 3CD, 3D3, 3DB, 3DN, 3DT, 3E7, 3E9, 3ED, 3EJ, 3EL, 3ER, 3F5, 3F7, 3FJ, 3FT, 3FV, 3G9, 3GN, 3GT, 3H1, 3H7, 3HF, 3HL, 3HR, 3IB, 3IN, 3IP, 3IT, 3JB, 3JH, 3JL, 3JT, 3K7, 3KF, 3KL, 3KR, 3LH, 3LN, 3LP, 3M3, 3MH, 3ML, 3MR, 3ND, 3NF, 3NP, 3O7, 3OB, 3OD, 3ON, 3P5, 3P9, 3PH, 3Q3, 3Q7, 3QD, 3QF, 3QJ, 3QP, 3QR, 3R7, 3RB, 3RV, 3SL, 3T1, 3T3, 3T7, 3TD, 3TJ, 3TL, 3TR, 3UH, 3UJ, 3UP, 3V9, 3VF, 3VR, 3VT, 403, 40F, 40V, 411, 415, 41B, 41P, 41T, 41V, 42H, 439, 43J, 43P, 43R, 445, 447, 44H, 44J, 44T, 453, 455, 45F, 45H, 45R, 461, 469, 477, 47H, 47J, 47T, 485, 48B, 48L, 497, 49D, 49P, 4A5, 4A7, 4AP, 4AV, 4B3, 4B9, 4BF, 4C1, 4C3, 4CD, 4CR, 4D1, 4D5, 4D7, 4DB, 4E3, 4E5, 4EH, 4EN, 4F7, 4FF, 4FL, 4FR, 4GD, 4GT, 4GV, 4H3, 4H9, 4HB, 4HH, 4HN, 4I1, 4I7, 4IJ, 4IV, 4JH, 4JJ, 4JP, 4JT, 4KF, 4KN, 4LF, 4LJ, 4LL, 4LP, 4LV, 4M1, 4MD, 4MH, 4MV, 4NT, 4O7, 4OD, 4OP, 4P7, 4PD, 4PN, 4Q3, 4Q5, 4Q9, 4QF, 4QN, 4QT, 4R7, 4R9, 4RD, 4RR, 4S1, 4S7, 4SB, 4SH, 4SJ, 4ST, 4SV, 4TF, 4TR, 4U3, 4UL, 4UP, 4UV, 4VB, 4VD, 4VJ, 4VP, 4VV, 50R, 511, 51F, 51J, 51R, 525, 52D, 52P, 53B, 53F, 53H, 53L, 54D, 54P, 54V, 551, 55H, 55N, 55T, 56B, 56L, 573, 577, 585, 58B, 58H, 58N, 58V, 595, 599, 59B, 59N, 59T, 5A1, 5A3, 5A9, 5AV, 5B5, 5B7, 5BB, 5BT, 5BV, 5C3, 5CF, 5CH, 5CN, 5CR, 5DL, 5DR, 5E1, 5E5, 5ED, 5EN, 5FN, 5G7, 5G9, 5GF, 5GJ, 5GL, 5GP, 5GR, 5H5, 5HJ, 5HP, 5HT, 5I5, 5IF, 5IL, 5J9, 5JD, 5JF, 5JL, 5KJ, 5KN, 5KV, 5L9, 5LF, 5LL, 5LT, 5M3, 5MF, 5MJ, 5MP, 5MR, 5N1, 5N5, 5NB, 5ND, 5NN, 5NP, 5O9, 5OF, 5P3, 5P7, 5PJ, 5Q1, 5QT, 5R3, 5RN, 5RR, 5SD, 5SL, 5SR, 5SV, 5T5, 5TJ, 5TP, 5TV, 5U9, 5UB, 5UL, 5V1, 5V9, 5VJ, 5VL, 5VV, 607, 60J, 60T, 61L, 61N, 61R, 623, 629, 62D, 62L, 637, 63H, 63N, 63T, 63V, 645, 64F, 64R, 64T, 657, 65D, 65J, 65P, 661, 667, 66H, 66N, 66P, 66V, 675, 67B, 67L, 67T, 68L, 68R, 69H, 69J, 6A5, 6A9, 6AH, 6AR, 6BP, 6C1, 6CJ, 6CN, 6CP, 6D3, 6D9, 6DB, 6DH, 6DL, 6E7, 6EF, 6ER, 6FD, 6FT, 6G3, 6G5, 6GH, 6GN, 6H1, 6H3, 6HD, 6HF, 6HL, 6HV, 6ID, 6IH, 6J9, 6JB, 6JR, 6JT, 6K7, 6K9, 6KJ, 6L7, 6LB, 6LD, 6LH, 6LP, 6M9, 6MF, 6ML, 6MN, 6N3, 6NJ, 6NR, 6NV, 6O5, 6P3, 6P5, 6PF, 6PH, 6PN, 6PR, 6Q1, 6Q7, 6QF, 6QL, 6QP, 6R5, 6RB, 6RJ, 6RV, 6S3, 6SH, 6ST, 6T7, 6TV, 6U5, 6UH, 6UN, 6UP, 6VF, 6VN, 709, 70J, 70P, 717, 71B, 71D, 71J, 71T, 725, 72B, 72F, 72L, 73J, 741, 74B, 74D, 74P, 753, 755, 75L, 75N, 769, 771, 77J, 77P, 789, 78R, 791, 793, 79L, 79P, 79V, 7A1, 7AB, 7AJ, 7AT, 7B3, 7B9, 7BH, 7BL, 7BR, 7BT, 7C7, 7C9, 7CL, 7CP, 7CV, 7D5, 7D7, 7DJ, 7DN, 7E5, 7EN, 7ER, 7F1, 7FL, 7FP, 7G1, 7G7, 7GB, 7GJ, 7GN, 7H5, 7HB, 7HF, 7HT, 7I9, 7ID, 7IF, 7JD, 7JH, 7K9, 7KF, 7KL, 7L1, 7LD, 7LR, 7M1, 7M5, 7M7, 7MB, 7MT, 7N3, 7NF, 7NN, 7NT, 7O1, 7OD, 7OF, 7OR, 7PP, 7Q9, 7QB, 7QH, 7R7, ...|
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|35|12, 16, 18, 1C, 1I, 1O, 1Q, 1W, 21, 23, 29, 2D, 2J, 2R, 2V, 2X, 32, 34, 38, 3M, 3Q, 3W, 3Y, 49, 4B, 4H, 4N, 4R, 4X, 54, 56, 5G, 5I, 5M, 5O, 61, 6D, 6H, 6J, 6N, 6T, 6V, 76, 7C, 7I, 7O, 7Q, 7W, 81, 83, 8D, 8R, 8V, 8X, 92, 9G, 9M, 9W, 9Y, A3, A9, AH, AN, AT, AX, B4, BC, BG, BO, BY, C1, CB, CD, CJ, CN, CT, D2, D6, D8, DC, DO, DW, E1, E9, ED, EJ, EV, EX, FG, FM, FW, G3, G9, GB, GH, GR, GX, H4, H6, HC, HI, HM, HO, I1, IB, ID, IH, IN, IT, IV, J8, JC, JI, JQ, K1, K9, KJ, KR, KX, L4, L8, LG, LM, LQ, LY, M3, MH, MR, N4, N6, NG, NI, NM, NO, NY, OD, OH, OJ, ON, P2, P6, P8, PC, PW, Q1, Q9, QJ, QR, QV, R2, R8, RM, RQ, RW, S3, SB, SH, ST, SX, T4, T6, TG, TI, TO, TY, U1, UB, UD, UJ, V2, V6, V8, VC, VI, VO, VW, W3, W9, WV, WX, X8, XG, XQ, XW, Y3, YB, YN, YR, YX, 104, 106, 10C, 10O, 10Y, 11H, 11J, 11N, 11T, 11V, 122, 126, 128, 12C, 12O, 12Q, 12W, 13V, 142, 148, 14G, 14Y, 159, 15N, 15R, 15T, 15X, 164, 16C, 16G, 16I, 16O, 171, 17B, 17D, 17H, 17J, 17N, 17T, 186, 18I, 18Q, 193, 199, 19D, 19J, 19R, 19V, 1A4, 1A8, 1AM, 1AQ, 1AW, 1AY, 1B3, 1B9, 1BB, 1BH, 1BR, 1CC, 1CI, 1CM, 1CO, 1DD, 1DH, 1DJ, 1DT, 1E6, 1E8, 1EI, 1EQ, 1EW, 1F3, 1F9, 1FR, 1FX, 1G2, 1G4, 1GG, 1GQ, 1H3, 1HB, 1HR, 1I6, 1IC, 1IG, 1II, 1IM, 1IO, 1IY, 1JB, 1JH, 1JN, 1K6, 1K8, 1KO, 1KQ, 1LD, 1LJ, 1LR, 1LX, 1M2, 1M4, 1M8, 1MG, 1MM, 1MW, 1MY, 1N9, 1NN, 1NX, 1O4, 1OG, 1OI, 1OM, 1OO, 1OY, 1PB, 1PD, 1PT, 1PV, 1Q2, 1Q6, 1Q8, 1QI, 1QQ, 1R9, 1RX, 1S2, 1S8, 1SG, 1SW, 1SY, 1T3, 1TB, 1TR, 1TT, 1TX, 1U6, 1UC, 1UI, 1UM, 1UY, 1V1, 1VN, 1VT, 1VV, 1W2, 1W6, 1WC, 1WQ, 1WW, 1X1, 1X3, 1X9, 1XD, 1XJ, 1XV, 1Y2, 1Y8, 1YM, 1YQ, 1YW, 209, 20H, 20N, 20R, 21I, 221, 22B, 22J, 22N, 22T, 22V, 232, 23O, 241, 243, 24J, 24R, 24V, 258, 25M, 25W, 25Y, 263, 26B, 26H, 26N, 26R, 26T, 26X, 274, 27C, 27G, 27I, 27O, 27Y, 281, 28B, 28J, 28N, 292, 29C, 29O, 29Q, 29W, 2A1, 2A3, 2AJ, 2AX, 2B2, 2B8, 2BG, 2BM, 2BQ, 2C9, 2CH, 2CR, 2CX, 2D4, 2DC, 2DM, 2DY, 2ED, 2EH, 2EN, 2ET, 2EV, 2FO, 2FQ, 2G1, 2G9, 2GD, 2GR, 2GV, 2H4, 2HG, 2HM, 2HY, 2I3, 2I9, 2IT, 2J4, 2J6, 2JM, 2KD, 2KH, 2KJ, 2KV, 2L2, 2L6, 2LI, 2LO, 2LW, 2M1, 2M9, 2MV, 2MX, 2N2, 2N4, 2NG, 2O9, 2OB, 2OH, 2ON, 2OT, 2OX, 2P4, 2P6, 2PI, 2PM, 2PY, 2Q1, 2QB, 2QD, 2QT, 2QV, 2RC, 2RI, 2S3, 2SJ, 2SR, 2SV, 2SX, 2T2, 2T4, 2TQ, 2TY, 2UB, 2UH, 2UR, 2UT, 2UX, 2V4, 2V6, 2VC, 2VM, 2VO, 2W1, 2WB, 2WD, 2WN, 2X2, 2X8, 2XC, 2XI, 2XQ, 2XW, 2Y3, 2YJ, 2YV, 2YX, 302, 30G, 30M, 30Q, 30Y, 319, 31H, 31N, 31T, 32G, 32M, 32O, 32Y, 33D, 33H, 33N, 346, 348, 34I, 34W, 351, 353, 35D, 35R, 35V, 364, 36M, 36Q, 36W, 36Y, 373, 379, 37B, 37N, 37R, 38C, 38Y, 39B, 39D, 39H, 39N, 39T, 39V, 3A2, 3AO, 3AQ, 3AW, 3BD, 3BJ, 3BV, 3BX, 3C4, 3CG, 3CW, 3CY, 3D3, 3D9, 3DN, 3DR, 3DT, 3EC, 3F1, 3FB, 3FH, 3FJ, 3FT, 3FV, 3G6, 3G8, 3GI, 3GO, 3GQ, 3H1, 3H3, 3HD, 3HJ, 3HR, 3IM, 3IW, 3IY, 3J9, 3JH, 3JN, 3JX, 3KG, 3KM, 3KY, 3LB, 3LD, 3LV, 3M2, 3M6, 3MC, 3MI, 3N1, 3N3, 3ND, 3NR, 3NX, 3O2, 3O4, 3O8, 3OW, 3OY, 3PB, 3PH, 3PX, 3Q6, 3QC, 3QI, 3R1, 3RH, 3RJ, 3RN, 3RT, 3RV, 3S2, 3S8, 3SI, 3SO, 3T1, 3TD, 3TV, 3TX, 3U4, 3U8, 3UQ, 3UY, 3VN, 3VR, 3VT, 3VX, 3W4, 3W6, 3WI, 3WM, 3X1, 3XV, 3Y6, 3YC, 3YO, 403, 409, 40J, 40V, 40X, 412, 418, 41G, 41M, 41W, 41Y, 423, 42H, 42N, 42T, 42X, 434, 436, 43G, 43I, 43Y, 44B, 44J, 452, 456, 45C, 45O, 45Q, 45W, 463, 469, 472, 478, 47M, 47Q, 47Y, 489, 48H, 48T, 49C, 49G, 49I, 49M, 4AB, 4AN, 4AT, 4AV, 4BC, 4BI, 4BO, 4C3, 4CD, 4CR, 4CV, 4DQ, 4DW, 4E3, 4E9, 4EH, 4EN, 4ER, 4ET, 4F6, 4FC, 4FG, 4FI, 4FO, 4GB, 4GH, 4GJ, 4GN, 4H6, 4H8, 4HC, 4HO, 4HQ, 4HW, 4I1, 4IR, 4IX, 4J4, 4J8, 4JG, 4JQ, 4KN, 4L4, 4L6, 4LC, 4LG, 4LI, 4LM, 4LO, 4LY, 4MD, 4MJ, 4MN, 4MV, 4N6, 4NC, 4NW, 4O1, 4O3, 4O9, 4P4, 4P8, 4PG, 4PQ, 4PW, 4Q3, 4QB, 4QH, 4QT, 4QX, 4R4, 4R6, 4RC, 4RG, 4RM, 4RO, 4RY, 4S1, 4SH, 4SN, 4T8, 4TC, 4TO, 4U3, 4UV, 4V2, 4VM, 4VQ, 4W9, 4WH, 4WN, 4WR, 4WX, 4XC, 4XI, 4XO, 4XY, 4Y1, 4YB, 4YN, 4YV, 506, 508, 50I, 50Q, 513, 51D, 522, 524, 528, 52G, 52M, 52Q, 52Y, 53H, 53R, 53X, 544, 546, 54C, 54M, 54Y, 551, 55B, 55H, 55N, 55T, 562, 568, 56I, 56O, 56Q, 56W, 573, 579, 57J, 57R, 58G, 58M, 599, 59B, 59T, 59X, 5A6, 5AG, 5BB, 5BJ, 5C2, 5C6, 5C8, 5CI, 5CO, 5CQ, 5CW, 5D1, 5DJ, 5DR, 5E4, 5EM, 5F3, 5F9, 5FB, 5FN, 5FT, 5G4, 5G6, 5GG, 5GI, 5GO, 5GY, 5HD, 5HH, 5I6, 5I8, 5IO, 5IQ, 5J1, 5J3, 5JD, 5JX, 5K2, 5K4, 5K8, 5KG, 5KW, 5L3, 5L9, 5LB, 5LN, 5M4, 5MC, 5MG, 5MM, 5NH, 5NJ, 5NT, 5NV, 5O2, 5O6, 5OC, 5OI, 5OQ, 5OW, 5P1, 5PD, 5PJ, 5PR, 5Q4, 5Q8, 5QM, 5QY, 5R9, 5RX, 5S4, 5SG, 5SM, 5SO, 5TB, 5TJ, 5U2, 5UC, 5UI, 5UW, 5V1, 5V3, 5V9, 5VJ, 5VR, 5VX, 5W2, 5W8, 5X3, 5XH, 5XR, 5XT, 5Y6, 5YG, 5YI, 5YY, 601, 60J, 618, 61Q, 61W, 62D, 62V, 632, 634, 63M, 63Q, 63W, 63Y, 649, 64H, 64R, 64X, 654, 65C, 65G, 65M, 65O, 65Y, 661, 66D, 66H, 66N, 66T, 66V, 678, 67C, 67Q, 689, 68D, 68J, 694, 698, 69G, 69M, 69Q, 69Y, 6A3, 6AH, 6AN, 6AR, 6B6, 6BI, 6BM, 6BO, 6CJ, 6CN, 6DC, 6DI, 6DO, 6E1, 6ED, 6ER, 6EX, 6F2, 6F4, 6F8, 6FQ, 6FW, 6G9, 6GH, 6GN, 6GR, 6H4, 6H6, 6HI, 6ID, 6IT, 6IV, 6J2, 6JO, ...|
|36|11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 101, 105, 107, 10B, 10N, 10P, 10V, 11T, 11Z, 125, 12D, 12V, 135, 13J, 13N, 13P, 13T, 13Z, 147, 14B, 14D, 14J, 14V, 155, 157, 15B, 15D, 15H, 15N, 15Z, 16B, 16J, 16V, 171, 175, 17B, 17J, 17N, 17V, 17Z, 18D, 18H, 18N, 18P, 18T, 18Z, 191, 197, 19H, 1A1, 1A7, 1AB, 1AD, 1B1, 1B5, 1B7, 1BH, 1BT, 1BV, 1C5, 1CD, 1CJ, 1CP, 1CV, 1DD, 1DJ, 1DN, 1DP, 1E1, 1EB, 1EN, 1EV, 1FB, 1FP, 1FV, 1FZ, 1G1, 1G5, 1G7, 1GH, 1GT, 1GZ, 1H5, 1HN, 1HP, 1I5, 1I7, 1IT, 1IZ, 1J7, 1JD, 1JH, 1JJ, 1JN, 1JV, 1K1, 1KB, 1KD, 1KN, 1L1, 1LB, 1LH, 1LT, 1LV, 1LZ, 1M1, 1MB, 1MN, 1MP, 1N5, 1N7, 1ND, 1NH, 1NJ, 1NT, 1O1, 1OJ, 1P7, 1PB, 1PH, 1PP, 1Q5, 1Q7, 1QB, 1QJ, 1QZ, 1R1, 1R5, 1RD, 1RJ, 1RP, 1RT, 1S5, 1S7, 1ST, 1SZ, 1T1, 1T7, 1TB, 1TH, 1TV, 1U1, 1U5, 1U7, 1UD, 1UH, 1UN, 1UZ, 1V5, 1VB, 1VP, 1VT, 1VZ, 1WB, 1WJ, 1WP, 1WT, 1XJ, 1Y1, 1YB, 1YJ, 1YN, 1YT, 1YV, 1Z1, 1ZN, 1ZZ, 201, 20H, 20P, 20T, 215, 21J, 21T, 21V, 21Z, 227, 22D, 22J, 22N, 22P, 22T, 22Z, 237, 23B, 23D, 23J, 23T, 23V, 245, 24D, 24H, 24V, 255, 25H, 25J, 25P, 25T, 25V, 26B, 26P, 26T, 26Z, 277, 27D, 27H, 27Z, 287, 28H, 28N, 28T, 291, 29B, 29N, 2A1, 2A5, 2AB, 2AH, 2AJ, 2BB, 2BD, 2BN, 2BV, 2BZ, 2CD, 2CH, 2CP, 2D1, 2D7, 2DJ, 2DN, 2DT, 2ED, 2EN, 2EP, 2F5, 2FV, 2FZ, 2G1, 2GD, 2GJ, 2GN, 2GZ, 2H5, 2HD, 2HH, 2HP, 2IB, 2ID, 2IH, 2IJ, 2IV, 2JN, 2JP, 2JV, 2K1, 2K7, 2KB, 2KH, 2KJ, 2KV, 2KZ, 2LB, 2LD, 2LN, 2LP, 2M5, 2M7, 2MN, 2MT, 2ND, 2NT, 2O1, 2O5, 2O7, 2OB, 2OD, 2OZ, 2P7, 2PJ, 2PP, 2PZ, 2Q1, 2Q5, 2QB, 2QD, 2QJ, 2QT, 2QV, 2R7, 2RH, 2RJ, 2RT, 2S7, 2SD, 2SH, 2SN, 2SV, 2T1, 2T7, 2TN, 2TZ, 2U1, 2U5, 2UJ, 2UP, 2UT, 2V1, 2VB, 2VJ, 2VP, 2VV, 2WH, 2WN, 2WP, 2WZ, 2XD, 2XH, 2XN, 2Y5, 2Y7, 2YH, 2YV, 2YZ, 2Z1, 2ZB, 2ZP, 2ZT, 301, 30J, 30N, 30T, 30V, 30Z, 315, 317, 31J, 31N, 327, 32T, 335, 337, 33B, 33H, 33N, 33P, 33V, 34H, 34J, 34P, 355, 35B, 35N, 35P, 35V, 367, 36N, 36P, 36T, 36Z, 37D, 37H, 37J, 381, 38P, 38Z, 395, 397, 39H, 39J, 39T, 39V, 3A5, 3AB, 3AD, 3AN, 3AP, 3AZ, 3B5, 3BD, 3C7, 3CH, 3CJ, 3CT, 3D1, 3D7, 3DH, 3DZ, 3E5, 3EH, 3ET, 3EV, 3FD, 3FJ, 3FN, 3FT, 3FZ, 3GH, 3GJ, 3GT, 3H7, 3HD, 3HH, 3HJ, 3HN, 3IB, 3ID, 3IP, 3IV, 3JB, 3JJ, 3JP, 3JV, 3KD, 3KT, 3KV, 3KZ, 3L5, 3L7, 3LD, 3LJ, 3LT, 3LZ, 3MB, 3MN, 3N5, 3N7, 3ND, 3NH, 3NZ, 3O7, 3OV, 3OZ, 3P1, 3P5, 3PB, 3PD, 3PP, 3PT, 3Q7, 3R1, 3RB, 3RH, 3RT, 3S7, 3SD, 3SN, 3SZ, 3T1, 3T5, 3TB, 3TJ, 3TP, 3TZ, 3U1, 3U5, 3UJ, 3UP, 3UV, 3UZ, 3V5, 3V7, 3VH, 3VJ, 3VZ, 3WB, 3WJ, 3X1, 3X5, 3XB, 3XN, 3XP, 3XV, 3Y1, 3Y7, 3YZ, 3Z5, 3ZJ, 3ZN, 3ZV, 405, 40D, 40P, 417, 41B, 41D, 41H, 425, 42H, 42N, 42P, 435, 43B, 43H, 43V, 445, 44J, 44N, 45H, 45N, 45T, 45Z, 467, 46D, 46H, 46J, 46V, 471, 475, 477, 47D, 47Z, 485, 487, 48B, 48T, 48V, 48Z, 49B, 49D, 49J, 49N, 4AD, 4AJ, 4AP, 4AT, 4B1, 4BB, 4C7, 4CN, 4CP, 4CV, 4CZ, 4D1, 4D5, 4D7, 4DH, 4DV, 4E1, 4E5, 4ED, 4EN, 4ET, 4FD, 4FH, 4FJ, 4FP, 4GJ, 4GN, 4GV, 4H5, 4HB, 4HH, 4HP, 4HV, 4I7, 4IB, 4IH, 4IJ, 4IP, 4IT, 4IZ, 4J1, 4JB, 4JD, 4JT, 4JZ, 4KJ, 4KN, 4KZ, 4LD, 4M5, 4MB, 4MV, 4MZ, 4NH, 4NP, 4NV, 4NZ, 4O5, 4OJ, 4OP, 4OV, 4P5, 4P7, 4PH, 4PT, 4Q1, 4QB, 4QD, 4QN, 4QV, 4R7, 4RH, 4S5, 4S7, 4SB, 4SJ, 4SP, 4ST, 4T1, 4TJ, 4TT, 4TZ, 4U5, 4U7, 4UD, 4UN, 4UZ, 4V1, 4VB, 4VH, 4VN, 4VT, 4W1, 4W7, 4WH, 4WN, 4WP, 4WV, 4X1, 4X7, 4XH, 4XP, 4YD, 4YJ, 4Z5, 4Z7, 4ZP, 4ZT, 501, 50B, 515, 51D, 51V, 51Z, 521, 52B, 52H, 52J, 52P, 52T, 53B, 53J, 53V, 54D, 54T, 54Z, 551, 55D, 55J, 55T, 55V, 565, 567, 56D, 56N, 571, 575, 57T, 57V, 58B, 58D, 58N, 58P, 58Z, 59J, 59N, 59P, 59T, 5A1, 5AH, 5AN, 5AT, 5AV, 5B7, 5BN, 5BV, 5BZ, 5C5, 5CZ, 5D1, 5DB, 5DD, 5DJ, 5DN, 5DT, 5DZ, 5E7, 5ED, 5EH, 5ET, 5EZ, 5F7, 5FJ, 5FN, 5G1, 5GD, 5GN, 5HB, 5HH, 5HT, 5HZ, 5I1, 5IN, 5IV, 5JD, 5JN, 5JT, 5K7, 5KB, 5KD, 5KJ, 5KT, 5L1, 5L7, 5LB, 5LH, 5MB, 5MP, 5MZ, 5N1, 5ND, 5NN, 5NP, 5O5, 5O7, 5OP, 5PD, 5PV, 5Q1, 5QH, 5QZ, 5R5, 5R7, 5RP, 5RT, 5RZ, 5S1, 5SB, 5SJ, 5ST, 5SZ, 5T5, 5TD, 5TH, 5TN, 5TP, 5TZ, 5U1, 5UD, 5UH, 5UN, 5UT, 5UV, 5V7, 5VB, 5VP, 5W7, 5WB, 5WH, 5X1, 5X5, 5XD, 5XJ, 5XN, 5XV, 5XZ, 5YD, 5YJ, 5YN, 5Z1, 5ZD, 5ZH, 5ZJ, 60D, 60H, 615, 61B, 61H, 61T, 625, 62J, 62P, 62T, 62V, 62Z, 63H, 63N, 63Z, 647, 64D, 64H, 64T, 64V, 657, 661, 66H, 66J, 66P, 67B, ...|
The *OEIS* sequences for the base *b* representations of the prime numbers (of course, also includes the primes ≤ *b*) are: https://oeis.org/A004676 (*b* = 2), https://oeis.org/A001363 (*b* = 3), https://oeis.org/A004678 (*b* = 4), https://oeis.org/A004679 (*b* = 5), https://oeis.org/A004680 (*b* = 6), https://oeis.org/A004681 (*b* = 7), https://oeis.org/A004682 (*b* = 8), https://oeis.org/A004683 (*b* = 9), https://oeis.org/A000040 (*b* = 10), there are no *OEIS* sequences for *b* > 10 since *OEIS* disallows the alpha digits and only allows decimal characters (i.e. the 10 Arabic numerals (https://en.wikipedia.org/wiki/Arabic_numerals, https://mathworld.wolfram.com/ArabicNumeral.html)), see https://oeis.org/wiki/Disallowed_sequences#Sequences_of_rational_integers_with_digits_other_than_0_to_9, but there is a short *OEIS* sequence https://oeis.org/A004684 for *b* = 11, which stops exactly before the first term with a nondecimal character (i.e. the prime 43, which is written "3A" in base *b* = 11) and stops with the prime 41, which is written "38" in base *b* = 11.
Some bases 2 ≤ *b* ≤ 36 are notable: (fortunately, all of these bases except *b* = 32 and *b* = 36 are solved, except the primality proving for the probable prime 57<sub>62668</sub> in base *b* = 11 and the probable primes C5<sub>23755</sub>C, 80<sub>32017</sub>111, 95<sub>197420</sub>, A3<sub>592197</sub>A in base *b* = 13 and the probable primes DB<sub>32234</sub>, 4<sub>72785</sub>DD, 3<sub>116137</sub>AF in base *b* = 16 and the probable prime I0<sub>24608</sub>D in base *b* = 30)
* *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* = 18: octodecimal (https://web.archive.org/web/20141218113446/https://en.wikipedia.org/wiki/Octodecimal, https://ja.wikipedia.org/wiki/%E5%8D%81%E5%85%AB%E9%80%B2%E6%B3%95 (in Japanese))
* *b* = 20: vigesimal (https://en.wikipedia.org/wiki/Vigesimal, https://mathworld.wolfram.com/Vigesimal.html, https://oeis.org/A055644)
* *b* = 24: quadrivigesimal (https://web.archive.org/web/20141218113447/https://en.wikipedia.org/wiki/Base_24, https://ja.wikipedia.org/wiki/%E4%BA%8C%E5%8D%81%E5%9B%9B%E9%80%B2%E6%B3%95 (in Japanese))
* *b* = 30: trigesimal (https://ja.wikipedia.org/wiki/%E4%B8%89%E5%8D%81%E9%80%B2%E6%B3%95 (in Japanese))
* *b* = 32: duotrigesimal (https://web.archive.org/web/20141218113451/https://en.wikipedia.org/wiki/Base_32, https://ja.wikipedia.org/wiki/%E4%B8%89%E5%8D%81%E4%BA%8C%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, https://t5k.org/bios/page.php?id=950) and http://www.numericana.com/answer/primes.htm (the set of the primes) **(warning: the related link "The *n*−1 and *n*+1 primality tests by Curtis Bright, INTP (2013-10-09)" in this article is wrong, the correct link is http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/)** and http://www.numericana.com/answer/factoring.htm (factoring into primes).
|addition|multiplication|
|---|---|
|subtraction|division|
|0|1|
|negation|reciprocal|
|the set {1}|the set of the prime numbers|
|less than|divides|
|1 + 1 + 1 + ... + 1 with *n* 1's|the prime factorization of *n* (e.g. 360 = 2<sup>3</sup> × 3<sup>2</sup> × 5)|
Addition (https://en.wikipedia.org/wiki/Addition, https://www.rieselprime.de/ziki/Addition, https://mathworld.wolfram.com/Addition.html) and multiplication (https://en.wikipedia.org/wiki/Multiplication, https://www.rieselprime.de/ziki/Multiplication, https://mathworld.wolfram.com/Multiplication.html) are the basic operations of arithmetic (https://en.wikipedia.org/wiki/Arithmetic, https://www.rieselprime.de/ziki/Arithmetic, https://mathworld.wolfram.com/Arithmetic.html) (which is also the basics of mathematics (https://en.wikipedia.org/wiki/Mathematics, https://www.rieselprime.de/ziki/Mathematics, https://mathworld.wolfram.com/Mathematics.html)). In the addition operation, the identity element (https://en.wikipedia.org/wiki/Identity_element, https://mathworld.wolfram.com/IdentityElement.html) is 0, and all natural numbers > 0 can be written as the sum of many 1's, and the number 1 cannot be broken up; in the multiplication operation, the identity element is 1, and all natural numbers > 1 can be written as the product of many prime numbers, and the prime numbers cannot be broken up. Also, primes are the natural numbers *n* > 1 such that if *n* divides (https://en.wikipedia.org/wiki/Divides, https://t5k.org/glossary/xpage/Divides.html, https://t5k.org/glossary/xpage/Divisor.html, https://www.rieselprime.de/ziki/Factor, https://mathworld.wolfram.com/Divides.html, https://mathworld.wolfram.com/Divisor.html, https://mathworld.wolfram.com/Divisible.html, http://www.numericana.com/answer/primes.htm#divisor) *x*×*y* (*x* and *y* are natural numbers), then *n* divides either *x* or *y* (or both). Also, prime numbers are the natural numbers *n* such that the ring (https://en.wikipedia.org/wiki/Ring_(mathematics), https://mathworld.wolfram.com/Ring.html) of integers modulo *n* (https://en.wikipedia.org/wiki/Integers_modulo_n, https://mathworld.wolfram.com/Mod.html) (i.e. the ring *Z*<sub>*n*</sub>) is a field (https://en.wikipedia.org/wiki/Field_(mathematics), https://mathworld.wolfram.com/Field.html) (also is a domain (https://en.wikipedia.org/wiki/Domain_(ring_theory), https://mathworld.wolfram.com/Domain.html), also is an integral domain (https://en.wikipedia.org/wiki/Integral_domain, https://mathworld.wolfram.com/IntegralDomain.html), also is a division ring (https://en.wikipedia.org/wiki/Division_ring), also has no zero divisors (https://en.wikipedia.org/wiki/Zero_divisor, https://mathworld.wolfram.com/ZeroDivisor.html) other than 0 (for the special case that *n* = 1, it is the zero ring (https://en.wikipedia.org/wiki/Zero_ring, https://mathworld.wolfram.com/TrivialRing.html))). Also, see https://t5k.org/ (The Prime Pages, https://en.wikipedia.org/wiki/PrimePages, https://www.rieselprime.de/ziki/The_Prime_Pages) and https://www.primegrid.com/ (Primegrid, https://en.wikipedia.org/wiki/PrimeGrid, https://www.rieselprime.de/ziki/PrimeGrid, https://t5k.org/bios/page.php?id=950) and http://www.numericana.com/answer/primes.htm (the set of the primes) and http://www.numericana.com/answer/factoring.htm (factoring into primes). Besides, "the set of the minimal elements of the base *b* representations of the prime numbers > *b* under the subsequence ordering" to "the set of the prime numbers (except *b* itself) digit strings with length > 1 in base *b*" to "the partially ordered binary relation by subsequence" is "the set of the prime numbers" to "the set of the integers > 1" to "the partially ordered binary relation by divisibility" (and indeed, the "> 1" in "the prime numbers (except *b* itself) digit strings with length > 1 in base *b*" can be corresponded to the "> 1" in "the integers > 1") (for the reason why *b* itself is excluded (when *b* is prime, if *b* is composite, then there is no difference to include the *b* itself or not), see the sections below and https://www.mersenneforum.org/showpost.php?p=531632&postcount=7, the main reason is that *b* is the *only* prime ending with 0), thus the main problem in this project is very important and beautiful.
Also, the main problem in this project is hard and interesting, since the distribution of the primes are mysterious and almost completely random (https://en.wikipedia.org/wiki/Random_number, https://mathworld.wolfram.com/RandomNumber.html), and there is still no known formula of primes (https://en.wikipedia.org/wiki/Formula_for_primes, https://t5k.org/glossary/xpage/FormulasForPrimes.html, https://mathworld.wolfram.com/PrimeFormulas.html, https://mathworld.wolfram.com/Prime-GeneratingPolynomial.html, https://t5k.org/notes/faq/p_n.html, https://cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell78.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_210.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL20/Toth2/toth32.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_211.pdf), https://arxiv.org/ftp/arxiv/papers/1901/1901.01849.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_212.pdf)) which is efficiently computable (https://en.wikipedia.org/wiki/Algorithmic_efficiency), but if the Riemann hypothesis (https://en.wikipedia.org/wiki/Riemann_hypothesis, https://t5k.org/glossary/xpage/RiemannHypothesis.html, https://mathworld.wolfram.com/RiemannHypothesis.html, https://t5k.org/notes/rh.html, http://www.numericana.com/answer/open.htm#rh) is true, then there is Mills' formula (https://en.wikipedia.org/wiki/Formula_for_primes#Mills'_formula, https://en.wikipedia.org/wiki/Mills%27_constant, https://t5k.org/glossary/xpage/MillsTheorem.html, https://t5k.org/glossary/xpage/MillsPrime.html, https://t5k.org/glossary/xpage/MillsConstant.html, https://mathworld.wolfram.com/MillsTheorem.html, https://mathworld.wolfram.com/MillsPrime.html, https://mathworld.wolfram.com/MillsConstant.html, https://t5k.org/notes/proofs/A3n.html, https://t5k.org/notes/MillsConstant.html, https://www.ams.org/journals/bull/1947-53-06/S0002-9904-1947-08849-2/S0002-9904-1947-08849-2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_312.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell78.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_210.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL20/Toth2/toth32.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_211.pdf), https://oeis.org/A051254, https://oeis.org/A108739, https://oeis.org/A051021) *floor*(*A*<sup>3<sup>*n*</sup></sup>), 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<sup>2<sup>2<sup>...<sup>2<sup>*α*</sup></sup></sup></sup></sup>), 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*)<sup>2</sup> − ((*g*×*k*+2×*g*+*k*+1)×(*h*+*j*)+*h*−*z*)<sup>2</sup> − (2×*n*+*p*+*q*+*z*−*e*)<sup>2</sup> − (16×(*k*+1)<sup>3</sup>×(*k*+2)×(*n*+1)<sup>2</sup>+1−*f*<sup>2</sup>)<sup>2</sup> − (*e*<sup>3</sup>×(*e*+2)×(*a*+1)<sup>2</sup>+1−*o*<sup>2</sup>)<sup>2</sup> − ((*a*<sup>2</sup>−1)×*y*<sup>2</sup>+1−*x*<sup>2</sup>)<sup>2</sup> − (16×*r*<sup>2</sup>×*y*<sup>4</sup>×(*a*<sup>2</sup>−1)+1−*u*<sup>2</sup>)<sup>2</sup> − (((*a*+*u*<sup>2</sup>×(*u*<sup>2</sup>−*a*))<sup>2</sup>−1)×(*n*+4×*d*×*y*)<sup>2</sup>+1−(*x*+*c*×*u*)<sup>2</sup>)<sup>2</sup> - (*n*+*l*+*v*−*y*)<sup>2</sup> − ((*a*<sup>2</sup>−1)×*l*<sup>2</sup>+1−*m*<sup>2</sup>)<sup>2</sup> − (*a*×*i*+*k*+1−*l*−*i*)<sup>2</sup> − (*p*+*l*×(*a*−*n*−1)+*b*×(2×*a*×*n*+2×*a*−*n*<sup>2</sup>−2×*n*−2)−*m*)<sup>2</sup> − (*q*+*y*×(*a*−*p*−1)+*s*×(2×*a*×*p*+2×*a*−*p*<sup>2</sup>−2×*p*−2)−*x*)<sup>2</sup> − (*z*+*p*×*l*×(*a*−*p*)+*t*×(2×*a*×*p*−*p*<sup>2</sup>−1)−*p*×*m*)<sup>2</sup>)
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 1.6×10<sup>45</sup>)
|subsequence ordering|divisibility ordering|
|---|---|
|the "prime numbers > *b*" digit strings" in base *b*|the integers > 1|
|the set of the minimal elements of the "prime numbers > *b*" digit strings" in base *b* under the subsequence ordering (which is exactly the target of this project)|the set of the minimal elements of the integers > 1 under the divisibility ordering (which is exactly the set of the prime numbers)|
|no common subsequence with length > 1|coprime (no common divisor > 1) (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime)|
|proper subsequence with length > 1|proper factor (https://en.wikipedia.org/wiki/Proper_factor, https://mathworld.wolfram.com/ProperFactor.html, https://mathworld.wolfram.com/ProperDivisor.html) > 1|
|longest common subsequence (https://en.wikipedia.org/wiki/Longest_common_subsequence)|greatest common divisor (https://en.wikipedia.org/wiki/Greatest_common_divisor, https://t5k.org/glossary/xpage/GCD.html, https://mathworld.wolfram.com/GreatestCommonDivisor.html, http://www.numericana.com/answer/numbers.htm#gcd)|
|shortest common supersequence (https://en.wikipedia.org/wiki/Shortest_common_supersequence)|least common multiple (https://en.wikipedia.org/wiki/Least_common_multiple, https://t5k.org/glossary/xpage/LCM.html, https://mathworld.wolfram.com/LeastCommonMultiple.html, http://www.numericana.com/answer/numbers.htm#lcm)|
|pairwise incomparable strings (no string is a subsequence of another string)|pairwise incomparable numbers (no number divides another number)|
This problem is an extension of the original minimal prime problem (https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_17.pdf), https://cs.uwaterloo.ca/~shallit/Papers/br10.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_18.pdf), https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_19.pdf), https://doi.org/10.1080/10586458.2015.1064048 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_20.pdf), https://scholar.colorado.edu/downloads/hh63sw661 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_16.pdf) **(warning: the datas for bases 8 and 10 have errors, the data for base 8 misses the prime 6101 and the data for base 10 misses the primes 9001 and 9049 and instead wrongly includes the primes 90001, 90469, and 9000049, and the correct values of *S*<sub>***m***</sub> 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*<sup>*γ*×(*b*−1)×*eulerphi*(*b*)</sup>, this article has this error is because it only search bases 2 ≤ *b* ≤ 10, and for the data of 2 ≤ *b* ≤ 10 for the original minimal problem, you may think that they are the same sizes of *b* (however, if you extend the data to *b* = 11, 13, 16, then you will know that they are not the same sizes of *b*), since bases *b* = 7 and *b* = 9 have very large differences of the "number of minimal primes base *b*" between the original minimal problem and this new minimal prime problem (*b* = 7: 9 v.s. 71, *b* = 9: 12 v.s. 151), and bases *b* = 5 and *b* = 8 and *b* = 9 have very large differences of the "length of the largest minimal prime base *b*" between the original minimal problem and this new minimal prime problem (*b* = 5: 5 v.s. 96, *b* = 8: 9 v.s. 221, *b* = 9: 4 v.s. 1161))**, https://github.com/curtisbright/mepn-data, https://github.com/curtisbright/mepn, https://github.com/RaymondDevillers/primes) to cover Conjectures 'R Us Sierpinski/Riesel conjectures base *b* (http://www.noprimeleftbehind.net/crus/, http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-stats.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-top20.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-proven.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, https://t5k.org/bios/page.php?id=1372, https://www.rieselprime.de/ziki/Conjectures_%27R_Us, https://srbase.my-firewall.org/sr5/, https://srbase.my-firewall.org/sr5/stats.php, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), http://www.bitman.name/math/article/2005 (in Italian)) with *k*-values < *b* (thus, the main problem in this project also covers the original minimal prime problem, the only numbers in the set of the original minimal prime problem and not in the set of the main problem in this project are exactly the primes ≤ *b*, and there are *primepi*(*b*) such primes (where *primepi* is the prime-counting function (https://en.wikipedia.org/wiki/Prime-counting_function, https://t5k.org/glossary/xpage/PrimeCountingFunction.html, https://mathworld.wolfram.com/PrimeCountingFunction.html, https://oeis.org/A000720, https://oeis.org/A006880, https://oeis.org/A007053, https://faculty.lynchburg.edu/~nicely/index.html, https://faculty.lynchburg.edu/~nicely/pi/pix_0000.htm, https://faculty.lynchburg.edu/~nicely/pi/pix_0001.htm, https://faculty.lynchburg.edu/~nicely/constell.zip, http://sweet.ua.pt/tos/primes.html, https://pzktupel.de/counting/PI_01.php, https://sweet.ua.pt/tos/bib/5.4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_452.pdf), https://arxiv.org/pdf/1503.01839.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_453.pdf))), and of course all of these primes are very easily to find), i.e. finding the smallest prime of the form *k*×*b*<sup>*n*</sup>+1 and *k*×*b*<sup>*n*</sup>−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://web.mit.edu/kenta/www/three/prime/dual-sierpinski/ezgxggdm/dualsierp.txt.gz, https://www.primegrid.com/download/5ob_all.html, http://www.bitman.name/math/article/1126 (in Italian), http://www.bitman.name/math/article/1125 (in Italian), https://www.mersenneforum.org/showpost.php?p=144991&postcount=1, https://www.mersenneforum.org/showthread.php?t=10761, https://www.mersenneforum.org/showthread.php?t=6545) Sierpinski/Riesel conjectures base *b* with *k*-values < *b*, i.e. finding the smallest prime of the form *b*<sup>*n*</sup>+*k* and *b*<sup>*n*</sup>−*k* (which are the dual forms of *k*×*b*<sup>*n*</sup>+1 and *k*×*b*<sup>*n*</sup>−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 *b*<sup>*n*</sup>+2, *b*<sup>*n*</sup>−2, *b*<sup>*n*</sup>+(*b*−1), *b*<sup>*n*</sup>−(*b*−1), 2×*b*<sup>*n*</sup>+1, 2×*b*<sup>*n*</sup>−1, (*b*−1)×*b*<sup>*n*</sup>+1, (*b*−1)×*b*<sup>*n*</sup>−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)×*b*<sup>*n*</sup>+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)×*b*<sup>*n*</sup>−1 if and only if neither *b*−1 nor *b*−2 is prime, but I want the problem covers finding the smallest prime of these forms for *all* bases *b*)). The original minimal prime base *b* problem does not cover Conjectures 'R Us Sierpinski/Riesel conjectures base *b* with conjectured *k* (http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177) < *b*, since in Riesel side, the prime is not minimal prime in original definition if either *k*−1 or *b*−1 (or both) is prime, and in Sierpinski side, the prime is not minimal prime in original definition if *k* is prime (e.g. 25×30<sup>34205</sup>−1 is not minimal prime in base 30 in original definition, since it is OT<sub>34205</sub> in base 30, and T (= 29 in decimal) is prime, but it is minimal prime in base 30 if only primes > base are counted), but this extended version of minimal prime base *b* problem does.
**(warning: the data in the Table 5 in https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf) has many errors, e.g. for base *b* = 53, *k* = 4 is remaining at *n* = 2500000 and *k* = 8 has first prime at *n* = 227183 (which should not be found when the article was written), but the table only lists {1816, 1838, 1862, 1892} for the "*k*s not yet eliminated" data for base *b* = 53, and for base *b* = 48, the correct "*k*s 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 "*k*s 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 *k*s 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*<sup>*r*</sup>" covers "finding the smallest prime of the form *k*×*b*<sup>*n*</sup>+1 and *k*×*b*<sup>*n*</sup>−1 and *b*<sup>*n*</sup>+*k* and *b*<sup>*n*</sup>−*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://www.mersenneforum.org/showpost.php?p=531632&postcount=7)
Besides, this problem is better than the original minimal prime problem since this problem is regardless whether 1 is considered as prime or not, i.e. no matter 1 is considered as prime or not prime (https://t5k.org/notes/faq/one.html, https://primefan.tripod.com/Prime1ProCon.html, https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_24.pdf), http://www.numericana.com/answer/numbers.htm#one), the sets in this problem are the same, while the sets in the original minimal prime problem are different, e.g. in base 10, if 1 is considered as prime, then the set in the original minimal prime problem is {1, 2, 3, 5, 7, 89, 409, 449, 499, 6469, 6949, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049}, while if 1 is not considered as prime, then the set in the original minimal prime problem is {2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049}, however, in base 10, the set in this problem is always {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, no matter 1 is considered as prime or not prime.
The third reason for excluding the single-digit primes is that they are trivial (https://en.wikipedia.org/wiki/Triviality_(mathematics), https://mathworld.wolfram.com/Trivial.html), like that Conjectures 'R Us Sierpinski/Riesel conjectures base *b* requires exponent *n* ≥ 1 for these primes (see https://www.mersenneforum.org/showpost.php?p=447679&postcount=27), *n* = 0 is not acceptable to avoid the trivial primes (e.g. 2×*b*<sup>*n*</sup>+1, 4×*b*<sup>*n*</sup>+1, 6×*b*<sup>*n*</sup>+1, 10×*b*<sup>*n*</sup>+1, 12×*b*<sup>*n*</sup>+1, 16×*b*<sup>*n*</sup>+1, 3×*b*<sup>*n*</sup>−1, 4×*b*<sup>*n*</sup>−1, 6×*b*<sup>*n*</sup>−1, 8×*b*<sup>*n*</sup>−1, 12×*b*<sup>*n*</sup>−1, 14×*b*<sup>*n*</sup>−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, https://oeis.org/A006880, https://oeis.org/A007053, https://faculty.lynchburg.edu/~nicely/index.html, https://faculty.lynchburg.edu/~nicely/pi/pix_0000.htm, https://faculty.lynchburg.edu/~nicely/pi/pix_0001.htm, https://faculty.lynchburg.edu/~nicely/constell.zip, http://sweet.ua.pt/tos/primes.html, https://pzktupel.de/counting/PI_01.php, https://sweet.ua.pt/tos/bib/5.4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_452.pdf), https://arxiv.org/pdf/1503.01839.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_453.pdf)), *omega* is the little prime omega function (https://en.wikipedia.org/wiki/Prime_omega_function, https://oeis.org/A001221))
The fifth reason (and the main reason) for excluding the primes ≤ *b* is that the possible last digits of a minimal prime in base *b* are exactly the last digits in base *b* such that there are infinitely many primes, by the Dirichlet's theorem (https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions, https://t5k.org/glossary/xpage/DirichletsTheorem.html, https://mathworld.wolfram.com/DirichletsTheorem.html, https://t5k.org/notes/Dirichlet.html, http://www.numericana.com/answer/primes.htm#dirichlet), thus in this problem, we completely need not to consider the last digits which are not coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to *b*, while in the original minimal prime problem, this is false for the primes dividing *b*.
The sixth reason is that (heuristically) the more one-digit primes are contained in the set, the less primes have to be considered (since all numbers that contain one of these digits cannot be contained in the minimal set), thus one-digit primes will make this problem much easier and more uninteresting (and when single-digit primes are excluded, all base *b* digits may appear in large minimal primes in base *b*, e.g. when base *b* = 19 searched to length 100000, all base 19 digits except 2 and 8 still appear in the list of the 23 unsolved families), the reason is the same as why the article https://nntdm.net/papers/nntdm-25/NNTDM-25-1-036-047.pdf deals only with the minimal sets for *eulerphi*(*n*) + *k* (where *eulerphi* is the Euler's totient function (https://en.wikipedia.org/wiki/Euler%27s_totient_function, https://t5k.org/glossary/xpage/EulersPhi.html, https://mathworld.wolfram.com/TotientFunction.html, http://www.numericana.com/answer/modular.htm#phi, http://www.javascripter.net/math/calculators/eulertotientfunction.htm, https://oeis.org/A000010)) with *k* ≤ 5, since for *k* = 6, *eulerphi*(*n*) + 6 contains only two one-digit numbers (the author of that article wishes that the problem become easier, while I wish that this problem become much harder, thus I exclude the single-digit primes).
The seventh reason is that in this problem, for the linear families *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) in base *b* which we search, *y* can be any base *b* digit except 1 (*y* can be 1 if and only if base *b* has no repunit primes), while in the original minimal prime problem, *y* cannot be any prime digit (and *y* can be 1 if and only if base *b* has no repunit primes).
The eighth reason is that for the numbers in the sets in this problem, the greatest common divisor (https://en.wikipedia.org/wiki/Greatest_common_divisor, https://t5k.org/glossary/xpage/GCD.html, https://mathworld.wolfram.com/GreatestCommonDivisor.html, http://www.numericana.com/answer/numbers.htm#gcd) of the digits of these numbers must be 1, and the last digit must be coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to the base (*b*), since a prime *p* (when written in base *b*) have both "greatest common divisor (https://en.wikipedia.org/wiki/Greatest_common_divisor, https://t5k.org/glossary/xpage/GCD.html, https://mathworld.wolfram.com/GreatestCommonDivisor.html, http://www.numericana.com/answer/numbers.htm#gcd) of the digits of these numbers is 1" and "last digit is coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to the base (*b*)" if and only if *p* > *b*, thus it is not true for the numbers in the sets in the original minimal prime problem.
The minimal elements (https://en.wikipedia.org/wiki/Minimal_element) of the set (https://en.wikipedia.org/wiki/Set_(mathematics), https://mathworld.wolfram.com/Set.html) of the prime numbers (https://en.wikipedia.org/wiki/Prime_number, https://t5k.org/glossary/xpage/Prime.html, https://www.rieselprime.de/ziki/Prime, https://mathworld.wolfram.com/PrimeNumber.html, https://www.numbersaplenty.com/set/prime_number/, http://www.numericana.com/answer/primes.htm#definition, http://irvinemclean.com/maths/pfaq2.htm, https://oeis.org/A000040, https://t5k.org/lists/small/1000.txt, https://t5k.org/lists/small/10000.txt, https://t5k.org/lists/small/100000.txt, https://t5k.org/lists/small/millions/) > *b* in base (https://en.wikipedia.org/wiki/Radix, https://t5k.org/glossary/xpage/Radix.html, https://www.rieselprime.de/ziki/Base, https://mathworld.wolfram.com/Radix.html) *b* under the subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html) ordering (https://en.wikipedia.org/wiki/Partially_ordered_set, https://mathworld.wolfram.com/PartialOrder.html, https://mathworld.wolfram.com/PartiallyOrderedSet.html) is interesting because all prime numbers > *b* in base *b* except the repunits (https://en.wikipedia.org/wiki/Repunit, https://en.wikipedia.org/wiki/List_of_repunit_primes, https://t5k.org/glossary/xpage/Repunit.html, https://t5k.org/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://www.rieselprime.de/ziki/Generalized_Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://pzktupel.de/Primetables/TableRepunit.php, https://pzktupel.de/Primetables/TableRepunitGen.php, https://pzktupel.de/Primetables/TableRepunitGen.txt, https://stdkmd.net/nrr/prime/prime_rp.htm, https://stdkmd.net/nrr/prime/prime_rp.txt, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html, https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/, https://web.archive.org/web/20120227163614/http://phi.redgolpe.com/5.asp, https://web.archive.org/web/20120227163508/http://phi.redgolpe.com/4.asp, https://web.archive.org/web/20120227163610/http://phi.redgolpe.com/3.asp, https://web.archive.org/web/20120227163512/http://phi.redgolpe.com/2.asp, https://web.archive.org/web/20120227163521/http://phi.redgolpe.com/1.asp, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, https://jpbenney.blogspot.com/2022/04/another-sequence-of-note.html, http://perplexus.info/show.php?pid=8661&cid=51696, https://benvitalenum3ers.wordpress.com/2013/07/24/repunit-11111111111111-in-other-bases/, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, http://www.bitman.name/math/article/380/231/ (in Italian), 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), 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")
|subsequence|substring|
|---|---|
|https://oeis.org/A071062|https://oeis.org/A033274|
|https://oeis.org/A130448|https://oeis.org/A238334|
|https://oeis.org/A039995|https://oeis.org/A039997|
|https://oeis.org/A039994|https://oeis.org/A039996|
|https://oeis.org/A094535|https://oeis.org/A093301|
|https://oeis.org/A350508|https://oeis.org/A038103|
|https://oeis.org/A354113|https://oeis.org/A354114|
|https://t5k.org/glossary/xpage/MinimalPrime.html|https://www.mersenneforum.org/showpost.php?p=235098&postcount=5|
|longest common subsequence (https://en.wikipedia.org/wiki/Longest_common_subsequence)|longest common substring (https://en.wikipedia.org/wiki/Longest_common_substring)|
The longest common subsequence problem (https://en.wikipedia.org/wiki/Longest_common_subsequence) and the longest common substring problem (https://en.wikipedia.org/wiki/Longest_common_substring) are two hard problems on strings (https://en.wikipedia.org/wiki/String_(computer_science), https://mathworld.wolfram.com/String.html), the former is NP-complete (https://en.wikipedia.org/wiki/NP-complete, https://mathworld.wolfram.com/NP-CompleteProblem.html) and NP-hard (https://en.wikipedia.org/wiki/NP-hard, https://mathworld.wolfram.com/NP-HardProblem.html), while the latter is not.
(we do not research substring in this project, because for substring ordering, "the set of the minimal elements of the base 10 representations of the prime numbers > 10" are very likely to be infinite (and thus almost unsolvable), since all primes of the form 1{0}3 (i.e. https://oeis.org/A159352, for the exponents see https://oeis.org/A049054, also see https://stdkmd.net/nrr/1/10003.htm) or 3{0}1 (i.e. https://oeis.org/A259866, for the exponents see https://oeis.org/A056807, also see https://stdkmd.net/nrr/3/30001.htm) are minimal elements of the base 10 representations of the prime numbers > 10 under the substring ordering, and there is likely infinitely many primes of the form 1{0}3 and infinitely many primes of the form 3{0}1, see https://web.archive.org/web/20100628035147/http://www.math.niu.edu/~rusin/known-math/98/exp_primes and https://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Mathematics/2023_September_25#Are_there_infinitely_many_primes_of_the_form_1000%E2%80%A60007.2C_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://www.mersenneforum.org/showpost.php?p=564786&postcount=3 and https://www.mersenneforum.org/showpost.php?p=461665&postcount=7 and https://www.mersenneforum.org/showpost.php?p=354505&postcount=5 and https://www.mersenneforum.org/showpost.php?p=344985&postcount=293 and https://www.mersenneforum.org/showpost.php?p=625978&postcount=1027, https://www.primegrid.com/forum_thread.php?id=5093&nowrap=true#66471 and https://www.primegrid.com/forum_thread.php?id=4935&nowrap=true#63813)
It is found that both "number of minimal primes base *b*" and "length of the largest minimal prime base *b*" are roughly (https://en.wikipedia.org/wiki/Asymptotic_analysis, https://t5k.org/glossary/xpage/AsymptoticallyEqual.html, https://mathworld.wolfram.com/Asymptotic.html) *e*<sup>*γ*×(*b*−1)×*eulerphi*(*b*)</sup>, where *e* = 2.7182818284... 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* = 0.5772156649... 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*<sup>*γ*×(*b*−1)×*eulerphi*(*b*)</sup> 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 *b*<sup>2</sup>), thus *e*<sup>*γ*×(*b*−1)×*eulerphi*(*b*)</sup> has exponential growth for *b*, and "largest minimal prime base *b*" is roughly *b*<sup>*e*<sup>*γ*×(*b*−1)×*eulerphi*(*b*)</sup></sup>, which has double exponential growth (https://en.wikipedia.org/wiki/Double_exponential_function) for *b*, however, for bases *b* = 7 and *b* = 15, the estimation of "length of the largest minimal prime base *b*" is much higher than the real value, since these two bases are very-high Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=50442&postcount=1, https://www.mersenneforum.org/showpost.php?p=50444&postcount=1, https://www.mersenneforum.org/showpost.php?p=201642&postcount=1, https://www.mersenneforum.org/attachment.php?attachmentid=13663&d=1451910741, https://github.com/happy5214/nash, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, https://web.archive.org/web/20230928115952/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, https://web.archive.org/web/20230928120009/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, https://web.archive.org/web/20230928120025/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, https://web.archive.org/web/20230928120047/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)) bases, these two bases are "primeful" as the Conjectures 'R Us Sierpinski/Riesel conjectures (http://www.noprimeleftbehind.net/crus/, http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-stats.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-top20.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-proven.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, https://t5k.org/bios/page.php?id=1372, https://www.rieselprime.de/ziki/Conjectures_%27R_Us, https://srbase.my-firewall.org/sr5/, https://srbase.my-firewall.org/sr5/stats.php, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), http://www.bitman.name/math/article/2005 (in Italian)) in bases *b* = 7 and *b* = 15, since they are of the form 2<sup>*r*</sup>−1, while for bases *b* = 5 and *b* = 11 and *b* = 14, the estimation of "length of the largest minimal prime base *b*" is lower than the real value, since they are low Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=50442&postcount=1, https://www.mersenneforum.org/showpost.php?p=50444&postcount=1, https://www.mersenneforum.org/showpost.php?p=201642&postcount=1, https://www.mersenneforum.org/attachment.php?attachmentid=13663&d=1451910741, https://github.com/happy5214/nash, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash, https://www.rieselprime.de/ziki/Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_2_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_2_Low-weight, https://www.rieselprime.de/ziki/Category:Riesel_5_Low-weight, https://www.rieselprime.de/ziki/Category:Proth_5_Low-weight, https://web.archive.org/web/20230928115952/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, https://web.archive.org/web/20230928120009/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_weights.txt, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf)) (or difficulty (https://stdkmd.net/nrr/prime/prime_difficulty.htm, https://stdkmd.net/nrr/prime/prime_difficulty.txt, https://web.archive.org/web/20240305201107/https://stdkmd.net/nrr/prime/primedifficulty.htm, https://web.archive.org/web/20240305201027/https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, https://web.archive.org/web/20230928120025/http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, https://web.archive.org/web/20230928120047/http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)) bases (as in Conjectures 'R Us Sierpinski/Riesel conjectures (http://www.noprimeleftbehind.net/crus/, http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-stats.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-top20.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-proven.htm, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/attachment.php?attachmentid=4415&d=1260969652, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, https://t5k.org/bios/page.php?id=1372, https://www.rieselprime.de/ziki/Conjectures_%27R_Us, https://srbase.my-firewall.org/sr5/, https://srbase.my-firewall.org/sr5/stats.php, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), http://www.bitman.name/math/article/2005 (in Italian)), bases *b* == 2 mod 3 are low Nash weight bases), but the estimation of "number of minimal primes base *b*" is always near to the real value. (there are also asymptotic analysis for other sets of primes in various bases *b*, such as the left-truncatable primes and the right-truncatable primes (https://en.wikipedia.org/wiki/Truncatable_prime, https://t5k.org/glossary/xpage/LeftTruncatablePrime.html, https://t5k.org/glossary/xpage/RightTruncatablePrime.html, https://mathworld.wolfram.com/TruncatablePrime.html, https://www.numbersaplenty.com/set/truncatable_prime/) in various bases *b*, see http://chesswanks.com/num/LTPs/ for the left-truncatable primes in bases *b* ≤ 120 and http://fatphil.org/maths/rtp/rtp.html for the right-truncatable primes in bases *b* ≤ 90, also see *OEIS* sequences https://oeis.org/A103443 (largest left-truncatable prime in base *b*) and https://oeis.org/A023107 (largest right-truncatable prime in base *b*) and https://oeis.org/A103463 (length of the largest left-truncatable prime in base *b*) and https://oeis.org/A103483 (length of the largest right-truncatable prime in base *b*) and https://oeis.org/A076623 (number of left-truncatable primes in base *b*) and https://oeis.org/A076586 (number of right-truncatable primes in base *b*))
**(of course, you can also try to extend the main problem in this project to bases 2 ≤ *b* ≤ 50 or 2 ≤ *b* ≤ 100 (I cannot imagine the effort needed for bases *b* around 500 or 1000, even if strong probable primes are allowed) (in fact, the *GMP* (https://gmplib.org/, https://en.wikipedia.org/wiki/GNU_Multiple_Precision_Arithmetic_Library) program supports bases 2 ≤ *b* ≤ 50, but I only ran for bases 2 ≤ *b* ≤ 36) (suggestion to use the character ":" to saparate the digits for bases *b* > 36 (and just use decimal to write the digits), just like https://baseconvert.com/ and https://baseconvert.com/high-precision), but warning: these problems will be extremely hard (especially the bases *b* such that (*b*−1)×*eulerphi*(*b*) (https://oeis.org/A062955) is larger)!!! The difficulty of base *b* is roughly (https://en.wikipedia.org/wiki/Asymptotic_analysis, https://t5k.org/glossary/xpage/AsymptoticallyEqual.html, https://mathworld.wolfram.com/Asymptotic.html) *e*<sup>*γ*×(*b*−1)×*eulerphi*(*b*)</sup>, and *e*<sup>*γ*×(*b*−1)×*eulerphi*(*b*)</sup> 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.
|*b*|number of possible first digits of a prime > *b* in base *b* (equal *b*−1, since all digits except 0 can be the first digit)|number of possible last digits of a prime > *b* in base *b* (equal *eulerphi*(*b*), since only digits coprime to *b* (i.e. the digits in the reduced residue system mod *b*) can be the last digit)|number of possible (first digit,last digit) combos of a prime > *b* in base *b* (equal (*b*−1)×*eulerphi*(*b*), by the rule of product), also the relative hardness for the "minimal prime problem" in base *b*|
|---|---|---|---|
|2|1|1|1|
|3|2|2|4|
|4|3|2|6|
|5|4|4|16|
|6|5|2|10|
|7|6|6|36|
|8|7|4|28|
|9|8|6|48|
|10|9|4|36|
|11|10|10|100|
|12|11|4|44|
|13|12|12|144|
|14|13|6|78|
|15|14|8|112|
|16|15|8|120|
|17|16|16|256|
|18|17|6|102|
|19|18|18|324|
|20|19|8|152|
|21|20|12|240|
|22|21|10|210|
|23|22|22|484|
|24|23|8|184|
|25|24|20|480|
|26|25|12|300|
|27|26|18|468|
|28|27|12|324|
|29|28|28|784|
|30|29|8|232|
|31|30|30|900|
|32|31|16|496|
|33|32|20|640|
|34|33|16|528|
|35|34|24|816|
|36|35|12|420|
**(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* < *b*<sup>2</sup> 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* < *b*<sup>2</sup> 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 *b*<sup>*n*</sup> and 2×*b*<sup>*n*</sup> 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*×*b*<sup>*n*</sup> and (*d*×*b*<sup>*n*</sup>) × (1+1/*d*) = (*d*+1)×*b*<sup>*n*</sup> (let the number *ε* in the formula be 1/*d*), which must necessarily start with the digit *d* in base *b*, also see https://t5k.org/notes/faq/BenfordsLaw.html, the primes do not follow the Benford's law (https://en.wikipedia.org/wiki/Benford%27s_law, https://t5k.org/glossary/xpage/BenfordsLaw.html, https://mathworld.wolfram.com/BenfordsLaw.html, https://www.mathpages.com/home/kmath302/kmath302.htm, https://t5k.org/notes/faq/BenfordsLaw.html, https://www.ams.org/publications/journals/notices/201702/rnoti-p132.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_261.pdf)) (only the prime factors of the numbers with exponential growth (https://en.wikipedia.org/wiki/Exponential_growth, https://mathworld.wolfram.com/ExponentialGrowth.html) (such as the repunits (https://en.wikipedia.org/wiki/Repunit, https://en.wikipedia.org/wiki/List_of_repunit_primes, https://t5k.org/glossary/xpage/Repunit.html, https://t5k.org/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://www.rieselprime.de/ziki/Generalized_Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://pzktupel.de/Primetables/TableRepunit.php, https://pzktupel.de/Primetables/TableRepunitGen.php, https://pzktupel.de/Primetables/TableRepunitGen.txt, https://stdkmd.net/nrr/prime/prime_rp.htm, https://stdkmd.net/nrr/prime/prime_rp.txt, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html, https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/, https://web.archive.org/web/20120227163614/http://phi.redgolpe.com/5.asp, https://web.archive.org/web/20120227163508/http://phi.redgolpe.com/4.asp, https://web.archive.org/web/20120227163610/http://phi.redgolpe.com/3.asp, https://web.archive.org/web/20120227163512/http://phi.redgolpe.com/2.asp, https://web.archive.org/web/20120227163521/http://phi.redgolpe.com/1.asp, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, https://jpbenney.blogspot.com/2022/04/another-sequence-of-note.html, http://perplexus.info/show.php?pid=8661&cid=51696, https://benvitalenum3ers.wordpress.com/2013/07/24/repunit-11111111111111-in-other-bases/, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, http://www.bitman.name/math/article/380/231/ (in Italian), 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), 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 (*b*<sup>*n*</sup>−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, https://t5k.org/notes/Dirichlet.html, http://www.numericana.com/answer/primes.htm#dirichlet), for every base *b* and every digit *d* coprime to *b*, there are infinitely many primes with *d* as the last digit in base *b* (since there are infinitely many primes == *d* mod *b* (https://en.wikipedia.org/wiki/Modular_arithmetic, https://en.wikipedia.org/wiki/Congruence_relation, https://en.wikipedia.org/wiki/Modulo, https://t5k.org/glossary/xpage/Congruence.html, https://t5k.org/glossary/xpage/CongruenceClass.html, https://t5k.org/glossary/xpage/Residue.html, https://mathworld.wolfram.com/Congruence.html, https://mathworld.wolfram.com/Congruent.html, https://mathworld.wolfram.com/Residue.html, https://mathworld.wolfram.com/MinimalResidue.html, https://mathworld.wolfram.com/Mod.html), which must necessarily end with the digit *d* in base *b*), and by the prime number theorem (https://en.wikipedia.org/wiki/Prime_number_theorem, https://t5k.org/glossary/xpage/PrimeNumberThm.html, https://mathworld.wolfram.com/PrimeNumberTheorem.html, https://t5k.org/howmany.html, http://www.numericana.com/answer/primes.htm#pnt, https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Tables_of_special_primes) (extended to arithmetic progression (https://en.wikipedia.org/wiki/Primes_in_arithmetic_progression, https://t5k.org/glossary/xpage/ArithmeticSequence.html, https://mathworld.wolfram.com/PrimeArithmeticProgression.html, https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem, https://mathworld.wolfram.com/Green-TaoTheorem.html, https://t5k.org/top20/page.php?id=14, https://t5k.org/primes/search.php?Comment=Arithmetic%20progression&OnList=all&Number=1000000&Style=HTML, https://www.primegrid.com/forum_thread.php?id=7022, https://www.primegrid.com/stats_ap26.php, https://www.pzktupel.de/JensKruseAndersen/aprecords.php, http://www.primerecords.dk/aprecords.htm, https://oeis.org/A133277, https://oeis.org/A113827, https://oeis.org/A005115, https://oeis.org/A093364, https://oeis.org/A133276, https://oeis.org/A033189, https://oeis.org/A113872, https://oeis.org/A033188, https://oeis.org/A231406, https://oeis.org/A113834, https://oeis.org/A088430)), all digits coprime to *b* have the same probability (i.e. probability 1/*eulerphi*(*b*)) for a random prime in base *b*, and for the smallest prime with last digit *d* in base *b* for the digits *d* coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to *b*, the largest of these primes are https://oeis.org/A038026 (the *d* which make these records are https://oeis.org/A038025), or https://oeis.org/A085420 (the numbers deleted the last digit (*d*) are https://oeis.org/A194943) if prime > *b* is required (like the main problem in this project), and the smallest prime > *b* which has last digit *d* in base *b* are https://oeis.org/A060940, this is related to Linnik's Theorem (https://en.wikipedia.org/wiki/Linnik%27s_theorem, https://mathworld.wolfram.com/LinniksTheorem.html, https://t5k.org/glossary/xpage/LinniksConstant.html, https://mathworld.wolfram.com/LinniksConstant.html), however, according to Chebyshev's bias (https://en.wikipedia.org/wiki/Chebyshev%27s_bias, https://mathworld.wolfram.com/ChebyshevBias.html, http://www.math.uiuc.edu/~ford/wwwpapers/lehman.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_59.pdf), https://dms.umontreal.ca/~andrew/PDF/PrimeRace.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_60.pdf), http://math101.guru/wp-content/uploads/2018/09/01-A3-Presentation-v7.3EN-no.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_67.pdf), https://arxiv.org/pdf/1910.08983.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_262.pdf), https://www.ams.org/journals/mcom/2004-73-247/S0025-5718-04-01649-7/S0025-5718-04-01649-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_263.pdf), https://arxiv.org/pdf/math/0010086.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_264.pdf), https://projecteuclid.org/euclid.em/1048515870 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_265.pdf), https://www.ams.org/journals/mcom/2000-69-230/S0025-5718-99-01105-9/S0025-5718-99-01105-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_266.pdf), https://www.ams.org/journals/mcom/1978-32-142/S0025-5718-1978-0476616-X/S0025-5718-1978-0476616-X.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_267.pdf), https://oeis.org/A007350, https://oeis.org/A007352, https://oeis.org/A199547, https://oeis.org/A306891, https://oeis.org/A321856, https://oeis.org/A066520, https://oeis.org/A321857, https://oeis.org/A321859, https://oeis.org/A071838, https://oeis.org/A320857, https://oeis.org/A321860, https://oeis.org/A321858, https://oeis.org/A112632, https://oeis.org/A038698, https://oeis.org/A321862, https://oeis.org/A321864, https://oeis.org/A321861, https://oeis.org/A320858, https://oeis.org/A321865, https://oeis.org/A321863, https://oeis.org/A275939, https://oeis.org/A306499, https://oeis.org/A306500, https://oeis.org/A329224, https://oeis.org/A306502, https://oeis.org/A306503, https://oeis.org/A329225), if *d*<sub>1</sub> 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*, *d*<sub>2</sub> is a quadratic nonresidue mod *b* (i.e. *d*<sub>1</sub> 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 *d*<sub>2</sub> cannot be), then for the primes ≤ *N* for a random positive integer *N*, the probability for the number of primes ending with *d*<sub>2</sub> in base *b* is more than the number of primes ending with *d*<sub>1</sub> 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* (thus, the sequences including https://oeis.org/A062332, https://oeis.org/A062333, https://oeis.org/A062334, https://oeis.org/A062335, etc. are infinite), also, the sum of the reciprocals of these primes diverges (https://en.wikipedia.org/wiki/Divergent_series, https://mathworld.wolfram.com/DivergentSeries.html), i.e. the set of them is a large set (https://en.wikipedia.org/wiki/Large_set_(combinatorics)), see https://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Mathematics/2023_October_1#Does_the_sum_of_the_reciprocals_of_all_primes_starting_with_7_and_ending_with_7_in_base_10_diverge.3F for the proof)**
The lengths of the minimal primes in base *b* (e.g. in decimal (base *b* = 10) the lengths of the 77 minimal primes are {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 7, 8, 8, 8, 8, 8, 12, 31}, respectively, and in base *b* = 16 the lengths of the 2347 minimal (probable) primes are {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23, 23, 23, 23, 23, 24, 24, 24, 25, 25, 25, 25, 25, 26, 26, 26, 27, 27, 27, 27, 29, 29, 29, 30, 30, 31, 32, 32, 32, 32, 32, 33, 33, 33, 33, 33, 34, 34, 35, 35, 36, 37, 37, 38, 38, 38, 38, 40, 40, 40, 41, 41, 42, 42, 44, 45, 45, 46, 47, 47, 49, 49, 49, 50, 50, 51, 54, 54, 56, 58, 60, 61, 62, 66, 67, 68, 73, 74, 79, 89, 101, 105, 105, 125, 130, 130, 132, 137, 146, 179, 186, 205, 210, 214, 220, 243, 249, 265, 265, 294, 307, 426, 547, 547, 796, 1053, 1066, 1519, 1717, 1965, 3545, 3703, 17806, 32235, 72787, 116139}, respectively) appear to follow the Zipf's law (https://en.wikipedia.org/wiki/Zipf%27s_law, https://mathworld.wolfram.com/ZipfsLaw.html) (which is similar to the Benford's law (https://en.wikipedia.org/wiki/Benford%27s_law, https://t5k.org/glossary/xpage/BenfordsLaw.html, https://mathworld.wolfram.com/BenfordsLaw.html, https://www.mathpages.com/home/kmath302/kmath302.htm, https://t5k.org/notes/faq/BenfordsLaw.html, https://www.ams.org/publications/journals/notices/201702/rnoti-p132.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_261.pdf)), for *n* ≥ 2, the number of *n*-digit minimal primes in base *b* is inversely proportional (https://en.wikipedia.org/wiki/Inversely_proportional, https://mathworld.wolfram.com/InverselyProportional.html) to *n*−1 (i.e. the expected value (https://en.wikipedia.org/wiki/Expected_value, https://mathworld.wolfram.com/ExpectationValue.html) is *c*/(*n*−1), where c is a fixed constant) (i.e. the graph of the points (*x*,*y* = the number of *x*-digit minimal primes in base *b*) in the *xy*-plane is near to the graph of *y* = *c*/(*x*−1) in the *xy*-plane for a fixed real number *c*), for any fixed base *b*, also, for *n* ≥ 1, the length of the *n*th 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 *x*th largest minimal prime in base *b*) in the *xy*-plane is near to the graph of *y* = *c*/*x* in the *xy*-plane for a fixed real number *c*), for any fixed base *b*. (for more information of the Zipf's law and the Benford's law, see https://oeis.org/A008952 and https://oeis.org/A008963 and https://oeis.org/A060956 and https://oeis.org/A241299 and https://oeis.org/A244059 and https://oeis.org/A363746 and http://www.cut-the-knot.org/do_you_know/zipfLaw.shtml and https://www.fq.math.ca/Scanned/13-4/webb.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_281.pdf) and https://www.fq.math.ca/Scanned/9-1/wlodarski2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_282.pdf) and https://arxiv.org/pdf/cond-mat/0412004.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_283.pdf))
We can use the sense of https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf) to say: (note that some of the left families may cover another left family, e.g. the base 19 left family 5{H}5 covers another base 19 left family 5{H}05, and if the smallest prime in family 5{H}5 in base 19 has length *n*, and the family 5{H}05 in base 19 has no prime with length ≤ *n*, then family 5{H}05 in base 19 can be removed from the unsolved families for base 19, however, if the smallest prime in family 5{H}5 in base 19 has length *n*, but the family 5{H}05 in base 19 is not tested to length *n* or more, then family 5{H}05 in base 19 should not be removed from the unsolved families for base 19, since a number in family 5{H}05 covers the prime in family 5{H}5 with length *n* if and only if the length of this number is ≥ *n*+1; besides, the base 19 left family FH0{H} covers another base 19 left family FHHH0{H}, and if the smallest prime in family FH0{H} in base 19 has length *n*, and the family FHHH0{H} in base 19 has no prime with length ≤ *n*+1, then family FHHH0{H} in base 19 can be removed from the unsolved families for base 19, however, if the smallest prime in family FH0{H} in base 19 has length *n*, but the family FHHH0{H} in base 19 is not tested to length *n*+1 or more, then family FHHH0{H} in base 19 should not be removed from the unsolved families for base 19, since a number in family FHHH0{H} covers the prime in family FH0{H} with length *n* if and only if the length of this number is ≥ *n*+2; besides, the base 21 left family {9}D covers another base 21 left family F{9}D, and if the smallest prime in family {9}D in base 21 has length *n*, and the family F{9}D in base 21 has no prime with length ≤ *n*, then family F{9}D in base 21 can be removed from the unsolved families for base 21, however, if the smallest prime in family {9}D in base 21 has length *n*, but the family F{9}D in base 21 is not tested to length *n* or more, then family F{9}D in base 21 should not be removed from the unsolved families for base 21, since a number in family F{9}D covers the prime in family {9}D with length *n* if and only if the length of this number is ≥ *n*+1 (if a family has no primes, then we say "the smallest prime in this family has length ∞ (https://en.wikipedia.org/wiki/Infinity, https://t5k.org/glossary/xpage/Infinite.html, https://mathworld.wolfram.com/Infinity.html) (instead of 0 or −1)", see http://gladhoboexpress.blogspot.com/2019/05/prime-sandwiches-made-with-one-derbread.html and http://chesswanks.com/seq/a306861.txt (for the *OEIS* sequence https://oeis.org/A306861) and http://chesswanks.com/seq/a269254.txt (for the *OEIS* sequence https://oeis.org/A269254) (since this is more convenient, e.g. the *n* of the smallest prime in the base 21 family 9<sub>*n*</sub>D, this family has been searched to *n* = 100000 with no prime or probable prime found, we can use ">100000" for the *n* of the smallest prime in the base 21 family 9<sub>*n*</sub>D (while for the *n* of the smallest prime in the base 13 family A3<sub>*n*</sub>A, it is 592199), ">100000" includes infinity (since infinity is > 100000) but does not includes 0 or −1, it is still possible that there is no prime in the base 21 family 9<sub>*n*</sub>D, although by the heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, https://web.archive.org/web/20230911032453/https://www.utm.edu/staff/caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf), https://arxiv.org/pdf/2103.04483.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_113.pdf)) above, this is very impossible, also "the smallest *n* ≥ 1 such that (*a*×*b*<sup>*n*</sup>+*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*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) is prime, and if there is no *n* ≥ 1 such that (*a*×*b*<sup>*n*</sup>+*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*×2<sup>*n*</sup>+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 (*b*<sup>*n*</sup>−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:
32. http://www.sosmath.com/tables/prime/prime.html (limit: first 1000 primes, up to 7919)
33. https://www.bigprimes.net/archive/prime (page limit: first 100 primes, up to 541, total limit: first 1000099 primes, up to 15487457)
34. https://web.archive.org/web/20201130071856/http://www.mathematical.com/primelist1to100kk.html (limit: first 664579 primes, up to 9999991)
35. https://web.archive.org/web/20191118082053/http://www.tsm-resources.com/alists/prim.html (limit: first 2000 primes, up to 17389)
36. https://web.archive.org/web/20090917191047/http://planetmath.org/encyclopedia/FirstThousandPositivePrimeNumbers.html (limit: first 1000 primes, up to 7919)
37. https://openlibrary.org/books/OL16553580M (limit: first 664999 primes, up to 10006721)
38. https://sites.google.com/view/the-first-25000-primes (limit: first 25000 primes, up to 287117)
39. https://faculty.lynchburg.edu/~nicely/gaps/gaplist.html (the longest list ever calculated, with all primes < 2<sup>64</sup> (but unlikely other lists here, the primes are not all stored))
40. https://en.wikipedia.org/wiki/List_of_prime_numbers#The_first_1000_prime_numbers (limit: first 1000 primes, up to 7919)
Also http://5.199.134.130/primes.tar.xz and http://5.199.134.130/primes.tar.xz.SHA256SUM and http://5.199.134.130/primes.tar.xz.par2 and http://5.199.134.130/primes.tar.xz.vol00+10.par2 and the .xz files in http://5.199.134.130/primes/ ("*n*.xz" for the definitely primes with *n* decimal digits for 19 ≤ *n* ≤ 9999, http://5.199.134.130/primes/above9999.xz for all definitely primes with ≥ 10000 decimal digits, http://5.199.134.130/primes/PRP.xz fpr all probable primes) and https://download.mersenne.ca/factordb_2024-03-21_primes.tar.xz.torrent for all primes and probable primes > 10<sup>18</sup> in *factordb* (i.e. the numbers in either http://factordb.com/listtype.php?t=4 or http://factordb.com/listtype.php?t=1 (and http://factordb.com/stat_1.php?prp)) (pixz is recommended for unpacking. "pixz -d < primes.tar.xz | tar -xv"), also http://factordb.com/dlprp.php?n=1000000 for 10000 random chosen small unproven probable primes in *factordb* (i.e. the numbers in http://factordb.com/listtype.php?t=1 and http://factordb.com/stat_1.php?prp), also http://factordb.com/dlc.php?n=1000000 for 5000 random chosen small composites with no known proper (prime or composite) factors in *factordb* (i.e. the numbers in http://factordb.com/listtype.php?t=3 and http://factordb.com/stat_1.php), also http://factordb.com/primobatch.php?digits=300&files=10&parts=1&start=Generate+zip and http://factordb.com/primobatch.php?digits=300&files=32000&parts=32&start=Generate+zip for the input files for *Primo* in *factordb*, also see http://factordb.com/distribution.php?size=20&start=0&lim=500000 and http://factordb.com/types_pie.php?large for the distribution of numbers by digits and status (http://factordb.com/status.html, http://factordb.com/distribution.php) in *factordb*, also http://5.199.134.130/certificates.tar.xz and http://5.199.134.130/certificates.tar.xz.SHA256SUM and http://5.199.134.130/certificates.tar.xz.par2 and http://5.199.134.130/certificates.tar.xz.vol00+10.par2 and the .xz files in http://5.199.134.130/certificates/ ("*n*.xz" for the primality certificates of the definitely primes with *n* decimal digits for 300 ≤ *n* ≤ 5679 and selected 5680 ≤ *n* ≤ 86453) for all primality certificates in *factordb*, also http://factordb.com/certtop.php for the primality certificates by user in *factordb*, also http://factordb.com/certtypetop.php for the primality certificates by software in *factordb*.
Lists of factorizations of small integers:
1. http://primefan.tripod.com/500factored.html (page limit: 500, total limit: 2500)
2. http://www.sosmath.com/tables/factor/factor.html (page limit: 200, total limit: 1000)
3. https://sites.google.com/view/prime-factorization-of-integer (limit: 25000)
4. https://web.archive.org/web/20060210182347/http://bearnol.is-a-geek.com/Panfur%20Project/ (limit: 20000000, however, currently the data is only available up to 1500000 in the archive pages, the data for numbers > 1500000 has no available archive pages) **(warning: this site does not factor the composite numbers 15, 51, 85, 91, 255, 435, 451, 561, 595, 679, 703, 771, 1105, 1261, 1285, 1351, 1387, ...)**
5. http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/ (limit: 10000)
6. http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/?infinity (no limit)
7. https://oeis.org/A027746/b027746.txt (flattened rows, the *N*th row ends with the term with index "the term *N* of https://oeis.org/A022559") (limit: 2048)
8. https://oeis.org/A112798/b112798.txt (replace the *n*th prime by *n*, flattened rows, the *N*th row ends with the term with index "the term *N* of https://oeis.org/A022559") (limit: 3275)
9. https://oeis.org/A027750/a027750.txt (all (prime or composite or unit) factors of *N*) (limit: 10000)
10. https://oeis.org/A027750/b027750.txt (all (prime or composite or unit) factors of *N*, flattened rows, the *N*th row ends with the term with index "the term *N* of https://oeis.org/A006218") (limit: 1000)
11. https://oeis.org/A054841/b054841.txt (the *n*th rightmost decimal digit represents the exponent of the *n*th prime, but only for the exponents ≤ 9, thus the number will be incorrect if *N* is divisible by a 10th power (> 1), thus, the number is incorrect only for 9 values of *N*, namely *N* = 1024, 2048, 3072, 4096, 5120, 6144, 7168, 8192, 9216, since the limit of *N* is 10000) (limit: 10000)
12. http://factorzone.tripod.com/factors.htm (all (prime or composite or unit) factors of *N*) (limit: 1100)
13. http://functions.wolfram.com/NumberTheoryFunctions/Divisors/03/02 (all (prime or composite or unit) factors of *N*) (limit: 50)
14. https://www.asahi-net.or.jp/~KC2H-MSM/cn/old/p01n002.txt (limit: 1001)
15. https://en.wikipedia.org/wiki/Table_of_prime_factors (limit: 1000)
16. https://en.wikipedia.org/wiki/Table_of_divisors (all (prime or composite or unit) factors of *N*) (limit: 1000)
17. http://factordb.com/index.php?query=n&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (from *factordb*) (the limit in the list is 199, but you can change the number to get the list of the factorizations of larger integers (at most 1000000000000000000000000000198, since the start value of *n* in *factordb* must be ≤ 999999999999999999999999999999))
18. http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=20001&FExp=1&TExp=1&c0=-&EN=&LM= (from an online factor database for numbers of the form *b*<sup>*n*</sup>±1) (the limit in the list is 20000, but you can change the number to get the list of the factorizations of larger integers (at most 1099999, since the base *b* in http://myfactors.mooo.com/ must be ≤ 1100000))
19. http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=20001&FExp=1&TExp=1&c0=%2B&EN=&LM= (from an online factor database for numbers of the form *b*<sup>*n*</sup>±1) (the limit in the list is 20002, but you can change the number to get the list of the factorizations of larger integers (at most 1100001, since the base *b* in http://myfactors.mooo.com/ must be ≤ 1100000))
Lists of small integers in various bases:
1. https://sites.google.com/view/integer-in-various-base (bases 2 ≤ *b* ≤ 36) (limit: 12500)
2. https://math.tools/table/bases-table/ (bases 2 ≤ *b* ≤ 35) (use lower case letters instead of upper case letters) (limit: 256)
3. https://math.tools/table/base/ (bases 2 ≤ *b* ≤ 64) (limit: 64, together with 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000)
4. https://linesegment.web.fc2.com/application/math/numbers/RadixConversion.html (in Japanese) (bases 2 ≤ *b* ≤ 36 and *b* = 62) (limit: 1024 for *b* = 2 and *b* = 4, 2187 for *b* = 3, 3125 for *b* = 5, *b*<sup>4</sup> for 6 ≤ *b* ≤ 8, 729 for *b* = 9, 3993 for *b* = 11, 3456 for *b* = 12, 4394 for *b* = 13, *b*<sup>3</sup> for 14 ≤ *b* ≤ 17, *b*<sup>2</sup> for 18 ≤ *b* ≤ 36, 3843 for *b* = 62)
5. https://web.archive.org/web/20240824165354/https://en.wikipedia.org/wiki/Table_of_bases (bases 2 ≤ *b* ≤ 36) (limit: 1296)
Also, programs related to this research: (some of these programs can also be downloaded in http://www.fermatsearch.org/download.php or https://www.mersenne.org/download/freeware.php or https://download.mersenne.ca/) (some of these programs need to use *GMP* (https://gmplib.org/))
Primality (or probable primality) testing (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/index.html) programs (https://www.rieselprime.de/ziki/Primality_testing_program):
1. *LLR* (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64)
2. *PFGW* (https://sourceforge.net/projects/openpfgw/, https://t5k.org/bios/page.php?id=175, https://www.rieselprime.de/ziki/PFGW, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/pfgw_win_4.0.3)
3. *Primo* (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309)
4. *Proth.exe* (https://t5k.org/programs/gallot/, https://t5k.org/bios/page.php?id=411, https://www.rieselprime.de/ziki/Proth.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/proth)
5. *CHG* (https://www.mersenneforum.org/attachment.php?attachmentid=21133&d=1571237465, https://t5k.org/bios/page.php?id=797, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/CHG)
Sieving (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.primegrid.com/stats_psp_sieve.php, https://www.primegrid.com/stats_pps_sieve.php, https://www.primegrid.com/stats_sr5_sieve.php, https://www.primegrid.com/stats_cw_sieve.php, https://www.primegrid.com/sieving/rsp/) programs (https://www.rieselprime.de/ziki/Sieving_program):
1. *SRSIEVE* (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, https://www.mersenneforum.org/showpost.php?p=631129&postcount=1, http://www.rieselprime.de/dl/CRUS_pack.zip, http://www.noprimeleftbehind.net/crus/sieve-programs.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve_1.8.2, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve-other-programs, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2cl.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srfile, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve2)
2. *MTSIEVE* (https://sourceforge.net/projects/mtsieve/, https://www.mersenneforum.org/rogue/mtsieve.html, https://download.mersenne.ca/mtsieve, https://t5k.org/bios/page.php?id=449, https://www.rieselprime.de/ziki/Mtsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/mtsieve_2.4.8)
3. *NewPGen* (https://t5k.org/programs/NewPGen/, https://t5k.org/bios/page.php?id=105, https://www.rieselprime.de/ziki/NewPGen, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/newpgen, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/newpgenlinux)
Integer factoring (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) programs (https://www.rieselprime.de/ziki/Factoring_program): (for more integer factoring programs see https://www.mersenneforum.org/showthread.php?t=3255)
1. *GMP*-*ECM* (https://web.archive.org/web/20210803045418/https://gforge.inria.fr/projects/ecm, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/ecm704dev-svn2990-win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/ecm704dev-svn2990-linux64, https://www.rieselprime.de/ziki/GMP-ECM)
2. *MSieve* (https://sourceforge.net/projects/msieve, https://download.mersenne.ca/msieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/msieve153_win64)
3. *GGNFS* (http://sourceforge.net/projects/ggnfs, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/GGNFS)
4. *CADO*-*NFS* (https://web.archive.org/web/20210506173015/http://cado-nfs.gforge.inria.fr/index.html, https://www.rieselprime.de/ziki/CADO-NFS, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/cado-nfs-2.3.0)
5. *YAFU* (http://bbuhrow.googlepages.com/home, https://github.com/bbuhrow/yafu)
6. *YTools* (https://github.com/bbuhrow/ytools)
7. *YSieve* (https://github.com/bbuhrow/ysieve)
For the files in this page:
* File "kernel *b*": Data for all known minimal primes in base *b*, expressed as base *b* strings
* File "sqkern *b*": Data for all known minimal primes in base *b*, expressed as base *b* strings, but with *x*(*y*^*n*)*z* format (which in the *README* files it is denoted as "*xy*<sub>*n*</sub>*z*") (*y*^*n* means *n* copies of *y*, e.g. 34(5^6)78 is 3455555578, this *n* is written in decimal) for the minimal primes in base *b* with (base *b*) length > 1000
* File "left *b*": *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) families in base *b* such that we were unable to determine if they contain a prime > *b* or not (i.e. *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) families in base *b* such that no prime member > *b* could be found, nor could the family be ruled out as only containing composites (only count the numbers > *b*)), we reduce these families by removing all trailing digits *y* from *x*, and removing all leading digits *y* from *z*, to make the families be easier, e.g. family 12333{3}33345 in base *b* is reduced to family 12{3}45 in base *b*, since they are in fact the same family, these families are sorted by "the length *n* number in these families, from the smallest number to the largest number, this *n* is large enough such that *n* replaced to any larger number does not affect the sorting" (e.g. for base 17, we sort with B{0}B3 -> B{0}DB -> {B}2BE -> {B}2E -> {B}E9 -> {B}EE, since in this case 7 digits is enough, B0000B3 < B0000DB < BBBB2BE < BBBBB2E < BBBBBE9 < BBBBBEE, if the 7 replaced to any larger number, this result of the sorting will not change **(edit: the base 17 family {B}2E was already solved (the probable prime found is B<sub>67103</sub>2E), and the base 17 family {B}2BE is now unneeded since this family is covered by the family {B}2E, but I just use some examples to show how I sort the families in the "left *b*" files)**) (just like https://stdkmd.net/nrr/1/ and https://stdkmd.net/nrr/2/ and https://stdkmd.net/nrr/3/ and https://stdkmd.net/nrr/4/ and https://stdkmd.net/nrr/5/ and https://stdkmd.net/nrr/6/ and https://stdkmd.net/nrr/7/ and https://stdkmd.net/nrr/8/ and https://stdkmd.net/nrr/9/ and https://stdkmd.net/nrr/cont/#form, but not completely like https://stdkmd.net/nrr/cert/1/ and https://stdkmd.net/nrr/cert/2/ and https://stdkmd.net/nrr/cert/3/ and https://stdkmd.net/nrr/cert/4/ and https://stdkmd.net/nrr/cert/5/ and https://stdkmd.net/nrr/cert/6/ and https://stdkmd.net/nrr/cert/7/ and https://stdkmd.net/nrr/cert/8/ and https://stdkmd.net/nrr/cert/9/, e.g. "1w77" ({1}77 in the notation for this project) should be between "11173" ({1}73 in the notation for this project) and "11179" ({1}79 in the notation for this project), "337w" (33{7} in the notation for this project) should be between "337w3" (33{7}3 in the notation for this project) and "337w9" (33{7}9 in the notation for this project), etc.) (note that some of the left families may cover another left family, e.g. the base 19 left family 5{H}5 covers another base 19 left family 5{H}05, and if the smallest prime in family 5{H}5 in base 19 has length *n*, and the family 5{H}05 in base 19 has no prime with length ≤ *n*, then family 5{H}05 in base 19 can be removed from the unsolved families for base 19, however, if the smallest prime in family 5{H}5 in base 19 has length *n*, but the family 5{H}05 in base 19 is not tested to length *n* or more, then family 5{H}05 in base 19 should not be removed from the unsolved families for base 19, since a number in family 5{H}05 covers the prime in family 5{H}5 with length *n* if and only if the length of this number is ≥ *n*+1; besides, the base 19 left family FH0{H} covers another base 19 left family FHHH0{H}, and if the smallest prime in family FH0{H} in base 19 has length *n*, and the family FHHH0{H} in base 19 has no prime with length ≤ *n*+1, then family FHHH0{H} in base 19 can be removed from the unsolved families for base 19, however, if the smallest prime in family FH0{H} in base 19 has length *n*, but the family FHHH0{H} in base 19 is not tested to length *n*+1 or more, then family FHHH0{H} in base 19 should not be removed from the unsolved families for base 19, since a number in family FHHH0{H} covers the prime in family FH0{H} with length *n* if and only if the length of this number is ≥ *n*+2; besides, the base 21 left family {9}D covers another base 21 left family F{9}D, and if the smallest prime in family {9}D in base 21 has length *n*, and the family F{9}D in base 21 has no prime with length ≤ *n*, then family F{9}D in base 21 can be removed from the unsolved families for base 21, however, if the smallest prime in family {9}D in base 21 has length *n*, but the family F{9}D in base 21 is not tested to length *n* or more, then family F{9}D in base 21 should not be removed from the unsolved families for base 21, since a number in family F{9}D covers the prime in family {9}D with length *n* if and only if the length of this number is ≥ *n*+1 (if a family has no primes, then we say "the smallest prime in this family has length ∞ (https://en.wikipedia.org/wiki/Infinity, https://t5k.org/glossary/xpage/Infinite.html, https://mathworld.wolfram.com/Infinity.html) (instead of 0 or −1)", see http://gladhoboexpress.blogspot.com/2019/05/prime-sandwiches-made-with-one-derbread.html and http://chesswanks.com/seq/a306861.txt (for the *OEIS* sequence https://oeis.org/A306861) and http://chesswanks.com/seq/a269254.txt (for the *OEIS* sequence https://oeis.org/A269254) (since this is more convenient, e.g. the *n* of the smallest prime in the base 21 family 9<sub>*n*</sub>D, this family has been searched to *n* = 100000 with no prime or probable prime found, we can use ">100000" for the *n* of the smallest prime in the base 21 family 9<sub>*n*</sub>D (while for the *n* of the smallest prime in the base 13 family A3<sub>*n*</sub>A, it is 592199), ">100000" includes infinity (since infinity is > 100000) but does not includes 0 or −1, it is still possible that there is no prime in the base 21 family 9<sub>*n*</sub>D, although by the heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, https://web.archive.org/web/20230911032453/https://www.utm.edu/staff/caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf), https://arxiv.org/pdf/2103.04483.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_113.pdf)) above, this is very impossible, also "the smallest *n* ≥ 1 such that (*a*×*b*<sup>*n*</sup>+*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*×*b*<sup>*n*</sup>+*c*)/*gcd*(*a*+*c*,*b*−1) is prime, and if there is no *n* ≥ 1 such that (*a*×*b*<sup>*n*</sup>+*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*×2<sup>*n*</sup>+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 (*b*<sup>*n*</sup>−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))
* File "special *b*": Non-linear families which cannot be ruled out by the "GMP.cc" program (https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/kGMP.cc), but you can either handle them by hand or analyse them with the "famk.cc" program (https://github.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/blob/main/code/famk.cc)