A Prime Game:

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

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

e.g.

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

These primes are called minimal primes.

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

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

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

The minimal elements (https://en.wikipedia.org/wiki/Minimal_element) (https://mathworld.wolfram.com/MaximalElement.html for maximal element, the dual of minimal element, unfortunely 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, https://oeis.org/A000040) which are > b written in the positional numeral system (https://en.wikipedia.org/wiki/Positional_numeral_system) with radix (https://en.wikipedia.org/wiki/Radix, https://t5k.org/glossary/xpage/Radix.html, https://www.rieselprime.de/ziki/Base, https://mathworld.wolfram.com/Radix.html) b, as digit (https://en.wikipedia.org/wiki/Numerical_digit, https://www.rieselprime.de/ziki/Digit, https://mathworld.wolfram.com/Digit.html) strings (https://en.wikipedia.org/wiki/String_(computer_science), https://mathworld.wolfram.com/String.html) under the subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html) ordering (https://en.wikipedia.org/wiki/Partially_ordered_set, https://mathworld.wolfram.com/PartialOrder.html, https://mathworld.wolfram.com/PartiallyOrderedSet.html), for 2 ≤ b ≤ 36 (I stop at base 36 since this base is a 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)), 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://reference.wolfram.com/language/ref/BaseForm.html, https://www.cut-the-knot.org/recurrence/word_primes.shtml, https://en.wikipedia.org/wiki/Base36, https://web.archive.org/web/20150320103231/https://en.wikipedia.org/wiki/Base_36, https://baseconvert.com/, https://baseconvert.com/high-precision, https://www.calculand.com/unit-converter/zahlen.php?og=Base+2-36&ug=1, http://www.unitconversion.org/unit_converter/numbers.html, http://www.unitconversion.org/unit_converter/numbers-ex.html, http://www.kwuntung.net/hkunit/base/base.php (in Chinese), https://linesegment.web.fc2.com/application/math/numbers/RadixConversion.html (in Japanese), also see https://t5k.org/notes/words.html for the English words which are prime numbers when viewed as a number base 36), using A−Z to represent digit values 10 to 35.

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

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

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

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 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):

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, prime numbers 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): every natural number greater than 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 of primes that is unique up to (https://en.wikipedia.org/wiki/Up_to) their order. 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 numbers n such that the ring (https://en.wikipedia.org/wiki/Ring_(mathematics), https://mathworld.wolfram.com/Ring.html) of integers modulo n (https://en.wikipedia.org/wiki/Integers_modulo_n, https://mathworld.wolfram.com/Mod.html) (i.e. the ring Z_n) is a field (https://en.wikipedia.org/wiki/Field_(mathematics), https://mathworld.wolfram.com/Field.html) (also is an integral domain (https://en.wikipedia.org/wiki/Integral_domain, https://mathworld.wolfram.com/IntegralDomain.html), also is a division ring (https://en.wikipedia.org/wiki/Division_ring), also has no zero divisors (https://en.wikipedia.org/wiki/Zero_divisor, https://mathworld.wolfram.com/ZeroDivisor.html) other than 0 (for the special case that n = 1, it is the zero ring (https://en.wikipedia.org/wiki/Zero_ring, https://mathworld.wolfram.com/TrivialRing.html))). Also, see https://t5k.org/ (The Prime Pages, https://en.wikipedia.org/wiki/PrimePages, https://www.rieselprime.de/ziki/The_Prime_Pages) and https://www.primegrid.com/ (Primegrid, https://en.wikipedia.org/wiki/PrimeGrid, https://www.rieselprime.de/ziki/PrimeGrid) and http://www.numericana.com/answer/primes.htm (the set of the primes) and http://www.numericana.com/answer/factoring.htm (factoring into primes). Besides, "the set of the minimal elements of the base b representations of the prime numbers > b under the subsequence ordering" to "the set of the prime numbers (except b itself) digit strings with length > 1 in base b" to "the partially ordered binary relation by subsequence" is "the set of the prime numbers" to "the set of the integers > 1" to "the partially ordered binary relation by divisibility" (and indeed, the "> 1" in "the prime numbers (except b itself) digit strings with length > 1 in base b" can be corresponded to the "> 1" in "the integers > 1") (for the reason why b itself is excluded (when b is prime, if b is composite, then there is no difference to include the b itself or not), see the sections below and https://mersenneforum.org/showpost.php?p=531632&postcount=7, the main reason is that b is the only prime ending with 0), thus the problem in this project is very important and beautiful.

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), 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, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://t5k.org/bios/page.php?id=1372, https://srbase.my-firewall.org/sr5/, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian)) with k-values < b, i.e. finding the smallest prime of the form k×bⁿ+1 and k×bⁿ−1 (or proving that such prime does not exist) for all k < b (also to cover dual (http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_1.pdf), https://www.rechenkraft.net/wiki/Five_or_Bust, https://oeis.org/A076336/a076336c.html, http://www.mit.edu/~kenta/three/prime/dual-sierpinski/ezgxggdm/dualsierp-excerpt.txt, http://mit.edu/kenta/www/three/prime/dual-sierpinski/ezgxggdm/dualsierp.txt.gz, https://mersenneforum.org/showpost.php?p=144991&postcount=1, https://mersenneforum.org/showthread.php?t=10761, https://mersenneforum.org/showthread.php?t=6545) Sierpinski/Riesel conjectures base b with k-values < b, i.e. finding the smallest prime of the form bⁿ+k and bⁿ−k (which are the dual forms of k×bⁿ+1 and k×bⁿ−1, respectively) (or proving that such prime does not exist) for all k < b) (also to cover finding the smallest prime of some classic forms (or proving that such prime does not exist), such as bⁿ+2, bⁿ−2, bⁿ+(b−1), bⁿ−(b−1), 2×bⁿ+1, 2×bⁿ−1, (b−1)×bⁿ+1, (b−1)×bⁿ−1, with n ≥ 1, for the same base b (of course, for some bases b the original minimal prime base b problem already covers finding the smallest prime of these forms, e.g. the original minimal prime base b problem covers finding the smallest prime of the form (b−1)×bⁿ+1 if and only if b−1 is not prime, and the original minimal prime base b problem covers finding the smallest prime of the form (b−1)×bⁿ−1 if and only if neither b−1 nor b−2 is prime, but I want the problem covers finding the smallest prime of these forms for all bases b)). The original minimal prime base b problem does not cover Conjectures ‘R Us Sierpinski/Riesel conjectures base b with conjectured k (http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt) < b, since in Riesel side, the prime is not minimal prime in original definition if either k−1 or b−1 (or both) is prime, and in Sierpinski side, the prime is not minimal prime in original definition if k is prime (e.g. 25×30³⁴²⁰⁵−1 is not minimal prime in base 30 in original definition, since it is OT₃₄₂₀₅ in base 30, and T (= 29 in decimal) is prime, but it is minimal prime in base 30 if only primes > base are counted), but this extended version of minimal prime base b problem does. (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")

(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://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf) and http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf) and https://mersenneforum.org/showpost.php?p=564786&postcount=3 and https://mersenneforum.org/showpost.php?p=461665&postcount=7 and https://mersenneforum.org/showpost.php?p=625978&postcount=1027)

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

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

The third reason for excluding the single-digit primes is that they are trivial like that Conjectures ‘R Us Sierpinski/Riesel conjectures base b requires exponent n ≥ 1 for these primes (see https://mersenneforum.org/showpost.php?p=447679&postcount=27), n = 0 is not acceptable to avoid the trivial primes (e.g. 2×bⁿ+1, 4×bⁿ+1, 6×bⁿ+1, 10×bⁿ+1, 12×bⁿ+1, 16×bⁿ+1, 3×bⁿ−1, 4×bⁿ−1, 6×bⁿ−1, 8×bⁿ−1, 12×bⁿ−1, 14×bⁿ−1, ... cannot be quickly eliminated with n = 0, or the conjectures become much more easy 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 more easy 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), 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, 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, 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.

It is found that both "number of minimal primes base b" and "length of the largest minimal prime base b" are roughly (https://en.wikipedia.org/wiki/Asymptotic_analysis, https://t5k.org/glossary/xpage/AsymptoticallyEqual.html, https://mathworld.wolfram.com/Asymptotic.html) e^{γ×(b−1)×eulerphi(b)}, where e = 2.718281828459... is the base of the natural logarithm (https://en.wikipedia.org/wiki/E_(mathematical_constant), https://mathworld.wolfram.com/e.html, https://oeis.org/A001113), γ = 0.577215664901 is the Euler–Mascheroni constant (https://en.wikipedia.org/wiki/Euler%27s_constant, https://t5k.org/glossary/xpage/Gamma.html, https://mathworld.wolfram.com/Euler-MascheroniConstant.html, https://oeis.org/A001620), eulerphi is Euler's totient function (https://en.wikipedia.org/wiki/Euler%27s_totient_function, https://t5k.org/glossary/xpage/EulersPhi.html, https://mathworld.wolfram.com/TotientFunction.html, http://www.numericana.com/answer/modular.htm#phi, https://oeis.org/A000010), you can see the condensed table for bases 2 ≤ b ≤ 36 in the bottom of this article, e^{γ×(b−1)×eulerphi(b)} is an exponential sequence (https://en.wikipedia.org/wiki/Exponential_growth, https://mathworld.wolfram.com/ExponentialGrowth.html) for (b−1)×eulerphi(b) (https://oeis.org/A062955), and since (b−1)×eulerphi(b) has polynomial growth (https://en.wikipedia.org/wiki/Polynomial, https://mathworld.wolfram.com/Polynomial.html) for b (since it is always between b−1 and b²), thus e^{γ×(b−1)×eulerphi(b)} has exponential growth for b, and "largest minimal prime base b" is roughly b^{eγ×(b−1)×eulerphi(b)}, which has double exponential growth (https://en.wikipedia.org/wiki/Double_exponential_function) for b. (there are also asymptotic analysis for other sets of primes in various bases b, such as the left-truncatable primes and the right-truncatable primes (https://en.wikipedia.org/wiki/Truncatable_prime, https://t5k.org/glossary/xpage/LeftTruncatablePrime.html, https://t5k.org/glossary/xpage/RightTruncatablePrime.html, https://mathworld.wolfram.com/TruncatablePrime.html, https://www.numbersaplenty.com/set/truncatable_prime/) in various bases b, see http://chesswanks.com/num/LTPs/ for the left-truncatable primes in bases b ≤ 120 and http://fatphil.org/maths/rtp/rtp.html for the right-truncatable primes in bases b ≤ 90)

The lengths of the minimal primes in base b appear to follow the Zipf's law (https://en.wikipedia.org/wiki/Zipf%27s_law, https://mathworld.wolfram.com/ZipfsLaw.html) (which is similar to the Benford's law (https://en.wikipedia.org/wiki/Benford%27s_law, https://t5k.org/glossary/xpage/BenfordsLaw.html, https://mathworld.wolfram.com/BenfordsLaw.html, https://www.mathpages.com/home/kmath302/kmath302.htm, https://t5k.org/notes/faq/BenfordsLaw.html)), for n ≥ 2, the number of n-digit minimal primes in base b is inversely proportional (https://en.wikipedia.org/wiki/Inversely_proportional, https://mathworld.wolfram.com/InverselyProportional.html) to n−1 (i.e. the expected value (https://en.wikipedia.org/wiki/Expected_value, https://mathworld.wolfram.com/ExpectationValue.html) is c/(n−1), where c is a fixed constant) (i.e. the graph of the points (x,y = the number of x-digit minimal primes in base b) in the xy-plane is near to the graph of y = c/(x−1) in the xy-plane for a fixed real number c), for any fixed base b, also, for n ≥ 1, the length of the nth largest minimal prime in base b is inversely proportional (https://en.wikipedia.org/wiki/Inversely_proportional, https://mathworld.wolfram.com/InverselyProportional.html) to n (i.e. the expected value (https://en.wikipedia.org/wiki/Expected_value, https://mathworld.wolfram.com/ExpectationValue.html) is c/n, where c is a fixed constant) (i.e. the graph of the points (x,y = the length of the xth largest minimal prime in base b) in the xy-plane is near to the graph of y = c/x in the xy-plane for a fixed real number c), for any fixed base b.

We can use the sense of http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf) to say: (note that some of the left families may cover another left family, e.g. the base 19 left family 5{H}5 covers another base 19 left family 5{H}05, and if the smallest prime in family 5{H}5 in base 19 has length n, and the family 5{H}05 in base 19 has no prime with length ≤ n, then family 5{H}05 in base 19 can be removed from the unsolved families for base 19, however, if the smallest prime in family 5{H}5 in base 19 has length n, but the family 5{H}05 in base 19 is not tested to length n or more, then family 5{H}05 in base 19 should not be removed from the unsolved families for base 19, since a number in family 5{H}05 covers the prime in family 5{H}5 with length n if and only if the length of this number is ≥ n+1; besides, the base 19 left family FH0{H} covers another base 19 left family FHHH0{H}, and if the smallest prime in family FH0{H} in base 19 has length n, and the family FHHH0{H} in base 19 has no prime with length ≤ n+1, then family FHHH0{H} in base 19 can be removed from the unsolved families for base 19, however, if the smallest prime in family FH0{H} in base 19 has length n, but the family FHHH0{H} in base 19 is not tested to length n+1 or more, then family FHHH0{H} in base 19 should not be removed from the unsolved families for base 19, since a number in family FHHH0{H} covers the prime in family FH0{H} with length n if and only if the length of this number is ≥ n+2; besides, the base 21 left family {9}D covers another base 21 left family F{9}D, and if the smallest prime in family {9}D in base 21 has length n, and the family F{9}D in base 21 has no prime with length ≤ n, then family F{9}D in base 21 can be removed from the unsolved families for base 21, however, if the smallest prime in family {9}D in base 21 has length n, but the family F{9}D in base 21 is not tested to length n or more, then family F{9}D in base 21 should not be removed from the unsolved families for base 21, since a number in family F{9}D covers the prime in family {9}D with length n if and only if the length of this number is ≥ n+1 (if a family has no primes, then we say "the smallest prime in this family has length ∞ (https://en.wikipedia.org/wiki/Infinity, https://t5k.org/glossary/xpage/Infinite.html, https://mathworld.wolfram.com/Infinity.html) (instead of 0 or −1)", see http://gladhoboexpress.blogspot.com/2019/05/prime-sandwiches-made-with-one-derbread.html and http://chesswanks.com/seq/a306861.txt (for the OEIS sequence https://oeis.org/A306861) and http://chesswanks.com/seq/a269254.txt (for the OEIS sequence https://oeis.org/A269254) (since this is more convenient, e.g. the n of the smallest prime in the base 13 family A3_nA, this family has been searched to n = 500000 with no prime or probable prime found, we can use ">500000" for the n of the smallest prime in the base 13 family A3_nA (while for the n of the smallest prime in the base 13 family 95_n, it is 197420), ">500000" includes infinity (since infinity is > 500000) but does not includes 0 or −1, it is still possible that there is no prime in the base 13 family A3_nA, although by the heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, http://www.utm.edu/~caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf)) above, this is very impossible, also "the smallest n ≥ 1 such that (a×bⁿ+c)/gcd(a+c,b−1) is prime" should be the infimum (https://en.wikipedia.org/wiki/Infimum, https://mathworld.wolfram.com/Infimum.html) of the set S of the numbers n ≥ 1 such that (a×bⁿ+c)/gcd(a+c,b−1) is prime, and if there is no n ≥ 1 such that (a×bⁿ+c)/gcd(a+c,b−1) is prime, then this set S is the empty set (https://en.wikipedia.org/wiki/Empty_set, https://mathworld.wolfram.com/EmptySet.html), and by the definition of "inf", the infimum of the empty set is ∞), ∞ is > any finite number, e.g. "the smallest n ≥ 1 such that k×2ⁿ+1 is prime" is ∞ for k = 78557, 157114, 271129, 271577, 314228, 322523, 327739, 482719, ..., while it is 31172165 for k = 10223 and 13018586 for k = 19249, another example is "the smallest n such that (bⁿ−1)/(b−1) is prime" is ∞ for b = 9, 25, 32, 49, 64, 81, 121, 125, 144, 169, 216, 225, 243, 289, 324, 343, 361, 441, 484, 529, 625, 729, 784, 841, 900, 961, 1000, ..., while it is 62903 for b = 691 and 41189 for b = 693))

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

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

This problem covers finding the smallest prime in these families in the same base b (or proving that such prime does not exist), since the smallest prime in these families (if exists) must be a minimal prime in base b: (while the original minimal prime problem does not cover some of these forms for some bases (or all bases) b)

(below (as well as the "left b" files), family "12{3}45" means sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...}, where the members are expressed as base b strings, like the numbers in https://stdkmd.net/nrr/aaaab.htm, https://stdkmd.net/nrr/abbbb.htm, https://stdkmd.net/nrr/aaaba.htm, https://stdkmd.net/nrr/abaaa.htm, https://stdkmd.net/nrr/abbba.htm, https://stdkmd.net/nrr/abbbc.htm, https://stdkmd.net/nrr/prime/primesize.txt, https://stdkmd.net/nrr/prime/primesize.zip, https://stdkmd.net/nrr/prime/primecount.htm, https://stdkmd.net/nrr/prime/primecount.txt, https://stdkmd.net/nrr/prime/primedifficulty.htm, https://stdkmd.net/nrr/prime/primedifficulty.txt, e.g. 1{3} (in decimal) is the numbers in https://stdkmd.net/nrr/1/13333.htm, and {1}3 (in decimal) is the numbers in https://stdkmd.net/nrr/1/11113.htm, and 1{2}3 (in decimal) is the numbers in https://stdkmd.net/nrr/1/12223.htm, also, superscripts always means exponents (https://en.wikipedia.org/wiki/Exponentiation, https://www.rieselprime.de/ziki/Exponent, https://mathworld.wolfram.com/Exponent.html, https://mathworld.wolfram.com/Power.html, https://mathworld.wolfram.com/Exponentiation.html), subscripts are always used to indicate repetitions of digits, e.g. 123₄567 = 123333567, all subscripts are written in decimal)

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

• x{0}y
• x{y} (unless y = 1) (see https://stdkmd.net/nrr/abbbb.htm)
• {x}y (unless x = 1) (see https://stdkmd.net/nrr/aaaab.htm)
• x{0}yz (unless there is a prime of the form x{0}y or x{0}z)
• xy{0}z (unless there is a prime of the form x{0}z or y{0}z)
• xy{x} (unless either x = 1 or there is a prime of the form y{x} (or both)) (see https://stdkmd.net/nrr/abaaa.htm)
• {x}yx (unless either x = 1 or there is a prime of the form {x}y (or both)) (see https://stdkmd.net/nrr/aaaba.htm)

The primes in forms x{y}, {x}y, xy{x}, {x}yx in base b are near-repdigit primes (https://t5k.org/glossary/xpage/NearRepdigitPrime.html, https://t5k.org/top20/page.php?id=15, https://pzktupel.de/Primetables/TableNRD.php, https://oeis.org/A164937, https://stdkmd.net/nrr/#factortables_nr, https://stdkmd.net/nrr/#factortables_np, https://stdkmd.net/nrr/aaaab.htm, https://stdkmd.net/nrr/abbbb.htm, https://stdkmd.net/nrr/aaaba.htm, https://stdkmd.net/nrr/abaaa.htm, https://stdkmd.net/nrr/aabaa.htm, https://stdkmd.net/nrr/records.htm#nrprime, https://stdkmd.net/nrr/records.htm#nrprp, https://stdkmd.net/nrr/records.htm#nrpprime, https://stdkmd.net/nrr/records.htm#nrpprp, http://factordb.com/tables.php?open=1, http://factordb.com/tables.php?open=3) in base b.

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

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

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

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

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

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

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

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

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

Case 1: p ends with 1.

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

Case 1.1: p begins with 2.

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

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

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

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

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

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

Case 1.2: p begins with 5.

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

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

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

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

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

Case 1.3: p begins with 8.

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

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

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

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

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

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

Case 1.4: p begins with 9.

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

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

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

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

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

Case 2: p ends with 3.

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

Case 3: p ends with 7.

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

Case 3.1: p begins with 2.

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

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

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

If y ∈ 8{0}, then p ∈ 28{0}7. But then p is divisible by 7, since for n ≥ 0 we have 7 × 40_n1 = 280_n7.

Case 3.2: p begins with 5.

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

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

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

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

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

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

Case 3.3: p begins with 7.

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

Case 3.4: p begins with 8.

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

Case 4: p ends with 9.

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

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

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

Case 4.1: p begins with 3.

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

Case 4.2: p begins with 4.

In this case we can write p = 4y9, where all digits of y are 0, 3, 6, or 9. If 0 ◁ y, then 409 ◁ p. If 3 ◁ y, then 43 ◁ p. If 9 ◁ y, then 499 ◁ p. Hence we may assume y ∈ {6}, and thus p ∈ 4{6}9. But then p is divisible by 7, since for n ≥ 0 we have 7 × 6_n7 = 46_n9.

Case 4.3: p begins with 6.

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

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

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

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

Case 4.4: p begins with 9.

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

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

I left it as an exercise for the reader to write the proof for bases b = 2, 3, 4, 5, 6, 7, 8, 9, 12, of course, the proof for base b = 2 is trivial, since all primes p > 2 must start and end with 1 in base 2, thus we must have 11 ◁ p, however, for some bases b like 24 (the currently largest "proven" base b, including the primality proving for the primes in the set), it is almost impossible to write the proof by hand, since base b = 24 has too many (3409) minimal primes to write the proof, thus the C++ program code (for computer to compute (https://en.wikipedia.org/wiki/Computing) the proof) is made.

(in fact, the fully proof should also include the primality proving (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/prove3.html, https://t5k.org/prove/prove4.html) for all primes in the set (like https://web.archive.org/web/20020809212051/http://www.users.globalnet.co.uk/~aads/C0034177.html (for the generalized repunit prime in base b = 3 with length 4177) and https://web.archive.org/web/20020701171455/http://www.users.globalnet.co.uk/~aads/C0066883.html (for the generalized repunit prime in base b = 6 with length 6883) and https://web.archive.org/web/20020809122706/http://www.users.globalnet.co.uk/~aads/C0071699.html (for the generalized repunit prime in base b = 7 with length 1699) and https://web.archive.org/web/20020809122635/http://www.users.globalnet.co.uk/~aads/C0101031.html (for the generalized repunit prime in base b = 10 with length 1031) and https://web.archive.org/web/20020809122237/http://www.users.globalnet.co.uk/~aads/C0114801.html (for the generalized repunit prime in base b = 11 with length 4801) and https://web.archive.org/web/20020809122947/http://www.users.globalnet.co.uk/~aads/C0130991.html (for the generalized repunit prime in base b = 13 with length 991) and https://web.archive.org/web/20020809124216/http://www.users.globalnet.co.uk/~aads/C0131021.html (for the generalized repunit prime in base b = 13 with length 1021) and https://web.archive.org/web/20020809125049/http://www.users.globalnet.co.uk/~aads/C0131193.html (for the generalized repunit prime in base b = 13 with length 1193) and https://web.archive.org/web/20020809124458/http://www.users.globalnet.co.uk/~aads/C0152579.html (for the generalized repunit prime in base b = 15 with length 2579) and https://web.archive.org/web/20020809124537/http://www.users.globalnet.co.uk/~aads/C0220857.html (for the generalized repunit prime in base b = 22 with length 857) and https://web.archive.org/web/20020809152611/http://www.users.globalnet.co.uk/~aads/C0315581.html (for the generalized repunit prime in base b = 31 with length 5581) and https://web.archive.org/web/20020809124929/http://www.users.globalnet.co.uk/~aads/C0351297.html (for the generalized repunit prime in base b = 35 with length 1297) and https://stdkmd.net/nrr/pock/ (for the near-repdigit primes, although the primes 2×10¹⁷⁵⁵−1 and 2×10³⁰²⁰−1 can be quickly proven prime using the N+1 primality proving (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2)) and http://xenon.stanford.edu/~tjw/pp/index.html (for the generalized repunit primes) and https://t5k.org/lists/single_primes/50005cert.txt (for the prime https://t5k.org/primes/page.php?id=12806) and https://www.alfredreichlg.de/10w7/cert/primo-10w7_27669.out (for the large prime factor of 10²⁷⁶⁶⁹+7) and https://www.alfredreichlg.de/10w7/cert/primo-10w7_15093.out (for the prime 10¹⁵⁰⁹³+7) and https://www.alfredreichlg.de/10w7/cert/primo-10w7_10393.out (for the large prime factor of 10¹⁰³⁹³+7) and https://homes.cerias.purdue.edu/~ssw/cun/third/proofs (for the larger prime factors of bⁿ±1 with 2 ≤ b ≤ 12) and https://web.archive.org/web/20150911225651/https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0104&L=nmbrthry&P=R1807&D=0 (for the prime https://t5k.org/primes/page.php?id=11084) and https://web.archive.org/web/20170515153924/http://bitc.bme.emory.edu/~lzhou/blogs/?p=263 (for the primes corresponding to https://oeis.org/A181980) and https://web.archive.org/web/20131020160719/http://www.primes.viner-steward.org/andy/E/33281741.html (for the prime https://t5k.org/primes/page.php?id=82858), or using an elliptic curve primality proving (https://t5k.org/prove/prove4_2.html, https://en.wikipedia.org/wiki/Elliptic_curve_primality, https://t5k.org/glossary/xpage/ECPP.html, https://mathworld.wolfram.com/EllipticCurvePrimalityProving.html, https://t5k.org/top20/page.php?id=27, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/~morain/Prgms/ecpp.english.html, https://www.multiprecision.org/cm/ecpp.html) implementation such as PRIMO (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309) or CM (https://www.multiprecision.org/cm/index.html, https://t5k.org/bios/page.php?id=5485, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/cm) to compute primality certificates (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php)) and the compositeness proving for all proper subsequence of all primes in the set (usually by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://oeis.org/A189172) (usually to 10⁹, this will covered by sieving (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php) for the numbers > 10¹⁰⁰⁰) or Fermat primality test (https://t5k.org/prove/prove2_2.html, https://en.wikipedia.org/wiki/Fermat_primality_test, https://en.wikipedia.org/wiki/Fermat_pseudoprime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Fermat_pseudoprimality_test, https://mathworld.wolfram.com/FermatPseudoprime.html, https://www.numbersaplenty.com/set/Poulet_number/, http://www.numericana.com/answer/pseudo.htm#pseudoprime, http://ntheory.org/data/psps.txt, https://sites.google.com/view/fermat-pseudoprime, https://sites.google.com/view/bases-fermat-pseudoprime, https://oeis.org/A001567, https://oeis.org/A005935, https://oeis.org/A005936, https://oeis.org/A005938, https://oeis.org/A052155, https://oeis.org/A083737, https://oeis.org/A083739, https://oeis.org/A083876, https://oeis.org/A181780, https://oeis.org/A063994, https://oeis.org/A194946, https://oeis.org/A195327, https://oeis.org/A002997, https://oeis.org/A191311, https://oeis.org/A090086, https://oeis.org/A007535) (usually base 2 and base 3)), but in the proof above we assume that we know whether a number is prime or not)

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

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

A repunit (in any base b) with length n can be prime only if n is prime, since otherwise b^k×m−1 is a binomial number (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) which can be factored algebraically (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://mersenneforum.org/showpost.php?p=196598&postcount=492, https://mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php (e.g. for the family 3{8} in base 9, type "4*9^n-1", and it will tell you that this form can be factored to (2×3ⁿ−1) × (2×3ⁿ+1)), https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/). In fact, if n = 2×m is even, then b^2×m−1 = (b^m−1) × (b^m+1).

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

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

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

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

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

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

Problem: Given strings x, z (may be empty), a digit y, and a base b, does there exist a prime number whose base-b expansion is of the form xy_nz for some n ≥ 0? (If we say "yes", then we should find a such prime (the smallest such prime may be very large, e.g. > 10²⁵⁰⁰⁰, and if so, then we should use (probable) primality testing (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/index.html) programs (https://www.rieselprime.de/ziki/Primality_testing_program) such as PFGW (https://sourceforge.net/projects/openpfgw/, https://t5k.org/bios/page.php?id=175, https://www.rieselprime.de/ziki/PFGW, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/pfgw_win_4.0.3) or LLR (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64) to find it, and before using these programs, we should use sieving (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php) programs (https://www.rieselprime.de/ziki/Sieving_program) such as srsieve (or sr1/2/5sieve) (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, http://www.rieselprime.de/dl/CRUS_pack.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2cl.exe) to remove the numbers either having small prime factors or having algebraic factors) and prove its primality (by N−1 primality test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1) or N+1 primality test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2) or elliptic curve primality proving (https://t5k.org/prove/prove4_2.html, https://en.wikipedia.org/wiki/Elliptic_curve_primality, https://t5k.org/glossary/xpage/ECPP.html, https://mathworld.wolfram.com/EllipticCurvePrimalityProving.html, https://t5k.org/top20/page.php?id=27, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/~morain/Prgms/ecpp.english.html, https://www.multiprecision.org/cm/ecpp.html) implementation such as PRIMO (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309) or CM (https://www.multiprecision.org/cm/index.html, https://t5k.org/bios/page.php?id=5485, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/cm) to compute primality certificates (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php)) (and if we want to solve the problem in this project, we should check whether this prime is the smallest such prime or not, i.e. prove all smaller numbers of the form xy_nz with n ≥ 0 are composite, usually by trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://oeis.org/A189172) or Fermat primality test (https://t5k.org/prove/prove2_2.html, https://en.wikipedia.org/wiki/Fermat_primality_test, https://en.wikipedia.org/wiki/Fermat_pseudoprime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Fermat_pseudoprimality_test, https://mathworld.wolfram.com/FermatPseudoprime.html, https://www.numbersaplenty.com/set/Poulet_number/, http://www.numericana.com/answer/pseudo.htm#pseudoprime, http://ntheory.org/data/psps.txt, https://sites.google.com/view/fermat-pseudoprime, https://sites.google.com/view/bases-fermat-pseudoprime, https://oeis.org/A001567, https://oeis.org/A005935, https://oeis.org/A005936, https://oeis.org/A005938, https://oeis.org/A052155, https://oeis.org/A083737, https://oeis.org/A083739, https://oeis.org/A083876, https://oeis.org/A181780, https://oeis.org/A063994, https://oeis.org/A194946, https://oeis.org/A195327, https://oeis.org/A002997, https://oeis.org/A191311, https://oeis.org/A090086, https://oeis.org/A007535)), and if we say "no", then we should prove that such prime does not exist, may by covering congruence (http://irvinemclean.com/maths/siercvr.htm, http://web.archive.org/web/20060925100410/http://www.glasgowg43.freeserve.co.uk/siernums.htm, https://web.archive.org/web/20061116164533/http://www.glasgowg43.freeserve.co.uk/brier2.htm, https://web.archive.org/web/20221230035324/https://sites.google.com/site/robertgerbicz/coveringsets, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/coveringsets, http://www.numericana.com/answer/primes.htm#sierpinski, http://irvinemclean.com/maths/sierpin.htm, http://irvinemclean.com/maths/sierpin2.htm, http://irvinemclean.com/maths/sierpin3.htm, http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://www.ams.org/journals/mcom/1975-29-129/S0025-5718-1975-0376583-0/S0025-5718-1975-0376583-0.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_27.pdf), https://www.ams.org/journals/mcom/1983-40-161/S0025-5718-1983-0679453-8/S0025-5718-1983-0679453-8.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_40.pdf), http://yves.gallot.pagesperso-orange.fr/papers/smallbrier.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_48.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://arxiv.org/pdf/2209.10646.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_52.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL14/Jones/jones12.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_80.pdf), https://web.archive.org/web/20081119135435/http://math.crg4.com/a094076.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_102.pdf), http://www.renyi.hu/~p_erdos/1950-07.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_103.pdf), http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_1.pdf), https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), http://www.primepuzzles.net/puzzles/puzz_614.htm, http://www.primepuzzles.net/problems/prob_029.htm, http://www.primepuzzles.net/problems/prob_030.htm, http://www.primepuzzles.net/problems/prob_036.htm, http://www.primepuzzles.net/problems/prob_049.htm, https://www.rieselprime.de/Related/LiskovetsGallot.htm, https://www.rieselprime.de/Related/RieselTwinSG.htm, https://stdkmd.net/nrr/coveringset.htm, https://stdkmd.net/nrr/9/91113.htm#prime_period, https://stdkmd.net/nrr/9/94449.htm#prime_period, https://stdkmd.net/nrr/9/95559.htm#prime_period, https://web.archive.org/web/20070220134129/http://www.users.globalnet.co.uk/~perry/maths/wildeprimes/wildeprimes.htm, https://www.rose-hulman.edu/~rickert/Compositeseq/, https://oeis.org/A244561, https://oeis.org/A244562, https://oeis.org/A244563, https://oeis.org/A244564, https://oeis.org/A244070, https://oeis.org/A244071, https://oeis.org/A244072, https://oeis.org/A244073, https://oeis.org/A257647, https://oeis.org/A258154, https://oeis.org/A289110, https://oeis.org/A257861, https://oeis.org/A306151, https://oeis.org/A305473, https://en.wikipedia.org/wiki/Covering_set, https://www.rieselprime.de/ziki/Covering_set, https://mathworld.wolfram.com/SierpinskisCompositeNumberTheorem.html) (i.e. finding a finite set (https://en.wikipedia.org/wiki/Finite_set, https://mathworld.wolfram.com/FiniteSet.html) S of primes p such that all numbers in a given family are divisible (https://en.wikipedia.org/wiki/Divides, https://t5k.org/glossary/xpage/Divides.html, https://t5k.org/glossary/xpage/Divisor.html, https://www.rieselprime.de/ziki/Factor, https://mathworld.wolfram.com/Divides.html, https://mathworld.wolfram.com/Divisor.html, https://mathworld.wolfram.com/Divisible.html, http://www.numericana.com/answer/primes.htm#divisor) by some element of S (this is equivalent to finding a positive integer N such that all numbers in a given family are not coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to N, this N is usually a factor of a small generalized repunit number (https://en.wikipedia.org/wiki/Repunit, https://t5k.org/glossary/xpage/Repunit.html, https://t5k.org/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://pzktupel.de/Primetables/TableRepunit.php, https://pzktupel.de/Primetables/TableRepunitGen.php, https://pzktupel.de/Primetables/TableRepunitGen.txt, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html, https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/, https://web.archive.org/web/20120227163614/http://phi.redgolpe.com/5.asp, https://web.archive.org/web/20120227163508/http://phi.redgolpe.com/4.asp, https://web.archive.org/web/20120227163610/http://phi.redgolpe.com/3.asp, https://web.archive.org/web/20120227163512/http://phi.redgolpe.com/2.asp, https://web.archive.org/web/20120227163521/http://phi.redgolpe.com/1.asp, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.fermatquotient.com/PrimSerien/GenRepuP.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/article/380/231/, http://www.bitman.name/math/table/379, http://www.bitman.name/math/table/488, https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_5.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf), https://t5k.org/top20/page.php?id=57, https://t5k.org/top20/page.php?id=16, https://oeis.org/A002275, https://oeis.org/A004022, https://oeis.org/A053696, https://oeis.org/A085104, https://oeis.org/A179625) in base b, e.g. all numbers in the family 2{5} in base 11 are not coprime to 6, gcd((5×11ⁿ−1)/2, 6) can only be 2 or 3, and cannot be 1, also equivalent to finding a prime p such that all numbers in a given family are not p-rough numbers (https://en.wikipedia.org/wiki/Rough_number, https://mathworld.wolfram.com/RoughNumber.html, https://oeis.org/A007310, https://oeis.org/A007775, https://oeis.org/A008364, https://oeis.org/A008365, https://oeis.org/A008366, https://oeis.org/A166061, https://oeis.org/A166063)), by modular arithmetic (https://en.wikipedia.org/wiki/Modular_arithmetic, https://en.wikipedia.org/wiki/Congruence_relation, https://en.wikipedia.org/wiki/Modulo, https://t5k.org/glossary/xpage/Congruence.html, https://t5k.org/glossary/xpage/CongruenceClass.html, https://t5k.org/glossary/xpage/Residue.html, https://mathworld.wolfram.com/Congruence.html, https://mathworld.wolfram.com/Congruent.html, https://mathworld.wolfram.com/Residue.html, https://mathworld.wolfram.com/MinimalResidue.html, https://mathworld.wolfram.com/Mod.html)), algebraic factorization (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://mersenneforum.org/showpost.php?p=196598&postcount=492, https://mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php (e.g. for the family 3{8} in base 9, type "4*9^n-1", and it will tell you that this form can be factored to (2×3ⁿ−1) × (2×3ⁿ+1)), https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) (which includes difference-of-two-squares factorization (https://en.wikipedia.org/wiki/Difference_of_two_squares) and sum/difference-of-two-cubes factorization (https://en.wikipedia.org/wiki/Sum_of_two_cubes) and difference-of-two-nth-powers factorization with n > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and sum/difference-of-two-nth-powers factorization with odd n > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and Aurifeuillean factorization (https://en.wikipedia.org/wiki/Aurifeuillean_factorization, https://www.rieselprime.de/ziki/Aurifeuillian_factor, https://mathworld.wolfram.com/AurifeuilleanFactorization.html, http://www.numericana.com/answer/numbers.htm#aurifeuille, http://pagesperso-orange.fr/colin.barker/lpa/cycl_fac.htm, http://list.seqfan.eu/oldermail/seqfan/2017-March/017363.html, http://myfactorcollection.mooo.com:8090/source/cyclo.cpp, http://myfactorcollection.mooo.com:8090/LCD_2_199, http://myfactorcollection.mooo.com:8090/LCD_2_998, https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean, https://www.unshlump.com/hcn/aurif.html, https://www.ams.org/journals/mcom/2006-75-253/S0025-5718-05-01766-7/S0025-5718-05-01766-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_138.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb127.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_164.pdf), https://www.jams.jp/scm/contents/Vol-2-3/2-3-16.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_167.pdf)) of x⁴+4×y⁴ or x⁶+27×y⁶), or combine of them (https://mersenneforum.org/showthread.php?t=11143, https://mersenneforum.org/showthread.php?t=10279, https://www.fq.math.ca/Scanned/33-3/izotov.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_46.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://oeis.org/A213353, https://oeis.org/A233469))

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

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

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

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

(this conjecture (https://en.wikipedia.org/wiki/Conjecture, https://t5k.org/glossary/xpage/Conjecture.html, https://mathworld.wolfram.com/Conjecture.html) is very important for the problem in this project, since if this conjecture is in fact false, then there will may be some unsolved families which in fact contain no primes, thus the problem in this project in corresponding bases b will may be unsolvable (at least in current technology, unless someone find a new theorem (i.e. other than covering congruence, algebraic factorization, or combine of them) to prove that some families contain no primes, but I do not think that this is possible), however, this conjecture is currently to far to prove, much far than the Bunyakovsky conjecture (https://en.wikipedia.org/wiki/Bunyakovsky_conjecture, https://mathworld.wolfram.com/BouniakowskyConjecture.html) and even the Schinzel's hypothesis H (https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H, https://mathworld.wolfram.com/SchinzelsHypothesis.html, http://www.numericana.com/answer/open.htm#h), besides, this conjecture is reasonable, since there is a heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, http://www.utm.edu/~caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf)) that all families which cannot be ruled out as only containing composites or only containing finitely many primes (by covering congruence, algebraic factorization, or combine of them) contain infinitely many primes (references: https://t5k.org/mersenne/heuristic.html, https://t5k.org/notes/faq/NextMersenne.html, https://t5k.org/glossary/xpage/Repunit.html, https://web.archive.org/web/20100628035147/http://www.math.niu.edu/~rusin/known-math/98/exp_primes, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf), https://mersenneforum.org/showthread.php?t=12327, https://oeis.org/A234285 (the comment by Farideh Firoozbakht, although this comment is not true, there is no prime for s = 509203 and s = −78557, s = 509203 has a covering set of {3, 5, 7, 13, 17, 241}, and s = −78557 has a covering set of {3, 5, 7, 13, 19, 37, 73}), https://mersenneforum.org/showpost.php?p=564786&postcount=3, https://mersenneforum.org/showpost.php?p=461665&postcount=7, https://mersenneforum.org/showpost.php?p=625978&postcount=1027, also the graphs https://t5k.org/gifs/lg_lg_Mn.gif (for the family {1} in base b = 2) and https://t5k.org/gifs/repunits.gif (for the family {1} in base b = 10) and https://mersenneforum.org/attachment.php?attachmentid=4010&d=1642088235 (for the family 2{0}1 in base b = 3)), since by the prime number theorem (https://en.wikipedia.org/wiki/Prime_number_theorem, https://t5k.org/glossary/xpage/PrimeNumberThm.html, https://mathworld.wolfram.com/PrimeNumberTheorem.html, https://t5k.org/howmany.html, http://www.numericana.com/answer/primes.htm#pnt, https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Tables_of_special_primes) the chance (https://en.wikipedia.org/wiki/Probability, https://mathworld.wolfram.com/Probability.html) that a random (https://en.wikipedia.org/wiki/Random_number, https://mathworld.wolfram.com/RandomNumber.html) n-digit base b number is prime is approximately (https://en.wikipedia.org/wiki/Asymptotic_analysis, https://t5k.org/glossary/xpage/AsymptoticallyEqual.html, https://mathworld.wolfram.com/Asymptotic.html) 1/n (more accurately, the chance is approximately 1/(n×ln(b)), where ln is the natural logarithm (https://en.wikipedia.org/wiki/Natural_logarithm, https://t5k.org/glossary/xpage/Log.html, https://mathworld.wolfram.com/NaturalLogarithm.html), i.e. the logarithm with base e = 2.718281828459... (https://en.wikipedia.org/wiki/E_(mathematical_constant), https://mathworld.wolfram.com/e.html)). If one conjectures the numbers x{y}z behave similarly (i.e. the numbers x{y}z is a pseudorandom sequence (https://en.wikipedia.org/wiki/Pseudorandomness, https://mathworld.wolfram.com/PseudorandomNumber.html, https://people.seas.harvard.edu/~salil/pseudorandomness/pseudorandomness-Aug12.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_197.pdf))) you would expect 1/1 + 1/2 + 1/3 + 1/4 + ... = ∞ (https://en.wikipedia.org/wiki/Harmonic_series_(mathematics), https://mathworld.wolfram.com/HarmonicSeries.html) primes of the form x{y}z (of course, this does not always happen, since some x{y}z families can be ruled out as only containing composites (only count the numbers > b) (by covering congruence, algebraic factorization, or combine of them), and every family has its own Nash weight (https://www.rieselprime.de/ziki/Nash_weight, http://irvinemclean.com/maths/nash.htm, http://www.brennen.net/primes/ProthWeight.html, https://www.mersenneforum.org/showthread.php?t=11844, https://www.mersenneforum.org/showthread.php?t=2645, https://www.mersenneforum.org/showthread.php?t=7213, https://www.mersenneforum.org/showthread.php?t=18818, https://www.mersenneforum.org/showpost.php?p=421186&postcount=19, 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, http://www.noprimeleftbehind.net/crus/vstats_new/riesel_weights.txt, 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), 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/primedifficulty.htm, https://stdkmd.net/nrr/prime/primedifficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/riesel_difficulty.txt, http://www.noprimeleftbehind.net/crus/vstats_new/sierpinski_difficulty.txt)), see https://mersenneforum.org/showpost.php?p=564786&postcount=3, families which can be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them) have Nash weight (or difficulty) 0, and families which cannot be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them) have positive Nash weight (or difficulty), but it is at least a reasonable conjecture in the absence of evidence to the contrary), there are approximately (https://en.wikipedia.org/wiki/Asymptotic_analysis, https://t5k.org/glossary/xpage/AsymptoticallyEqual.html, https://mathworld.wolfram.com/Asymptotic.html) (e^γ×W×N−1/1−1/2−1/3−...−1/(length(x)+length(z)−1))/ln(b) primes in the family x{y}z in base b with length ≤ N (where e = 2.718281828459... is the base of the natural logarithm (https://en.wikipedia.org/wiki/E_(mathematical_constant), https://mathworld.wolfram.com/e.html, https://oeis.org/A001113), γ = 0.577215664901 is the Euler–Mascheroni constant (https://en.wikipedia.org/wiki/Euler%27s_constant, https://t5k.org/glossary/xpage/Gamma.html, https://mathworld.wolfram.com/Euler-MascheroniConstant.html, https://oeis.org/A001620), W is the Nash weight (or difficulty) of the family x{y}z in base b (W = 0 if and only if the family x{y}z in base b can be proven to only contain composites or only contain finitely many primes (by covering congruence, algebraic factorization, or combine of them)), ln is the natural logarithm (https://en.wikipedia.org/wiki/Natural_logarithm, https://t5k.org/glossary/xpage/Log.html, https://mathworld.wolfram.com/NaturalLogarithm.html) (i.e. the logarithm with base e = 2.718281828459... (https://en.wikipedia.org/wiki/E_(mathematical_constant), https://mathworld.wolfram.com/e.html))).

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

This conjecture will imply:

• There are infinitely many Mersenne primes (i.e. primes of the form 2^p−1 with prime p) (https://en.wikipedia.org/wiki/Mersenne_prime, https://t5k.org/glossary/xpage/MersenneNumber.html, https://t5k.org/glossary/xpage/Mersennes.html, https://www.rieselprime.de/ziki/Mersenne_number, https://www.rieselprime.de/ziki/Mersenne_prime, https://mathworld.wolfram.com/MersenneNumber.html, https://mathworld.wolfram.com/MersennePrime.html, https://pzktupel.de/Primetables/TableMersenne.php, https://t5k.org/top20/page.php?id=4, https://www.mersenne.org/, https://www.mersenne.org/primes/, https://www.mersenne.ca/, https://t5k.org/mersenne/) (https://oeis.org/A001348, https://oeis.org/A000668, https://oeis.org/A000043)
• There are infinitely many Fermat primes (i.e. primes of the form 2²ⁿ+1) (https://en.wikipedia.org/wiki/Fermat_number, https://t5k.org/glossary/xpage/FermatNumber.html, https://www.rieselprime.de/ziki/Fermat_number, https://mathworld.wolfram.com/FermatNumber.html, https://mathworld.wolfram.com/FermatPrime.html, https://pzktupel.de/Primetables/TableFermat.php, http://www.prothsearch.com/fermat.html, http://www.fermatsearch.org/, https://64ordle.au/fermat/) (https://oeis.org/A000215, https://oeis.org/A019434)
• There are infinitely many generalized repunit primes (i.e. primes of the form (b^p−1)/(b−1) with prime p) (https://en.wikipedia.org/wiki/Repunit, https://t5k.org/glossary/xpage/Repunit.html, https://t5k.org/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://pzktupel.de/Primetables/TableRepunit.php, https://pzktupel.de/Primetables/TableRepunitGen.php, https://pzktupel.de/Primetables/TableRepunitGen.txt, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html, https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/, https://web.archive.org/web/20120227163614/http://phi.redgolpe.com/5.asp, https://web.archive.org/web/20120227163508/http://phi.redgolpe.com/4.asp, https://web.archive.org/web/20120227163610/http://phi.redgolpe.com/3.asp, https://web.archive.org/web/20120227163512/http://phi.redgolpe.com/2.asp, https://web.archive.org/web/20120227163521/http://phi.redgolpe.com/1.asp, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, http://www.bitman.name/math/article/380/231/, http://www.bitman.name/math/table/379, https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf), https://t5k.org/top20/page.php?id=57, https://t5k.org/top20/page.php?id=16, https://oeis.org/A002275, https://oeis.org/A004022, https://oeis.org/A053696, https://oeis.org/A085104, https://oeis.org/A179625) to every base b ≥ 2 which is not a perfect power (i.e. of the form m^r with r > 1) (https://oeis.org/A001597, https://en.wikipedia.org/wiki/Perfect_power, https://mathworld.wolfram.com/PerfectPower.html, https://www.numbersaplenty.com/set/perfect_power/) (https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A246005, https://oeis.org/A065854, https://oeis.org/A279068, https://oeis.org/A065813, https://oeis.org/A128164, https://oeis.org/A285642)
• There are infinitely many generalized Wagstaff primes (i.e. primes of the form (b^p+1)/(b+1) with odd prime p) (https://en.wikipedia.org/wiki/Wagstaff_prime, https://t5k.org/glossary/xpage/WagstaffPrime.html, https://mathworld.wolfram.com/WagstaffPrime.html, https://pzktupel.de/Primetables/TableWagstaff.php, https://pzktupel.de/Primetables/TableWagstaffGen.php, https://pzktupel.de/Primetables/TableWagstaffGen.txt, https://web.archive.org/web/20211031110623/http://mprime.s3-website.us-west-1.amazonaws.com/wagstaff/, http://www.fermatquotient.com/PrimSerien/GenRepuP.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, http://www.bitman.name/math/table/488, https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_5.pdf), https://t5k.org/top20/page.php?id=67, https://oeis.org/A007583, https://oeis.org/A000979, https://oeis.org/A000978, https://oeis.org/A059054, https://oeis.org/A059055) to every base b ≥ 2 which is neither a perfect odd power (i.e. of the form m^r with odd r > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) nor of the form 4×m⁴ (https://oeis.org/A141046) (https://oeis.org/A084742, https://oeis.org/A084741, https://oeis.org/A126659, https://oeis.org/A065507)
• There are infinitely many generalized Fermat primes (i.e. primes of the form b²ⁿ+1 with even b) (https://t5k.org/glossary/xpage/GeneralizedFermatNumber.html, https://t5k.org/glossary/xpage/GeneralizedFermatPrime.html, https://www.rieselprime.de/ziki/Generalized_Fermat_number, https://mathworld.wolfram.com/GeneralizedFermatNumber.html, https://pzktupel.de/Primetables/TableFermatGFBB.php, https://web.archive.org/web/20230323021722/https://pzktupel.de/Primetables/TableFermatGF09.php, https://web.archive.org/web/20230323021722/https://pzktupel.de/Primetables/TableFermatGF10.php, https://pzktupel.de/Primetables/TableFermatGF11.php, https://pzktupel.de/Primetables/TableFermatGF12.php, https://pzktupel.de/Primetables/TableFermatGF13.php, https://pzktupel.de/Primetables/TableFermatGF14.php, https://pzktupel.de/Primetables/TableFermatGF15.php, https://pzktupel.de/Primetables/TableFermatGF16.php, https://pzktupel.de/Primetables/TableFermatGF17.php, https://pzktupel.de/Primetables/TableFermatGF1820.php, http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, http://yves.gallot.pagesperso-orange.fr/primes/index.html, http://yves.gallot.pagesperso-orange.fr/primes/results.html, http://yves.gallot.pagesperso-orange.fr/primes/stat.html, http://www.primegrid.com/forum_thread.php?id=3980, http://www.primegrid.com/stats_genefer.php, https://t5k.org/top20/page.php?id=12, http://www.prothsearch.com/fermat.html, http://www.prothsearch.com/GFN06.html, http://www.prothsearch.com/GFN10.html, http://www.prothsearch.com/GFN12.html, http://www.prothsearch.com/GFNfacs.html, http://www.prothsearch.com/GFNsmall.html, https://oeis.org/A006314, https://oeis.org/A006313, https://oeis.org/A006315, https://oeis.org/A006316, https://oeis.org/A056994, https://oeis.org/A056995, https://oeis.org/A057465, https://oeis.org/A057002, https://oeis.org/A088361, https://oeis.org/A088362, https://oeis.org/A226528, https://oeis.org/A226529, https://oeis.org/A226530, https://oeis.org/A251597, https://oeis.org/A253854, https://oeis.org/A244150, https://oeis.org/A243959, https://oeis.org/A321323) to every even base b ≥ 2 which is not a perfect odd power (i.e. of the form m^r with odd r > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) (https://oeis.org/A228101, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A123669)
• There are infinitely many generalized half-Fermat primes (i.e. primes of the form (b²ⁿ+1)/2 with odd b) (http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt (in German), http://www.prothsearch.com/GFN03.html, http://www.prothsearch.com/GFN05.html, http://www.prothsearch.com/GFN07.html, http://www.prothsearch.com/GFN11.html, http://www.prothsearch.com/GFNfacs.html, http://www.prothsearch.com/GFNsmall.html) to every odd base b ≥ 2 which is not a perfect odd power (i.e. of the form m^r with odd r > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265)
• There are infinitely many Williams primes of the first kind (i.e. primes of the form (b−1)×bⁿ−1) (https://harvey563.tripod.com/wills.txt, https://www.rieselprime.de/ziki/Williams_prime_MM_least, https://www.rieselprime.de/ziki/Williams_prime_MM_table, https://pzktupel.de/Primetables/TableWilliams1.php, https://sites.google.com/view/williams-primes, http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_9.pdf), https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_10.pdf), http://www.bitman.name/math/table/484 (in Italian)) to every base b ≥ 2 (https://oeis.org/A122396)
• There are infinitely many Williams primes of the second kind (i.e. primes of the form (b−1)×bⁿ+1) (https://www.rieselprime.de/ziki/Williams_prime_MP_least, https://www.rieselprime.de/ziki/Williams_prime_MP_table, https://pzktupel.de/Primetables/TableWilliams2.php, https://sites.google.com/view/williams-primes, http://www.bitman.name/math/table/477 (in Italian)) to every base b ≥ 2 (https://oeis.org/A305531, https://oeis.org/A087139) (warning: this may be false, (b−1)×bⁿ+1 may be able to be proven to only contain composites by combine of covering congruence and algebraic factorization when b−1 is either a perfect odd power (i.e. of the form m^r with odd r > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) or of the form 4×m⁴ (https://oeis.org/A141046), but the smallest such base b will be very large, however, this is at least true for bases b such that b−1 is neither a perfect odd power (i.e. of the form m^r with odd r > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) nor of the form 4×m⁴ (https://oeis.org/A141046), also at least true for bases b ≤ 10⁶)
• There are infinitely many Williams primes of the third kind (i.e. primes of the form (b+1)×bⁿ−1) (https://www.rieselprime.de/ziki/Williams_prime_PM_least, https://www.rieselprime.de/ziki/Williams_prime_PM_table, https://pzktupel.de/Primetables/TableWilliams3.php, https://sites.google.com/view/williams-primes, http://www.bitman.name/math/table/471 (in Italian)) to every base b ≥ 2
• There are infinitely many Williams primes of the fourth kind (i.e. primes of the form (b+1)×bⁿ+1) (https://www.rieselprime.de/ziki/Williams_prime_PP_least, https://www.rieselprime.de/ziki/Williams_prime_PP_table, https://pzktupel.de/Primetables/TableWilliams4.php, https://sites.google.com/view/williams-primes, http://www.bitman.name/math/table/474 (in Italian)) to every base b ≥ 2 which is not == 1 mod 3 (warning: this may be false, (b+1)×bⁿ+1 may be able to be proven to only contain composites by covering congruence, like the case of 2×bⁿ+1 and bⁿ+2 for b = 201446503145165177, which has a covering set {3, 5, 17, 257, 641, 65537, 6700417}, however, this is at least true for bases b ≤ 10⁶)
• There are infinitely many dual Williams primes of the first kind (i.e. primes of the form bⁿ−(b−1)) (https://sites.google.com/view/williams-primes, https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435 (in Italian)) to every base b ≥ 2 (https://oeis.org/A113516, https://oeis.org/A343589)
• There are infinitely many dual Williams primes of the second kind (i.e. primes of the form bⁿ+(b−1)) (https://sites.google.com/view/williams-primes) to every base b ≥ 2 (https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179) (warning: this may be false, bⁿ+(b−1) may be able to be proven to only contain composites by combine of covering congruence and algebraic factorization when b−1 is either a perfect odd power (i.e. of the form m^r with odd r > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) or of the form 4×m⁴ (https://oeis.org/A141046), but the smallest such base b will be very large, however, this is at least true for bases b such that b−1 is neither a perfect odd power (i.e. of the form m^r with odd r > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) nor of the form 4×m⁴ (https://oeis.org/A141046), also at least true for bases b ≤ 10⁶)
• There are infinitely many dual Williams primes of the third kind (i.e. primes of the form bⁿ−(b+1)) (https://sites.google.com/view/williams-primes) to every base b ≥ 2 (https://oeis.org/A178250)
• There are infinitely many dual Williams primes of the fourth kind (i.e. primes of the form bⁿ+(b+1)) (https://sites.google.com/view/williams-primes) to every base b ≥ 2 which is not == 1 mod 3 (https://oeis.org/A346149, https://oeis.org/A346154) (warning: this may be false, bⁿ+(b+1) may be able to be proven to only contain composites by covering congruence, like the case of 2×bⁿ+1 and bⁿ+2 for b = 201446503145165177, which has a covering set {3, 5, 17, 257, 641, 65537, 6700417}, however, this is at least true for bases b ≤ 10⁶)
• 78557 is the smallest Sierpinski number (i.e. odd numbers k such that k×2ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base2.txt, 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.prothsearch.com/sierp.html, http://www.primegrid.com/forum_thread.php?id=1647, http://www.primegrid.com/forum_thread.php?id=972, http://www.primegrid.com/forum_thread.php?id=1750, http://www.primegrid.com/forum_thread.php?id=5758, http://www.primegrid.com/stats_sob_llr.php, http://www.primegrid.com/stats_psp_llr.php, http://www.primegrid.com/stats_esp_llr.php, https://web.archive.org/web/20160405211049/http://www.seventeenorbust.com/, https://en.wikipedia.org/wiki/Sierpi%C5%84ski_number, https://t5k.org/glossary/xpage/SierpinskiNumber.html, https://www.rieselprime.de/ziki/Sierpi%C5%84ski_problem, https://www.rieselprime.de/ziki/Proth_2_Sierpinski, https://mathworld.wolfram.com/SierpinskiNumberoftheSecondKind.html, https://en.wikipedia.org/wiki/Seventeen_or_Bust, https://www.rieselprime.de/ziki/Seventeen_or_Bust, https://web.archive.org/web/20190929190947/https://primes.utm.edu/glossary/xpage/ColbertNumber.html, https://mathworld.wolfram.com/ColbertNumber.html, http://www.numericana.com/answer/primes.htm#sierpinski, http://www.bitman.name/math/article/204 (in Italian), https://www.ams.org/journals/mcom/1983-40-161/S0025-5718-1983-0679453-8/S0025-5718-1983-0679453-8.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_40.pdf), https://www.fq.math.ca/Scanned/33-3/izotov.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_46.pdf), http://www.digizeitschriften.de/download/PPN378850199_0015/PPN378850199_0015___log24.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_213.pdf), https://www.ams.org/journals/mcom/1981-37-155/S0025-5718-1981-0616376-2/S0025-5718-1981-0616376-2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_214.pdf), https://www.ams.org/journals/mcom/1983-41-164/S0025-5718-1983-0717710-7/S0025-5718-1983-0717710-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_215.pdf), http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://oeis.org/A123159, https://oeis.org/A076336, https://oeis.org/A078680, https://oeis.org/A078683, https://oeis.org/A033809, https://oeis.org/A040076, https://oeis.org/A225721, https://oeis.org/A050921, https://oeis.org/A046067, https://oeis.org/A057025, https://oeis.org/A057192, https://oeis.org/A057247)
• 509203 is the smallest Riesel number (i.e. odd numbers k such that k×2ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base2-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base2.zip, http://www.prothsearch.com/rieselprob.html, http://www.primegrid.com/forum_thread.php?id=1731, http://www.primegrid.com/stats_trp_llr.php, https://web.archive.org/web/20061021153313/http://stats.rieselsieve.com//queue.php, https://en.wikipedia.org/wiki/Riesel_number, https://t5k.org/glossary/xpage/RieselNumber.html, https://www.rieselprime.de/ziki/Riesel_problem, https://www.rieselprime.de/ziki/Riesel_2_Riesel, https://mathworld.wolfram.com/RieselNumber.html, http://www.bitman.name/math/article/203 (in Italian), http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://oeis.org/A273987, https://oeis.org/A076337, https://oeis.org/A101036, https://oeis.org/A050412, https://oeis.org/A052333, https://oeis.org/A108129, https://oeis.org/A040081, https://oeis.org/A038699, https://oeis.org/A046069, https://oeis.org/A057026, https://oeis.org/A128979, https://oeis.org/A101050)
• 271129 is the second-smallest Sierpinski number (i.e. odd numbers k such that k×2ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base2.txt, 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.prothsearch.com/sierp.html, http://www.primegrid.com/forum_thread.php?id=1647, http://www.primegrid.com/forum_thread.php?id=972, http://www.primegrid.com/forum_thread.php?id=1750, http://www.primegrid.com/forum_thread.php?id=5758, http://www.primegrid.com/stats_sob_llr.php, http://www.primegrid.com/stats_psp_llr.php, http://www.primegrid.com/stats_esp_llr.php, https://web.archive.org/web/20160405211049/http://www.seventeenorbust.com/, https://en.wikipedia.org/wiki/Sierpi%C5%84ski_number, https://t5k.org/glossary/xpage/SierpinskiNumber.html, https://www.rieselprime.de/ziki/Sierpi%C5%84ski_problem, https://www.rieselprime.de/ziki/Proth_2_Sierpinski, https://mathworld.wolfram.com/SierpinskiNumberoftheSecondKind.html, https://en.wikipedia.org/wiki/Seventeen_or_Bust, https://www.rieselprime.de/ziki/Seventeen_or_Bust, https://web.archive.org/web/20190929190947/https://primes.utm.edu/glossary/xpage/ColbertNumber.html, https://mathworld.wolfram.com/ColbertNumber.html, http://www.numericana.com/answer/primes.htm#sierpinski, http://www.bitman.name/math/article/204 (in Italian), https://www.ams.org/journals/mcom/1983-40-161/S0025-5718-1983-0679453-8/S0025-5718-1983-0679453-8.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_40.pdf), https://www.fq.math.ca/Scanned/33-3/izotov.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_46.pdf), http://www.digizeitschriften.de/download/PPN378850199_0015/PPN378850199_0015___log24.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_213.pdf), https://www.ams.org/journals/mcom/1981-37-155/S0025-5718-1981-0616376-2/S0025-5718-1981-0616376-2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_214.pdf), https://www.ams.org/journals/mcom/1983-41-164/S0025-5718-1983-0717710-7/S0025-5718-1983-0717710-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_215.pdf), http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://oeis.org/A123159, https://oeis.org/A076336, https://oeis.org/A078680, https://oeis.org/A078683, https://oeis.org/A033809, https://oeis.org/A040076, https://oeis.org/A225721, https://oeis.org/A050921, https://oeis.org/A046067, https://oeis.org/A057025, https://oeis.org/A057192, https://oeis.org/A057247)
• 762701 is the second-smallest Riesel number (i.e. odd numbers k such that k×2ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base2-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base2.zip, http://www.prothsearch.com/rieselprob.html, http://www.primegrid.com/forum_thread.php?id=1731, http://www.primegrid.com/stats_trp_llr.php, https://web.archive.org/web/20061021153313/http://stats.rieselsieve.com//queue.php, https://en.wikipedia.org/wiki/Riesel_number, https://t5k.org/glossary/xpage/RieselNumber.html, https://www.rieselprime.de/ziki/Riesel_problem, https://www.rieselprime.de/ziki/Riesel_2_Riesel, https://mathworld.wolfram.com/RieselNumber.html, http://www.bitman.name/math/article/203 (in Italian), http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://oeis.org/A273987, https://oeis.org/A076337, https://oeis.org/A101036, https://oeis.org/A050412, https://oeis.org/A052333, https://oeis.org/A108129, https://oeis.org/A040081, https://oeis.org/A038699, https://oeis.org/A046069, https://oeis.org/A057026, https://oeis.org/A128979, https://oeis.org/A101050)
• 125050976086 is the smallest generalized Sierpinski number to base 3 (i.e. numbers k such that gcd(k+1, 3−1) = 1 and k×3ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base3-reserve.htm, http://www.noprimeleftbehind.net/crus/remain-sierp-base3.zip, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base3-10M.zip, http://www.noprimeleftbehind.net/crus/prime-sierp-base3-gt-25K.zip, 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), https://oeis.org/A123159, https://oeis.org/A291437, https://oeis.org/A291438)
• 63064644938 is the smallest generalized Riesel number to base 3 (i.e. numbers k such that gcd(k−1, 3−1) = 1 and k×3ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base3-reserve.htm, http://www.noprimeleftbehind.net/crus/remain-riesel-base3.zip, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base3-10M.zip, http://www.noprimeleftbehind.net/crus/prime-riesel-base3-gt-25K.zip, https://oeis.org/A273987)
• 66741 is the smallest generalized Sierpinski number to base 4 (i.e. numbers k such that gcd(k+1, 4−1) = 1 and k×4ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base4.txt, 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), https://oeis.org/A123159, https://oeis.org/A251057, https://oeis.org/A256002, http://www.rieselprime.de/Related/LiskovetsGallot.htm, http://www.primepuzzles.net/problems/prob_036.htm)
• 39939 is the smallest non-square generalized Riesel number to base 4 (i.e. numbers k such that gcd(k−1, 4−1) = 1 and k×4ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base4.txt, https://oeis.org/A273987, https://oeis.org/A251757, http://www.rieselprime.de/Related/LiskovetsGallot.htm, http://www.primepuzzles.net/problems/prob_036.htm)
• 159986 is the smallest generalized Sierpinski number to base 5 (i.e. numbers k such that gcd(k+1, 5−1) = 1 and k×5ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base5-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base5.txt, 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), https://oeis.org/A123159, http://www.primegrid.com/forum_thread.php?id=5087, http://www.primegrid.com/stats_sr5_llr.php, https://web.archive.org/web/20131016004333/http://www.sr5.psp-project.de/, https://web.archive.org/web/20111018190410/http://www.sr5.psp-project.de/s5stats.html, https://oeis.org/A345698)
• 346802 is the smallest generalized Riesel number to base 5 (i.e. numbers k such that gcd(k−1, 5−1) = 1 and k×5ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base5-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base5.zip, https://oeis.org/A273987, http://www.primegrid.com/forum_thread.php?id=5087, http://www.primegrid.com/stats_sr5_llr.php, https://web.archive.org/web/20131016004333/http://www.sr5.psp-project.de/, https://web.archive.org/web/20111018190339/http://www.sr5.psp-project.de/r5stats.html, https://oeis.org/A345403)
• 174308 is the smallest generalized Sierpinski number to base 6 (i.e. numbers k such that gcd(k+1, 6−1) = 1 and k×6ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base6.zip, 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), https://oeis.org/A123159, https://oeis.org/A244549, https://oeis.org/A250204)
• 84687 is the smallest generalized Riesel number to base 6 (i.e. numbers k such that gcd(k−1, 6−1) = 1 and k×6ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base6.txt, https://oeis.org/A273987, https://oeis.org/A244351, https://oeis.org/A250205)
• 1112646039348 is the smallest generalized Sierpinski number to base 7 (i.e. numbers k such that gcd(k+1, 7−1) = 1 and k×7ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base7-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base7-10M.zip, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base7-prime.htm, 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), https://oeis.org/A123159)
• 408034255082 is the smallest generalized Riesel number to base 7 (i.e. numbers k such that gcd(k−1, 7−1) = 1 and k×7ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base7-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base7-10M.zip, http://www.noprimeleftbehind.net/crus/prime-riesel-base7-gt-25K.txt, https://oeis.org/A273987)
• 47 is the smallest non-cube generalized Sierpinski number to base 8 (i.e. numbers k such that gcd(k+1, 8−1) = 1 and k×8ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base8.txt, 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), https://oeis.org/A123159) (solved, largest prime is 31×8²⁰+1)
• 14 is the smallest non-cube generalized Riesel number to base 8 (i.e. numbers k such that gcd(k−1, 8−1) = 1 and k×8ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base8.txt, https://oeis.org/A273987) (solved, largest prime is 11×8¹⁸−1)
• 2344 is the smallest generalized Sierpinski number to base 9 (i.e. numbers k such that gcd(k+1, 9−1) = 1 and k×9ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base9.txt, 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), https://oeis.org/A123159)
• 74 is the smallest non-square generalized Riesel number to base 9 (i.e. numbers k such that gcd(k−1, 9−1) = 1 and k×9ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base9.txt, https://oeis.org/A273987) (solved, largest prime is 24×9⁸−1)
• 9175 is the smallest generalized Sierpinski number to base 10 (i.e. numbers k such that gcd(k+1, 10−1) = 1 and k×10ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base10.txt, 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), https://oeis.org/A123159, https://oeis.org/A243969)
• 10176 is the smallest generalized Riesel number to base 10 (i.e. numbers k such that gcd(k−1, 10−1) = 1 and k×10ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base10.txt, https://oeis.org/A273987, https://oeis.org/A243974)
• 1490 is the smallest generalized Sierpinski number to base 11 (i.e. numbers k such that gcd(k+1, 11−1) = 1 and k×11ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base11.txt, 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), https://oeis.org/A123159) (solved, largest prime is 958×11³⁰⁰⁵⁴⁴+1)
• 862 is the smallest generalized Riesel number to base 11 (i.e. numbers k such that gcd(k−1, 11−1) = 1 and k×11ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base11.txt, https://oeis.org/A273987) (solved, largest prime is 62×11²⁶²⁰²−1)
• 521 is the smallest generalized Sierpinski number to base 12 (i.e. numbers k such that gcd(k+1, 12−1) = 1 and k×12ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base12.txt, 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), https://oeis.org/A123159)
• 376 is the smallest non-(m² with m == 5, 8 mod 13 or 3×m² with m == 3, 10 mod 13) generalized Riesel number to base 12 (i.e. numbers k such that gcd(k−1, 12−1) = 1 and k×12ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base12.txt, https://oeis.org/A273987) (solved, largest prime is 157×12²⁸⁵−1)
• 132 is the smallest generalized Sierpinski number to base 13 (i.e. numbers k such that gcd(k+1, 13−1) = 1 and k×13ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base13.txt, 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), https://oeis.org/A123159) (solved, largest prime is 48×13⁶²⁶⁷+1)
• 302 is the smallest generalized Riesel number to base 13 (i.e. numbers k such that gcd(k−1, 13−1) = 1 and k×13ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base13.txt, https://oeis.org/A273987) (solved, largest prime is 288×13¹⁰⁹²¹⁷−1)
• 4 is the smallest generalized Sierpinski number to base 14 (i.e. numbers k such that gcd(k+1, 14−1) = 1 and k×14ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base14.txt, 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), https://oeis.org/A123159) (solved, largest prime is 1×14²+1)
• 4 is the smallest generalized Riesel number to base 14 (i.e. numbers k such that gcd(k−1, 14−1) = 1 and k×14ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base14.txt, https://oeis.org/A273987) (solved, largest prime is 2×14⁴−1)
• 91218919470156 is the smallest generalized Sierpinski number to base 15 (i.e. numbers k such that gcd(k+1, 15−1) = 1 and k×15ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base15-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base15-10M.zip, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base15-prime.htm, 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), https://oeis.org/A123159)
• 36370321851498 is the smallest generalized Riesel number to base 15 (i.e. numbers k such that gcd(k−1, 15−1) = 1 and k×15ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base15-reserve.htm, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base15-10M.zip, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base15-prime.htm, https://oeis.org/A273987)
• 66741 is the smallest non-(4×m⁴) generalized Sierpinski number to base 16 (i.e. numbers k such that gcd(k+1, 16−1) = 1 and k×16ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base16.txt, 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), https://oeis.org/A123159)
• 33965 is the smallest non-square generalized Riesel number to base 16 (i.e. numbers k such that gcd(k−1, 16−1) = 1 and k×16ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base16.txt, https://oeis.org/A273987)
• 278 is the smallest generalized Sierpinski number to base 17 (i.e. numbers k such that gcd(k+1, 17−1) = 1 and k×17ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base17.txt, 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), https://oeis.org/A123159)
• 86 is the smallest generalized Riesel number to base 17 (i.e. numbers k such that gcd(k−1, 17−1) = 1 and k×17ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base17.txt, https://oeis.org/A273987) (solved, largest prime is 44×17⁶⁴⁸⁸−1)
• 398 is the smallest generalized Sierpinski number to base 18 (i.e. numbers k such that gcd(k+1, 18−1) = 1 and k×18ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base18.txt, 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), https://oeis.org/A123159)
• 246 is the smallest generalized Riesel number to base 18 (i.e. numbers k such that gcd(k−1, 18−1) = 1 and k×18ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base18.txt, https://oeis.org/A273987) (solved, largest prime is 151×18⁴¹⁸−1)
• 765174 is the smallest generalized Sierpinski number to base 19 (i.e. numbers k such that gcd(k+1, 19−1) = 1 and k×19ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base19.zip, 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), https://oeis.org/A123159)
• 1119866 is the smallest non-(m² with m == 2, 3 mod 5 or 19×m² with m == 2, 3 mod 13) generalized Riesel number to base 19 (i.e. numbers k such that gcd(k−1, 19−1) = 1 and k×19ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base19.zip, https://oeis.org/A273987)
• 8 is the smallest generalized Sierpinski number to base 20 (i.e. numbers k such that gcd(k+1, 20−1) = 1 and k×20ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base20.txt, 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), https://oeis.org/A123159) (solved, largest prime is 6×20¹⁵+1)
• 8 is the smallest generalized Riesel number to base 20 (i.e. numbers k such that gcd(k−1, 20−1) = 1 and k×20ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base20.txt, https://oeis.org/A273987) (solved, largest prime is 2×20¹⁰−1)
• 1002 is the smallest generalized Sierpinski number to base 21 (i.e. numbers k such that gcd(k+1, 21−1) = 1 and k×21ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base21.txt, 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), https://oeis.org/A123159) (solved, largest prime is 118×21¹⁹⁸⁴⁹+1)
• 560 is the smallest generalized Riesel number to base 21 (i.e. numbers k such that gcd(k−1, 21−1) = 1 and k×21ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base21.txt, https://oeis.org/A273987) (solved, largest prime is 64×21²⁸⁶⁷−1)
• 6694 is the smallest generalized Sierpinski number to base 22 (i.e. numbers k such that gcd(k+1, 22−1) = 1 and k×22ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base22.txt, 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), https://oeis.org/A123159)
• 4461 is the smallest generalized Riesel number to base 22 (i.e. numbers k such that gcd(k−1, 22−1) = 1 and k×22ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base22.txt, https://oeis.org/A273987)
• 182 is the smallest generalized Sierpinski number to base 23 (i.e. numbers k such that gcd(k+1, 23−1) = 1 and k×23ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base23.txt, 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), https://oeis.org/A123159) (solved, largest prime is 68×23³⁶⁵²³⁹+1)
• 476 is the smallest generalized Riesel number to base 23 (i.e. numbers k such that gcd(k−1, 23−1) = 1 and k×23ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base23.txt, https://oeis.org/A273987)
• 30651 is the smallest generalized Sierpinski number to base 24 (i.e. numbers k such that gcd(k+1, 24−1) = 1 and k×24ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base24.txt, 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), https://oeis.org/A123159)
• 32336 is the smallest non-(m² with m == 2, 3 mod 5 or 6×m² with m == 1, 4 mod 5) generalized Riesel number to base 24 (i.e. numbers k such that gcd(k−1, 24−1) = 1 and k×24ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base24.txt, https://oeis.org/A273987)
• 262638 is the smallest generalized Sierpinski number to base 25 (i.e. numbers k such that gcd(k+1, 25−1) = 1 and k×25ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base25.zip, 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), https://oeis.org/A123159)
• 346802 is the smallest non-square generalized Riesel number to base 25 (i.e. numbers k such that gcd(k−1, 25−1) = 1 and k×25ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base25.zip, https://oeis.org/A273987)
• 221 is the smallest generalized Sierpinski number to base 26 (i.e. numbers k such that gcd(k+1, 26−1) = 1 and k×26ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base26.txt, 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), https://oeis.org/A123159)
• 149 is the smallest generalized Riesel number to base 26 (i.e. numbers k such that gcd(k−1, 26−1) = 1 and k×26ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base26.txt, https://oeis.org/A273987) (solved, largest prime is 115×26⁵²⁰²⁷⁷−1)
• 538 is the smallest non-cube generalized Sierpinski number to base 27 (i.e. numbers k such that gcd(k+1, 27−1) = 1 and k×27ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base27.txt, 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), https://oeis.org/A123159)
• 804 is the smallest non-cube generalized Riesel number to base 27 (i.e. numbers k such that gcd(k−1, 27−1) = 1 and k×27ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base27.txt, https://oeis.org/A273987)
• 4554 is the smallest generalized Sierpinski number to base 28 (i.e. numbers k such that gcd(k+1, 28−1) = 1 and k×28ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base28.txt, 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), https://oeis.org/A123159)
• 9078 is the smallest generalized Riesel number to base 28 (i.e. numbers k such that gcd(k−1, 28−1) = 1 and k×28ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base28.txt, https://oeis.org/A273987)
• 4 is the smallest generalized Sierpinski number to base 29 (i.e. numbers k such that gcd(k+1, 29−1) = 1 and k×29ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base29.txt, 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), https://oeis.org/A123159) (solved, largest prime is 2×29¹+1)
• 4 is the smallest generalized Riesel number to base 29 (i.e. numbers k such that gcd(k−1, 29−1) = 1 and k×29ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base29.txt, https://oeis.org/A273987) (solved, largest prime is 2×29¹³⁶−1)
• 867 is the smallest generalized Sierpinski number to base 30 (i.e. numbers k such that gcd(k+1, 30−1) = 1 and k×30ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base30.txt, 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), https://oeis.org/A123159)
• 4928 is the smallest generalized Riesel number to base 30 (i.e. numbers k such that gcd(k−1, 30−1) = 1 and k×30ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base30.txt, https://oeis.org/A273987) other than 1369
• 6360528 is the smallest generalized Sierpinski number to base 31 (i.e. numbers k such that gcd(k+1, 31−1) = 1 and k×31ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base31.zip, 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), https://oeis.org/A123159)
• 134718 is the smallest generalized Riesel number to base 31 (i.e. numbers k such that gcd(k−1, 31−1) = 1 and k×31ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base31.txt, https://oeis.org/A273987)
• 10 is the smallest non-5th-power generalized Sierpinski number to base 32 (i.e. numbers k such that gcd(k+1, 32−1) = 1 and k×32ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base32.txt, 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), https://oeis.org/A123159)
• 10 is the smallest non-5th-power generalized Riesel number to base 32 (i.e. numbers k such that gcd(k−1, 32−1) = 1 and k×32ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base32.txt, https://oeis.org/A273987) (solved, largest prime is 3×32¹¹−1)
• 1854 is the smallest generalized Sierpinski number to base 33 (i.e. numbers k such that gcd(k+1, 33−1) = 1 and k×33ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base33.txt, 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), https://oeis.org/A123159) (solved, largest prime is 766×33⁶¹⁰⁴¹²+1)
• 764 is the smallest non-(m² with m == 4, 13 mod 17 or 33×m² with m == 4, 13 mod 17) generalized Riesel number to base 33 (i.e. numbers k such that gcd(k−1, 33−1) = 1 and k×33ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base33.txt, https://oeis.org/A273987) (solved, largest prime is 732×33¹⁹⁰¹¹−1)
• 6 is the smallest generalized Sierpinski number to base 34 (i.e. numbers k such that gcd(k+1, 34−1) = 1 and k×34ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base34.txt, 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), https://oeis.org/A123159) (solved, largest prime is 1×34⁴+1)
• 6 is the smallest non-(m² with m == 2, 3 mod 5 or 34×m² with m == 2, 3 mod 5) generalized Riesel number to base 34 (i.e. numbers k such that gcd(k−1, 34−1) = 1 and k×34ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base34.txt, https://oeis.org/A273987) (solved, largest prime is 5×34²−1)
• 214018 is the smallest generalized Sierpinski number to base 35 (i.e. numbers k such that gcd(k+1, 35−1) = 1 and k×35ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base35.txt, 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), https://oeis.org/A123159)
• 287860 is the smallest generalized Riesel number to base 35 (i.e. numbers k such that gcd(k−1, 35−1) = 1 and k×35ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base35.zip, https://oeis.org/A273987)
• 1886 is the smallest generalized Sierpinski number to base 36 (i.e. numbers k such that gcd(k+1, 36−1) = 1 and k×36ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/all-ks-sierp-base36.txt, 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), https://oeis.org/A123159)
• 116364 is the smallest non-square generalized Riesel number to base 36 (i.e. numbers k such that gcd(k−1, 36−1) = 1 and k×36ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.noprimeleftbehind.net/crus/all-ks-riesel-base36.txt, https://oeis.org/A273987)
• 78557 is the smallest dual Sierpinski number (i.e. odd numbers k such that 2ⁿ+k is composite for all n ≥ 1) (http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_1.pdf), https://www.rechenkraft.net/wiki/Five_or_Bust, https://oeis.org/A076336/a076336c.html, http://www.mit.edu/~kenta/three/prime/dual-sierpinski/ezgxggdm/dualsierp-excerpt.txt, http://mit.edu/kenta/www/three/prime/dual-sierpinski/ezgxggdm/dualsierp.txt.gz, https://mersenneforum.org/showthread.php?t=10761, https://oeis.org/A067760, https://oeis.org/A123252, https://oeis.org/A094076, https://oeis.org/A139758) (solved if we allow probable primes, largest (probable) prime is 2^9092392+40291)
• 509203 is the smallest dual Riesel number (i.e. odd numbers k such that 2ⁿ−k is composite for all n ≥ 1 such that 2ⁿ > k) (https://mersenneforum.org/showthread.php?t=6545, https://oeis.org/A096502, https://oeis.org/A096822, https://oeis.org/A101462)
• 271129 is the second-smallest dual Sierpinski number (i.e. odd numbers k such that 2ⁿ+k is composite for all n ≥ 1) (http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_1.pdf), https://www.rechenkraft.net/wiki/Five_or_Bust, https://oeis.org/A076336/a076336c.html, http://www.mit.edu/~kenta/three/prime/dual-sierpinski/ezgxggdm/dualsierp-excerpt.txt, http://mit.edu/kenta/www/three/prime/dual-sierpinski/ezgxggdm/dualsierp.txt.gz, https://mersenneforum.org/showthread.php?t=10761, https://oeis.org/A067760, https://oeis.org/A123252, https://oeis.org/A094076, https://oeis.org/A139758)
• 762701 is the second-smallest dual Riesel number (i.e. odd numbers k such that 2ⁿ−k is composite for all n ≥ 1 such that 2ⁿ > k) (https://mersenneforum.org/showthread.php?t=6545, https://oeis.org/A096502, https://oeis.org/A096822, https://oeis.org/A101462)
• 125050976086 is the smallest generalized dual Sierpinski number to base 3 (i.e. numbers k such that gcd(k, 3) = 1 and gcd(k+1, 3−1) = 1 and 3ⁿ+k is composite for all n ≥ 1)
• 63064644938 is the smallest generalized dual Riesel number to base 3 (i.e. numbers k such that gcd(k, 3) = 1 and gcd(k−1, 3−1) = 1 and 3ⁿ−k is composite for all n ≥ 1 such that 3ⁿ > k)
• 159986 is the smallest generalized dual Sierpinski number to base 5 (i.e. numbers k such that gcd(k, 5) = 1 and gcd(k+1, 5−1) = 1 and 5ⁿ+k is composite for all n ≥ 1)
• 346802 is the smallest generalized dual Riesel number to base 5 (i.e. numbers k such that gcd(k, 5) = 1 and gcd(k−1, 5−1) = 1 and 5ⁿ−k is composite for all n ≥ 1 such that 5ⁿ > k)
• 1112646039348 is the smallest generalized dual Sierpinski number to base 7 (i.e. numbers k such that gcd(k, 7) = 1 and gcd(k+1, 7−1) = 1 and 7ⁿ+k is composite for all n ≥ 1)
• 408034255082 is the smallest generalized dual Riesel number to base 7 (i.e. numbers k such that gcd(k, 7) = 1 and gcd(k−1, 7−1) = 1 and 7ⁿ−k is composite for all n ≥ 1 such that 7ⁿ > k)
• 1490 is the smallest generalized dual Sierpinski number to base 11 (i.e. numbers k such that gcd(k, 11) = 1 and gcd(k+1, 11−1) = 1 and 11ⁿ+k is composite for all n ≥ 1)
• 862 is the smallest generalized dual Riesel number to base 11 (i.e. numbers k such that gcd(k, 11) = 1 and gcd(k−1, 11−1) = 1 and 11ⁿ−k is composite for all n ≥ 1 such that 11ⁿ > k)
• 132 is the smallest generalized dual Sierpinski number to base 13 (i.e. numbers k such that gcd(k, 13) = 1 and gcd(k+1, 13−1) = 1 and 13ⁿ+k is composite for all n ≥ 1) (solved, largest prime is 13⁴¹⁶+120)
• 302 is the smallest generalized dual Riesel number to base 13 (i.e. numbers k such that gcd(k, 13) = 1 and gcd(k−1, 13−1) = 1 and 13ⁿ−k is composite for all n ≥ 1 such that 13ⁿ > k)
• 278 is the smallest generalized dual Sierpinski number to base 17 (i.e. numbers k such that gcd(k, 17) = 1 and gcd(k+1, 17−1) = 1 and 17ⁿ+k is composite for all n ≥ 1)
• 86 is the smallest generalized dual Riesel number to base 17 (i.e. numbers k such that gcd(k, 17) = 1 and gcd(k−1, 17−1) = 1 and 17ⁿ−k is composite for all n ≥ 1 such that 17ⁿ > k) (solved, largest prime is 17¹⁸−80)
• 765174 is the smallest generalized dual Sierpinski number to base 19 (i.e. numbers k such that gcd(k, 19) = 1 and gcd(k+1, 19−1) = 1 and 19ⁿ+k is composite for all n ≥ 1)
• 1119866 is the smallest non-(m² with m == 2, 3 mod 5 or 19×m² with m == 2, 3 mod 13) generalized dual Riesel number to base 19 (i.e. numbers k such that gcd(k, 19) = 1 and gcd(k−1, 19−1) = 1 and 19ⁿ−k is composite for all n ≥ 1 such that 19ⁿ > k)
• 182 is the smallest generalized dual Sierpinski number to base 23 (i.e. numbers k such that gcd(k, 23) = 1 and gcd(k+1, 23−1) = 1 and 23ⁿ+k is composite for all n ≥ 1) (solved, largest prime is 23¹⁹²⁶+82)
• 476 is the smallest generalized dual Riesel number to base 23 (i.e. numbers k such that gcd(k, 23) = 1 and gcd(k−1, 23−1) = 1 and 23ⁿ−k is composite for all n ≥ 1 such that 23ⁿ > k)
• 4 is the smallest generalized dual Sierpinski number to base 29 (i.e. numbers k such that gcd(k, 29) = 1 and gcd(k+1, 29−1) = 1 and 29ⁿ+k is composite for all n ≥ 1) (solved, largest prime is 29¹+2)
• 4 is the smallest generalized dual Riesel number to base 29 (i.e. numbers k such that gcd(k, 29) = 1 and gcd(k−1, 29−1) = 1 and 29ⁿ−k is composite for all n ≥ 1 such that 29ⁿ > k) (solved, largest prime is 29²−2)
• 6360528 is the smallest generalized dual Sierpinski number to base 31 (i.e. numbers k such that gcd(k, 31) = 1 and gcd(k+1, 31−1) = 1 and 31ⁿ+k is composite for all n ≥ 1)
• 134718 is the smallest generalized dual Riesel number to base 31 (i.e. numbers k such that gcd(k, 31) = 1 and gcd(k−1, 31−1) = 1 and 31ⁿ−k is composite for all n ≥ 1 such that 31ⁿ > k)
• 201446503145165177 is the smallest reverse Sierpinski base to k = 2 (i.e. bases b such that gcd(2+1, b−1) = 1 and 2×bⁿ+1 is composite for all n ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://mersenneforum.org/showthread.php?t=6918, https://mersenneforum.org/showthread.php?t=19725, https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354, https://oeis.org/A119624, https://oeis.org/A253178, https://oeis.org/A098872)
• There are no reverse Riesel bases to k = 2 (i.e. bases b such that gcd(2−1, b−1) = 1 and 2×bⁿ−1 is composite for all n ≥ 1) (https://mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217, https://oeis.org/A119591, https://oeis.org/A098873, https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
• There are no reverse Sierpinski bases to k = 3 (i.e. bases b such that gcd(3+1, b−1) = 1 and 3×bⁿ+1 is composite for all n ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354, https://oeis.org/A098877)
• There are no reverse Riesel bases to k = 3 (i.e. bases b such that gcd(3−1, b−1) = 1 and 3×bⁿ−1 is composite for all n ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354, https://oeis.org/A098876)
• 14 is the smallest reverse Sierpinski base to k = 4 (i.e. bases b such that gcd(4+1, b−1) = 1 and 4×bⁿ+1 is composite for all n ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) (solved, largest prime is 4×12²+1)
• 14 is the smallest non-square reverse Riesel base to k = 4 (i.e. bases b such that gcd(4−1, b−1) = 1 and 4×bⁿ−1 is composite for all n ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) (solved, largest prime is 4×12¹−1)
• 140324348 is the smallest reverse Sierpinski base to k = 5 (i.e. bases b such that gcd(5+1, b−1) = 1 and 5×bⁿ+1 is composite for all n ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
• There are no reverse Riesel bases to k = 5 (i.e. bases b such that gcd(5−1, b−1) = 1 and 5×bⁿ−1 is composite for all n ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
• 34 is the smallest reverse Sierpinski base to k = 6 (i.e. bases b such that gcd(6+1, b−1) = 1 and 6×bⁿ+1 is composite for all n ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) (solved, largest prime is 6×20¹⁵+1)
• 34 is the smallest non-(6×m² with m == 2, 3 mod 5) reverse Riesel base to k = 6 (i.e. bases b such that gcd(6−1, b−1) = 1 and 6×bⁿ−1 is composite for all n ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) (solved, largest prime is 6×27²−1)
• There are no reverse Sierpinski bases to k = 7 (i.e. bases b such that gcd(7+1, b−1) = 1 and 7×bⁿ+1 is composite for all n ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
• 9162668342 is the smallest reverse Riesel base to k = 7 (i.e. bases b such that gcd(7−1, b−1) = 1 and 7×bⁿ−1 is composite for all n ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
• 20 is the smallest non-cube reverse Sierpinski base to k = 8 (i.e. bases b such that gcd(8+1, b−1) = 1 and 8×bⁿ+1 is composite for all n ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) (solved, largest prime is 8×6⁴+1)
• 20 is the smallest non-cube reverse Riesel base to k = 8 (i.e. bases b such that gcd(8−1, b−1) = 1 and 8×bⁿ−1 is composite for all n ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) (solved, largest prime is 8×7⁴−1)
• 177744 is the smallest reverse Sierpinski base to k = 9 (i.e. bases b such that gcd(9+1, b−1) = 1 and 9×bⁿ+1 is composite for all n ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
• There are no reverse Riesel bases to k = 9 (i.e. bases b such that gcd(9−1, b−1) = 1 and 9×bⁿ−1 is composite for all n ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) except the even square bases b and the bases b == 4 mod 10
• 32 is the smallest reverse Sierpinski base to k = 10 (i.e. bases b such that gcd(10+1, b−1) = 1 and 10×bⁿ+1 is composite for all n ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) (solved, largest prime is 10×17¹³⁵⁶+1)
• 32 is the smallest reverse Riesel base to k = 10 (i.e. bases b such that gcd(10−1, b−1) = 1 and 10×bⁿ−1 is composite for all n ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) (solved, largest prime is 10×17¹¹⁷−1)
• 14 is the smallest reverse Sierpinski base to k = 11 (i.e. bases b such that gcd(11+1, b−1) = 1 and 11×bⁿ+1 is composite for all n ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) (solved, largest prime is 11×12³+1)
• 14 is the smallest reverse Riesel base to k = 11 (i.e. bases b such that gcd(11−1, b−1) = 1 and 11×bⁿ−1 is composite for all n ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) (solved, largest prime is 11×8¹⁸−1)
• 142 is the smallest reverse Sierpinski base to k = 12 (i.e. bases b such that gcd(12+1, b−1) = 1 and 12×bⁿ+1 is composite for all n ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
• 142 is the smallest reverse Riesel base to k = 12 (i.e. bases b such that gcd(12−1, b−1) = 1 and 12×bⁿ−1 is composite for all n ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) (solved, largest prime is 12×98³⁵⁹⁹−1)
• 20 is the smallest reverse Sierpinski base to k = 13 (i.e. bases b such that gcd(13+1, b−1) = 1 and 13×bⁿ+1 is composite for all n ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) (solved, largest prime is 13×18¹⁰+1)
• 20 is the smallest reverse Riesel base to k = 13 (i.e. bases b such that gcd(13−1, b−1) = 1 and 13×bⁿ−1 is composite for all n ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) (solved, largest prime is 13×12²−1)
• 38 is the smallest reverse Sierpinski base to k = 14 (i.e. bases b such that gcd(14+1, b−1) = 1 and 14×bⁿ+1 is composite for all n ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) (solved, largest prime is 14×23⁵+1)
• 8 is the smallest reverse Riesel base to k = 14 (i.e. bases b such that gcd(14−1, b−1) = 1 and 14×bⁿ−1 is composite for all n ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) (solved, largest prime is 14×5²−1)
• There are no reverse Sierpinski bases to k = 15 (i.e. bases b such that gcd(15+1, b−1) = 1 and 15×bⁿ+1 is composite for all n ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
• 8241218 is the smallest reverse Riesel base to k = 15 (i.e. bases b such that gcd(15−1, b−1) = 1 and 15×bⁿ−1 is composite for all n ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
• 38 is the smallest reverse Sierpinski base to k = 16 (i.e. bases b such that gcd(16+1, b−1) = 1 and 16×bⁿ+1 is composite for all n ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
• 50 is the smallest non-square reverse Riesel base to k = 16 (i.e. bases b such that gcd(16−1, b−1) = 1 and 16×bⁿ−1 is composite for all n ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) other than 33 (solved, largest prime is 16×39³⁵−1)
• 201446503145165177 is the smallest dual reverse Sierpinski base to k = 2 (i.e. bases b such that gcd(2, b) = 1 and gcd(2+1, b−1) = 1 and bⁿ+2 is composite for all n ≥ 1) (https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067)
• There are no dual reverse Riesel bases to k = 2 (i.e. bases b such that gcd(2, b) = 1 and gcd(2−1, b−1) = 1 and bⁿ−2 is composite for all n ≥ 1 such that bⁿ > 2) (https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A255707, https://oeis.org/A084714, https://oeis.org/A250200, https://oeis.org/A292201)
• There are no dual reverse Sierpinski bases to k = 3 (i.e. bases b such that gcd(3, b) = 1 and gcd(3+1, b−1) = 1 and bⁿ+3 is composite for all n ≥ 1)
• There are no dual reverse Riesel bases to k = 3 (i.e. bases b such that gcd(3, b) = 1 and gcd(3−1, b−1) = 1 and bⁿ−3 is composite for all n ≥ 1 such that bⁿ > 3)
• 140324348 is the smallest dual reverse Sierpinski base to k = 5 (i.e. bases b such that gcd(5, b) = 1 and gcd(5+1, b−1) = 1 and bⁿ+5 is composite for all n ≥ 1)
• There are no dual reverse Riesel bases to k = 5 (i.e. bases b such that gcd(5, b) = 1 and gcd(5−1, b−1) = 1 and bⁿ−5 is composite for all n ≥ 1 such that bⁿ > 5)
• There are no dual reverse Sierpinski bases to k = 7 (i.e. bases b such that gcd(7, b) = 1 and gcd(7+1, b−1) = 1 and bⁿ+7 is composite for all n ≥ 1)
• 9162668342 is the smallest dual reverse Riesel base to k = 7 (i.e. bases b such that gcd(7, b) = 1 and gcd(7−1, b−1) = 1 and bⁿ−7 is composite for all n ≥ 1 such that bⁿ > 7)
• 14 is the smallest dual reverse Sierpinski base to k = 11 (i.e. bases b such that gcd(11, b) = 1 and gcd(11+1, b−1) = 1 and bⁿ+11 is composite for all n ≥ 1) (solved, largest prime is 12¹+11)
• 74 is the smallest dual reverse Riesel base to k = 11 (i.e. bases b such that gcd(11, b) = 1 and gcd(11−1, b−1) = 1 and bⁿ−11 is composite for all n ≥ 1 such that bⁿ > 11) (solved, largest prime is 68⁶−11) (note that for b = 14, the only one prime of the form 14ⁿ−11 with n ≥ 1 is 14¹−11 = 3)
• 20 is the smallest dual reverse Sierpinski base to k = 13 (i.e. bases b such that gcd(13, b) = 1 and gcd(13+1, b−1) = 1 and bⁿ+13 is composite for all n ≥ 1) (solved, largest prime is 14¹⁶+13)
• 38 is the smallest dual reverse Riesel base to k = 13 (i.e. bases b such that gcd(13, b) = 1 and gcd(13−1, b−1) = 1 and bⁿ−13 is composite for all n ≥ 1 such that bⁿ > 13) (solved, largest prime is 14³−13) (note that for b = 20, the only one prime of the form 20ⁿ−13 with n ≥ 1 is 20¹−13 = 7)

We call families of the form x{y}z (where x and z are strings (may be empty) of digits in base b, y is a digit in base b) "linear" families. Our algorithm then proceeds as follows:

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

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

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

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

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

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

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

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

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

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

etc.

The process of exploring/examining/splitting a family can be concisely expressed in a tree of decompositions.

We should first make data up to linear families (i.e. only linear families left) (see https://github.com/curtisbright/mepn-data/commit/7acfa0656d3c6b759f95a031f475a30f7664a122 for the original minimal prime problem in bases 2 ≤ b ≤ 26), then searching each left linear family to certain limit of length (say length 1000) (just like the new base script for Conjectures ‘R Us Sierpinski/Riesel conjectures base b (http://www.noprimeleftbehind.net/crus/, http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-stats.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-top20.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-proven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://t5k.org/bios/page.php?id=1372, https://srbase.my-firewall.org/sr5/, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian)), see http://www.noprimeleftbehind.net/crus/new-bases-5.1.txt and https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/CRUS_pack/scripts/new-bases-5.1.txt, also see https://github.com/curtisbright/mepn-data/commit/e6b2b806f341e9dc5b96662edba2caf3220c98b7 for the original minimal prime problem in bases 2 ≤ b ≤ 28), then find the smallest prime in each left linear family (use SRSIEVE (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, http://www.rieselprime.de/dl/CRUS_pack.zip, https://t5k.org/bios/page.php?id=905, https://www.rieselprime.de/ziki/Srsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srsieve_1.1.4, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr1sieve_1.4.6, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/sr2sieve_2.0.0, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/srbsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/mtsieve_2.4.8/srsieve2cl.exe) to sieve, then use LLR (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64) or PFGW (https://sourceforge.net/projects/openpfgw/, https://t5k.org/bios/page.php?id=175, https://www.rieselprime.de/ziki/PFGW, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/pfgw_win_4.0.3) to test the probable-primality of the remain numbers, then use PRIMO (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309) to prove the primality of the probable primes < 10²⁵⁰⁰⁰).

Some x{y}z (where x and z are strings (may be empty) of digits in base b, y is a digit in base b) families can be proven to contain no primes > b, by covering congruence (http://irvinemclean.com/maths/siercvr.htm, http://web.archive.org/web/20060925100410/http://www.glasgowg43.freeserve.co.uk/siernums.htm, https://web.archive.org/web/20061116164533/http://www.glasgowg43.freeserve.co.uk/brier2.htm, https://web.archive.org/web/20221230035324/https://sites.google.com/site/robertgerbicz/coveringsets, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/coveringsets, http://www.numericana.com/answer/primes.htm#sierpinski, http://irvinemclean.com/maths/sierpin.htm, http://irvinemclean.com/maths/sierpin2.htm, http://irvinemclean.com/maths/sierpin3.htm, http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://www.ams.org/journals/mcom/1975-29-129/S0025-5718-1975-0376583-0/S0025-5718-1975-0376583-0.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_27.pdf), https://www.ams.org/journals/mcom/1983-40-161/S0025-5718-1983-0679453-8/S0025-5718-1983-0679453-8.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_40.pdf), http://yves.gallot.pagesperso-orange.fr/papers/smallbrier.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_48.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://arxiv.org/pdf/2209.10646.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_52.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL14/Jones/jones12.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_80.pdf), https://web.archive.org/web/20081119135435/http://math.crg4.com/a094076.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_102.pdf), http://www.renyi.hu/~p_erdos/1950-07.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_103.pdf), http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_1.pdf), https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), http://www.primepuzzles.net/puzzles/puzz_614.htm, http://www.primepuzzles.net/problems/prob_029.htm, http://www.primepuzzles.net/problems/prob_030.htm, http://www.primepuzzles.net/problems/prob_036.htm, http://www.primepuzzles.net/problems/prob_049.htm, https://www.rieselprime.de/Related/LiskovetsGallot.htm, https://www.rieselprime.de/Related/RieselTwinSG.htm, https://stdkmd.net/nrr/coveringset.htm, https://stdkmd.net/nrr/9/91113.htm#prime_period, https://stdkmd.net/nrr/9/94449.htm#prime_period, https://stdkmd.net/nrr/9/95559.htm#prime_period, https://web.archive.org/web/20070220134129/http://www.users.globalnet.co.uk/~perry/maths/wildeprimes/wildeprimes.htm, https://www.rose-hulman.edu/~rickert/Compositeseq/, https://oeis.org/A244561, https://oeis.org/A244562, https://oeis.org/A244563, https://oeis.org/A244564, https://oeis.org/A244070, https://oeis.org/A244071, https://oeis.org/A244072, https://oeis.org/A244073, https://oeis.org/A257647, https://oeis.org/A258154, https://oeis.org/A289110, https://oeis.org/A257861, https://oeis.org/A306151, https://oeis.org/A305473, https://en.wikipedia.org/wiki/Covering_set, https://www.rieselprime.de/ziki/Covering_set, https://mathworld.wolfram.com/SierpinskisCompositeNumberTheorem.html) (i.e. finding a finite set (https://en.wikipedia.org/wiki/Finite_set, https://mathworld.wolfram.com/FiniteSet.html) S of primes p such that all numbers in a given family are divisible (https://en.wikipedia.org/wiki/Divides, https://t5k.org/glossary/xpage/Divides.html, https://t5k.org/glossary/xpage/Divisor.html, https://www.rieselprime.de/ziki/Factor, https://mathworld.wolfram.com/Divides.html, https://mathworld.wolfram.com/Divisor.html, https://mathworld.wolfram.com/Divisible.html, http://www.numericana.com/answer/primes.htm#divisor) by some element of S (this is equivalent to finding a positive integer N such that all numbers in a given family are not coprime (https://en.wikipedia.org/wiki/Coprime_integers, https://t5k.org/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime) to N, this N is usually a factor of a small generalized repunit number (https://en.wikipedia.org/wiki/Repunit, https://t5k.org/glossary/xpage/Repunit.html, https://t5k.org/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://pzktupel.de/Primetables/TableRepunit.php, https://pzktupel.de/Primetables/TableRepunitGen.php, https://pzktupel.de/Primetables/TableRepunitGen.txt, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html, https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/, https://web.archive.org/web/20120227163614/http://phi.redgolpe.com/5.asp, https://web.archive.org/web/20120227163508/http://phi.redgolpe.com/4.asp, https://web.archive.org/web/20120227163610/http://phi.redgolpe.com/3.asp, https://web.archive.org/web/20120227163512/http://phi.redgolpe.com/2.asp, https://web.archive.org/web/20120227163521/http://phi.redgolpe.com/1.asp, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.fermatquotient.com/PrimSerien/GenRepuP.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/article/380/231/, http://www.bitman.name/math/table/379, http://www.bitman.name/math/table/488, https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_5.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf), https://t5k.org/top20/page.php?id=57, https://t5k.org/top20/page.php?id=16, https://oeis.org/A002275, https://oeis.org/A004022, https://oeis.org/A053696, https://oeis.org/A085104, https://oeis.org/A179625) in base b, e.g. all numbers in the family 2{5} in base 11 are not coprime to 6, gcd((5×11ⁿ−1)/2, 6) can only be 2 or 3, and cannot be 1, also equivalent to finding a prime p such that all numbers in a given family are not p-rough numbers (https://en.wikipedia.org/wiki/Rough_number, https://mathworld.wolfram.com/RoughNumber.html, https://oeis.org/A007310, https://oeis.org/A007775, https://oeis.org/A008364, https://oeis.org/A008365, https://oeis.org/A008366, https://oeis.org/A166061, https://oeis.org/A166063)), by modular arithmetic (https://en.wikipedia.org/wiki/Modular_arithmetic, https://en.wikipedia.org/wiki/Congruence_relation, https://en.wikipedia.org/wiki/Modulo, https://t5k.org/glossary/xpage/Congruence.html, https://t5k.org/glossary/xpage/CongruenceClass.html, https://t5k.org/glossary/xpage/Residue.html, https://mathworld.wolfram.com/Congruence.html, https://mathworld.wolfram.com/Congruent.html, https://mathworld.wolfram.com/Residue.html, https://mathworld.wolfram.com/MinimalResidue.html, https://mathworld.wolfram.com/Mod.html)), algebraic factorization (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://mersenneforum.org/showpost.php?p=196598&postcount=492, https://mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php (e.g. for the family 3{8} in base 9, type "4*9^n-1", and it will tell you that this form can be factored to (2×3ⁿ−1) × (2×3ⁿ+1)), https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) (which includes difference-of-two-squares factorization (https://en.wikipedia.org/wiki/Difference_of_two_squares) and sum/difference-of-two-cubes factorization (https://en.wikipedia.org/wiki/Sum_of_two_cubes) and difference-of-two-nth-powers factorization with n > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and sum/difference-of-two-nth-powers factorization with odd n > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and Aurifeuillean factorization (https://en.wikipedia.org/wiki/Aurifeuillean_factorization, https://www.rieselprime.de/ziki/Aurifeuillian_factor, https://mathworld.wolfram.com/AurifeuilleanFactorization.html, http://www.numericana.com/answer/numbers.htm#aurifeuille, http://pagesperso-orange.fr/colin.barker/lpa/cycl_fac.htm, http://list.seqfan.eu/oldermail/seqfan/2017-March/017363.html, http://myfactorcollection.mooo.com:8090/source/cyclo.cpp, http://myfactorcollection.mooo.com:8090/LCD_2_199, http://myfactorcollection.mooo.com:8090/LCD_2_998, https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean, https://www.unshlump.com/hcn/aurif.html, https://www.ams.org/journals/mcom/2006-75-253/S0025-5718-05-01766-7/S0025-5718-05-01766-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_138.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb127.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_164.pdf), https://www.jams.jp/scm/contents/Vol-2-3/2-3-16.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_167.pdf)) of x⁴+4×y⁴ or x⁶+27×y⁶), or combine of them (https://mersenneforum.org/showthread.php?t=11143, https://mersenneforum.org/showthread.php?t=10279, https://www.fq.math.ca/Scanned/33-3/izotov.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_46.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://oeis.org/A213353, https://oeis.org/A233469), for me, there is nothing worse then searching a family for a long time that later is proven to contain no primes (e.g. we searched the base 16 families {C}D and {C}DD to length 10000 and searched the base 21 family B0{H}6H to length 20000 and searched the base 30 family A{0}9J to length 10000 in the past, and later we proved that the base 16 families {C}D and {C}DD both have Aurifeuillean factorization of x⁴+4×y⁴ and the base 21 family B0{H}6H has a single trivial factor 4637 and the base 30 family A{0}9J has a covering set of {7, 13, 19, 31} with period 6, this is because the original program cannot remove the families which can be ruled out as only containing composites by these conditions: "Aurifeuillean factorization of x⁴+4×y⁴" and "single trivial prime factor > b²" and "covering congruence with period > 4", now the program can remove the families which can be ruled out as only containing composites by these conditions: "single trivial prime factor < b³" and "covering congruence with period ≤ 24" and "difference of r-th powers with r ≤ 5" and "Aurifeuillean factorization of x⁴+4×y⁴").