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 delete the digit 4, to get the prime 19 * Write down the prime 439 → I can delete the digit 9, to get the prime 43 * Write down the prime 857 → I can delete zero digits, to get the prime 857 * Write down the prime 2081 → I can delete the digit 0, to get the prime 281 * Write down the largest known double Mersenne prime 170141183460469231731687303715884105727 (2¹²⁷−1) → I can delete all digits except the third-leftmost 1 and the second-rightmost 3, to get the prime 13 * Write down the largest known Fermat prime 65537 → I can delete the 6 and the 3, to get the prime 557 (also I can choose to delete the 6 and two 5's, to get the prime 37) (also I can choose to delete two 5's and the 3, to get the prime 67) (also I can choose to delete 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 delete 17 1's, to get the prime 11 * Write down the prime 1000000000000000000000000000000000000000000000000000000000007 (which is the next prime after 10⁶⁰) → I can delete all 0's, to get the prime 17 * Write down the prime 95801 → I can delete the 9, to get the prime 5801 * Write down the prime 946969 → I can delete the first 9 and two 6's, to get the prime 499 * Write down the prime 90000000581 → I can delete five 0's, the 5, and the 8, to get the prime 9001 * Write down the prime 8555555555555555555551 → I can delete the 8 and nine 5's, to get the prime 555555555551 Now we extend this prime game to bases 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://primes.utm.edu/glossary/xpage/Prime.html, https://www.rieselprime.de/ziki/Prime, https://mathworld.wolfram.com/PrimeNumber.html) 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://primes.utm.edu/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), 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, ..., 9 and A, B, ..., Z, 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://baseconvert.com/, https://www.calculand.com/unit-converter/zahlen.php?og=Base+2-36&ug=1, also see https://primes.utm.edu/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. By the theorem that there are no infinite (https://en.wikipedia.org/wiki/Infinite_set, https://primes.utm.edu/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 ordering (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, https://primes.utm.edu/notes/proofs/infinite/, https://primes.utm.edu/notes/proofs/infinite/euclids.html, https://primes.utm.edu/notes/proofs/infinite/topproof.html, https://primes.utm.edu/notes/proofs/infinite/goldbach.html, https://primes.utm.edu/notes/proofs/infinite/kummers.html, https://primes.utm.edu/notes/proofs/infinite/Saidak.html), there must be only finitely such minimal elements in every base *b*. 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://primes.utm.edu/glossary/xpage/FundamentalTheorem.html, https://mathworld.wolfram.com/FundamentalTheoremofArithmetic.html): 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) 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://primes.utm.edu/glossary/xpage/Divides.html, https://mathworld.wolfram.com/Divides.html) *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))). 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 article is very important and beautiful. |subsequence ordering|divisibility ordering| |---|---| |the "prime numbers > *b*" digit strings in base *b*|the integers > 1| |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 (which is exactly the target of this project)|the set of the prime numbers| |no common subsequence with length > 1|coprime (no common divisor > 1) (https://en.wikipedia.org/wiki/Coprime_integers, https://primes.utm.edu/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html)| |proper subsequence with length > 1|proper factor (https://en.wikipedia.org/wiki/Proper_factor, https://mathworld.wolfram.com/ProperFactor.html, https://mathworld.wolfram.com/ProperDivisor.html) > 1| |longest common subsequence (https://en.wikipedia.org/wiki/Longest_common_subsequence_problem)|greatest common divisor (https://en.wikipedia.org/wiki/Greatest_common_divisor, https://mathworld.wolfram.com/GreatestCommonDivisor.html)| |shortest common supersequence (https://en.wikipedia.org/wiki/Shortest_common_supersequence_problem)|least common multiple (https://en.wikipedia.org/wiki/Least_common_multiple, https://mathworld.wolfram.com/LeastCommonMultiple.html)| |pairwise incomparable strings (no string is a subsequence of another string)|pairwise incomparable numbers (no number divides another number)| This problem is an extension of the original minimal prime problem (https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_17.pdf), https://cs.uwaterloo.ca/~shallit/Papers/br10.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_18.pdf), https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_19.pdf), https://doi.org/10.1080/10586458.2015.1064048 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_20.pdf), https://scholar.colorado.edu/downloads/hh63sw661 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_16.pdf), 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, 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)) with *k*-values < *b*, i.e. finding the smallest prime of the form *k*×*b*^*n*+1 and *k*×*b*^*n*−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/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 *b*^*n*+*k* and *b*^*n*−*k* (which are the dual forms of *k*×*b*^*n*+1 and *k*×*b*^*n*−1, respectively) (or proving that such prime does not exist) for all *k* < *b*) (also to cover finding the smallest prime of some classic forms (or proving that such prime does not exist), such as *b*^*n*+2, *b*^*n*−2, *b*^*n*+(*b*−1), *b*^*n*−(*b*−1), 2×*b*^*n*+1, 2×*b*^*n*−1, (*b*−1)×*b*^*n*+1, (*b*−1)×*b*^*n*−1, with *n* ≥ 1, for the same base *b* (of course, for some bases *b* the original minimal prime base *b* problem already covers finding the smallest prime of these forms, e.g. the original minimal prime base *b* problem covers finding the smallest prime of the form (*b*−1)×*b*^*n*+1 if and only if *b*−1 is not prime, and the original minimal prime base *b* problem covers finding the smallest prime of the form (*b*−1)×*b*^*n*−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, http://www.noprimeleftbehind.net/crus/vstats/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/vstats/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") |subsequence|substring| |---|---| |https://oeis.org/A071062|https://oeis.org/A033274| |https://oeis.org/A130448|https://oeis.org/A238334| |https://oeis.org/A039995|https://oeis.org/A039997| |https://oeis.org/A039994|https://oeis.org/A039996| |https://oeis.org/A094535|https://oeis.org/A093301| |https://oeis.org/A350508|https://oeis.org/A038103| |https://oeis.org/A354113|https://oeis.org/A354114| |longest common subsequence problem (https://en.wikipedia.org/wiki/Longest_common_subsequence_problem)|longest common substring problem (https://en.wikipedia.org/wiki/Longest_common_substring_problem) 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://primes.utm.edu/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×*b*^*n*+1, 4×*b*^*n*+1, 6×*b*^*n*+1, 10×*b*^*n*+1, 12×*b*^*n*+1, 16×*b*^*n*+1, 3×*b*^*n*−1, 4×*b*^*n*−1, 6×*b*^*n*−1, 8×*b*^*n*−1, 12×*b*^*n*−1, 14×*b*^*n*−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), 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://primes.utm.edu/glossary/xpage/EulersPhi.html, https://mathworld.wolfram.com/TotientFunction.html, 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. This problem covers finding the smallest prime of these forms in the same base *b* (or proving that such prime does not exist): (while the original minimal prime problem does not cover some of these forms for some bases (or all bases) *b*) |family|smallest allowed *n*|*OEIS* sequences for the smallest *n* such that this form is prime|references |---|---|---|---| |(*b*^*n*−1)/(*b*−1)|2|https://oeis.org/A084740
https://oeis.org/A084738 (corresponding primes)
https://oeis.org/A065854 (prime *b*)
https://oeis.org/A279068 (prime *b*, corresponding primes)
https://oeis.org/A128164 (*n* = 2 not allowed)
https://oeis.org/A285642 (*n* = 2 not allowed, corresponding primes)|http://www.fermatquotient.com/PrimSerien/GenRepu.txt
https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html
http://www.primenumbers.net/Henri/us/MersFermus.htm
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)| |*b*^*n*+1|1|https://oeis.org/A228101 (*log*₂ of *n*)
https://oeis.org/A079706
https://oeis.org/A084712 (corresponding primes)
https://oeis.org/A123669 (*n* = 1 not allowed, corresponding primes)|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| |(*b*^*n*+1)/2 (for odd *b*)|2||http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt| |2×*b*^*n*+1|1|https://oeis.org/A119624
https://oeis.org/A253178 (only bases *b* which have possible primes)
https://oeis.org/A098872 (*b* divisible by 6)|https://mersenneforum.org/showthread.php?t=6918
https://mersenneforum.org/showthread.php?t=19725 (*b* == 11 mod 12)| |2×*b*^*n*−1|1|https://oeis.org/A119591
https://oeis.org/A098873 (*b* divisible by 6)|https://mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217| |*b*^*n*+2|1|https://oeis.org/A138066
https://oeis.org/A084713 (corresponding primes)
https://oeis.org/A138067 (*n* = 1 not allowed)| |*b*^*n*−2|2|https://oeis.org/A250200
https://oeis.org/A255707 (*n* = 1 allowed)
https://oeis.org/A084714 (*n* = 1 allowed, corresponding primes)
https://oeis.org/A292201 (prime *b*, *n* = 1 allowed)|https://www.primepuzzles.net/puzzles/puzz_887.htm (*n* = 1 allowed)| |3×*b*^*n*+1|1|https://oeis.org/A098877 (*b* divisible by 6)|| |3×*b*^*n*−1|1|https://oeis.org/A098876 (*b* divisible by 6)|| |10×*b*^*n*+1|1|https://oeis.org/A088782
https://oeis.org/A088622 (corresponding primes)|| |2×*b*^*n*+3|1||https://www.primegrid.com/forum_thread.php?id=9538| |*b*^*n*/2+1 (for even *b*)|2||https://www.primegrid.com/forum_thread.php?id=9538| |(*b*−1)×*b*^*n*+1|1|https://oeis.org/A305531
https://oeis.org/A087139 (prime *b*, *n* replaced by *n*+1)|https://www.rieselprime.de/ziki/Williams_prime_MP_least
https://www.rieselprime.de/ziki/Williams_prime_MP_table
https://sites.google.com/view/williams-primes
http://www.bitman.name/math/table/477| |(*b*−1)×*b*^*n*−1|1|https://oeis.org/A122396 (prime *b*, *n* replaced by *n*+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://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| |*b*^*n*+(*b*−1)|1|https://oeis.org/A076845
https://oeis.org/A076846 (corresponding primes)
https://oeis.org/A078178 (*n* = 1 not allowed)
https://oeis.org/A078179 (*n* = 1 not allowed, corresponding primes)|https://sites.google.com/view/williams-primes| |*b*^*n*−(*b*−1)|2|https://oeis.org/A113516
https://oeis.org/A343589 (corresponding primes)|https://sites.google.com/view/williams-primes
https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html (prime *b*)
http://www.bitman.name/math/table/435 (prime *b*)| |*k*×*b*^*n*+1 for all 2 ≤ *k* ≤ 12|1||https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n
https://mersenneforum.org/showthread.php?t=10354| |*k*×*b*^*n*−1 for all 2 ≤ *k* ≤ 12|1||https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n
https://mersenneforum.org/showthread.php?t=10354| (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) 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) 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: * 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. * 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) 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 2^*n*−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*, a digit *y*, and a base *b*, does there exist a prime number whose base-*b* expansion is of the form *xy*_*n**z* for some *n* ≥ 0? (If we say "yes", then we should find such a prime (the smallest such prime may be very large, e.g. > 10²⁵⁰⁰⁰, and if so, then we should use primality testing programs such as *PFGW* or *LLR* to find it, and before using these programs, we should use sieving programs such as *srsieve* (or *sr*1/2/5*sieve*) to remove the numbers either having small prime factors or having algebraic factors) and prove its primality (and if we want to solve the problem in this article, we should check whether this prime is the smallest such prime or not, i.e. prove all smaller numbers of the form *xy*_*n**z* with *n* ≥ 0 are composite, usually by trial division or Fermat primality test), and if we say "no", then we should prove that such prime does not exist, may by covering congruence, algebraic factorization, or combine of them) 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://primes.utm.edu/glossary/xpage/FermatNumber.html, https://www.rieselprime.de/ziki/Fermat_number, https://mathworld.wolfram.com/FermatNumber.html, https://mathworld.wolfram.com/FermatPrime.html) (of the form 2^{2^*n*}+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 2^*n*+1 is prime then *n* must be a power of two, a prime of the form *x*{*y*}*z* 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://primes.utm.edu/glossary/xpage/MersenneNumber.html, https://primes.utm.edu/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) (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 2^*n*−1 is prime then *n* must be a prime, a prime of the form *x*{*y*}*z* in base *b* must be a new Mersenne prime. (for the references of Fermat primes and Mersenne primes, see http://www.prothsearch.com/fermat.html and https://www.mersenne.org/primes/, respectively) 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://sites.google.com/site/robertgerbicz/coveringsets, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/coveringsets, 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), http://www.primepuzzles.net/problems/prob_029.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://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://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://primes.utm.edu/glossary/xpage/Divides.html, https://mathworld.wolfram.com/Divides.html) 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://primes.utm.edu/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html) to *N*, e.g. all numbers in the family 2{5} in base 11 are not coprime to 6, gcd((5×11^*n*−1)/2, 6) can only be 2 or 3, and cannot be 1)), algebraic factorization (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=790838198#Recognizable_patterns, https://mathworld.wolfram.com/PolynomialFactorization.html, https://stdkmd.net/nrr/repunit/repunitnote.htm, 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/4/40001.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/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, https://brnikat.com/nums/cullen_woodall/algebraic.txt, 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 (2×3^*n*−1) × (2×3^*n*+1))) (which includes difference-of-two-squares factorization (https://en.wikipedia.org/wiki/Difference_of_two_squares) and sum/difference-of-two-*n*th-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://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) of *x*⁴+4*y*⁴), 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)), e.g. (only list the families which all numbers do not contain "prime > *b*" subsequence) (see post https://mersenneforum.org/showpost.php?p=594923&postcount=231 for the factor pattern for some of these families) (for the case of covering congruence, we can show that the corresponding numbers are > all elements in the sets if the corresponding numbers are > *b*, thus these factorizations are nontrivial; and for the case of algebraic factorization, we can show that both factors are > 1 if the corresponding numbers are > *b*, thus these factorizations are nontrivial; for the case of combine of them, we can show that for the part of covering congruence, the corresponding numbers are > all elements in the sets if the corresponding numbers are > *b*, and for the part of algebraic factorization, both factors are > 1 if the corresponding numbers are > *b*, thus these factorizations are nontrivial) (You can see the factorization (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html) of the numbers in these families in *factordb* (http://factordb.com/), you have to convert them to algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form, and you will find that *all* numbers in these families have status (http://factordb.com/status.html, http://factordb.com/distribution.php) either "FF" or "CF", and no numbers in these families have status (http://factordb.com/status.html, http://factordb.com/distribution.php) "C" (i.e. in http://factordb.com/listtype.php?t=3), e.g. for the family 3{0}95 in base 13, its algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form is 3×13^*n*+2+122, and in *factordb* you will find that all numbers in this family are divisible by some element of {5,7,17}, see http://factordb.com/index.php?query=3*13%5E%28n%2B2%29%2B122&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show; for the family {7}D in base 21, its algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form is (7×21^*n*+1+113)/20, and in *factordb* you will find that all numbers in this family are divisible by some element of {2,13,17}, see http://factordb.com/index.php?query=%287*21%5E%28n%2B1%29%2B113%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (note: for this family *n* = 0 is not allowed, since we only consider the numbers > base); for the family 30{F}A0F in base 16, its algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form is 49×16^*n*+3−1521, and in *factordb* you will find that no numbers in this family have a prime factor with decimal length > ((the decimal length of the number + 1)/2), and all numbers in this family have two nearly equal (prime or composite) factors, see http://factordb.com/index.php?query=49*16%5E%28n%2B3%29-1521&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show; for the family 5{1} in base 25, its algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form is (121×25^*n*−1)/24, and in *factordb* you will find that no numbers in this family have a prime factor with decimal length > ((the decimal length of the number + 1)/2), and all numbers in this family have two nearly equal (prime or composite) factors, see http://factordb.com/index.php?query=%28121*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (note: for this family *n* = 0 is not allowed, since we only consider the numbers > base); for the family {D}5 in base 14, its algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form is 14^*n*+1−9, and in *factordb* you will find that all numbers with even *n* in this family are divisible by 5, and you will find that no numbers with odd *n* in this family have a prime factor with decimal length > ((the decimal length of the number + 1)/2), and all numbers with odd *n* in this family have two nearly equal (prime or composite) factors, see http://factordb.com/index.php?query=14%5E%28n%2B1%29-9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (note: for this family *n* = 0 is not allowed, since we only consider the numbers > base); for the family 7{9} in base 17, its algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form is (121×17^*n*−9)/16, and in *factordb* you will find that all numbers with odd *n* in this family are divisible by 2, and you will find that no numbers with even *n* in this family have a prime factor with decimal length > ((the decimal length of the number + 1)/2), and all numbers with even *n* in this family have two nearly equal (prime or composite) factors, see http://factordb.com/index.php?query=%28121*17%5En-9%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (note: for this family *n* = 0 is not allowed, since we only consider the numbers > base). In contrast, you can see the factorization of the numbers in unsolved families in base *b* (which are listed in the "left *b*" file) in *factordb*, you will find some numbers in these families which have neither small prime factors (say < 10¹⁶) nor two nearly equal (prime or composite) factors, also you will find some numbers in these families which have no known proper factor (https://en.wikipedia.org/wiki/Proper_factor, https://mathworld.wolfram.com/ProperFactor.html, https://mathworld.wolfram.com/ProperDivisor.html) > 1 (i.e. you will find some numbers in these families with status (http://factordb.com/status.html, http://factordb.com/distribution.php) "C" (instead of "CF" or "FF") (i.e. in http://factordb.com/listtype.php?t=3) in *factordb* (http://factordb.com/)), and they have positive Nash weight (https://www.rieselprime.de/ziki/Nash_weight, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash) (or difficulty (https://stdkmd.net/nrr/prime/primedifficulty.htm, https://stdkmd.net/nrr/prime/primedifficulty.txt)), and they have prime candidates, we can use the sense of http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://stdkmd.net/nrr/1/10003.htm#prime_period, https://stdkmd.net/nrr/3/30001.htm#prime_period, https://stdkmd.net/nrr/1/13333.htm#prime_period, https://stdkmd.net/nrr/3/33331.htm#prime_period, https://stdkmd.net/nrr/1/11113.htm#prime_period, https://stdkmd.net/nrr/3/31111.htm#prime_period, https://mersenneforum.org/showpost.php?p=138737&postcount=24, https://mersenneforum.org/showpost.php?p=153508&postcount=147, to show this) (for the examples of non-simple families, see https://stdkmd.net/nrr/prime/primecount3.htm and https://stdkmd.net/nrr/prime/primecount3.txt (only base 10 families), non-simple families usually have small primes if they cannot be ruled out as only containing composites by covering congruence, see the section above) (for the factorization of the numbers in these families, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html) may be used, they have *SNFS* polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm) |*b*|family|algebraic ((*a*×*b*^*n*+*c*)/*d*) form of this family (*n* is the number of digits in the "{}", also the lower bound of *n* to make the numbers > *b*)
(note: *d* divides *gcd*(*a*+*c*,*b*−1), but *d* need not be *gcd*(*a*+*c*,*b*−1), *d* = *gcd*(*a*+*c*,*b*−1) if and only if the numbers in the family are not divisible by some prime factor of *b*−1, i.e. the numbers in the family are coprime to *b*−1)|why this family contain no primes > *b*|factorization of the numbers in this family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*)| |---|---|---|---|---| |10|2{0}1|2×10^*n*+1+1 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=2*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |10|2{0}7|2×10^*n*+1+7 (*n* ≥ 0)|always divisible by 3
(in fact, always divisible by 9)|http://factordb.com/index.php?query=2*10%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |10|5{0}1|5×10^*n*+1+1 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=5*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |10|5{0}7|5×10^*n*+1+7 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=5*10%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |10|8{0}1|8×10^*n*+1+1 (*n* ≥ 0)|always divisible by 3
(in fact, always divisible by 9)|http://factordb.com/index.php?query=8*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |10|8{0}7|8×10^*n*+1+7 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=8*10%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |10|28{0}7|28×10^*n*+1+7 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=28*10%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |10|4{6}9|(14×10^*n*+1+7)/3 (*n* ≥ 0)|always divisible by 7|http://factordb.com/index.php?query=%2814*10%5E%28n%2B1%29%2B7%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |10|families ending with 0, 2, 4, 6, or 8|–|always divisible by 2|–| |10|families ending with 0 or 5|–|always divisible by 5|–| |10|{0,3,6,9}|–|always divisible by 3
(non-simple family)|–| |10|{0,7}|–|always divisible by 7
(non-simple family)|–| |any base (*b*)|families ending with digits *d* which are not coprime to *b*|–|always divisible by *gcd*(*d*,*b*)|–| |any base (*b*)|families whose digits all have a common factor *d* > 1|–|always divisible by *d*|–| |3|1{0}1|3^*n*+1+1 (*n* ≥ 0)|always divisible by 2|http://factordb.com/index.php?query=3%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |4|2{0}1|2×4^*n*+1+1 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=2*4%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |5|1{0}1|5^*n*+1+1 (*n* ≥ 0)|always divisible by 2|http://factordb.com/index.php?query=5%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |5|1{0}3|5^*n*+1+3 (*n* ≥ 0)|always divisible by 2
(in fact, always divisible by 4)|http://factordb.com/index.php?query=5%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |5|3{0}1|3×5^*n*+1+1 (*n* ≥ 0)|always divisible by 2
(in fact, always divisible by 4)|http://factordb.com/index.php?query=3*5%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |5|11{0}3|6×5^*n*+1+3 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=6*5%5E%28n%2B1%29%2B3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |5|3{0}11|3×5^*n*+2+6 (*n* ≥ 0)|always divisible by 3|http://factordb.com/index.php?query=3*5%5E%28n%2B2%29%2B6&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |6|4{0}1|4×6^*n*+1+1 (*n* ≥ 0)|always divisible by 5|http://factordb.com/index.php?query=4*6%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |7|1{0}1|always divisible by 2| |7|1{0}3|always divisible by 2| |7|1{0}5|always divisible by 2
(in fact, always divisible by 6)| |7|3{0}1|always divisible by 2| |7|3{0}5|always divisible by 2| |7|5{0}1|always divisible by 2
(in fact, always divisible by 6)| |7|5{0}3|always divisible by 2| |7|1{0}2|always divisible by 3| |7|2{0}1|always divisible by 3| |7|4{0}5|always divisible by 3| |7|5{0}4|always divisible by 3| |7|1{0}1{0}1|always divisible by 3
(non-simple family)| |7|1{0}3{0}5|always divisible by 3
(non-simple family)| |7|1{0}5{0}3|always divisible by 3
(non-simple family)| |7|3{0}1{0}5|always divisible by 3
(non-simple family)| |7|3{0}5{0}1|always divisible by 3
(non-simple family)| |7|5{0}1{0}3|always divisible by 3
(non-simple family)| |7|5{0}3{0}1|always divisible by 3
(non-simple family)| |7|1{0}1{0}1{0}1|always divisible by 2
(non-simple family)| |7|1{0}1{0}2|always divisible by 2
(non-simple family)| |7|1{0}2{0}1|always divisible by 2
(non-simple family)| |7|2{0}1{0}1|always divisible by 2
(non-simple family)| |7|4{0}5{0}5|always divisible by 2
(non-simple family)| |7|5{0}4{0}5|always divisible by 2
(non-simple family)| |7|5{0}5{0}4|always divisible by 2
(non-simple family)| |8|2{0}5|always divisible by 7| |8|4{0}3|always divisible by 7| |8|6{0}1|always divisible by 7| |8|44{0}3|always divisible by 3| |8|6{0}11|always divisible by 3| |9|{7}62|always divisible by 7| |9|2{7}5|always divisible by 23| |9|5{7}2|always divisible by 47| |11|2{5}3|always divisible by 5
(in fact, always divisible by 25)| |11|3{5}2|always divisible by 5
(in fact, always divisible by 35)| |11|3{7}4|always divisible by 37| |11|4{7}3|always divisible by 47| |12|A{0}21|always divisible by 5| |13|C{A}5|always divisible by 7| |14|40{4}9|always divisible by 61| |15|9{6}8|always divisible by 11| |16|2{C}3|always divisible by 7| |21|B0{H}6H|always divisible by 4637| |28|4{O}9|always divisible by 11| |28|D{6}R|always divisible by 17| |28|N{6}R|always divisible by 11| |9|{1}5|(9^*n*+1+31)/8 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%289%5E%28n%2B1%29%2B31%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|{1}61|(9^*n*+2+359)/8 (*n* ≥ 0)|always divisible by some element of {2,5}
divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%289%5E%28n%2B2%29%2B359%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|2{7}|(23×9^*n*−7)/8 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%2823*9%5En-7%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|{3}5|(3×9^*n*+1+13)/8 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%283*9%5E%28n%2B1%29%2B13%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|{3}8|(3×9^*n*+1+37)/8 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%283*9%5E%28n%2B1%29%2B37%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|{3}05|(3×9^*n*+2−203)/8 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%283*9%5E%28n%2B2%29-203%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|5{1}|(41×9^*n*−1)/8 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%2841*9%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|5{7}|(47×9^*n*−7)/8 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%2847*9%5En-7%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|6{1}|(49×9^*n*−1)/8 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%2849*9%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|{7}2|(7×9^*n*+1−47)/8 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is even, divisible by 5 if *n* is odd|http://factordb.com/index.php?query=%287*9%5E%28n%2B1%29-47%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|{7}5|(7×9^*n*+1−23)/8 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%287*9%5E%28n%2B1%29-23%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|{7}05|(3×9^*n*+2−527)/8 (*n* ≥ 1)|always divisible by some element of {2,5}
divisible by 2 if *n* is odd, divisible by 5 if *n* is even|http://factordb.com/index.php?query=%287*9%5E%28n%2B2%29-527%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|{1}6{1}|–|always divisible by some element of {2,5}
(non-simple family)
divisible by 2 if the length is odd, divisible by 5 if the length is even|–| |9|{7}2{7}|–|always divisible by some element of {2,5}
(non-simple family)
divisible by 2 if the length is odd, divisible by 5 if the length is even|–| |9|5{0}{7}|–|always divisible by some element of {2,5}
(non-simple family)
divisible by 2 if the number of 7's is odd, divisible by 5 if the number of 7's is even|–| |9|{3}{0}5|–|always divisible by some element of {2,5}
(non-simple family)
divisible by 2 if the number of 3's is odd, divisible by 5 if the number of 3's is even|–| |9|{7}{0}5|–|always divisible by some element of {2,5}
(non-simple family)
divisible by 2 if the number of 7's is odd, divisible by 5 if the number of 7's is even|–| |11|2{5}|(5×11^*n*−1)/2 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd|http://factordb.com/index.php?query=%285*11%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|3{5}|(7×11^*n*−1)/2 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is odd, divisible by 3 if *n* is even|http://factordb.com/index.php?query=%287*11%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|3{7}|(37×11^*n*−7)/10 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is odd, divisible by 3 if *n* is even|http://factordb.com/index.php?query=%2837*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|4{7}|(47×11^*n*−7)/10 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd|http://factordb.com/index.php?query=%2847*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|8{5}|(17×11^*n*−1)/2 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd|http://factordb.com/index.php?query=%2817*11%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|9{5}|(19×11^*n*−1)/2 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is odd, divisible by 3 if *n* is even|http://factordb.com/index.php?query=%2819*11%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|9{7}|(97×11^*n*−7)/10 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is odd, divisible by 3 if *n* is even|http://factordb.com/index.php?query=%2897*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|A{7}|(107×11^*n*−7)/10 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd|http://factordb.com/index.php?query=%28107*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|{5}2|(11^*n*+1−7)/2 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd|http://factordb.com/index.php?query=%2811%5E%28n%2B1%29-7%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|{5}3|(11^*n*+1−5)/2 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is odd, divisible by 3 if *n* is even|http://factordb.com/index.php?query=%2811%5E%28n%2B1%29-5%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|{7}3|(7×11^*n*+1−47)/10 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is odd, divisible by 3 if *n* is even|http://factordb.com/index.php?query=%287*11%5E%28n%2B1%29-47%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|{7}4|(7×11^*n*+1−37)/10 (*n* ≥ 1)|always divisible by some element of {2,3}
divisible by 2 if *n* is even, divisible by 3 if *n* is odd|http://factordb.com/index.php?query=%287*11%5E%28n%2B1%29-37%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|{5}8|always divisible by some element of {2,3}
divisible by 2 if the length is odd, divisible by 3 if the length is even| |11|{5}9|always divisible by some element of {2,3}
divisible by 2 if the length is even, divisible by 3 if the length is odd| |11|{7}9|always divisible by some element of {2,3}
divisible by 2 if the length is even, divisible by 3 if the length is odd| |11|{7}A|always divisible by some element of {2,3}
divisible by 2 if the length is odd, divisible by 3 if the length is even| |11|3{0}{5}|always divisible by some element of {2,3}
(non-simple family)
divisible by 2 if the number of 5's is odd, divisible by 3 if the number of 5's is even| |11|{5}{0}3|always divisible by some element of {2,3}
(non-simple family)
divisible by 2 if the number of 5's is odd, divisible by 3 if the number of 5's is even| |14|4{0}1|always divisible by some element of {3,5}
divisible by 3 if the length is even, divisible by 5 if the length is odd| |14|B{0}1|always divisible by some element of {3,5}
divisible by 3 if the length is odd, divisible by 5 if the length is even| |14|3{D}|always divisible by some element of {3,5}
divisible by 3 if the length is odd, divisible by 5 if the length is even| |14|A{D}|always divisible by some element of {3,5}
divisible by 3 if the length is even, divisible by 5 if the length is odd| |14|1{0}B|always divisible by some element of {3,5}
divisible by 3 if the length is odd, divisible by 5 if the length is even| |14|{D}3|always divisible by some element of {3,5}
divisible by 3 if the length is odd, divisible by 5 if the length is even| |14|{4}9|always divisible by some element of {3,5}
divisible by 3 if the length is odd, divisible by 5 if the length is even| |14|{8}5|always divisible by some element of {3,5}
divisible by 3 if the length is even, divisible by 5 if the length is odd| |20|8{0}1|always divisible by some element of {3,7}
divisible by 3 if the length is odd, divisible by 7 if the length is even| |20|D{0}1|always divisible by some element of {3,7}
divisible by 3 if the length is even, divisible by 7 if the length is odd| |20|7{J}|always divisible by some element of {3,7}
divisible by 3 if the length is even, divisible by 7 if the length is odd| |20|C{J}|always divisible by some element of {3,7}
divisible by 3 if the length is odd, divisible by 7 if the length is even| |20|1{0}D|always divisible by some element of {3,7}
divisible by 3 if the length is even, divisible by 7 if the length is odd| |20|{J}7|always divisible by some element of {3,7}
divisible by 3 if the length is even, divisible by 7 if the length is odd| |25|D{1}|always divisible by some element of {2,13}
divisible by 2 if the length is even, divisible by 13 if the length is odd| |25|E{1}|always divisible by some element of {2,13}
divisible by 2 if the length is odd, divisible by 13 if the length is even| |25|1E{1}|always divisible by some element of {2,13}
divisible by 2 if the length is odd, divisible by 13 if the length is even| |25|1F{1}|always divisible by some element of {2,13}
divisible by 2 if the length is even, divisible by 13 if the length is odd| |32|A{0}1|always divisible by some element of {3,11}
divisible by 3 if the length is even, divisible by 11 if the length is odd| |32|N{0}1|always divisible by some element of {3,11}
divisible by 3 if the length is odd, divisible by 11 if the length is even| |32|9{V}|always divisible by some element of {3,11}
divisible by 3 if the length is odd, divisible by 11 if the length is even| |32|M{V}|always divisible by some element of {3,11}
divisible by 3 if the length is even, divisible by 11 if the length is odd| |32|1{0}N|always divisible by some element of {3,11}
divisible by 3 if the length is odd, divisible by 11 if the length is even| |32|{V}9|always divisible by some element of {3,11}
divisible by 3 if the length is odd, divisible by 11 if the length is even| |34|6{0}1|always divisible by some element of {5,7}
divisible by 5 if the length is even, divisible by 7 if the length is odd| |34|5{X}|always divisible by some element of {5,7}
divisible by 5 if the length is odd, divisible by 7 if the length is even| |34|S{X}|always divisible by some element of {5,7}
divisible by 5 if the length is even, divisible by 7 if the length is odd| |34|{X}5|always divisible by some element of {5,7}
divisible by 5 if the length is odd, divisible by 7 if the length is even| |8|6{4}7|always divisible by some element of {3,5,13}
divisible by 3 if the length is odd, divisible by 5 if the length is == 2 mod 4, divisible by 13 if the length is == 0 mod 4
(special example, as the numbers with length ≥ 222 in this family contain "prime > b" subsequence, this prime is 4₂₂₀7)| |13|3{0}95|always divisible by some element of {5,7,17}
divisible by 7 if the length is even, divisible by 5 if the length is == 1 mod 4, divisible by 17 if the length is == 3 mod 4| |13|95{0}3|always divisible by some element of {5,7,17}
divisible by 7 if the length is odd, divisible by 5 if the length is == 2 mod 4, divisible by 17 if the length is == 0 mod 4| |16|{4}D|always divisible by some element of {3,7,13}
divisible by 3 if the length is == 0 mod 3, divisible by 7 if the length is == 2 mod 3, divisible by 13 if the length is == 1 mod 3| |16|{8}F|always divisible by some element of {3,7,13}
divisible by 3 if the length is == 1 mod 3, divisible by 7 if the length is == 0 mod 3, divisible by 13 if the length is == 2 mod 3| |17|7F{0}D|always divisible by some element of {3,5,29}
divisible by 3 if the length is even, divisible by 5 if the length is == 1 mod 4, divisible by 29 if the length is == 3 mod 4| |17|D{0}7F|always divisible by some element of {3,5,29}
divisible by 3 if the length is odd, divisible by 5 if the length is == 2 mod 4, divisible by 29 if the length is == 0 mod 4| |21|{7}D|always divisible by some element of {2,13,17}
divisible by 2 if the length is even, divisible by 13 if the length is == 1 mod 4, divisible by 17 if the length is == 3 mod 4| |23|7L{0}1|always divisible by some element of {3,5,53}
divisible by 3 if the length is even, divisible by 5 if the length is == 1 mod 4, divisible by 53 if the length is == 3 mod 4| |23|1{0}7L|always divisible by some element of {3,5,53}
divisible by 3 if the length is odd, divisible by 5 if the length is == 2 mod 4, divisible by 53 if the length is == 0 mod 4| |23|{D}GA|always divisible by some element of {2,5,7,37,79}
divisible by 2 if the length is even, divisible by 5 if the length is == 3 mod 4, divisible by 7 if the length is == 2 mod 3, divisible by 37 if the length is == 9 mod 12, divisible by 79 if the length is == 1 mod 3| |23|L{5}L|always divisible by some element of {2,5,7,13,37}
divisible by 2 if the length is even, divisible by 5 if the length is == 3 mod 4, divisible by 7 if the length is == 2 mod 3, divisible by 13 if the length is == 3 mod 6, divisible by 37 if the length is == 1 mod 12| |27|JP{0}1|always divisible by some element of {5,7,73}
divisible by 7 if the length is even, divisible by 5 if the length is == 1 mod 4, divisible by 73 if the length is == 3 mod 4| |27|1{0}JP|always divisible by some element of {5,7,73}
divisible by 7 if the length is odd, divisible by 5 if the length is == 2 mod 4, divisible by 73 if the length is == 0 mod 4| |27|J{0}2|always divisible by some element of {5,7,73}
divisible by 7 if the length is odd, divisible by 5 if the length is == 2 mod 4, divisible by 73 if the length is == 0 mod 4| |29|{2}{5}|always divisible by some element of {2,3,5}
(non-simple family)
divisible by 2 if the number of 5's is even, divisible by 3 if the number of 2's and the number of 5's are both even or both odd, divisible by 5 if the number of 2's is even| |30|A{0}9J|always divisible by some element of {7,13,19,31}
divisible by 7 if the length is == 0 mod 3, divisible by 13 if the length is == 1 mod 6, divisible by 19 if the length is == 2 mod 3, divisible by 31 if the length is even| |31|O{5}|always divisible by some element of {2,3,7,19}
divisible by 2 if the length is odd, divisible by 3 if the length is == 1 mod 3, divisible by 7 if the length is 2 mod 6, divisible by 19 if the length is == 0 mod 6| |32|8{0}V|always divisible by some element of {3,5,41}
divisible by 3 if the length is odd, divisible by 5 if the length is == 0 mod 4, divisible by 41 if the length is == 2 mod 4| |32|{G}L|always divisible by some element of {3,5,41}
divisible by 3 if the length is odd, divisible by 5 if the length is == 0 mod 4, divisible by 41 if the length is == 2 mod 4| |9|{1}|difference-of-squares factorization
(9^*n*−1)/8 = (3^*n*−1) × (3^*n*+1) / 8| |8|1{0}1|sum-of-cubes factorization
8^*n*+1 = (2^*n*+1) × (4^*n*−2^*n*+1)| |9|3{1}|difference-of-squares factorization
(25×9^*n*−1)/8 = (5×3^*n*−1) × (5×3^*n*+1) / 8| |9|3{8}|difference-of-squares factorization
4×9^*n*−1 = (2×3^*n*−1) × (2×3^*n*+1)| |9|{8}5|difference-of-squares factorization
9^*n*−4 = (3^*n*−2) × (3^*n*+2)| |9|3{8}35|difference-of-squares factorization
4×9^*n*−49 = (2×3^*n*−7) × (2×3^*n*+7)| |16|8{F}|difference-of-squares factorization
9×16^*n*−1 = (3×4^*n*−1) × (3×4^*n*+1)| |16|{F}7|difference-of-squares factorization
16^*n*−9 = (4^*n*−3) × (4^*n*+3)| |16|{4}1|difference-of-squares factorization
(4×16^*n*−49)/15 = (2×4^*n*−7) × (2×4^*n*+7) / 15| |16|B{4}1|difference-of-squares factorization
(169×16^*n*−49)/15 = (13×4^*n*−7) × (13×4^*n*+7) / 15| |16|1{5}|difference-of-squares factorization
(4×16^*n*−1)/3 = (2×4^*n*−1) × (2×4^*n*+1) / 3| |16|8{5}|difference-of-squares factorization
(25×16^*n*−1)/3 = (5×4^*n*−1) × (5×4^*n*+1) / 3| |16|10{5}|difference-of-squares factorization
(49×16^*n*−1)/3 = (7×4^*n*−1) × (7×4^*n*+1) / 3| |16|A1{5}|difference-of-squares factorization
(484×16^*n*−1)/3 = (22×4^*n*−1) × (22×4^*n*+1) / 3| |16|7{3}|difference-of-squares factorization
(36×16^*n*−1)/5 = (6×4^*n*−1) × (6×4^*n*+1) / 5| |16|3{F}AF|difference-of-squares factorization
4×16^*n*−81 = (2×4^*n*−9) × (2×4^*n*+9)| |16|30{F}AF|difference-of-squares factorization
49×16^*n*−81 = (7×4^*n*−9) × (7×4^*n*+9)| |16|3{F}A0F|difference-of-squares factorization
4×16^*n*−1521 = (2×4^*n*−39) × (2×4^*n*+39)| |16|30{F}A0F|difference-of-squares factorization
49×16^*n*−1521 = (7×4^*n*−39) × (7×4^*n*+39)| |16|{3}23|difference-of-squares factorization
(16^*n*−81)/5 = (4^*n*−9) × (4^*n*+9) / 5
(in fact, difference-of-4th-powers factorization)
(16^*n*−81)/5 = (2^*n*−3) × (2^*n*+3) × (4^*n*+9) / 5| |16|{5}45|difference-of-squares factorization
(16^*n*−49)/3 = (4^*n*−7) × (4^*n*+7) / 3| |16|{C}B|difference-of-squares factorization
(4×16^*n*−9)/5 = (2×4^*n*−3) × (2×4^*n*+3) / 5| |16|{C}D|Aurifeuillian factorization of *x*⁴+4×*y*⁴
(4×16^*n*+1)/5 = (2×4^*n*−2×2^*n*+1) × (2×4^*n*+2×2^*n*+1) / 5| |16|{C}DD|Aurifeuillian factorization of *x*⁴+4×*y*⁴
(4×16^*n*+81)/5 = (2×4^*n*−6×2^*n*+9) × (2×4^*n*+6×2^*n*+9) / 5| |25|{1}|difference-of-squares factorization
(25^*n*−1)/24 = (5^*n*−1) × (5^*n*+1) / 24| |25|2{1}|difference-of-squares factorization
(49×25^*n*−1)/24 = (7×5^*n*−1) × (7×5^*n*+1) / 24| |25|5{1}|difference-of-squares factorization
(121×25^*n*−1)/24 = (11×5^*n*−1) × (11×5^*n*+1) / 24| |25|7{1}|difference-of-squares factorization
(169×25^*n*−1)/24 = (13×5^*n*−1) × (13×5^*n*+1) / 24| |25|C{1}|difference-of-squares factorization
(289×25^*n*−1)/24 = (17×5^*n*−1) × (17×5^*n*+1) / 24| |25|F{1}|difference-of-squares factorization
(361×25^*n*−1)/24 = (19×5^*n*−1) × (19×5^*n*+1) / 24| |25|M{1}|difference-of-squares factorization
(529×25^*n*−1)/24 = (23×5^*n*−1) × (23×5^*n*+1) / 24| |25|27{1}|difference-of-squares factorization
(1369×25^*n*−1)/24 = (37×5^*n*−1) × (37×5^*n*+1) / 24| |25|7C{1}|difference-of-squares factorization
(4489×25^*n*−1)/24 = (67×5^*n*−1) × (67×5^*n*+1) / 24| |25|D5{1}|difference-of-squares factorization
(7921×25^*n*−1)/24 = (89×5^*n*−1) × (89×5^*n*+1) / 24| |25|1{3}|difference-of-squares factorization
(9×25^*n*−1)/8 = (3×5^*n*−1) × (3×5^*n*+1) / 8| |25|1{8}|difference-of-squares factorization
(4×25^*n*−1)/3 = (2×5^*n*−1) × (2×5^*n*+1) / 3| |25|5{8}|difference-of-squares factorization
(16×25^*n*−1)/3 = (4×5^*n*−1) × (4×5^*n*+1) / 3| |25|A{3}|difference-of-squares factorization
(81×25^*n*−1)/8 = (9×5^*n*−1) × (9×5^*n*+1) / 8| |25|L{8}|difference-of-squares factorization
(64×25^*n*−1)/3 = (8×5^*n*−1) × (8×5^*n*+1) / 3| |25|{3}2|difference-of-squares factorization
(25^*n*−9)/8 = (5^*n*−3) × (5^*n*+3) / 8| |25|{8}3|difference-of-squares factorization
(25^*n*−16)/3 = (5^*n*−4) × (5^*n*+4) / 3| |25|{8}7|difference-of-squares factorization
(25^*n*−4)/3 = (5^*n*−2) × (5^*n*+2) / 3| |25|{3}2I|difference-of-squares factorization
(25^*n*−81)/8 = (5^*n*−9) × (5^*n*+9) / 8| |25|{8}5I|difference-of-squares factorization
(25^*n*−196)/3 = (5^*n*−14) × (5^*n*+14) / 3| |25|{8}7C|difference-of-squares factorization
(25^*n*−64)/3 = (5^*n*−8) × (5^*n*+8) / 3| |27|8{0}1|sum-of-cubes factorization
8×27^*n*+1 = (2×3^*n*+1) × (4×9^*n*−2×3^*n*+1)| |27|1{0}8|sum-of-cubes factorization
27^*n*+8 = (3^*n*+2) × (9^*n*−2×3^*n*+4)| |27|{D}E|sum-of-cubes factorization
(27^*n*+1)/2 = (3^*n*+1) × (9^*n*−3^*n*+1) / 2| |27|7{Q}|difference-of-cubes factorization
8×27^*n*−1 = (2×3^*n*−1) × (4×9^*n*+2×3^*n*+1)| |27|{Q}J|difference-of-cubes factorization
27^*n*−8 = (3^*n*−2) × (9^*n*+2×3^*n*+4)| |27|9{G}|difference-of-cubes factorization
(125×27^*n*−8)/13 = (5×3^*n*−2) × (25×9^*n*+10×3^*n*+4) / 13| |32|1{0}1|sum-of-5th-powers factorization
32^*n*+1 = (2^*n*+1) × (16^*n*−8^*n*+4^*n*−2^*n*+1)| |32|{1}|difference-of-5th-powers factorization
(32^*n*−1)/31 = (2^*n*−1) × (16^*n*+8^*n*+4^*n*+2^*n*+1) / 31| |36|3{7}|difference-of-squares factorization
(16×36^*n*−1)/5 = (4×6^*n*−1) × (4×6^*n*+1) / 5| |36|3{Z}|difference-of-squares factorization
4×36^*n*−1 = (2×6^*n*−1) × (2×6^*n*+1)| |36|8{Z}|difference-of-squares factorization
9×36^*n*−1 = (3×6^*n*−1) × (3×6^*n*+1)| |36|O{Z}|difference-of-squares factorization
25×36^*n*−1 = (5×6^*n*−1) × (5×6^*n*+1)| |36|{Z}B|difference-of-squares factorization
36^*n*−25 = (6^*n*−5) × (6^*n*+5)| |36|8{Z}B|difference-of-squares factorization
9×36^*n*−25 = (3×6^*n*−5) × (3×6^*n*+5)| |36|F{Z}B|difference-of-squares factorization
16×36^*n*−25 = (4×6^*n*−5) × (4×6^*n*+5)| |36|{Z}RZ|difference-of-squares factorization
36^*n*−289 = (6^*n*−17) × (6^*n*+17)| |36|F{Z}RZ|difference-of-squares factorization
16×36^*n*−289 = (4×6^*n*−17) × (4×6^*n*+17)| |36|O{Z}RZ|difference-of-squares factorization
25×36^*n*−289 = (5×6^*n*−17) × (5×6^*n*+17)| |36|O{5}|difference-of-squares factorization
(169×36^*n*−1)/7 = (13×6^*n*−1) × (13×6^*n*+1) / 7| |36|O{7}|difference-of-squares factorization
(121×36^*n*−1)/5 = (11×6^*n*−1) × (11×6^*n*+1) / 5| |36|{9}1|difference-of-squares factorization
(9×36^*n*−289)/35 = (3×6^*n*−17) × (3×6^*n*+17) / 35| |36|T{9}1|difference-of-squares factorization
(1024×36^*n*−289)/35 = (32×6^*n*−17) × (32×6^*n*+17) / 35| |36|{G}D|difference-of-squares factorization
(16×36^*n*−121)/35 = (4×6^*n*−11) × (4×6^*n*+11) / 35| |36|{G}8D|difference-of-squares factorization
(16×36^*n*−10201)/35 = (4×6^*n*−101) × (4×6^*n*+101) / 35| |36|R{G}D|difference-of-squares factorization
(961×36^*n*−121)/35 = (31×6^*n*−11) × (31×6^*n*+11) / 35| |36|3{G}8D|difference-of-squares factorization
(121×36^*n*−10201)/35 = (11×6^*n*−101) × (11×6^*n*+101) / 35| |36|R{G}8D|difference-of-squares factorization
(961×36^*n*−10201)/35 = (31×6^*n*−101) × (31×6^*n*+101) / 35| |36|{K}H|difference-of-squares factorization
(4×36^*n*−25)/7 = (2×6^*n*−5) × (2×6^*n*+5) / 7| |36|{K}IH|difference-of-squares factorization
(4×36^*n*−529)/7 = (2×6^*n*−23) × (2×6^*n*+23) / 7| |36|B{K}H|difference-of-squares factorization
(81×36^*n*−25)/7 = (9×6^*n*−5) × (9×6^*n*+5) / 7| |36|3{K}IH|difference-of-squares factorization
(25×36^*n*−529)/7 = (5×6^*n*−23) × (5×6^*n*+23) / 7| |36|B{K}IH|difference-of-squares factorization
(81×36^*n*−529)/7 = (9×6^*n*−23) × (9×6^*n*+23) / 7| |36|{S}J|difference-of-squares factorization
(4×36^*n*−49)/5 = (2×6^*n*−7) × (2×6^*n*+7) / 5| |36|{S}IJ|difference-of-squares factorization
(4×36^*n*−1849)/5 = (2×6^*n*−43) × (2×6^*n*+43) / 5| |36|1{S}J|difference-of-squares factorization
(9×36^*n*−49)/5 = (3×6^*n*−7) × (3×6^*n*+7) / 5| |36|C{S}J|difference-of-squares factorization
(64×36^*n*−49)/5 = (8×6^*n*−7) × (8×6^*n*+7) / 5| |36|X{S}J|difference-of-squares factorization
(169×36^*n*−49)/5 = (13×6^*n*−7) × (13×6^*n*+7) / 5| |36|1{S}GJ|difference-of-squares factorization
(9×36^*n*−2209)/5 = (3×6^*n*−47) × (3×6^*n*+47) / 5| |36|9{S}GJ|difference-of-squares factorization
(49×36^*n*−2209)/5 = (7×6^*n*−47) × (7×6^*n*+47) / 5| |36|C{S}GJ|difference-of-squares factorization
(64×36^*n*−2209)/5 = (8×6^*n*−47) × (8×6^*n*+47) / 5| |36|X{S}GJ|difference-of-squares factorization
(169×36^*n*−2209)/5 = (13×6^*n*−47) × (13×6^*n*+47) / 5| |36|1{S}IJ|difference-of-squares factorization
(9×36^*n*−1849)/5 = (3×6^*n*−43) × (3×6^*n*+43) / 5| |36|9{S}IJ|difference-of-squares factorization
(49×36^*n*−1849)/5 = (7×6^*n*−43) × (7×6^*n*+43) / 5| |14|8{D}|combine of factor 5 and difference-of-squares factorization
even length is divisible by 5, odd length has factorization 9×14^2×*n*−1 = (3×14^*n*−1) × (3×14^*n*+1)| |12|{B}9B|combine of factor 13 and difference-of-squares factorization
odd length is divisible by 13, even length has factorization 12^2×*n*−25 = (12^*n*−5) × (12^*n*+5)| |14|{D}5|combine of factor 5 and difference-of-squares factorization
odd length is divisible by 5, even length has factorization 14^2×*n*−9 = (14^*n*−3) × (14^*n*+3)| |17|1{9}|combine of factor 2 and difference-of-squares factorization
even length is divisible by 2, odd length has factorization (25×17^2×*n*−9)/16 = (5×17^*n*−3) × (5×17^*n*+3) / 16| |17|7{9}|combine of factor 2 and difference-of-squares factorization
even length is divisible by 2, odd length has factorization (121×17^2×*n*−9)/16 = (11×17^*n*−3) × (11×17^*n*+3) / 16| |17|{9}2|combine of factor 2 and difference-of-squares factorization
odd length is divisible by 2, even length has factorization (9×17^2×*n*−121)/16 = (3×17^*n*−11) × (3×17^*n*+11) / 16| |17|{9}8|combine of factor 2 and difference-of-squares factorization
odd length is divisible by 2, even length has factorization (9×17^2×*n*−25)/16 = (3×17^*n*−5) × (3×17^*n*+5) / 16| |19|1{6}|combine of factor 5 and difference-of-squares factorization
even length is divisible by 5, odd length has factorization (4×19^2×*n*−1)/3 = (2×19^*n*−1) × (2×19^*n*+1) / 3| |19|{6}5|combine of factor 5 and difference-of-squares factorization
odd length is divisible by 5, even length has factorization (19^2×*n*−4)/3 = (19^*n*−2) × (19^*n*+2) / 3| |19|7{2}|combine of factor 5 and difference-of-squares factorization
even length is divisible by 5, odd length has factorization (64×19^2×*n*−1)/9 = (8×19^*n*−1) × (8×19^*n*+1) / 9| |19|89{6}|combine of factor 5 and difference-of-squares factorization
even length is divisible by 5, odd length has factorization (484×19^2×*n*−1)/3 = (22×19^*n*−1) × (22×19^*n*+1) / 3| |24|3{N}|combine of factor 5 and difference-of-squares factorization
even length is divisible by 5, odd length has factorization 4×24^2×*n*−1 = (2×24^*n*−1) × (2×24^*n*+1)| |24|5{N}|combine of factor 5 and difference-of-squares factorization
odd length is divisible by 5, even length has factorization 6×24^2×*n*+1−1 = (12×24^*n*−1) × (12×24^*n*+1)| |24|8{N}|combine of factor 5 and difference-of-squares factorization
even length is divisible by 5, odd length has factorization 9×24^2×*n*−1 = (3×24^*n*−1) × (3×24^*n*+1)| |24|{6}1|combine of factor 5 and difference-of-squares factorization
even length is divisible by 5, odd length has factorization (6×24^2×*n*+1−121)/23 = (12×24^*n*−11) × (12×24^*n*+11) / 23| |24|{N}LN|combine of factor 5 and difference-of-squares factorization
odd length is divisible by 5, even length has factorization 24^2×*n*−49 = (24^*n*−7) × (24^*n*+7)| |33|F{W}|combine of factor 17 and difference-of-squares factorization
even length is divisible by 17, odd length has factorization 16×33^2×*n*−1 = (4×33^*n*−1) × (4×33^*n*+1)| |33|{W}H|combine of factor 17 and difference-of-squares factorization
odd length is divisible by 17, even length has factorization 33^2×*n*−16 = (33^*n*−4) × (33^*n*+4)| |33|3{P}|combine of factor 2 and difference-of-squares factorization
even length is divisible by 2, odd length has factorization (121×33^2×*n*−25)/32 = (11×33^*n*−5) × (11×33^*n*+5) / 32| |33|D{P}|combine of factor 2 and difference-of-squares factorization
even length is divisible by 2, odd length has factorization (441×33^2×*n*−25)/32 = (21×33^*n*−5) × (21×33^*n*+5) / 32| |33|{9}4|combine of factor 2 and difference-of-squares factorization
even length is divisible by 2, odd length has factorization (9×33^2×*n*−169)/32 = (3×33^*n*−13) × (3×33^*n*+13) / 32| |34|1{B}|combine of factor 5 and difference-of-squares factorization
even length is divisible by 5, odd length has factorization (4×34^2×*n*−1)/3 = (2×34^*n*−1) × (2×34^*n*+1) / 3| |34|8{X}|combine of factor 5 and difference-of-squares factorization
even length is divisible by 5, odd length has factorization 9×34^2×*n*−1 = (3×34^*n*−1) × (3×34^*n*+1)| |34|{X}P|combine of factor 5 and difference-of-squares factorization
odd length is divisible by 5, even length has factorization 34^2×*n*−9 = (34^*n*−3) × (34^*n*+3)| Also families which contain only one very small prime > *b*: |*b*|family|why this family contains only one prime > *b*| |---|---|---| |27|2{0}J|always divisible by some element of {5,7,73}
but 2J is prime, and 2J is the only prime > *b* in this family
divisible by 7 if the length is odd, divisible by 5 if the length is == 0 mod 4, divisible by 73 if the length is == 2 mod 4| |4|{1}|difference-of-squares factorization
but 11 is prime, and 11 is the only prime > *b* in this family
(4^*n*−1)/3 = (2^*n*−1) × (2^*n*+1) / 3| |8|{1}|difference-of-cubes factorization
but 111 is prime, and 111 is the only prime > *b* in this family
(8^*n*−1)/7 = (2^*n*−1) × (4^*n*+2^*n*+1) / 7| |16|{1}|difference-of-squares factorization
but 11 is prime, and 11 is the only prime > *b* in this family
(16^*n*−1)/15 = (4^*n*−1) × (4^*n*+1) / 15| |27|{1}|difference-of-cubes factorization
but 111 is prime, and 111 is the only prime > *b* in this family
(27^*n*−1)/26 = (3^*n*−1) × (9^*n*+3^*n*+1) / 26| |27|{G}7|difference-of-cubes factorization
but G7 is prime, and G7 is the only prime > *b* in this family
(8×27^*n*−125)/13 = (2×3^*n*−5) × (4×9^*n*+10×3^*n*+25) / 13| |36|{1}|difference-of-squares factorization
but 11 is prime, and 11 is the only prime > *b* in this family
(36^*n*−1)/35 = (6^*n*−1) × (6^*n*+1) / 35| Some *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) families could not be proven to contain no primes > *b* (by covering congruence, algebraic factorization, or combine of them) but no primes > *b* could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to test the primality of larger and larger numbers of such form and hope a prime is eventually discovered. Many *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) families contain no small primes > *b* even though they do contain very large primes. e.g. the smallest prime in base 13 family 9{5} is 95₁₉₇₄₂₀ (http://www.primenumbers.net/prptop/searchform.php?form=%28113*13%5E197420-5%29%2F12&action=Search, http://factordb.com/index.php?id=1100000003943359311), its algebraic form is (113×13¹⁹⁷⁴²⁰−5)/12, when written in decimal it contains 219916 digits; and the smallest prime in base 16 family {3}AF is 3₁₁₆₁₃₇AF (http://www.primenumbers.net/prptop/searchform.php?form=%2816%5E116139%2B619%29%2F5&action=Search, http://factordb.com/index.php?id=1100000003851731988), its algebraic form is (16¹¹⁶¹³⁹+619)/5, when written in decimal it contains 139845 digits; and the smallest prime in base 23 family 9{E} is 9E₈₀₀₈₇₃ (http://www.primenumbers.net/prptop/searchform.php?form=%28106*23%5E800873-7%29%2F11&action=Search, http://factordb.com/index.php?id=1100000000782858648), its algebraic form is (106×23⁸⁰⁰⁸⁷³−7)/11, when written in decimal it contains 1090573 digits; and the smallest prime in base 25 family 71JD{0}1 is 71JD0₄₅₈₅₄₉1 (http://primes.utm.edu/primes/page.php?id=111834, http://factordb.com/index.php?id=1100000002341496334), its algebraic form is 110488×25⁴⁵⁸⁵⁵⁰+1, when written in decimal it contains 641031 digits (this number can be proven prime by *N*−1 test (https://primes.utm.edu/prove/prove3_1.html, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored); and the smallest prime in base 32 family NU{0}1 is NU0₆₆₁₈₆₃1 (https://primes.utm.edu/primes/page.php?id=134216, http://factordb.com/index.php?id=1100000003813355148), its algebraic form is 766×32⁶⁶¹⁸⁶⁴+1, when written in decimal it contains 996208 digits (this number can be proven prime by *N*−1 test (https://primes.utm.edu/prove/prove3_1.html, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored); and the smallest prime in base 36 family {P}SZ is P₈₁₉₉₃SZ (http://www.primenumbers.net/prptop/searchform.php?form=%285*36%5E81995%2B821%29%2F7&action=Search, http://factordb.com/index.php?id=1100000002394962083), its algebraic form is (5×36⁸¹⁹⁹⁵+821)/7, when written in decimal it contains 127609 digits. (technically, probable (https://en.wikipedia.org/wiki/Probabilistic_algorithm) primality tests (https://primes.utm.edu/prove/prove2.html) were used to show these for the numbers which cannot be proven prime by *N*−1 test (https://primes.utm.edu/prove/prove3_1.html, http://factordb.com/nmoverview.php?method=1) or *N*+1 test (https://primes.utm.edu/prove/prove3_2.html, http://factordb.com/nmoverview.php?method=2), i.e. for the ordinary primes (https://primes.utm.edu/glossary/xpage/OrdinaryPrime.html) (which have a very small chance of making an error (https://primes.utm.edu/notes/prp_prob.html, https://www.ams.org/journals/mcom/1989-53-188/S0025-5718-1989-0982368-4/S0025-5718-1989-0982368-4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_22.pdf))) because all known primality tests (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://primes.utm.edu/prove/prove3.html, https://primes.utm.edu/prove/prove4.html) run far too slowly (longer than the life expectancy of human (https://en.wikipedia.org/wiki/Life_expectancy) for numbers > 10¹⁰⁰⁰⁰⁰, and longer than the age of the universe (https://en.wikipedia.org/wiki/Age_of_the_universe) for numbers > 10⁵⁰⁰⁰⁰⁰, even if we can do 10⁹ bitwise operations (https://en.wikipedia.org/wiki/Bitwise_operation) per second (https://en.wikipedia.org/wiki/Second) to run on these numbers) to run on numbers of these sizes unless either *N*−1 (https://primes.utm.edu/prove/prove3_1.html, http://factordb.com/nmoverview.php?method=1) or *N*+1 (https://primes.utm.edu/prove/prove3_2.html, http://factordb.com/nmoverview.php?method=2) (or both) (unfortunely, none of Wikipedia, Prime Wiki, Mathworld has article for *N*−1 primality test or *N*+1 primality test, but a similar article for Pocklington primality test: https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, also see the article for the cyclotomy primality test: https://primes.utm.edu/glossary/xpage/Cyclotomy.html) can be ≥ 1/3 factored (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html) or can be ≥ 1/4 factored and the number is not very large (say not > 10¹⁰⁰⁰⁰⁰), or *N*^*n*−1 can be ≥ 1/3 factored for a small *n*. If either *N*−1 or *N*+1 (or both) can be ≥ 1/2 factored, then we can use the Pocklington *N*−1 primality test (https://primes.utm.edu/prove/prove3_1.html, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1) (the *N*−1 case) or the Morrison *N*+1 primality test (https://primes.utm.edu/prove/prove3_2.html, http://factordb.com/nmoverview.php?method=2) (the *N*+1 case); if either *N*−1 or *N*+1 (or both) can be ≥ 1/3 factored, then we can use the Brillhart-Lehmer-Selfridge primality test (https://www.ams.org/journals/mcom/1975-29-130/S0025-5718-1975-0384673-1/S0025-5718-1975-0384673-1.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_23.pdf), https://en.wikipedia.org/wiki/Pocklington_primality_test#Extensions_and_variants); if either *N*−1 or *N*+1 (or both) can be ≥ 1/4 factored but neither can be ≥ 1/3 factored, then we need to use *CHG* (https://mersenneforum.org/attachment.php?attachmentid=21133&d=1571237465, https://primes.utm.edu/bios/page.php?id=797, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/CHG) to prove its primality (see https://mersenneforum.org/showpost.php?p=528149&postcount=3 and https://mersenneforum.org/showpost.php?p=603181&postcount=438), however, unlike Brillhart-Lehmer-Selfridge primality test for the numbers *N* such that *N*−1 or *N*+1 (or both) can be ≥ 1/3 factored can run for arbitrarily large numbers *N* (thus, there are no unproven probable primes *N* such that *N*−1 or *N*+1 (or both) can be ≥ 1/3 factored), *CHG* for the numbers *N* such that either *N*−1 or *N*+1 (or both) can be ≥ 1/4 factored but neither can be ≥ 1/3 factored cannot run for very large *N* (say > 10¹⁰⁰⁰⁰⁰), for the examples of the numbers which are proven prime by *CHG*, see https://primes.utm.edu/primes/page.php?id=126454, https://primes.utm.edu/primes/page.php?id=131964, https://primes.utm.edu/primes/page.php?id=123456, https://primes.utm.edu/primes/page.php?id=130933, https://stdkmd.net/nrr/cert/1/ (search for "CHG"), https://stdkmd.net/nrr/cert/2/ (search for "CHG"), https://stdkmd.net/nrr/cert/3/ (search for "CHG"), https://stdkmd.net/nrr/cert/4/ (search for "CHG"), https://stdkmd.net/nrr/cert/5/ (search for "CHG"), https://stdkmd.net/nrr/cert/6/ (search for "CHG"), https://stdkmd.net/nrr/cert/7/ (search for "CHG"), https://stdkmd.net/nrr/cert/8/ (search for "CHG"), https://stdkmd.net/nrr/cert/9/ (search for "CHG"), however, *factordb* (http://factordb.com/) lacks the ability to verify *CHG* proofs, see https://mersenneforum.org/showpost.php?p=608362&postcount=165; if neither *N*−1 nor *N*+1 can be ≥ 1/4 factored but *N*^*n*−1 can be ≥ 1/3 factored for a small *n*, then we can use the cyclotomy primality test (https://primes.utm.edu/glossary/xpage/Cyclotomy.html, https://primes.utm.edu/prove/prove3_3.html, http://factordb.com/nmoverview.php?method=3)) The numbers in *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) families are of the form (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) for some fixed *a*, *b*, *c* such that *a* ≥ 1, *b* ≥ 2 (*b* is the base), *c* ≠ 0, *gcd*(*a*,*c*) = 1, *gcd*(*b*,*c*) = 1. Except in the special case *c* = ±1 and *gcd*(*a*+*c*,*b*−1) = 1, when *n* is large the known primality tests (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://primes.utm.edu/prove/prove3.html, https://primes.utm.edu/prove/prove4.html) for such a number are too inefficient to run. In this case one must resort to a probable (https://en.wikipedia.org/wiki/Probabilistic_algorithm) primality test (https://primes.utm.edu/prove/prove2.html) such as a Miller–Rabin primality test (https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test, https://primes.utm.edu/glossary/xpage/MillersTest.html, https://www.rieselprime.de/ziki/Miller-Rabin_pseudoprimality_test, https://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html, http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm) or a Baillie–PSW primality test (https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test, https://mathworld.wolfram.com/Baillie-PSWPrimalityTest.html), unless a divisor of the number can be found. Since we are testing many numbers in an exponential sequence, it is possible to use a sieving process (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html) to find divisors rather than using trial division (https://en.wikipedia.org/wiki/Trial_division, https://primes.utm.edu/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html). To do this, we made use of Geoffrey Reynolds' *SRSIEVE* software (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, http://www.rieselprime.de/dl/CRUS_pack.zip, https://primes.utm.edu/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). This program uses the baby-step giant-step algorithm to find all primes *p* which divide *a*×*b*^*n*+*c* where *p* and *n* lie in a specified range (also, this program was updated so that it also removes the *n* such that *a*×*b*^*n*+*c* has algebraic factors (e.g. difference-of-two-squares factorization (https://en.wikipedia.org/wiki/Difference_of_two_squares) and sum/difference-of-two-*n*th-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://myfactorcollection.mooo.com:8090/source/cyclo.cpp, http://myfactorcollection.mooo.com:8090/LCD_2_199, http://myfactorcollection.mooo.com:8090/LCD_2_998) of *x*⁴+4*y*⁴), see https://mersenneforum.org/showpost.php?p=452132&postcount=66 and https://mersenneforum.org/showthread.php?t=21916 and https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/srsieve_1.1.4/algebraic.c (note: for the sequence (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1), the case of "Mersenne number" in https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/srsieve_1.1.4/algebraic.c is the case which *a* is rational power of *b*, *c* = −1 and the case which *a* is rational power of *b*, *c* = 1, *gcd*(*a*+*c*,*b*−1) ≥ 3, and the case of "GFN" in https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/srsieve_1.1.4/algebraic.c is the case which *a* is rational power of *b*, *c* = 1, *gcd*(*a*+*c*,*b*−1) is either 1 or 2)). Since this program cannot handle the general case (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) when *gcd*(*a*+*c*,*b*−1) > 1 we only used it to sieve the sequence *a*×*b*^*n*+*c* for primes *p* not dividing *gcd*(*a*+*c*,*b*−1), and initialized the list of candidates to not include *n* for which there is some prime *p* dividing *gcd*(*a*+*c*,*b*−1) for which *p* dividing (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1). The program had to be modified slightly to remove a check which would prevent it from running in the case when *a*, *b*, and *c* were all odd (since then 2 divides *a*×*b*^*n*+*c*, but 2 may not divide (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) (see https://github.com/curtisbright/mepn-data/commit/1b55b353f46c707bbe52897573914128b3303960). Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software *LLR* by Jean Penné (http://jpenne.free.fr/index2.html, https://primes.utm.edu/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://primes.utm.edu/bios/page.php?id=175, https://www.rieselprime.de/ziki/PFGW, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/pfgw_win_4.0.3). Although undocumented, it is possible to run these two programs on numbers of the form (*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1) when *gcd*(*a*+*c*,*b*−1) > 1, so this program required no modifications. A script was also written which allowed one to run srsieve while *LLR* or *PFGW* was testing the remaining candidates, so that when a divisor was found by srsieve on a number which had not yet been tested by *LLR* or *PFGW* it would be removed from the list of candidates. For the primes < 10²⁵⁰⁰⁰ for the solved or near-solved bases (bases *b* with ≤ 10 unsolved families, i.e. bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36), we employed *PRIMO* by Marcel Martin (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://primes.utm.edu/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), an elliptic curve primality proving (https://primes.utm.edu/prove/prove4_2.html, https://en.wikipedia.org/wiki/Elliptic_curve_primality, https://primes.utm.edu/glossary/xpage/ECPP.html, https://mathworld.wolfram.com/EllipticCurvePrimalityProving.html, https://primes.utm.edu/top20/page.php?id=27) implementation, to compute primality certificates (https://en.wikipedia.org/wiki/Primality_certificate, https://primes.utm.edu/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, http://factordb.com/certoverview.php) for the candidates for minimal prime base *b* which are > 10²⁹⁹ and neither *N*−1 nor *N*+1 can be ≥ 1/3 factored (need *CHG* proof if either *N*−1 or *N*+1 (or both) can be ≥ 1/4 factored but neither can be ≥ 1/3 factored, but *factordb* (http://factordb.com/) lacks the ability to verify *CHG* proofs, see https://mersenneforum.org/showpost.php?p=608362&postcount=165). We have completely solved this problem for bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24 (i.e. we have found all minimal primes in these bases and proved that they are all such primes and proved that they are definitely primes (https://en.wikipedia.org/wiki/Provable_prime, http://factordb.com/listtype.php?t=4) (i.e. not merely probable primes)) (thus, currently we can complete the classification of the minimal primes in these bases, and the "minimal prime problem" for these bases are theorems (https://en.wikipedia.org/wiki/Theorem, https://mathworld.wolfram.com/Theorem.html, https://primes.utm.edu/notes/proofs/)), also we have completely solved this problem for bases *b* = 11, 16, 22, 30 if we allow probable primes (https://en.wikipedia.org/wiki/Probable_prime, https://primes.utm.edu/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Probable_prime, https://mathworld.wolfram.com/ProbablePrime.html, http://factordb.com/listtype.php?t=1) > 10²⁵⁰⁰⁰ in place of proven primes, besides, we have completely solved this problem for bases *b* = 13, 17, 19, 21, 23, 25, 26, 27, 28, 32, 34, 36 (if we allow strong probable primes in place of proven primes) except the families *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) listed in the "left *b*" files (see the condensed table below for the searching limit of these families) (thus, currently the "minimal prime problem" for these bases are still unsolved problems (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://primes.utm.edu/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, https://primes.utm.edu/notes/conjectures/)). We are unable to determine if the families *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) listed in the "left *b*" files (see the condensed table below for the searching limit of these families) contain a prime (only count the numbers > *b*) or not (even if we allow strong probable primes), i.e. these families have no known prime (or strong probable prime) members > *b*, nor can they be ruled out as only containing composites (only count the numbers > *b*) (by covering congruence, algebraic factorization, or combine of them), i.e. whether these families contain a prime or a strong probable prime (only count the numbers > *b*) are open problems (https://en.wikipedia.org/wiki/Open_problem, https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics, https://primes.utm.edu/glossary/xpage/OpenQuestion.html, https://mathworld.wolfram.com/UnsolvedProblems.html, https://primes.utm.edu/notes/conjectures/), and all of these families are expected to contain a prime > *b* (in fact, expected to contain infinitely many primes), since there is a heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://primes.utm.edu/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html) 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://primes.utm.edu/mersenne/heuristic.html, https://primes.utm.edu/notes/faq/NextMersenne.html, https://web.archive.org/web/20100628035147/http://www.math.niu.edu/~rusin/known-math/98/exp_primes), since by the prime number theorem (https://en.wikipedia.org/wiki/Prime_number_theorem, https://primes.utm.edu/glossary/xpage/PrimeNumberThm.html, https://mathworld.wolfram.com/PrimeNumberTheorem.html, https://primes.utm.edu/howmany.html, 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://primes.utm.edu/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://primes.utm.edu/glossary/xpage/Log.html, https://mathworld.wolfram.com/NaturalLogarithm.html)). If one conjectures the numbers *x*{*y*}*z* behave similarly 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, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/allnash) (or difficulty (https://stdkmd.net/nrr/prime/primedifficulty.htm, https://stdkmd.net/nrr/prime/primedifficulty.txt)), 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 also unproven probable primes (however, in this project our results assume that they are in fact primes, since they are > 10²⁵⁰⁰⁰ and the probability that they are in fact composite is < 10⁻²⁰⁰⁰, see https://primes.utm.edu/notes/prp_prob.html and https://www.ams.org/journals/mcom/1989-53-188/S0025-5718-1989-0982368-4/S0025-5718-1989-0982368-4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_22.pdf)), the unproven probable primes for bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 (the solved or near-solved bases, i.e. the bases *b* with ≤ 10 unsolved families) are (together with the factorization of the numbers in their corresponding families): (you can click the "show" in the *factordb* page to see these unproven probable primes written in base 10 and base *b* (for base *b*, change the "10" in "Digits (Base 10)" box to "*b*", support bases 2 ≤ *b* ≤ 36), also you can click the "*N*−1" or the "*N*+1" (open the "Primality proving" box) to see the factorization of *N*−1 and *N*+1) (for the factorization of the numbers in these families, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html) may be used, they have *SNFS* polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm) |*b*|index of this minimal prime in base *b* (assuming the primality of all probable primes in base *b*)|base-*b* form of this unproven probable prime|algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form of this unproven probable prime|*factordb* entry of this unproven probable prime|*Primo* input file of this unproven probable prime|factorization of the numbers in corresponding family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*)| |---|---|---|---|---|---|---| |11|1068|57₆₂₆₆₈|(57×11⁶²⁶⁶⁸−7)/10|http://factordb.com/index.php?id=1100000003573679860|http://factordb.com/cert.php?id=1100000003573679860&inputfile|http://factordb.com/index.php?query=%2857*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3194|C5₂₃₇₅₅C|(149×13²³⁷⁵⁶+79)/12|http://factordb.com/index.php?id=1100000003590647776|http://factordb.com/cert.php?id=1100000003590647776&inputfile|http://factordb.com/index.php?query=%28149*13%5E%28n%2B1%29%2B79%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3195|80₃₂₀₁₇111|8×13³²⁰²⁰+183|http://factordb.com/index.php?id=1100000000490878060|http://factordb.com/cert.php?id=1100000000490878060&inputfile|http://factordb.com/index.php?query=8*13%5E%28n%2B3%29%2B183&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3196|95₁₉₇₄₂₀|(113×13¹⁹⁷⁴²⁰−5)/12|http://factordb.com/index.php?id=1100000003943359311|(no *Primo* input file, since this unproven probable prime is too large (> 10¹⁴⁹⁹⁹⁹) to be PRP-tested in *factordb*, and *factordb* does not have *Primo* input file for numbers with status (http://factordb.com/status.html, http://factordb.com/distribution.php) "U" (i.e. in http://factordb.com/listtype.php?t=2), factordb only has *Primo* input file for numbers with status "PRP" (i.e. in http://factordb.com/listtype.php?t=1))|http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2345|DB₃₂₂₃₄|(206×16³²²³⁴−11)/15|http://factordb.com/index.php?id=1100000002383583629|http://factordb.com/cert.php?id=1100000002383583629&inputfile|http://factordb.com/index.php?query=%28206*16%5En-11%29%2F15&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2346|4₇₂₇₈₅DD|(4×16⁷²⁷⁸⁷+2291)/15|http://factordb.com/index.php?id=1100000003615909841|http://factordb.com/cert.php?id=1100000003615909841&inputfile|http://factordb.com/index.php?query=%284*16%5E%28n%2B2%29%2B2291%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2347|3₁₁₆₁₃₇AF|(16¹¹⁶¹³⁹+619)/5|http://factordb.com/index.php?id=1100000003851731988|http://factordb.com/cert.php?id=1100000003851731988&inputfile|http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |22|8003|BK₂₂₀₀₁5|(251×22²²⁰⁰²−335)/21|http://factordb.com/index.php?id=1100000003594696838|http://factordb.com/cert.php?id=1100000003594696838&inputfile|http://factordb.com/index.php?query=%28251*22%5E%28n%2B1%29-335%29%2F21&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25250|5₁₉₃₉₁6F|(26¹⁹³⁹³+179)/5|http://factordb.com/index.php?id=1100000003850151202|http://factordb.com/cert.php?id=1100000003850151202&inputfile|http://factordb.com/index.php?query=%2826%5E%28n%2B2%29%2B179%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25251|7₂₀₂₇₉OL|(7×26²⁰²⁸¹+11393)/25|http://factordb.com/index.php?id=1100000003892628605|http://factordb.com/cert.php?id=1100000003892628605&inputfile|http://factordb.com/index.php?query=%287*26%5E%28n%2B2%29%2B11393%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25252|LD0₂₀₉₇₅7|559×26²⁰⁹⁷⁶+7|http://factordb.com/index.php?id=1100000003892628658|http://factordb.com/cert.php?id=1100000003892628658&inputfile|http://factordb.com/index.php?query=559*26%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|25253|6K₂₃₃₀₀5|(34×26²³³⁰¹−79)/5|http://factordb.com/index.php?id=1100000003892628745|http://factordb.com/cert.php?id=1100000003892628745&inputfile|http://factordb.com/index.php?query=%2834*26%5E%28n%2B1%29-79%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25526|N6₂₄₀₅₁LR|(209×28²⁴⁰⁵³+3967)/9|http://factordb.com/index.php?id=1100000003879667576|http://factordb.com/cert.php?id=1100000003879667576&inputfile|http://factordb.com/index.php?query=%28209*28%5E%28n%2B2%29%2B3967%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25527|5OA₃₁₂₃₈F|(4438×28³¹²³⁹+125)/27|http://factordb.com/index.php?id=1100000003880455200|http://factordb.com/cert.php?id=1100000003880455200&inputfile|http://factordb.com/index.php?query=%284438*28%5E%28n%2B1%29%2B125%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|25528|O4O₉₄₅₃₅9|(6092×28⁹⁴⁵³⁶−143)/9|http://factordb.com/index.php?id=1100000000808118231|http://factordb.com/cert.php?id=1100000000808118231&inputfile|http://factordb.com/index.php?query=%286092*28%5E%28n%2B1%29-143%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |30|2618|I0₂₄₆₀₈D|18×30²⁴⁶⁰⁹+13|http://factordb.com/index.php?id=1100000003593967511|http://factordb.com/cert.php?id=1100000003593967511&inputfile|http://factordb.com/index.php?query=18*30%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35284|7K₂₆₅₆₇Z|(53×36²⁶⁵⁶⁸+101)/7|http://factordb.com/index.php?id=1100000003896952461|http://factordb.com/cert.php?id=1100000003896952461&inputfile|http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35285|S0₇₅₀₀₇8H|28×36⁷⁵⁰⁰⁹+305|http://factordb.com/index.php?id=1100000004020085177|http://factordb.com/cert.php?id=1100000004020085177&inputfile|http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|35286|P₈₁₉₉₃SZ|(5×36⁸¹⁹⁹⁵+821)/7|http://factordb.com/index.php?id=1100000002394962083|http://factordb.com/cert.php?id=1100000002394962083&inputfile|http://factordb.com/index.php?query=%285*36%5E%28n%2B2%29%2B821%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| All these numbers are strong probable primes (https://en.wikipedia.org/wiki/Strong_pseudoprime, https://primes.utm.edu/glossary/xpage/StrongPRP.html, https://mathworld.wolfram.com/StrongPseudoprime.html) to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61 (see https://oeis.org/A014233), and strong Lucas probable primes (https://en.wikipedia.org/wiki/Lucas_pseudoprime#Strong_Lucas_pseudoprimes, https://mathworld.wolfram.com/StrongLucasPseudoprime.html) with parameters (*P*, *Q*) defined by Selfridge's Method *A* (see https://oeis.org/A217255), and trial factored to 10¹⁶ (thus, all these numbers are Baillie–PSW probable primes. The unsolved families for bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 (the solved or near-solved bases, i.e. the bases *b* with ≤ 10 unsolved families) and the factorization of the numbers in these families: (for the factorization of the numbers in these families, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html) may be used, they have *SNFS* polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm) |*b*|base-*b* form of the unsolved family|algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form of the unsolved family|current searching limit of length of this family|factorization of the numbers in this family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*)| |---|---|---|---|---| |13|A{3}A|(41×13^*n*+1+27)/4|358000|http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|85{M}B|(5347×26^*n*+1−297)/25|100000|http://factordb.com/index.php?query=%285347*26%5E%28n%2B1%29-297%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|{A}6F|(2×26^*n*+2−497)/5|100000|http://factordb.com/index.php?query=%282*26%5E%28n%2B2%29-497%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|{H}MH|(17×26^*n*+2+3233)/25|100000|http://factordb.com/index.php?query=%2817*26%5E%28n%2B2%29%2B3233%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |26|{I}GL|(18×26^*n*+2−1243)/25|100000|http://factordb.com/index.php?query=%2818*26%5E%28n%2B2%29-1243%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |28|O{A}F|(658×28^*n*+1+125)/27|543203|http://factordb.com/index.php?query=(658*28^(n%2B1)%2B125)/27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|B{0}EUV|11×36^*n*+3+19255|100000|http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|HM{0}N|634×36^*n*+1+23|100000|http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|N{0}YYN|23×36^*n*+3+45311|100000|http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |36|O{L}Z|(123×36^*n*+1+67)/5|100000|http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| The large proven primes (> 10²⁹⁹) for bases *b* = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 (the solved or near-solved bases, i.e. the bases *b* with ≤ 10 unsolved families) and their primality certificates (https://en.wikipedia.org/wiki/Primality_certificate, https://primes.utm.edu/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, http://factordb.com/certoverview.php) and the factorization of the numbers in their corresponding families: (you can click the "show" in the *factordb* page to see these primes written in base 10 and base *b* (for base *b*, change the "10" in "Digits (Base 10)" box to "*b*", support bases 2 ≤ *b* ≤ 36), also you can click the "*N*−1" or the "*N*+1" (open the "Primality proving" box) to see the factorization of *N*−1 and *N*+1) (for the factorization of the numbers in these families, the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html) may be used, they have *SNFS* polynomials (https://www.rieselprime.de/ziki/SNFS_polynomial_selection), just like factorization of the numbers in https://stdkmd.net/nrr/aaaab.htm and https://stdkmd.net/nrr/abbbb.htm and https://stdkmd.net/nrr/aaaba.htm and https://stdkmd.net/nrr/abaaa.htm and https://stdkmd.net/nrr/abbba.htm and https://stdkmd.net/nrr/abbbc.htm and http://mklasson.com/factors/index.php and https://cs.stanford.edu/people/rpropper/math/factors/3n-2.txt, see https://stdkmd.net/nrr/records.htm and https://stdkmd.net/nrr/wanted.htm) |*b*|index of this minimal prime in base *b*|base-*b* form of this minimal prime|algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form of this minimal prime|*factordb* entry of this minimal prime|primality certificate for this minimal prime|factorization of the numbers in corresponding family (*n* is the number of digits in the "{}", start with the smallest *n* making the number > *b*)| |---|---|---|---|---|---|---| |9|149|76₃₂₉2|(31×9³³⁰−19)/4|http://factordb.com/index.php?id=1100000002359003642|http://factordb.com/cert.php?id=1100000002359003642|http://factordb.com/index.php?query=%2831*9%5E%28n%2B1%29-19%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|150|27₆₈₆07|(23×9⁶⁸⁸−511)/8|http://factordb.com/index.php?id=1100000002495467486|http://factordb.com/cert.php?id=1100000002495467486|http://factordb.com/index.php?query=%2823*9%5E%28n%2B2%29-511%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |9|151|30₁₁₅₈11|3×9¹¹⁶⁰+10|http://factordb.com/index.php?id=1100000002376318423|http://factordb.com/cert.php?id=1100000002376318423|http://factordb.com/index.php?query=3*9%5E%28n%2B2%29%2B10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|1065|A₇₁₃58|11⁷¹⁵−58|http://factordb.com/index.php?id=1100000003576826487|http://factordb.com/cert.php?id=1100000003576826487|http://factordb.com/index.php?query=11%5E%28n%2B2%29-58&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|1066|7₇₅₉44|(7×11⁷⁶¹−367)/10|http://factordb.com/index.php?id=1100000002505568840|http://factordb.com/cert.php?id=1100000002505568840|http://factordb.com/index.php?query=%287*11%5E%28n%2B2%29-367%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |11|1067|557₁₀₁₁|(607×11¹⁰¹¹−7)/10|http://factordb.com/index.php?id=1100000002361376522|http://factordb.com/cert.php?id=1100000002361376522|http://factordb.com/index.php?query=%28607*11%5En-7%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3165|50₂₇₀44|5×13²⁷²+56|http://factordb.com/index.php?id=1100000002632397005|http://factordb.com/cert.php?id=1100000002632397005|http://factordb.com/index.php?query=5*13%5E%28n%2B2%29%2B56&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3166|9₂₇₁095|(3×13²⁷⁴−6103)/4|http://factordb.com/index.php?id=1100000003590431654|http://factordb.com/cert.php?id=1100000003590431654|http://factordb.com/index.php?query=%283*13%5E%28n%2B3%29-6103%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3167|10₂₈₆7771|13²⁹⁰+16654|http://factordb.com/index.php?id=1100000003590431633|http://factordb.com/cert.php?id=1100000003590431633|http://factordb.com/index.php?query=13%5E%28n%2B4%29%2B16654&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3168|9₃₀₈1|(3×13³⁰⁹−35)/4|http://factordb.com/index.php?id=1100000000840126705|proven prime by *N*−1 test (https://primes.utm.edu/prove/prove3_1.html, http://factordb.com/nmoverview.php?method=1), since *N*−1 is 39/4×(13³⁰⁸−1), thus factor *N*−1 is equivalent to factor 13³⁰⁸−1, and for the factorization of 13³⁰⁸−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN=|http://factordb.com/index.php?query=%283*13%5E%28n%2B1%29-35%29%2F4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3169|B₃₄₁C4|(11×13³⁴³+61)/12|http://factordb.com/index.php?id=1100000003590431618|http://factordb.com/cert.php?id=1100000003590431618|http://factordb.com/index.php?query=%2811*13%5E%28n%2B2%29%2B61%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3170|8B₃₄₃|(107×13³⁴³−11)/12|http://factordb.com/index.php?query=%28107*13%5En-11%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3171|710₃₇₁111|92×13³⁷⁴+183|http://factordb.com/index.php?id=1100000003590431609|http://factordb.com/cert.php?id=1100000003590431609|http://factordb.com/index.php?query=92*13%5E%28n%2B3%29%2B183&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3172|75₃₇₅7|(89×13³⁷⁶+19)/12|http://factordb.com/index.php?id=1100000003590431596|http://factordb.com/cert.php?id=1100000003590431596|http://factordb.com/index.php?query=%2889*13%5E%28n%2B1%29%2B19%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3173|9B0₃₉₁9|128×13³⁹²+9|http://factordb.com/index.php?id=1100000002632396790|http://factordb.com/cert.php?id=1100000002632396790|http://factordb.com/index.php?query=128*13%5E%28n%2B1%29%2B9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3174|7B0B₃₉₇|(15923×13³⁹⁷−11)/12|http://factordb.com/index.php?id=1100000003590431574|http://factordb.com/cert.php?id=1100000003590431574|http://factordb.com/index.php?query=%2815923*13%5En-11%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3175|10₄₁₄93|13⁴¹⁶+120|http://factordb.com/index.php?id=1100000002523249240|http://factordb.com/cert.php?id=1100000002523249240|http://factordb.com/index.php?query=13%5E%28n%2B2%29%2B120&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3176|81010₄₁₅1|17746×13⁴¹⁶+1|http://factordb.com/index.php?id=1100000003590431555|proven prime by *N*−1 test (https://primes.utm.edu/prove/prove3_1.html, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored|http://factordb.com/index.php?query=17746*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3177|8110₄₃₅1|1366×13⁴³⁶+1|http://factordb.com/index.php?id=1100000002373259109|proven prime by *N*−1 test (https://primes.utm.edu/prove/prove3_1.html, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored|http://factordb.com/index.php?query=1366*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3178|B7₄₈₆|(139×13⁴⁸⁶−7)/12|http://factordb.com/index.php?id=1100000002321015892|http://factordb.com/cert.php?id=1100000002321015892|http://factordb.com/index.php?query=%28139*13%5En-7%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3179|B₅₆₃C|(11×13⁵⁶⁴+1)/12|http://factordb.com/index.php?id=1100000000000217927|proven prime by *N*−1 test (https://primes.utm.edu/prove/prove3_1.html, http://factordb.com/nmoverview.php?method=1), since *N*−1 is 11/12×(13⁵⁶⁴−1), thus factor *N*−1 is equivalent to factor 13⁵⁶⁴−1, and for the factorization of 13⁵⁶⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN=|http://factordb.com/index.php?query=%2811*13%5E%28n%2B1%29%2B1%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3180|1B₅₇₆|(23×13⁵⁷⁶−11)/12|http://factordb.com/index.php?id=1100000002321021456|proven prime by *N*−1 test (https://primes.utm.edu/prove/prove3_1.html, http://factordb.com/nmoverview.php?method=1), since *N*−1 is 23/12×(13⁵⁷⁶−1), thus factor *N*−1 is equivalent to factor 13⁵⁷⁶−1, and for the factorization of 13⁵⁷⁶−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN=|http://factordb.com/index.php?query=%2823*13%5En-11%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3181|80₆₉₃87|8×13⁶⁹⁵+111|http://factordb.com/index.php?id=1100000002615636527|http://factordb.com/cert.php?id=1100000002615636527|http://factordb.com/index.php?query=8*13%5E%28n%2B2%29%2B111&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3182|CC5₇₁₃|(2021×13⁷¹³−5)/12|http://factordb.com/index.php?id=1100000002615627353|http://factordb.com/cert.php?id=1100000002615627353|http://factordb.com/index.php?query=%282021*13%5En-5%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3183|B₈₃₄74|(11×13⁸³⁶−719)/12|http://factordb.com/index.php?id=1100000003590430871|http://factordb.com/cert.php?id=1100000003590430871|http://factordb.com/index.php?query=%2811*13%5E%28n%2B2%29-719%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3184|9₉₆₈B|(3×13⁹⁶⁹+5)/4|http://factordb.com/index.php?id=1100000000258566244|http://factordb.com/cert.php?id=1100000000258566244|http://factordb.com/index.php?query=%283*13%5E%28n%2B1%29%2B5%29%2F4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3185|10₁₂₉₅181|13¹²⁹⁸+274|http://factordb.com/index.php?id=1100000002615445013|http://factordb.com/cert.php?id=1100000002615445013|http://factordb.com/index.php?query=13%5E%28n%2B3%29%2B274&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3186|9₁₃₆₂5|(3×13¹³⁶³−19)/4|http://factordb.com/index.php?id=1100000002321017776|http://factordb.com/cert.php?id=1100000002321017776|http://factordb.com/index.php?query=%283*13%5E%28n%2B1%29-19%29%2F4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3187|7₁₅₀₄1|(7×13¹⁵⁰⁵−79)/12|http://factordb.com/index.php?id=1100000002320890755|http://factordb.com/cert.php?id=1100000002320890755|http://factordb.com/index.php?query=%287*13%5E%28n%2B1%29-79%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3188|930₁₅₅₁1|120×13¹⁵⁵²+1|http://factordb.com/index.php?id=1100000000765961452|proven prime by *N*−1 test (https://primes.utm.edu/prove/prove3_1.html, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored|http://factordb.com/index.php?query=120*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3189|720₂₂₉₇2|93×13²²⁹⁸+2|http://factordb.com/index.php?id=1100000002632396910|http://factordb.com/cert.php?id=1100000002632396910|http://factordb.com/index.php?query=93*13%5E%28n%2B1%29%2B2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3190|1770₂₇₀₃17|267×13²⁷⁰⁵+20|http://factordb.com/index.php?id=1100000003590430825|http://factordb.com/cert.php?id=1100000003590430825|http://factordb.com/index.php?query=267*13%5E%28n%2B2%29%2B20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3191|390₆₂₆₆1|48×13⁶²⁶⁷+1|http://factordb.com/index.php?id=1100000000765961441|proven prime by *N*−1 test (https://primes.utm.edu/prove/prove3_1.html, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored|http://factordb.com/index.php?query=48*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3192|B0₆₅₄₀BBA|11×13⁶⁵⁴³+2012|http://factordb.com/index.php?id=1100000002616382906|http://factordb.com/cert.php?id=1100000002616382906|http://factordb.com/index.php?query=11*13%5E%28n%2B3%29%2B2012&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |13|3193|C₁₀₆₃₁92|13¹⁰⁶³³−50|http://factordb.com/index.php?id=1100000003590493750|http://factordb.com/cert.php?id=1100000003590493750|http://factordb.com/index.php?query=13%5E%28n%2B2%29-50&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |14|649|34D₇₀₈|47×14⁷⁰⁸−1|http://factordb.com/index.php?id=1100000001540144903|proven prime by *N*+1 test (https://primes.utm.edu/prove/prove3_2.html, http://factordb.com/nmoverview.php?method=2), since *N*+1 is trivially fully factored|http://factordb.com/index.php?query=47*14%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |14|650|4D₁₉₆₉₈|5×14¹⁹⁶⁹⁸−1|http://factordb.com/index.php?id=1100000000884560233|proven prime by *N*+1 test (https://primes.utm.edu/prove/prove3_2.html, http://factordb.com/nmoverview.php?method=2), since *N*+1 is trivially fully factored|http://factordb.com/index.php?query=5*14%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2328|880₂₄₆7|136×16²⁴⁷+7|http://factordb.com/index.php?id=1100000002468140199|proven prime by *N*−1 test (https://primes.utm.edu/prove/prove3_1.html, http://factordb.com/nmoverview.php?method=1), *N*−1 is 2³×3×7×13×25703261×(289-digit prime)|http://factordb.com/index.php?query=136*16%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2329|D4₂₆₃D|(199×16²⁶⁴+131)/15|http://factordb.com/index.php?id=1100000002468170238|http://factordb.com/cert.php?id=1100000002468170238|http://factordb.com/index.php?query=%28199*16%5E%28n%2B1%29%2B131%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2330|E0₂₆₁4DD|14×16²⁶⁴+1245|http://factordb.com/index.php?id=1100000003588388352|http://factordb.com/cert.php?id=1100000003588388352|http://factordb.com/index.php?query=14*16%5E%28n%2B3%29%2B1245&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2331|8C0₂₉₀ED|140×16²⁹²+237|http://factordb.com/index.php?id=1100000003588388307|http://factordb.com/cert.php?id=1100000003588388307|http://factordb.com/index.php?query=140*16%5E%28n%2B2%29%2B237&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2332|DA₃₀₅5|(41×16³⁰⁶−17)/3|http://factordb.com/index.php?id=1100000003588388284|http://factordb.com/cert.php?id=1100000003588388284|http://factordb.com/index.php?query=%2841*16%5E%28n%2B1%29-17%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2333|CE80₄₂₂D|3304×16⁴²³+13|http://factordb.com/index.php?id=1100000003588388257|http://factordb.com/cert.php?id=1100000003588388257|http://factordb.com/index.php?query=3304*16%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2334|5F₅₄₄6F|6×16⁵⁴⁶−145|http://factordb.com/index.php?id=1100000002604723967|http://factordb.com/cert.php?id=1100000002604723967|http://factordb.com/index.php?query=6*16%5E%28n%2B2%29-145&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2335|88F₅₄₅|137×16⁵⁴⁵−1|http://factordb.com/index.php?id=1100000000413679658|proven prime by *N*+1 test (https://primes.utm.edu/prove/prove3_2.html, http://factordb.com/nmoverview.php?method=2), since *N*+1 is trivially fully factored|http://factordb.com/index.php?query=137*16%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2336|BE0₇₉₂BB|190×16⁷⁹⁴+187|http://factordb.com/index.php?id=1100000003588387938|http://factordb.com/cert.php?id=1100000003588387938|http://factordb.com/index.php?query=190*16%5E%28n%2B2%29%2B187&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2337|D9₁₀₅₂|(68×16¹⁰⁵²−3)/5|http://factordb.com/index.php?id=1100000002321036020|http://factordb.com/cert.php?id=1100000002321036020|http://factordb.com/index.php?query=%2868*16%5En-3%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2338|FAF₁₀₆₂45|251×16¹⁰⁶⁴−187|http://factordb.com/index.php?id=1100000003588387610|http://factordb.com/cert.php?id=1100000003588387610|http://factordb.com/index.php?query=251*16%5E%28n%2B2%29-187&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2339|F8₁₅₁₇F|(233×16¹⁵¹⁸+97)/15|http://factordb.com/index.php?id=1100000000633744824|http://factordb.com/cert.php?id=1100000000633744824|http://factordb.com/index.php?query=%28233*16%5E%28n%2B1%29%2B97%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2340|20₁₇₁₃321|2×16¹⁷¹⁶+801|http://factordb.com/index.php?id=1100000003588386735|http://factordb.com/cert.php?id=1100000003588386735|http://factordb.com/index.php?query=2*16%5E%28n%2B3%29%2B801&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2341|300F₁₉₆₀AF|769×16¹⁹⁶²−81|http://factordb.com/index.php?id=1100000003588368750|http://factordb.com/cert.php?id=1100000003588368750|http://factordb.com/index.php?query=769*16%5E%28n%2B2%29-81&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2342|90₃₅₄₂91|9×16³⁵⁴⁴+145|http://factordb.com/index.php?id=1100000000633424191|http://factordb.com/cert.php?id=1100000000633424191|http://factordb.com/index.php?query=9*16%5E%28n%2B2%29%2B145&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2343|5BC₃₇₀₀D|(459×16³⁷⁰¹+1)/5|http://factordb.com/index.php?id=1100000000993764322|http://factordb.com/cert.php?id=1100000000993764322|http://factordb.com/index.php?query=%28459*16%5E%28n%2B1%29%2B1%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |16|2344|D0B₁₇₈₀₄|(3131×16¹⁷⁸⁰⁴−11)/15|http://factordb.com/index.php?id=1100000003589278511|http://factordb.com/cert.php?id=1100000003589278511|http://factordb.com/index.php?query=%283131*16%5En-11%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |18|547|80₂₉₈B|8×18²⁹⁹+11|http://factordb.com/index.php?id=1100000002355574745|http://factordb.com/cert.php?id=1100000002355574745|http://factordb.com/index.php?query=8*18%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |18|548|H₇₆₆FH|18⁷⁶⁸−37|http://factordb.com/index.php?id=1100000003590430490|http://factordb.com/cert.php?id=1100000003590430490|http://factordb.com/index.php?query=18%5E%28n%2B2%29-37&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |18|549|C0₆₂₆₈C5|12×18⁶²⁷⁰+221|http://factordb.com/index.php?id=1100000003590442437|http://factordb.com/cert.php?id=1100000003590442437|http://factordb.com/index.php?query=12*18%5E%28n%2B2%29%2B221&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show| |20|3301|H₂₄₇A0H|(17×20²⁵⁰−59677)/19|http://factordb.com/index.php?id=1100000003590502619|http://factordb.com/cert.php?id=1100000003590502619| |20|3302|7₂₄₉A7|(7×20²⁵¹+1133)/19|http://factordb.com/index.php?id=1100000003590502602|http://factordb.com/cert.php?id=1100000003590502602| |20|3303|J7₂₇₀|(368×20²⁷⁰−7)/19|http://factordb.com/index.php?id=1100000002325395462|http://factordb.com/cert.php?id=1100000002325395462| |20|3304|J₃₃₀CCC7|20³³⁴−58953|http://factordb.com/index.php?id=1100000003590502572|http://factordb.com/cert.php?id=1100000003590502572| |20|3305|40₃₈₇404B|4×20³⁹¹+32091|http://factordb.com/index.php?id=1100000003590502563|http://factordb.com/cert.php?id=1100000003590502563| |20|3306|EC0₄₂₉7|292×20⁴³⁰+7|http://factordb.com/index.php?id=1100000002633348702|http://factordb.com/cert.php?id=1100000002633348702| |20|3307|G₄₄₇99|(16×20⁴⁴⁹−2809)/19|http://factordb.com/index.php?id=1100000000840126753|http://factordb.com/cert.php?id=1100000000840126753| |20|3308|3A₅₂₇3|(67×20⁵²⁸−143)/19|http://factordb.com/index.php?id=1100000003590502531|http://factordb.com/cert.php?id=1100000003590502531| |20|3309|E₅₆₆C7|(14×20⁵⁶⁸−907)/19|http://factordb.com/index.php?id=1100000003590502516|http://factordb.com/cert.php?id=1100000003590502516| |20|3310|JCJ₆₂₉|393×20⁶²⁹−1|http://factordb.com/index.php?id=1100000001559454258|proven prime by *N*+1 test (https://primes.utm.edu/prove/prove3_2.html, http://factordb.com/nmoverview.php?method=2), since *N*+1 is trivially fully factored| |20|3311|J₆₅₅05J|20⁶⁵⁸−7881|http://factordb.com/index.php?id=1100000003590502490|http://factordb.com/cert.php?id=1100000003590502490| |20|3312|50₁₁₆₃AJ|5×20¹¹⁶⁵+219|http://factordb.com/index.php?id=1100000003590502412|http://factordb.com/cert.php?id=1100000003590502412| |20|3313|CD₂₄₄₉|(241×20²⁴⁴⁹−13)/19|http://factordb.com/index.php?id=1100000002325393915|http://factordb.com/cert.php?id=1100000002325393915| |20|3314|G0₆₂₆₉D|16×20⁶²⁷⁰+13|http://factordb.com/index.php?id=1100000003590539457|http://factordb.com/cert.php?id=1100000003590539457| |22|7984|I7G0₂₅₄H|8882×22²⁵⁵+17|http://factordb.com/index.php?id=1100000003591372788|http://factordb.com/cert.php?id=1100000003591372788| |22|7985|D0₂₅₅5EEF|13×22²⁵⁹+60339|http://factordb.com/index.php?id=1100000003591371932|http://factordb.com/cert.php?id=1100000003591371932| |22|7986|IK₃₂₂F|(398×22³²³−125)/21|http://factordb.com/index.php?id=1100000000840384145|http://factordb.com/cert.php?id=1100000000840384145| |22|7987|C0₃₄₀G9|12×22³⁴²+361|http://factordb.com/index.php?id=1100000000840384159|http://factordb.com/cert.php?id=1100000000840384159| |22|7988|77E₃₄₈K7|(485×22³⁵⁰+373)/3|http://factordb.com/index.php?id=1100000003591369779|http://factordb.com/cert.php?id=1100000003591369779| |22|7989|J₃₇₉KJ|(19×22³⁸¹+443)/21|http://factordb.com/index.php?id=1100000003591369027|http://factordb.com/cert.php?id=1100000003591369027| |22|7990|J₃₈₈EJ|(19×22³⁹⁰−2329)/21|http://factordb.com/index.php?id=1100000003591367729|http://factordb.com/cert.php?id=1100000003591367729| |22|7991|DJ₄₀₀|(292×22⁴⁰⁰−19)/21|http://factordb.com/index.php?id=1100000002325880110|http://factordb.com/cert.php?id=1100000002325880110| |22|7992|E₄₀₄K7|(2×22⁴⁰⁶+373)/3|http://factordb.com/index.php?id=1100000003591366298|http://factordb.com/cert.php?id=1100000003591366298| |22|7993|66F₄₅₃B3|(971×22⁴⁵⁵−705)/7|http://factordb.com/index.php?id=1100000003591365809|http://factordb.com/cert.php?id=1100000003591365809| |22|7994|L0₄₅₄B63|21×22⁴⁵⁷+5459|http://factordb.com/index.php?id=1100000003591365331|http://factordb.com/cert.php?id=1100000003591365331| |22|7995|L₄₈₃G3|22⁴⁸⁵−129|http://factordb.com/index.php?id=1100000003591364730|http://factordb.com/cert.php?id=1100000003591364730| |22|7996|E60₄₉₆L|314×22⁴⁹⁷+21|http://factordb.com/index.php?id=1100000000632703239|http://factordb.com/cert.php?id=1100000000632703239| |22|7997|I₆₂₆AF|(6×22⁶²⁸−1259)/7|http://factordb.com/index.php?id=1100000000632724334|http://factordb.com/cert.php?id=1100000000632724334| |22|7998|K0₇₆₀EC1|20×22⁷⁶³+7041|http://factordb.com/index.php?id=1100000000632724415|http://factordb.com/cert.php?id=1100000000632724415| |22|7999|J0₇₆₇IGGJ|19×22⁷⁷¹+199779|http://factordb.com/index.php?id=1100000003591362567|http://factordb.com/cert.php?id=1100000003591362567| |22|8000|7₉₅₉K7|(22⁹⁶¹+857)/3|http://factordb.com/index.php?id=1100000003591361817|http://factordb.com/cert.php?id=1100000003591361817| |22|8001|L₂₃₈₅KE7|22²³⁸⁸−653|http://factordb.com/index.php?id=1100000003591360774|http://factordb.com/cert.php?id=1100000003591360774| |22|8002|7₃₈₁₅2L|(22³⁸¹⁷−289)/3|http://factordb.com/index.php?id=1100000003591359839|http://factordb.com/cert.php?id=1100000003591359839| |24|3400|I0₂₄₁I5|18×24²⁴³+437|http://factordb.com/index.php?id=1100000002633360037|http://factordb.com/cert.php?id=1100000002633360037| |24|3401|D0₂₅₉KKD|13×24²⁶²+12013|http://factordb.com/index.php?id=1100000003593270725|http://factordb.com/cert.php?id=1100000003593270725| |24|3402|C7₂₉₈|(283×24²⁹⁸−7)/23|http://factordb.com/index.php?id=1100000002326181235|http://factordb.com/cert.php?id=1100000002326181235| |24|3403|20₃₁₃7|2×24³¹⁴+7|http://factordb.com/index.php?id=1100000002355610241|http://factordb.com/cert.php?id=1100000002355610241| |24|3404|BC0₃₃₁B|276×24³³²+11|http://factordb.com/index.php?id=1100000002633359842|http://factordb.com/cert.php?id=1100000002633359842| |24|3405|N₂₆₄₄LLN|24²⁶⁴⁷−1201|http://factordb.com/index.php?id=1100000003593270089|http://factordb.com/cert.php?id=1100000003593270089| |24|3406|D₂₆₉₈LD|(13×24²⁷⁰⁰+4403)/23|http://factordb.com/index.php?id=1100000003593269876|http://factordb.com/cert.php?id=1100000003593269876| |24|3407|A0₂₉₅₁8ID|10×24²⁹⁵⁴+5053|http://factordb.com/index.php?id=1100000003593269654|http://factordb.com/cert.php?id=1100000003593269654| |24|3408|88N₅₉₅₁|201×24⁵⁹⁵¹−1|http://factordb.com/index.php?id=1100000003593275880|proven prime by *N*+1 test (https://primes.utm.edu/prove/prove3_2.html, http://factordb.com/nmoverview.php?method=2), since *N*+1 is trivially fully factored| |24|3409|N00N₈₁₂₉LN|13249×24⁸¹³¹−49|http://factordb.com/index.php?id=1100000003593391606|http://factordb.com/cert.php?id=1100000003593391606| |26|25174|OL0₂₁₄M9|645×26²¹⁶+581|http://factordb.com/index.php?id=1100000000840631576|proven prime by *N*−1 test (https://primes.utm.edu/prove/prove3_1.html, http://factordb.com/nmoverview.php?method=1), *N*−1 is 2²×5²×7×223×42849349×(296-digit prime) |26|25175|1A₂₁₉P|(7×26²²⁰+73)/5|http://factordb.com/index.php?id=1100000000840631595|http://factordb.com/cert.php?id=1100000000840631595| |26|25176|A₂₂₃DP|(2×26²²⁵+463)/5|http://factordb.com/index.php?id=1100000003850155262|http://factordb.com/cert.php?id=1100000003850155262| |26|25177|6J₂₂₅|(169×26²²⁵−19)/25|http://factordb.com/index.php?id=1100000002328050895|http://factordb.com/cert.php?id=1100000002328050895| |26|25178|O₂₂₈5|(24×26²²⁹−499)/5|http://factordb.com/index.php?id=1100000002328059255|http://factordb.com/cert.php?id=1100000002328059255| |26|25179|K0₂₃₀K0IP|20×26²³⁴+352013|http://factordb.com/index.php?id=1100000000840631669|http://factordb.com/cert.php?id=1100000000840631669| |26|25180|B0₂₃₆OB|11×26²³⁸+635|http://factordb.com/index.php?id=1100000002634136234|http://factordb.com/cert.php?id=1100000002634136234| |28|25485|JN₂₀₆|(536×28²⁰⁶−23)/27|proven prime by *N*−1 test (https://primes.utm.edu/prove/prove3_1.html, http://factordb.com/nmoverview.php?method=1), *N*−1 is 2×1061×1171×74311×(289-digit prime)| |28|25486|3₂₁₁M9|(28²¹³+4841)/9|http://factordb.com/index.php?id=1100000003850161936|http://factordb.com/cert.php?id=1100000003850161936| |28|25487|HD0₂₁₃D|489×28²¹⁴+13|http://factordb.com/index.php?id=1100000003850161937|http://factordb.com/cert.php?id=1100000003850161937| |28|25488|64O₂₁₇9|(1556×28²¹⁸−143)/9|http://factordb.com/index.php?id=1100000000840840215|http://factordb.com/cert.php?id=1100000000840840215| |28|25489|G0₂₁₇A0N|16×28²²⁰+7863|http://factordb.com/index.php?id=1100000003850161938|http://factordb.com/cert.php?id=1100000003850161938| |28|25490|55OA₂₂₆F|(110278×28²²⁷+125)/27|http://factordb.com/index.php?id=1100000003850161939|http://factordb.com/cert.php?id=1100000003850161939| |28|25491|L0₂₂₉Q3|21×28²³¹+731|http://factordb.com/cert.php?id=1100000003850161940| |28|25492|B0₂₃₁7ID|11×28²³⁴+6005|http://factordb.com/cert.php?id=1100000003850161941| |28|25493|PM₂₃₃B|(697×28²³⁴−319)/27|http://factordb.com/cert.php?id=1100000003850161942| |28|25494|K0₂₃₈OF|20×28²⁴⁰+687|http://factordb.com/cert.php?id=1100000000840840142| |28|25495|I₂₆₂E3|(2×28²⁶⁴−383)/3|http://factordb.com/cert.php?id=1100000003850161943| |28|25496|C5A₂₇₃F|(9217×28²⁷⁴+125)/27|http://factordb.com/cert.php?id=1100000003850161944| |28|25497|J0₂₇₆IMB|19×28²⁷⁹+14739|http://factordb.com/cert.php?id=1100000003850161945| |28|25498|F0₂₈₂QAP|15×28²⁸⁵+20689|http://factordb.com/cert.php?id=1100000000840840006| |28|25499|M0₂₉₆KKN|22×28²⁹⁹+16263|http://factordb.com/cert.php?id=1100000003850161946| |28|25500|C₃₁₀43|(4×28³¹²−2101)/9|http://factordb.com/cert.php?id=1100000003850161947| |28|25501|RN₃₁₉|(752×28³¹⁹−23)/27|http://factordb.com/cert.php?id=1100000002611723967| |28|25502|CA₃₂₀F|(334×28³²¹+125)/27|http://factordb.com/cert.php?id=1100000000840839995| |28|25503|D6₃₂₆LR|(119×28³²⁸+3967)/9|http://factordb.com/cert.php?id=1100000003850161948| |28|25504|B₃₅₀AB|(11×28³⁵²−767)/27|http://factordb.com/cert.php?id=1100000003850161949| |28|25505|GA0₃₅₅N|458×28³⁵⁶+23|http://factordb.com/cert.php?id=1100000003850161950| |28|25506|A0₃₅₆P7P|10×28³⁵⁹+19821|http://factordb.com/cert.php?id=1100000003850161951| |28|25507|J₃₆₃H|(19×28³⁶⁴−73)/27|http://factordb.com/cert.php?id=1100000002611724460| |28|25508|4B₃₈₁|(119×28³⁸¹−11)/27|http://factordb.com/cert.php?id=1100000002611724588| |28|25509|EB0₄₀₅1|403×28⁴⁰⁶+1|proven prime by *N*−1 test (https://primes.utm.edu/prove/prove3_1.html), since *N*−1 is trivially fully factored| |28|25510|AN₄₆₁|(293×28⁴⁶¹−23)/27|http://factordb.com/cert.php?id=1100000002611724556| |28|25511|4O₆₁₄09|(44×28⁶¹⁶−6191)/9|http://factordb.com/cert.php?id=1100000000840839989| |28|25512|2D₆₄₁|(67×28⁶⁴¹−13)/27|http://factordb.com/cert.php?id=1100000002611725341| |28|25513|70₇₄₈M5|7×28⁷⁵⁰+621|http://factordb.com/cert.php?id=1100000003850161956| |28|25514|4A0₈₀₄B|122×28⁸⁰⁵+11|http://factordb.com/cert.php?id=1100000003850161957| |28|25515|LK₉₂₅F|(587×28⁹²⁶−155)/27|http://factordb.com/cert.php?id=1100000000840839978| |28|25516|J0₁₀₇₁AC5|19×28¹⁰⁷⁴+8181|http://factordb.com/cert.php?id=1100000003850161959| |28|25517|J0₁₂₅₂J5|19×28¹²⁵⁴+537|http://factordb.com/cert.php?id=1100000003850161963| |28|25518|5₁₃₀₄6F|(5×28¹³⁰⁶+1021)/27|http://factordb.com/cert.php?id=1100000003850161964| |28|25519|5₁₃₃₂P8P|(5×28¹³³⁵+426163)/27|http://factordb.com/cert.php?id=1100000003850161965| |28|25520|5I₁₃₇₀F|(17×28¹³⁷¹−11)/3|http://factordb.com/cert.php?id=1100000003850161972| |28|25521|A₁₄₂₃6F|(10×28¹⁴²⁵−2899)/27|http://factordb.com/cert.php?id=1100000000840839947| |28|25522|G0₁₈₉₉AN|16×28¹⁹⁰¹+303|http://factordb.com/cert.php?id=1100000003850161973| |28|25523|5₃₇₄₆8P|(5×28³⁷⁴⁸+2803)/27|http://factordb.com/cert.php?id=1100000003850161974| |28|25524|QO₄₂₃₉69|(242×28⁴²⁴¹−4679)/9|http://factordb.com/cert.php?id=1100000000840839934| |28|25525|D0₅₂₆₇77D|13×28⁵²⁷⁰+5697|http://factordb.com/cert.php?id=1100000003850151420| |30|2613|AN₂₀₆|(313×30²⁰⁶−23)/29|http://factordb.com/cert.php?id=1100000002327651073| |30|2614|M₂₄₁QB|(22×30²⁴³+3139)/29|http://factordb.com/cert.php?id=1100000003593408295| |30|2615|M0₅₄₇SS7|22×30⁵⁵⁰+26047|http://factordb.com/cert.php?id=1100000003593407988| |30|2616|C0₁₀₂₂1|12×30¹⁰²³+1|proven prime by *N*−1 test (https://primes.utm.edu/prove/prove3_1.html, http://factordb.com/nmoverview.php?method=1), since *N*−1 is trivially fully factored| |30|2617|5₄₈₈₂J|(5×30⁴⁸⁸³+401)/29|http://factordb.com/cert.php?id=1100000002327649423| |30|2619|OT₃₄₂₀₅|25×30³⁴²⁰⁵−1|proven prime by *N*+1 test (https://primes.utm.edu/prove/prove3_2.html, http://factordb.com/nmoverview.php?method=2), since *N*+1 is trivially fully factored| Condensed table for bases 2 ≤ *b* ≤ 36: (the bases *b* = 11, 13, 16, 17, 19, 21\~23, 25\~36 data assumes the primality of the probable primes) (This data assumes that a number > 10²⁵⁰⁰⁰ which has passed the Miller–Rabin primality tests to all prime bases *p* < 64 and has passed the Baillie–PSW primality test and has trial factored to 10¹⁶ is in fact prime, since in some cases (e.g. *b* = 11) a candidate for minimal prime base *b* is too large to be proven prime rigorously) |*b*|number of minimal primes base *b*|base-*b* form of the top 10 known minimal prime base *b* (write "*d*_*n*" if there are 5 or more (*n*) consecutive same digits *d*)|length of the top 10 known minimal prime base *b*|algebraic ((*a*×*b*^*n*+*c*)/*gcd*(*a*+*c*,*b*−1)) form of the top 10 known minimal prime base *b*|number of unsolved families in base *b*|searching limit of length for the unsolved families in base *b* (if there are different searching limits for the unsolved families in base *b*, choose the lowest searching limit) |---|---|---|---|---|---|---| |2|1|11|2|3|0|–| |3|3|111
21
12|3
2
2|13
7
5|0|–| |4|5|221
31
23
13
11|3
2
2
2
2|41
13
11
7
5|0|–| |5|22|10₉₃13
300031
44441
33331
33001
30301
14444
10103
3101
414|96
6
5
5
5
5
5
5
4
3|5⁹⁵+8
9391
3121
2341
2251
1951
1249
653
401
109|0|–| |6|11|40041
4441
4401
51
45
35
31
25
21
15|5
4
4
2
2
2
2
2
2
2|5209
1033
1009
31
29
23
19
17
13
11|0|–| |7|71|3₁₆1
510₇1
3₆01
1100021
531101
351101
300053
150001
100121
40054|17
10
8
7
6
6
6
6
6
5|(7¹⁷−5)/2
36×7⁸+1
(7⁸−47)/2
134471
91631
62819
50459
28813
16871
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5₁₃25
7₁₂1
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74₇1
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5₅025
5550525
5500525
4₅77|221
15
13
11
9
9
8
7
7
7|(4×8²²¹+17)/7
(5×8¹⁵−173)/7
8¹³−7
(28669×8⁷−25)/7
(53×8⁸−25)/7
(4×8⁹−25)/7
(5×8⁸−2413)/7
1495381
1474901
(4×8⁷+185)/7|0|–| |9|151|30₁₁₅₈11
27₆₈₆07
76₃₂₉2
561₃₆
10₂₅57
30₂₀51
8₁₉335
727₁₅07
51₁₃61
10₁₂507|1161
689
331
38
28
23
22
19
16
15|3×9¹¹⁶⁰+10
(23×9⁶⁸⁸−511)/8
(31×9³³⁰−19)/4
(409×9³⁶−1)/8
9²⁷+52
3×9²²+46
9²²−454
(527×9¹⁷−511)/8
(41×9¹⁵+359)/8
9¹⁴+412|0|–| |10|77|50₂₈27
5₁₁1
805₅1
66600049
66000049
60₅49
220₅1
5200007
946669
666649|31
12
8
8
8
8
8
7
6
6|5×10³⁰+27
(5×10¹²−41)/9
(725×10⁶−41)/9
66600049
66000049
6×10⁷+49
22×10⁶+1
5200007
946669
666649|0|–| |11|1068|57₆₂₆₆₈
557₁₀₁₁
7₇₅₉44
A₇₁₃58
85₂₂₀05
507₂₀₆
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50₁₂₆57
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326₁₂₂|62669
1013
761
715
223
208
163
129
128
124|(57×11⁶²⁶⁶⁸−7)/10
(607×11¹⁰¹¹−7)/10
(7×11⁷⁶¹−367)/10
11⁷¹⁵−58
(17×11²²²−111)/2
(557×11²⁰⁶−7)/10
(11¹⁶³−57)/2
5×11¹²⁸+62
11¹²⁷+56
(178×11¹²²−3)/5|0|–| |12|106|40₃₉77
B0₂₇9B
B₆99B
AA0₅1
B00099B
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A00065
44AAA1
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30
9
8
7
7
6
6
6
5|4×12⁴¹+91
11×12²⁹+119
12⁹−313
130×12⁶+1
32847239
32555521
2985817
2488397
1097113
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80₃₂₀₁₇111
C5₂₃₇₅₅C
C₁₀₆₃₁92
B0₆₅₄₀BBA
390₆₂₆₆1
1770₂₇₀₃17
720₂₂₉₇2
930₁₅₅₁1
7₁₅₀₄1|197421
32021
23757
10633
6544
6269
2708
2300
1554
1505|(113×13¹⁹⁷⁴²⁰−5)/12
8×13³²⁰²⁰+183
(149×13²³⁷⁵⁶+79)/12
13¹⁰⁶³³−50
11×13⁶⁵⁴³+2012
48×13⁶²⁶⁷+1
267×13²⁷⁰⁵+20
93×13²²⁹⁸+2
120×13¹⁵⁵²+1
(7×13¹⁵⁰⁵−79)/12|1|358000| |14|650|4D₁₉₆₉₈
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8₈₆B
40₈₃49
8C₇₉3
18₇₉B
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4₆₃09
A₅₉3|19699
710
144
87
86
81
81
80
65
60|5×14¹⁹⁶⁹⁸−1
47×14⁷⁰⁸−1
9×14¹⁴³−79
(8×14⁸⁷+31)/13
4×14⁸⁵+65
(116×14⁸⁰−129)/13
(21×14⁸⁰+31)/13
(89×14⁷⁹−1649)/13
(4×14⁶⁵−667)/13
(10×14⁶⁰−101)/13|0|–| |15|1284|7₁₅₅97
E₁₄₅397
96₁₀₄08
7₇₃CE
7₅₉CCE
50₃₃17
EB₃₁
6330₂₆1
7050₂₄B
B70₂₄1|157
148
107
75
62
36
32
30
28
27|(15¹⁵⁷+59)/2
15¹⁴⁸−2558
(66×15¹⁰⁶−619)/7
(15⁷⁵+163)/2
(15⁶²+2413)/2
5×15³⁵+22
(207×15³¹−11)/14
1398×15²⁷+1
1580×15²⁵+11
172×15²⁵+1|0|–| |16|2347|3₁₁₆₁₃₇AF
4₇₂₇₈₅DD
DB₃₂₂₃₄
D0B₁₇₈₀₄
5BC₃₇₀₀D
90₃₅₄₂91
300F₁₉₆₀AF
20₁₇₁₃321
F8₁₅₁₇F
FAF₁₀₆₂45|116139
72787
32235
17806
3703
3545
1965
1717
1519
1066|(16¹¹⁶¹³⁹+619)/5
(4×16⁷²⁷⁸⁷+2291)/15
(206×16³²²³⁴−11)/15
(3131×16¹⁷⁸⁰⁴−11)/15
(459×16³⁷⁰¹+1)/5
9×16³⁵⁴⁴+145
769×16¹⁹⁶²−81
2×16¹⁷¹⁶+801
(233×16¹⁵¹⁸+97)/15
251×16¹⁰⁶⁴−187|0|–| |17|10409\~10427|B₆₇₁₀₃2E
570₅₁₃₁₀1
E9B₄₄₇₃₂
D0GD₃₇₀₉₆
G7₃₂₀₇₂F
150₂₄₃₂₅D
347₁₆₀₇₄
B30₁₃₀₇₇D
9D0₁₀₃₉₈5
10₉₀₁₉1F|67105
51313
44734
37099
32074
24328
16076
13080
10401
9022|(11×17⁶⁷¹⁰⁵−2411)/16
92×17⁵¹³¹¹+1
(3963×17⁴⁴⁷³²−11)/16
(60381×17³⁷⁰⁹⁶−13)/16
(263×17³²⁰⁷³+121)/16
22×17²⁴³²⁶+13
(887×17¹⁶⁰⁷⁴−7)/16
190×17¹³⁰⁷⁸+13
166×17¹⁰³⁹⁹+5
17⁹⁰²¹+32|18|100000| |18|549|C0₆₂₆₈C5
H₇₆₆FH
80₂₉₈B
C0₁₁₆F5
HD₉₃
GG0₃₀1
CF₃₀5
B₁₉6B
CCF₁₄5
7₁₄G7|6271
768
300
119
94
33
32
21
17
16|12×18⁶²⁷⁰+221
18⁷⁶⁸−37
8×18²⁹⁹+11
12×18¹¹⁸+275
(302×18⁹³−13)/17
304×18³¹+1
(219×18³¹−185)/17
(11×18²¹−1541)/17
(3891×18¹⁵−185)/17
(7×18¹⁶+2747)/17|0|–| |19|31412\~31435|H₈₆₂₉₁6
D90₇₃₀₄₆9
4F0₄₉₈₄₇6
2₄₈₂₂₄7
2₄₅₈₈₆7A
90₄₂₉₉₄G
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3335₃₁₀₈₈
B₂₆₅₈₈FG
10₂₂₇₉₀7717|86292
73049
49850
48225
45888
42996
36273
31091
26590
22795|(17×19⁸⁶²⁹²−215)/18
256×19⁷³⁰⁴⁷+9
91×19⁴⁹⁸⁴⁸+6
(19⁴⁸²²⁵+44)/9
(19⁴⁵⁸⁸⁸+926)/9
9×19⁴²⁹⁹⁵+16
(245×19³⁶²⁷²−11)/18
(20579×19³¹⁰⁸⁸−5)/18
(11×19²⁶⁵⁹⁰+1447)/18
19²²⁷⁹⁴+50566|23|100000| |20|3314|G0₆₂₆₉D
CD₂₄₄₉
50₁₁₆₃AJ
J₆₅₅05J
JCJ₆₂₉
E₅₆₆C7
3A₅₂₇3
G₄₄₇99
EC0₄₂₉7
40₃₈₇404B|6271
2450
1166
658
631
568
529
449
432
392|16×20⁶²⁷⁰+13
(241×20²⁴⁴⁹−13)/19
5×20¹¹⁶⁵+219
20⁶⁵⁸−7881
393×20⁶²⁹−1
(14×20⁵⁶⁸−907)/19
(67×20⁵²⁸−143)/19
(16×20⁴⁴⁹−2809)/19
292×20⁴³⁰+7
4×20³⁹¹+32091|0|–| |21|13382\~13394|40₄₇₃₃₃9G
B90₄₅₀₁₉E5
HD₃₇₄₁₄
BD₃₅₀₂₇B
990₃₃₂₃₉99H
5₃₀₆₀₆FEK
43₂₉₂₃₆B
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9₂₁₁₂₆0D
5D0₁₉₈₄₈1|47336
45023
37415
35029
33244
30609
29238
23306
21128
19851|4×21⁴⁷³³⁵+205
240×21⁴⁵⁰²¹+299
(353×21³⁷⁴¹⁴−13)/20
(233×21³⁵⁰²⁸−53)/20
198×21³³²⁴²+4175
(21³⁰⁶⁰⁹+18455)/4
(83×21²⁹²³⁷+157)/20
(19×21²³³⁰⁶−5479)/20
(9×21²¹¹²⁸−3709)/20
118×21¹⁹⁸⁴⁹+1|12|50000| |22|8003|BK₂₂₀₀₁5
7₃₈₁₅2L
L₂₃₈₅KE7
7₉₅₉K7
J0₇₆₇IGGJ
K0₇₆₀EC1
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E60₄₉₆L
L₄₈₃G3
L0₄₅₄B63|22003
3817
2388
961
772
764
628
499
485
458|(251×22²²⁰⁰²−335)/21
(22³⁸¹⁷−289)/3
22²³⁸⁸−653
(22⁹⁶¹+857)/3
19×22⁷⁷¹+199779
20×22⁷⁶³+7041
(6×22⁶²⁸−1259)/7
314×22⁴⁹⁷+21
22⁴⁸⁵−129
21×22⁴⁵⁷+5459|0|–| |23|65168\~65268|9₄₇₉₆₈7
H₃₈₉₉₄29
L35I₃₆₈₅₈
L₃₅₈₈₄D₅
L97₃₅₃₃₃
3D₃₄₈₅₄G
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HHLH0₃₂₈₂₃H
5553₃₁₉₅₄
J₃₁₅₄₃A4|47969
38996
36861
35889
35335
34856
34434
32828
31957
31545|(9×23⁴⁷⁹⁶⁹−53)/22
(17×23³⁸⁹⁹⁶−7783)/22
(123022×23³⁶⁸⁵⁸−9)/11
(21×23³⁵⁸⁸⁹−8×23⁵−13)/22
(10831×23³⁵³³³−7)/22
(79×23³⁴⁸⁵⁵+53)/22
268×23³⁴⁴³²+13
216332×23³²⁸²⁴+17
(60833×23³¹⁹⁵⁴−3)/22
(19×23³¹⁵⁴⁵−4903)/22|100|50000 |24|3409|N00N₈₁₂₉LN
88N₅₉₅₁
A0₂₉₅₁8ID
D₂₆₉₈LD
N₂₆₄₄LLN
BC0₃₃₁B
20₃₁₃7
C7₂₉₈
D0₂₅₉KKD
I0₂₄₁I5|8134
5953
2955
2700
2647
334
315
299
263
244|13249×24⁸¹³¹−49
201×24⁵⁹⁵¹−1
10×24²⁹⁵⁴+5053
(13×24²⁷⁰⁰+4403)/23
24²⁶⁴⁷−1201
276×24³³²+11
2×24³¹⁴+7
(283×24²⁹⁸−7)/23
13×24²⁶²+12013
18×24²⁴³+437|0|–| |25|133625~133724|5J₄₆₇₂₈
JD10₄₆₀₃₇D07
4F₄₂₇₈₃OO
D₄₁₆₆₇G
GHN0₄₀₄₄₄H
5₃₇₉₈₁A8
DH0H₃₅₇₇₃
50₃₄₁₅₁HHBB
H₃₂₆₈₃FH
M21₃₁₇₄₁|46729
46043
42786
41668
40448
37983
35776
34156
32685
31743|(139×25⁴⁶⁷²⁸−19)/24
12201×25⁴⁶⁰⁴⁰+8132
(37×25⁴²⁷⁸⁵+1867)/8
(13×25⁴¹⁶⁶⁸+59)/24
10448×25⁴⁰⁴⁴⁵+17
(5×25³⁷⁹⁸³+3067)/24
(205217×25³⁵⁷⁷³−17)/24
5×25³⁴¹⁵⁵+276536
(17×25³²⁶⁸⁵−1217)/24
(13249×25³¹⁷⁴¹−1)/24|99|50000| |26|25255\~25259|M0₆₁₁₈₆2BB
J0₄₄₃₀₃KCB
6K₂₃₃₀₀5
LD0₂₀₉₇₅7
7₂₀₂₇₉OL
5₁₉₃₉₁6F
9GDK₁₅₉₂₀P
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K0₄₃₆₄I5
J₄₂₂₂P|61190
44307
23302
20978
20281
19393
15924
8773
4367
4223|22×26⁶¹¹⁸⁹+1649
19×26⁴⁴³⁰⁶+13843
(34×26²³³⁰¹−79)/5
559×26²⁰⁹⁷⁶+7
(7×26²⁰²⁸¹+11393)/25
(26¹⁹³⁹³+179)/5
(32569×26¹⁵⁹²¹+21)/5
(22×26⁸⁷⁷³+53)/25
20×26⁴³⁶⁶+473
(19×26⁴²²³+131)/25|4|100000| |27|102848~102896|ME₄₉₆₄₀9G
PH0₄₇₈₉₀1
QF₄₇₁₆₅AF5
J0₄₀₇₉₁PD
510₃₉₁₆₄I07
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153F₃₅₈₃₁5
L₃₅₅₆₄GLG
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BF₂₇₅₁₄8|49643
47893
47169
40794
39169
36333
35835
35567
33404
27516|(293×27⁴⁹⁶⁴²−1736)/13
692×27⁴⁷⁸⁹¹+1
(691×27⁴⁷¹⁶⁸−95045)/26
19×27⁴⁰⁷⁹³+688
136×27³⁹¹⁶⁷+13129
17222×27³⁶³³⁰+23
(22557×27³⁵⁸³²−275)/26
(21×27³⁵⁵⁶⁷−94921)/26
698×27³³⁴⁰²+19
(301×27²⁷⁵¹⁵−197)/26|48|50000| |28|25528\~25529|O4O₉₄₅₃₅9
5OA₃₁₂₃₈F
N6₂₄₀₅₁LR
D0₅₂₆₇77D
QO₄₂₃₉69
5₃₇₄₆8P
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A₁₄₂₃6F
5I₁₃₇₀F
5₁₃₃₂P8P|94538
31241
24054
5271
4242
3748
1902
1425
1372
1335|(6092×28⁹⁴⁵³⁶−143)/9
(4438×28³¹²³⁹+125)/27
(209×28²⁴⁰⁵³+3967)/9
13×28⁵²⁷⁰+5697
(242×28⁴²⁴¹−4679)/9
(5×28³⁷⁴⁸+2803)/27
16×28¹⁹⁰¹+303
(10×28¹⁴²⁵−2899)/27
(17×28¹³⁷¹−11)/3
(5×28¹³³⁵+426163)/27|1|543202| |29||||||| |30|2619|OT₃₄₂₀₅
I0₂₄₆₀₈D
5₄₈₈₂J
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M₂₄₁QB
AN₂₀₆
50₁₆₄B
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J₉₄QQJ|34206
24610
4883
1024
551
243
207
166
155
97|25×30³⁴²⁰⁵−1
18×30²⁴⁶⁰⁹+13
(5×30⁴⁸⁸³+401)/29
12×30¹⁰²³+1
22×30⁵⁵⁰+26047
(22×30²⁴³+3139)/29
(313×30²⁰⁶−23)/29
5×30¹⁶⁵+11
(19×30¹⁵⁵+6071)/29
(19×30⁹⁷+188771)/29|0|–| |31||||||| |32|168832\~169017|V090₁₈₈₆₇D
T0₁₈₇₆₂F
DM₁₈₀₀₄L
F₁₇₇₈₃H
V₁₇₇₅₃33
A₁₇₆₅₀I5
80₁₇₁₈₆MJ
V6₁₇₁₀₇9
V₁₆₇₅₅O3
C0₁₆₇₃₇AAA9|18871
18764
18006
17784
17755
17652
17189
17109
16757
16742|31753×32¹⁸⁸⁶⁸+13
29×32¹⁸⁷⁶³+15
(425×32¹⁸⁰⁰⁵−53)/31
(15×32¹⁷⁷⁸⁴+47)/31
32¹⁷⁷⁵⁵−925
(10×32¹⁷⁶⁵²+7771)/31
8×32¹⁷¹⁸⁸+723
(967×32¹⁷¹⁰⁸+87)/31
32¹⁶⁷⁵⁷−253
12×32¹⁶⁷⁴¹+338249|185|20000| |33|279960\~280095|F₁₉₃₅₅6UW
F₁₉₀₇₈A3K
BU₁₈₉₃₄RP
WWC₁₈₅₉₉H
L₁₈₂₄₇BO7
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MF₁₆₉₀₁35
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T9₁₆₅₆₆U4
70₁₆₅₅₄7V|19358
19081
18937
18602
18250
17335
16904
16797
16569
16557|(15×33¹⁹³⁵⁸−297263)/32
(15×33¹⁹⁰⁸¹−186767)/32
(191×33¹⁸⁹³⁶−1679)/16
(8707×33¹⁸⁶⁰⁰+37)/8
(21×33¹⁸²⁵⁰−345781)/32
(27×33¹⁷³³⁵−16411)/32
(719×33¹⁶⁹⁰³−13007)/32
(413×33¹⁶⁷⁹⁶−31037)/32
(937×33¹⁶⁵⁶⁸+22007)/32
7×33¹⁶⁵⁵⁶+262|135|20000| |34|184772\~184833|UKN₄₉₈₄₅
I₄₆₉₄₆8FF
M₄₅₃₁₀UIF
QG₄₄₆₆₃L
W0₄₃₆₆₉MKN
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F₃₄₀₁₃X5
K0₃₂₉₀₁E1
K7₃₂₀₂₁|49847
46949
45313
44665
43673
41736
38243
34015
32904
32022|(34343×34⁴⁹⁸⁴⁵−23)/33
(6×34⁴⁶⁹⁴⁹−128321)/11
(2×34⁴⁵³¹³+27313)/3
(874×34⁴⁴⁶⁶⁴+149)/33
32×34⁴³⁶⁷²+26135
792×34⁴¹⁷³⁴+1
20×34³⁸²⁴²+22119
(5×34³⁴⁰¹⁵+6617)/11
20×34³²⁹⁰³+477
(667×34³²⁰²¹−7)/33|61|50000| |35||||||| |36|35286\~35290|P₈₁₉₉₃SZ
S0₇₅₀₀₇8H
7K₂₆₅₆₇Z
J₁₀₁₁₇LJ
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T09₄₆₁₈1
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OZ₃₉₃₂AZ|81995
75010
26569
10119
7261
6180
5780
4621
4564
3935|(5×36⁸¹⁹⁹⁵+821)/7
28×36⁷⁵⁰⁰⁹+305
(53×36²⁶⁵⁶⁸+101)/7
(19×36¹⁰¹¹⁹+2501)/35
1137×36⁷²⁵⁹+19
528×36⁶¹⁷⁸+31
16×36⁵⁷⁷⁹−1163
(36549×36⁴⁶¹⁹−289)/35
(979×36⁴⁵⁶³−629)/35
25×36³⁹³⁴−901|4|100000| Links for top (probable) primes: * https://primes.utm.edu/primes/lists/all.txt (top definitely primes) * https://primes.utm.edu/primes/lists/all.zip (top definitely primes, zipped file) * https://primes.utm.edu/primes/lists/all.pdf (top definitely primes, pdf version) * https://primes.utm.edu/primes/lists/all_pdf.zip (top definitely primes, pdf version, zipped file) * https://primes.utm.edu/primes/lists/short.txt (definitely primes with ≥ 800000 decimal digits) * https://primes.utm.edu/primes/lists/short.pdf (definitely primes with ≥ 800000 decimal digits, pdf version) * https://primes.utm.edu/primes/lists/short_pdf.zip (definitely primes with ≥ 800000 decimal digits, pdf version, zipped file) * https://primes.utm.edu/primes/search.php?OnList=all&Number=1000000&Style=HTML (all numbers in the list of top definitely primes) * https://primes.utm.edu/primes/download.php (index page of top definitely primes) * https://primes.utm.edu/largest.html (the information page of top definitely primes) * https://primes.utm.edu/notes/by_year.html (the information page of the largest known prime by year) * https://primes.utm.edu/notes/faq/why.html (the information page of why do people find these large primes) * https://primes.utm.edu/primes/search.php (search page of top definitely primes) * https://primes.utm.edu/primes/search.php?Advanced=1 (advanced search page of top definitely primes) * https://primes.utm.edu/primes/search_proth.php (search page of top definitely primes of the form *k*×*b*^*n*±1) * https://primes.utm.edu/primes/status.php (verification status page of top definitely primes) * https://primes.utm.edu/top20/index.php (the top 20 definitely primes of certain selected forms) * https://primes.utm.edu/bios/submission.php (submit page of top definitely primes) * https://primes.utm.edu/bios/newprover.php (submit page of top definitely primes, create a new prover account) * https://primes.utm.edu/bios/newcode.php (submit page of top definitely primes, create a new prover code) * https://primes.utm.edu/bios/index.php (index of the provers and programs and projects of top definitely primes) * http://www.primenumbers.net/prptop/prptop.php (top probable primes) * http://www.primenumbers.net/prptop/latest.php (recently found top probable primes) * http://www.primenumbers.net/prptop/searchform.php (search page of top probable primes) * http://www.primenumbers.net/prptop/searchform.php?form=%3F&action=Search (all numbers in the list of top probable primes) * http://www.primenumbers.net/prptop/submit.php (submit page of top probable primes) * http://www.primenumbers.net/prptop/topdisc.php (index of the provers of top probable primes) Other researches for the digits of the primes: Left-truncatable primes (https://en.wikipedia.org/wiki/Truncatable_prime, https://primes.utm.edu/glossary/xpage/LeftTruncatablePrime.html, https://mathworld.wolfram.com/TruncatablePrime.html), i.e. every nonempty suffix is prime: 1. http://primerecords.dk/left-truncatable.txt (base 10) 2. http://chesswanks.com/num/LTPs/ (bases 3 to 120) 3. https://www.ams.org/journals/mcom/1977-31-137/S0025-5718-1977-0427213-2/S0025-5718-1977-0427213-2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_28.pdf) (bases 3 to 11) 4. https://oeis.org/A103443 (largest left-truncatable prime in base *b*) 5. https://oeis.org/A103463 (length of the largest left-truncatable prime in base *b*) 6. https://oeis.org/A076623 (number of left-truncatable primes in base *b*) Right-truncatable primes (https://en.wikipedia.org/wiki/Truncatable_prime, https://primes.utm.edu/glossary/xpage/RightTruncatablePrime.html, https://mathworld.wolfram.com/TruncatablePrime.html), i.e. every nonempty prefix is prime: 1. http://primerecords.dk/right-truncatable.txt (base 10) 2. http://fatphil.org/maths/rtp/rtp.html (bases 3 to 90) 3. https://www.ams.org/journals/mcom/1977-31-137/S0025-5718-1977-0427213-2/S0025-5718-1977-0427213-2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_28.pdf) (bases 3 to 15) 4. https://oeis.org/A023107 (largest right-truncatable prime in base *b*) 5. https://oeis.org/A103483 (length of the largest right-truncatable prime in base *b*) 6. https://oeis.org/A076586 (number of right-truncatable primes in base *b*) Other researches for the minimal elements of other subsets of positive integers written in the positional numeral system with radix *b*, as digit strings with subsequence ordering: Primes == 1 mod 4: 1. https://www.primepuzzles.net/puzzles/puzz_178.htm 2. https://oeis.org/A111055 Primes == 3 mod 4: 1. https://www.primepuzzles.net/puzzles/puzz_178.htm 2. https://oeis.org/A111056 Palindromic primes: 1. https://www.primepuzzles.net/puzzles/puzz_178.htm 2. https://oeis.org/A114835 Composites: 1. https://oeis.org/A071070 Squares: 1. http://recursed.blogspot.com/2006/12/prime-game.html 2. https://oeis.org/A130448 Powers of 2: 1. https://oeis.org/A071071/a071071.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_13.pdf) 2. https://oeis.org/A071071 Multiples of 3: 1. https://oeis.org/A071073 Multiples of 4: 1. https://oeis.org/A071072 Other sets: 1. https://arxiv.org/pdf/1607.01548.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_14.pdf) (sums of three squares, quadratic residues mod 6, quadratic residues mod 7, range of Euler’s totient function, range of "Euler’s totient function + 3", range of Dedekind psi function, perfect numbers) 2. https://nntdm.net/papers/nntdm-25/NNTDM-25-1-036-047.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_15.pdf) (range of "Euler’s totient function + *n*", for 0 ≤ *n* ≤ 5) Tools about this research: (in fact, you can also use *Wolfram Alpha* (https://www.wolframalpha.com/) or online *Magma* calculator (http://magma.maths.usyd.edu.au/calc/) or *Pari*/*GP* (https://pari.math.u-bordeaux.fr/) or *Wolfram Mathematica* (https://www.wolfram.com/mathematica/) or *Maple* (https://www.maplesoft.com/)) Prime checkers: 1. https://primes.utm.edu/curios/includes/primetest.php 2. https://www.numberempire.com/primenumbers.php 3. http://www.numbertheory.org/php/lucas.html 4. http://www.javascripter.net/faq/numberisprime.htm 5. http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm 6. http://www.javascripter.net/math/calculators/100digitbigintcalculator.htm (just type *x* and click "prime?") 7. https://www.bigprimes.net/primalitytest 8. https://www.archimedes-lab.org/primOmatic.html 9. http://www.sonic.net/~undoc/java/PrimeCalc.html 10. http://www.proftnj.com/calcprem.htm (in French) (use the box "Rechercher si un nombre est premier" and click "Rechercher") 11. https://primes.utm.edu/nthprime/ (calculate the *n*th prime) 12. http://factordb.com/nextprime.php (calculate the next (probable) prime above *N*) Integer factorizers: 1. https://www.numberempire.com/numberfactorizer.php 2. https://www.alpertron.com.ar/ECM.HTM 3. http://www.javascripter.net/math/calculators/primefactorscalculator.htm 4. https://betaprojects.com/calculators/prime_factors.html 5. https://www.emathhelp.net/calculators/pre-algebra/prime-factorization-calculator/ 6. http://www.numbertheory.org/php/factor.html 7. https://primefan.tripod.com/Factorer.html 8. https://www.calculatorsoup.com/calculators/math/prime-factors.php 9. https://www.calculator.net/prime-factorization-calculator.html 10. http://www.se16.info/js/factor.htm 11. http://math.fau.edu/Richman/mla/factor-f.htm 12. http://www.rsok.com/~jrm/factor.html 13. http://www.brennen.net/primes/FactorApplet.html (need run with Java) 14. https://web.archive.org/web/20161004191531/http://britton.disted.camosun.bc.ca/jbprimefactor.htm 15. http://wims.unice.fr/~wims/en_tool~algebra~factor.en.html 16. http://www.analyzemath.com/Calculators_3/prime_factors.html 17. http://www.proftnj.com/calcprem.htm (in French) (use the box "Factoriser un nombre" and click "Factoriser") Base converters: 1. https://baseconvert.com/ 2. https://www.calculand.com/unit-converter/zahlen.php 3. https://www.dcode.fr/base-n-convert 4. https://www.cut-the-knot.org/Curriculum/Algorithms/BaseConversion.shtml 5. http://www.tonymarston.net/php-mysql/converter.php 6. http://math.fau.edu/Richman/mla/convert.htm 7. https://web.archive.org/web/20190629223750/http://thedevtoolkit.com/tools/base_conversion 8. http://www.kwuntung.net/hkunit/base/base.php (in Chinese) 9. https://linesegment.web.fc2.com/application/math/numbers/RadixConversion.html (in Japanese) Expression generators: 1. https://stdkmd.net/nrr/exprgen.htm (only support base 10 forms) 2. https://www.numberempire.com/simplifyexpression.php (e.g. for the form 5{7} in base 11, type "5\*11^n+7\*(11^n-1)/10") Lists of small primes: 1. https://primes.utm.edu/lists/small/1000.txt 2. https://primes.utm.edu/lists/small/10000.txt 3. https://primes.utm.edu/lists/small/100000.txt 4. https://primes.utm.edu/lists/small/millions/ 5. https://oeis.org/A000040/a000040.txt 6. https://oeis.org/A000040/b000040_1.txt 7. https://oeis.org/A000040/a000040_1B.7z 8. https://metanumbers.com/prime-numbers 9. https://www.calculatorsoup.com/calculators/math/prime-numbers.php 10. https://www2.cs.arizona.edu/icon/oddsends/primes.htm 11. https://cdn1.byjus.com/wp-content/uploads/2021/10/Prime-Numbers-from-1-to-1000.png 12. http://noe-education.org/D11102.php (in French) 13. https://web.archive.org/web/20060513054350/http://www.walter-fendt.de/m14i/primes_i.htm (in Italian) 14. https://primefan.tripod.com/500Primes1.html 15. https://www.gutenberg.org/files/65/65.txt 16. http://www.primos.mat.br/indexen.html 17. https://www.walter-fendt.de/html5/men/primenumbers_en.htm 18. http://www.rsok.com/~jrm/printprimes.html 19. http://www.numbertheory.org/php/prime_generator.html 20. https://jocelyn.quizz.chat/np/cache/index.html (in French) 21. http://www.sosmath.com/tables/prime/prime.html 22. https://www.bigprimes.net/archive/prime 23. https://web.archive.org/web/20201130071856/http://www.mathematical.com/primelist1to100kk.html 24. https://web.archive.org/web/20191118082053/http://www.tsm-resources.com/alists/prim.html 25. https://web.archive.org/web/20090917191047/http://planetmath.org/encyclopedia/FirstThousandPositivePrimeNumbers.html 26. https://faculty.lynchburg.edu/~nicely/gaps/gaplist.html (the longest list ever calculated, with all primes < 2⁶⁴ (but unlikely other lists here, the primes are not all stored), see https://primes.utm.edu/notes/faq/LongestList.html) 27. https://en.wikipedia.org/wiki/List_of_prime_numbers#The_first_1000_prime_numbers Lists of factorizations of small integers: 1. http://primefan.tripod.com/500factored.html 2. http://www.sosmath.com/tables/factor/factor.html 3. http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/ 4. https://oeis.org/A027750/a027750.txt (all (prime or composite or unit) factors of *N*) 5. http://factorzone.tripod.com/factors.htm (all (prime or composite or unit) factors of *N*) 6. http://functions.wolfram.com/NumberTheoryFunctions/Divisors/03/02 (all (prime or composite or unit) factors of *N*) 7. https://en.wikipedia.org/wiki/Table_of_prime_factors 8. https://en.wikipedia.org/wiki/Table_of_divisors (all (prime or composite or unit) factors of *N*) Lists of small integers in various bases: 1. https://en.wikipedia.org/wiki/Table_of_bases Also, programs related to this research: (some of these programs can also be downloaded in http://www.fermatsearch.org/download.php or https://www.mersenne.org/download/freeware.php) Primality (or probable primality) testing (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://primes.utm.edu/prove/index.html) programs (https://www.rieselprime.de/ziki/Primality_testing_program): 1. *LLR* (http://jpenne.free.fr/index2.html, https://primes.utm.edu/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64) 2. *PFGW* (https://sourceforge.net/projects/openpfgw/, https://primes.utm.edu/bios/page.php?id=175, https://www.rieselprime.de/ziki/PFGW, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/pfgw_win_4.0.3) 3. *PRIMO* (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://primes.utm.edu/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309) 4. *CHG* (https://mersenneforum.org/attachment.php?attachmentid=21133&d=1571237465, https://primes.utm.edu/bios/page.php?id=797, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/CHG) Sieving (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html) programs (https://www.rieselprime.de/ziki/Sieving_program): 1. *SRSIEVE* (https://www.bc-team.org/app.php/dlext/?cat=3, http://web.archive.org/web/20160922072340/https://sites.google.com/site/geoffreywalterreynolds/programs/, http://www.rieselprime.de/dl/CRUS_pack.zip, https://primes.utm.edu/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) 2. *MTSIEVE* (https://sourceforge.net/projects/mtsieve/, https://primes.utm.edu/bios/page.php?id=449, https://www.rieselprime.de/ziki/Mtsieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/mtsieve_2.3.3) Integer factoring (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html) programs (https://www.rieselprime.de/ziki/Factoring_program): 1. *GMP*-*ECM* (https://web.archive.org/web/20210803045418/https://gforge.inria.fr/projects/ecm, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/ecm704dev-svn2990-win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/ecm704dev-svn2990-linux64, https://www.rieselprime.de/ziki/GMP-ECM) 2. *MSIEVE* (https://sourceforge.net/projects/msieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/msieve153_win64) 3. *GGNFS* (http://sourceforge.net/projects/ggnfs, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/GGNFS) 4. *CADO*-*NFS* (https://web.archive.org/web/20210506173015/http://cado-nfs.gforge.inria.fr/index.html, https://www.rieselprime.de/ziki/CADO-NFS, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/cado-nfs-2.3.0) For the files in this page: * File "kernel *b*": Data for all known minimal primes in base *b*, expressed as base *b* strings * File "left *b*": *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) families in base *b* such that we were unable to determine if they contain a prime > *b* or not (i.e. *x*{*y*}*z* (where *x* and *z* are strings (may be empty) of digits in base *b*, *y* is a digit in base *b*) families in base *b* such that no prime member > *b* could be found, nor could the family be ruled out as only containing composites (only count the numbers > *b*)), these families are sorted by "the length *n* number in these families, from the smallest number to the largest number, this *n* is large enough such that *n* replaced to any larger number does not affect the sorting" (e.g. for base 17, we sort with B{0}B3 -> B{0}DB -> {B}2BE -> {B}2E -> {B}E9 -> {B}EE, since in this case 7 digits is enough, B0000B3 < B0000DB < BBBB2BE < BBBBB2E < BBBBBE9 < BBBBBEE, if the 7 replaced to any larger number, this result of the sorting will not change) See my article about this research: https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub