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 Zn) 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×bn+1 and k×bn−1 (or proving that such prime does not exist) for all k < b (also to cover dual (http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_1.pdf), https://www.rechenkraft.net/wiki/Five_or_Bust, https://oeis.org/A076336/a076336c.html, http://www.mit.edu/~kenta/three/prime/dual-sierpinski/ezgxggdm/dualsierp-excerpt.txt, http://mit.edu/kenta/www/three/prime/dual-sierpinski/ezgxggdm/dualsierp.txt.gz, https://mersenneforum.org/showthread.php?t=10761, https://mersenneforum.org/showthread.php?t=6545) Sierpinski/Riesel conjectures base b with k-values < b, i.e. finding the smallest prime of the form bn+k and bn−k (which are the dual forms of k×bn+1 and k×bn−1, respectively) (or proving that such prime does not exist) for all k < b) (also to cover finding the smallest prime of some classic forms (or proving that such prime does not exist), such as bn+2, bn−2, bn+(b−1), bn−(b−1), 2×bn+1, 2×bn−1, (b−1)×bn+1, (b−1)×bn−1, with n ≥ 1, for the same base b (of course, for some bases b the original minimal prime base b problem already covers finding the smallest prime of these forms, e.g. the original minimal prime base b problem covers finding the smallest prime of the form (b−1)×bn+1 if and only if b−1 is not prime, and the original minimal prime base b problem covers finding the smallest prime of the form (b−1)×bn−1 if and only if neither b−1 nor b−2 is prime, but I want the problem covers finding the smallest prime of these forms for all bases b)). The original minimal prime base b problem does not cover Conjectures ‘R Us Sierpinski/Riesel conjectures base b with conjectured k (http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, 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")
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×bn+1, 4×bn+1, 6×bn+1, 10×bn+1, 12×bn+1, 16×bn+1, 3×bn−1, 4×bn−1, 6×bn−1, 8×bn−1, 12×bn−1, 14×bn−1, ... cannot be quickly eliminated with n = 0, or the conjectures become much more easy and uninteresting), for the same reason, this minimal prime puzzle requires ≥ b (i.e. ≥ 2 digits) for these primes, single-digit primes are not acceptable to avoid the trivial primes (e.g. families containing digit 2, 3, 5, 7, B, D, H, J, N, T, V, ... cannot be quickly eliminated with the single-digit prime, or the conjectures become much more easy and uninteresting).
The fourth reason for excluding the primes ≤ b is that starting with b+1 makes the formula of the number of possible (first digit,last digit) combo of a minimal prime in base b more simple and smooth number (https://en.wikipedia.org/wiki/Smooth_number, https://mathworld.wolfram.com/SmoothNumber.html), 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)
(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 2n−2 strings of length n in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case Y contains only one digit, this family is of the form x{y}z, and there is only a single string of each length > (the length of x + the length of z), and it is not known if the following decision problem (https://en.wikipedia.org/wiki/Decision_problem, https://mathworld.wolfram.com/DecisionProblem.html) is recursively solvable:
Problem: Given strings x, z, a digit y, and a base b, does there exist a prime number whose base-b expansion is of the form xynz 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 sr1/2/5sieve) 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 xynz 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 22n+1) other than the known ones (corresponding to n = 0, 1, 2, 3, 4). To see this, take b = 2, x = 1, y = 0, and z = 0161. Since if 2n+1 is prime then n must be a power of two, 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 2p−1 with prime p). To see this, take b = 2, x = 𝜆 (the empty string (https://en.wikipedia.org/wiki/Empty_string)), y = 1, and z = 1n+1, where n is the exponent of the Mersenne prime which we want to know whether it is the largest Mersenne prime or not. Since if 2n−1 is prime then n must be a prime, 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((5×11n−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 (2×3n−1) × (2×3n+1))) (which includes difference-of-two-squares factorization (https://en.wikipedia.org/wiki/Difference_of_two_squares) and sum/difference-of-two-nth-powers factorization with odd n > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and Aurifeuillean factorization (https://en.wikipedia.org/wiki/Aurifeuillean_factorization, https://www.rieselprime.de/ziki/Aurifeuillian_factor, https://mathworld.wolfram.com/AurifeuilleanFactorization.html, http://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 x4+4y4), 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×bn+c)/gcd(a+c,b−1)) form, and you will find that all numbers in these families have status (http://factordb.com/status.html, http://factordb.com/distribution.php) either "FF" or "CF", and no numbers in these families have status (http://factordb.com/status.html, http://factordb.com/distribution.php) "C" (i.e. in http://factordb.com/listtype.php?t=3), e.g. for the family 3{0}95 in base 13, its algebraic ((a×bn+c)/gcd(a+c,b−1)) form is 3×13n+2+122, and in factordb you will find that all numbers in this family are divisible by some element of {5,7,17}, see http://factordb.com/index.php?query=3*13%5E%28n%2B2%29%2B122&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show; for the family {7}D in base 21, its algebraic ((a×bn+c)/gcd(a+c,b−1)) form is (7×21n+1+113)/20, and in factordb you will find that all numbers in this family are divisible by some element of {2,13,17}, see http://factordb.com/index.php?query=%287*21%5E%28n%2B1%29%2B113%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (note: for this family n = 0 is not allowed, since we only consider the numbers > base); for the family 30{F}A0F in base 16, its algebraic ((a×bn+c)/gcd(a+c,b−1)) form is 49×16n+3−1521, and in factordb you will find that no numbers in this family have a prime factor with decimal length > ((the decimal length of the number + 1)/2), and all numbers in this family have two nearly equal (prime or composite) factors, see http://factordb.com/index.php?query=49*16%5E%28n%2B3%29-1521&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show; for the family 5{1} in base 25, its algebraic ((a×bn+c)/gcd(a+c,b−1)) form is (121×25n−1)/24, and in factordb you will find that no numbers in this family have a prime factor with decimal length > ((the decimal length of the number + 1)/2), and all numbers in this family have two nearly equal (prime or composite) factors, see http://factordb.com/index.php?query=%28121*25%5En-1%29%2F24&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (note: for this family n = 0 is not allowed, since we only consider the numbers > base); for the family {D}5 in base 14, its algebraic ((a×bn+c)/gcd(a+c,b−1)) form is 14n+1−9, and in factordb you will find that all numbers with even n in this family are divisible by 5, and you will find that no numbers with odd n in this family have a prime factor with decimal length > ((the decimal length of the number + 1)/2), and all numbers with odd n in this family have two nearly equal (prime or composite) factors, see http://factordb.com/index.php?query=14%5E%28n%2B1%29-9&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (note: for this family n = 0 is not allowed, since we only consider the numbers > base); for the family 7{9} in base 17, its algebraic ((a×bn+c)/gcd(a+c,b−1)) form is (121×17n−9)/16, and in factordb you will find that all numbers with odd n in this family are divisible by 2, and you will find that no numbers with even n in this family have a prime factor with decimal length > ((the decimal length of the number + 1)/2), and all numbers with even n in this family have two nearly equal (prime or composite) factors, see http://factordb.com/index.php?query=%28121*17%5En-9%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (note: for this family n = 0 is not allowed, since we only consider the numbers > base). In contrast, you can see the factorization of the numbers in unsolved families in base b (which are listed in the "left b" file) in factordb, you will find some numbers in these families which have neither small prime factors (say < 1016) nor two nearly equal (prime or composite) factors, also you will find some numbers in these families which have no known proper factor (https://en.wikipedia.org/wiki/Proper_factor, https://mathworld.wolfram.com/ProperFactor.html, https://mathworld.wolfram.com/ProperDivisor.html) > 1 (i.e. you will find some numbers in these families with status (http://factordb.com/status.html, http://factordb.com/distribution.php) "C" (instead of "CF" or "FF") (i.e. in http://factordb.com/listtype.php?t=3) 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×bn+c)/d) form of this family (n is the number of digits in the "{}", also the lower bound of n to make the numbers > b) (note: d divides gcd(a+c,b−1), but d need not be gcd(a+c,b−1), d = gcd(a+c,b−1) if and only if the numbers in the family are not divisible by some prime factor of b−1, i.e. the numbers in the family are coprime to b−1) |
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 | 2×10n+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 | 2×10n+1+7 (n ≥ 0) | always divisible by 3 (in fact, always divisible by 9) |
http://factordb.com/index.php?query=2*10%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show |
10 | 5{0}1 | 5×10n+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 | 5×10n+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 | 8×10n+1+1 (n ≥ 0) | always divisible by 3 (in fact, always divisible by 9) |
http://factordb.com/index.php?query=8*10%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show |
10 | 8{0}7 | 8×10n+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 | 28×10n+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 | (14×10n+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 | 3n+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 | 2×4n+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 | 5n+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 | 5n+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 | 3×5n+1+1 (n ≥ 0) | always divisible by 2 (in fact, always divisible by 4) |
http://factordb.com/index.php?query=3*5%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show |
5 | 11{0}3 | 6×5n+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 | 3×5n+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 | 4×6n+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 | (9n+1+31)/8 (n ≥ 1) | always divisible by some element of {2,5} divisible by 2 if n is odd, divisible by 5 if n is even |
http://factordb.com/index.php?query=%289%5E%28n%2B1%29%2B31%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show |
9 | {1}61 | (9n+2+359)/8 (n ≥ 0) | always divisible by some element of {2,5} divisible by 2 if n is odd, divisible by 5 if n is even |
http://factordb.com/index.php?query=%289%5E%28n%2B2%29%2B359%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show |
9 | 2{7} | (23×9n−7)/8 (n ≥ 1) | always divisible by some element of {2,5} divisible by 2 if n is even, divisible by 5 if n is odd |
http://factordb.com/index.php?query=%2823*9%5En-7%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show |
9 | {3}5 | (3×9n+1+13)/8 (n ≥ 1) | always divisible by some element of {2,5} divisible by 2 if n is odd, divisible by 5 if n is even |
http://factordb.com/index.php?query=%283*9%5E%28n%2B1%29%2B13%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show |
9 | {3}8 | (3×9n+1+37)/8 (n ≥ 1) | always divisible by some element of {2,5} divisible by 2 if n is even, divisible by 5 if n is odd |
http://factordb.com/index.php?query=%283*9%5E%28n%2B1%29%2B37%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show |
9 | {3}05 | (3×9n+2−203)/8 (n ≥ 1) | always divisible by some element of {2,5} divisible by 2 if n is odd, divisible by 5 if n is even |
http://factordb.com/index.php?query=%283*9%5E%28n%2B2%29-203%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show |
9 | 5{1} | (41×9n−1)/8 (n ≥ 1) | always divisible by some element of {2,5} divisible by 2 if n is odd, divisible by 5 if n is even |
http://factordb.com/index.php?query=%2841*9%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show |
9 | 5{7} | (47×9n−7)/8 (n ≥ 1) | always divisible by some element of {2,5} divisible by 2 if n is odd, divisible by 5 if n is even |
http://factordb.com/index.php?query=%2847*9%5En-7%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show |
9 | 6{1} | (49×9n−1)/8 (n ≥ 1) | always divisible by some element of {2,5} divisible by 2 if n is even, divisible by 5 if n is odd |
http://factordb.com/index.php?query=%2849*9%5En-1%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show |
9 | {7}2 | (7×9n+1−47)/8 (n ≥ 1) | always divisible by some element of {2,5} divisible by 2 if n is even, divisible by 5 if n is odd |
http://factordb.com/index.php?query=%287*9%5E%28n%2B1%29-47%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show |
9 | {7}5 | (7×9n+1−23)/8 (n ≥ 1) | always divisible by some element of {2,5} divisible by 2 if n is odd, divisible by 5 if n is even |
http://factordb.com/index.php?query=%287*9%5E%28n%2B1%29-23%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show |
9 | {7}05 | (3×9n+2−527)/8 (n ≥ 1) | always divisible by some element of {2,5} divisible by 2 if n is odd, divisible by 5 if n is even |
http://factordb.com/index.php?query=%287*9%5E%28n%2B2%29-527%29%2F8&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show |
9 | {1}6{1} | – | 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} | (5×11n−1)/2 (n ≥ 1) | always divisible by some element of {2,3} divisible by 2 if n is even, divisible by 3 if n is odd |
http://factordb.com/index.php?query=%285*11%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show |
11 | 3{5} | (7×11n−1)/2 (n ≥ 1) | always divisible by some element of {2,3} divisible by 2 if n is odd, divisible by 3 if n is even |
http://factordb.com/index.php?query=%287*11%5En-1%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show |
11 | 3{7} | (37×11n−7)/10 (n ≥ 1) | always divisible by some element of {2,3} divisible by 2 if n is odd, divisible by 3 if n is even |
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} | (47×11n−7)/10 (n ≥ 1) | always divisible by some element of {2,3} divisible by 2 if n is even, divisible by 3 if n is odd |
http://factordb.com/index.php?query=%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} | (17×11n−1)/2 (n ≥ 1) | always divisible by some element of {2,3} divisible by 2 if n is even, divisible by 3 if n is odd |
http://factordb.com/index.php?query=%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} | (19×11n−1)/2 (n ≥ 1) | always divisible by some element of {2,3} divisible by 2 if n is odd, divisible by 3 if n is even |
http://factordb.com/index.php?query=%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} | (97×11n−7)/10 (n ≥ 1) | always divisible by some element of {2,3} divisible by 2 if n is odd, divisible by 3 if n is even |
http://factordb.com/index.php?query=%2897*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show |
11 | A{7} | (107×11n−7)/10 (n ≥ 1) | always divisible by some element of {2,3} divisible by 2 if n is even, divisible by 3 if n is odd |
http://factordb.com/index.php?query=%28107*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show |
11 | {5}2 | (11n+1−7)/2 (n ≥ 1) | always divisible by some element of {2,3} divisible by 2 if n is even, divisible by 3 if n is odd |
http://factordb.com/index.php?query=%2811%5E%28n%2B1%29-7%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show |
11 | {5}3 | (11n+1−5)/2 (n ≥ 1) | always divisible by some element of {2,3} divisible by 2 if n is odd, divisible by 3 if n is even |
http://factordb.com/index.php?query=%2811%5E%28n%2B1%29-5%29%2F2&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show |
11 | {7}3 | (7×11n+1−47)/10 (n ≥ 1) | always divisible by some element of {2,3} divisible by 2 if n is odd, divisible by 3 if n is even |
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 | (7×11n+1−37)/10 (n ≥ 1) | always divisible by some element of {2,3} divisible by 2 if n is even, divisible by 3 if n is odd |
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 |
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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 |
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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 |
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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 |
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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) |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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9 | {1} | difference-of-squares factorization (9n−1)/8 = (3n−1) × (3n+1) / 8 |
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8 | 1{0}1 | sum-of-cubes factorization 8n+1 = (2n+1) × (4n−2n+1) |
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9 | 3{1} | difference-of-squares factorization (25×9n−1)/8 = (5×3n−1) × (5×3n+1) / 8 |
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9 | 3{8} | difference-of-squares factorization 4×9n−1 = (2×3n−1) × (2×3n+1) |
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9 | {8}5 | difference-of-squares factorization 9n−4 = (3n−2) × (3n+2) |
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9 | 3{8}35 | difference-of-squares factorization 4×9n−49 = (2×3n−7) × (2×3n+7) |
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16 | 8{F} | difference-of-squares factorization 9×16n−1 = (3×4n−1) × (3×4n+1) |
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16 | {F}7 | difference-of-squares factorization 16n−9 = (4n−3) × (4n+3) |
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16 | {4}1 | difference-of-squares factorization (4×16n−49)/15 = (2×4n−7) × (2×4n+7) / 15 |
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16 | B{4}1 | difference-of-squares factorization (169×16n−49)/15 = (13×4n−7) × (13×4n+7) / 15 |
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16 | 1{5} | difference-of-squares factorization (4×16n−1)/3 = (2×4n−1) × (2×4n+1) / 3 |
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16 | 8{5} | difference-of-squares factorization (25×16n−1)/3 = (5×4n−1) × (5×4n+1) / 3 |
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16 | 10{5} | difference-of-squares factorization (49×16n−1)/3 = (7×4n−1) × (7×4n+1) / 3 |
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16 | A1{5} | difference-of-squares factorization (484×16n−1)/3 = (22×4n−1) × (22×4n+1) / 3 |
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16 | 7{3} | difference-of-squares factorization (36×16n−1)/5 = (6×4n−1) × (6×4n+1) / 5 |
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16 | 3{F}AF | difference-of-squares factorization 4×16n−81 = (2×4n−9) × (2×4n+9) |
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16 | 30{F}AF | difference-of-squares factorization 49×16n−81 = (7×4n−9) × (7×4n+9) |
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16 | 3{F}A0F | difference-of-squares factorization 4×16n−1521 = (2×4n−39) × (2×4n+39) |
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16 | 30{F}A0F | difference-of-squares factorization 49×16n−1521 = (7×4n−39) × (7×4n+39) |
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16 | {3}23 | difference-of-squares factorization (16n−81)/5 = (4n−9) × (4n+9) / 5 (in fact, difference-of-4th-powers factorization) (16n−81)/5 = (2n−3) × (2n+3) × (4n+9) / 5 |
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16 | {5}45 | difference-of-squares factorization (16n−49)/3 = (4n−7) × (4n+7) / 3 |
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16 | {C}B | difference-of-squares factorization (4×16n−9)/5 = (2×4n−3) × (2×4n+3) / 5 |
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16 | {C}D | Aurifeuillian factorization of x4+4×y4 (4×16n+1)/5 = (2×4n−2×2n+1) × (2×4n+2×2n+1) / 5 |
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16 | {C}DD | Aurifeuillian factorization of x4+4×y4 (4×16n+81)/5 = (2×4n−6×2n+9) × (2×4n+6×2n+9) / 5 |
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25 | {1} | difference-of-squares factorization (25n−1)/24 = (5n−1) × (5n+1) / 24 |
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25 | 2{1} | difference-of-squares factorization (49×25n−1)/24 = (7×5n−1) × (7×5n+1) / 24 |
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25 | 5{1} | difference-of-squares factorization (121×25n−1)/24 = (11×5n−1) × (11×5n+1) / 24 |
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25 | 7{1} | difference-of-squares factorization (169×25n−1)/24 = (13×5n−1) × (13×5n+1) / 24 |
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25 | C{1} | difference-of-squares factorization (289×25n−1)/24 = (17×5n−1) × (17×5n+1) / 24 |
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25 | F{1} | difference-of-squares factorization (361×25n−1)/24 = (19×5n−1) × (19×5n+1) / 24 |
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25 | M{1} | difference-of-squares factorization (529×25n−1)/24 = (23×5n−1) × (23×5n+1) / 24 |
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25 | 27{1} | difference-of-squares factorization (1369×25n−1)/24 = (37×5n−1) × (37×5n+1) / 24 |
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25 | 7C{1} | difference-of-squares factorization (4489×25n−1)/24 = (67×5n−1) × (67×5n+1) / 24 |
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25 | D5{1} | difference-of-squares factorization (7921×25n−1)/24 = (89×5n−1) × (89×5n+1) / 24 |
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25 | 1{3} | difference-of-squares factorization (9×25n−1)/8 = (3×5n−1) × (3×5n+1) / 8 |
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25 | 1{8} | difference-of-squares factorization (4×25n−1)/3 = (2×5n−1) × (2×5n+1) / 3 |
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25 | 5{8} | difference-of-squares factorization (16×25n−1)/3 = (4×5n−1) × (4×5n+1) / 3 |
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25 | A{3} | difference-of-squares factorization (81×25n−1)/8 = (9×5n−1) × (9×5n+1) / 8 |
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25 | L{8} | difference-of-squares factorization (64×25n−1)/3 = (8×5n−1) × (8×5n+1) / 3 |
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25 | {3}2 | difference-of-squares factorization (25n−9)/8 = (5n−3) × (5n+3) / 8 |
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25 | {8}3 | difference-of-squares factorization (25n−16)/3 = (5n−4) × (5n+4) / 3 |
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25 | {8}7 | difference-of-squares factorization (25n−4)/3 = (5n−2) × (5n+2) / 3 |
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25 | {3}2I | difference-of-squares factorization (25n−81)/8 = (5n−9) × (5n+9) / 8 |
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25 | {8}5I | difference-of-squares factorization (25n−196)/3 = (5n−14) × (5n+14) / 3 |
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25 | {8}7C | difference-of-squares factorization (25n−64)/3 = (5n−8) × (5n+8) / 3 |
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27 | 8{0}1 | sum-of-cubes factorization 8×27n+1 = (2×3n+1) × (4×9n−2×3n+1) |
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27 | 1{0}8 | sum-of-cubes factorization 27n+8 = (3n+2) × (9n−2×3n+4) |
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27 | {D}E | sum-of-cubes factorization (27n+1)/2 = (3n+1) × (9n−3n+1) / 2 |
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27 | 7{Q} | difference-of-cubes factorization 8×27n−1 = (2×3n−1) × (4×9n+2×3n+1) |
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27 | {Q}J | difference-of-cubes factorization 27n−8 = (3n−2) × (9n+2×3n+4) |
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27 | 9{G} | difference-of-cubes factorization (125×27n−8)/13 = (5×3n−2) × (25×9n+10×3n+4) / 13 |
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32 | 1{0}1 | sum-of-5th-powers factorization 32n+1 = (2n+1) × (16n−8n+4n−2n+1) |
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32 | {1} | difference-of-5th-powers factorization (32n−1)/31 = (2n−1) × (16n+8n+4n+2n+1) / 31 |
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36 | 3{7} | difference-of-squares factorization (16×36n−1)/5 = (4×6n−1) × (4×6n+1) / 5 |
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36 | 3{Z} | difference-of-squares factorization 4×36n−1 = (2×6n−1) × (2×6n+1) |
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36 | 8{Z} | difference-of-squares factorization 9×36n−1 = (3×6n−1) × (3×6n+1) |
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36 | O{Z} | difference-of-squares factorization 25×36n−1 = (5×6n−1) × (5×6n+1) |
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36 | {Z}B | difference-of-squares factorization 36n−25 = (6n−5) × (6n+5) |
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36 | 8{Z}B | difference-of-squares factorization 9×36n−25 = (3×6n−5) × (3×6n+5) |
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36 | F{Z}B | difference-of-squares factorization 16×36n−25 = (4×6n−5) × (4×6n+5) |
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36 | {Z}RZ | difference-of-squares factorization 36n−289 = (6n−17) × (6n+17) |
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36 | F{Z}RZ | difference-of-squares factorization 16×36n−289 = (4×6n−17) × (4×6n+17) |
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36 | O{Z}RZ | difference-of-squares factorization 25×36n−289 = (5×6n−17) × (5×6n+17) |
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36 | O{5} | difference-of-squares factorization (169×36n−1)/7 = (13×6n−1) × (13×6n+1) / 7 |
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36 | O{7} | difference-of-squares factorization (121×36n−1)/5 = (11×6n−1) × (11×6n+1) / 5 |
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36 | {9}1 | difference-of-squares factorization (9×36n−289)/35 = (3×6n−17) × (3×6n+17) / 35 |
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36 | T{9}1 | difference-of-squares factorization (1024×36n−289)/35 = (32×6n−17) × (32×6n+17) / 35 |
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36 | {G}D | difference-of-squares factorization (16×36n−121)/35 = (4×6n−11) × (4×6n+11) / 35 |
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36 | {G}8D | difference-of-squares factorization (16×36n−10201)/35 = (4×6n−101) × (4×6n+101) / 35 |
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36 | R{G}D | difference-of-squares factorization (961×36n−121)/35 = (31×6n−11) × (31×6n+11) / 35 |
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36 | 3{G}8D | difference-of-squares factorization (121×36n−10201)/35 = (11×6n−101) × (11×6n+101) / 35 |
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36 | R{G}8D | difference-of-squares factorization (961×36n−10201)/35 = (31×6n−101) × (31×6n+101) / 35 |
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36 | {K}H | difference-of-squares factorization (4×36n−25)/7 = (2×6n−5) × (2×6n+5) / 7 |
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36 | {K}IH | difference-of-squares factorization (4×36n−529)/7 = (2×6n−23) × (2×6n+23) / 7 |
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36 | B{K}H | difference-of-squares factorization (81×36n−25)/7 = (9×6n−5) × (9×6n+5) / 7 |
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36 | 3{K}IH | difference-of-squares factorization (25×36n−529)/7 = (5×6n−23) × (5×6n+23) / 7 |
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36 | B{K}IH | difference-of-squares factorization (81×36n−529)/7 = (9×6n−23) × (9×6n+23) / 7 |
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36 | {S}J | difference-of-squares factorization (4×36n−49)/5 = (2×6n−7) × (2×6n+7) / 5 |
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36 | {S}IJ | difference-of-squares factorization (4×36n−1849)/5 = (2×6n−43) × (2×6n+43) / 5 |
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36 | 1{S}J | difference-of-squares factorization (9×36n−49)/5 = (3×6n−7) × (3×6n+7) / 5 |
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36 | C{S}J | difference-of-squares factorization (64×36n−49)/5 = (8×6n−7) × (8×6n+7) / 5 |
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36 | X{S}J | difference-of-squares factorization (169×36n−49)/5 = (13×6n−7) × (13×6n+7) / 5 |
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36 | 1{S}GJ | difference-of-squares factorization (9×36n−2209)/5 = (3×6n−47) × (3×6n+47) / 5 |
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36 | 9{S}GJ | difference-of-squares factorization (49×36n−2209)/5 = (7×6n−47) × (7×6n+47) / 5 |
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36 | C{S}GJ | difference-of-squares factorization (64×36n−2209)/5 = (8×6n−47) × (8×6n+47) / 5 |
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36 | X{S}GJ | difference-of-squares factorization (169×36n−2209)/5 = (13×6n−47) × (13×6n+47) / 5 |
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36 | 1{S}IJ | difference-of-squares factorization (9×36n−1849)/5 = (3×6n−43) × (3×6n+43) / 5 |
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36 | 9{S}IJ | difference-of-squares factorization (49×36n−1849)/5 = (7×6n−43) × (7×6n+43) / 5 |
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14 | 8{D} | combine of factor 5 and difference-of-squares factorization even length is divisible by 5, odd length has factorization 9×142×n−1 = (3×14n−1) × (3×14n+1) |
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12 | {B}9B | combine of factor 13 and difference-of-squares factorization odd length is divisible by 13, even length has factorization 122×n−25 = (12n−5) × (12n+5) |
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14 | {D}5 | combine of factor 5 and difference-of-squares factorization odd length is divisible by 5, even length has factorization 142×n−9 = (14n−3) × (14n+3) |
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17 | 1{9} | combine of factor 2 and difference-of-squares factorization even length is divisible by 2, odd length has factorization (25×172×n−9)/16 = (5×17n−3) × (5×17n+3) / 16 |
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17 | 7{9} | combine of factor 2 and difference-of-squares factorization even length is divisible by 2, odd length has factorization (121×172×n−9)/16 = (11×17n−3) × (11×17n+3) / 16 |
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17 | {9}2 | combine of factor 2 and difference-of-squares factorization odd length is divisible by 2, even length has factorization (9×172×n−121)/16 = (3×17n−11) × (3×17n+11) / 16 |
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17 | {9}8 | combine of factor 2 and difference-of-squares factorization odd length is divisible by 2, even length has factorization (9×172×n−25)/16 = (3×17n−5) × (3×17n+5) / 16 |
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19 | 1{6} | combine of factor 5 and difference-of-squares factorization even length is divisible by 5, odd length has factorization (4×192×n−1)/3 = (2×19n−1) × (2×19n+1) / 3 |
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19 | {6}5 | combine of factor 5 and difference-of-squares factorization odd length is divisible by 5, even length has factorization (192×n−4)/3 = (19n−2) × (19n+2) / 3 |
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19 | 7{2} | combine of factor 5 and difference-of-squares factorization even length is divisible by 5, odd length has factorization (64×192×n−1)/9 = (8×19n−1) × (8×19n+1) / 9 |
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19 | 89{6} | combine of factor 5 and difference-of-squares factorization even length is divisible by 5, odd length has factorization (484×192×n−1)/3 = (22×19n−1) × (22×19n+1) / 3 |
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24 | 3{N} | combine of factor 5 and difference-of-squares factorization even length is divisible by 5, odd length has factorization 4×242×n−1 = (2×24n−1) × (2×24n+1) |
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24 | 5{N} | combine of factor 5 and difference-of-squares factorization odd length is divisible by 5, even length has factorization 6×242×n+1−1 = (12×24n−1) × (12×24n+1) |
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24 | 8{N} | combine of factor 5 and difference-of-squares factorization even length is divisible by 5, odd length has factorization 9×242×n−1 = (3×24n−1) × (3×24n+1) |
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24 | {6}1 | combine of factor 5 and difference-of-squares factorization even length is divisible by 5, odd length has factorization (6×242×n+1−121)/23 = (12×24n−11) × (12×24n+11) / 23 |
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24 | {N}LN | combine of factor 5 and difference-of-squares factorization odd length is divisible by 5, even length has factorization 242×n−49 = (24n−7) × (24n+7) |
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33 | F{W} | combine of factor 17 and difference-of-squares factorization even length is divisible by 17, odd length has factorization 16×332×n−1 = (4×33n−1) × (4×33n+1) |
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33 | {W}H | combine of factor 17 and difference-of-squares factorization odd length is divisible by 17, even length has factorization 332×n−16 = (33n−4) × (33n+4) |
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33 | 3{P} | combine of factor 2 and difference-of-squares factorization even length is divisible by 2, odd length has factorization (121×332×n−25)/32 = (11×33n−5) × (11×33n+5) / 32 |
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33 | D{P} | combine of factor 2 and difference-of-squares factorization even length is divisible by 2, odd length has factorization (441×332×n−25)/32 = (21×33n−5) × (21×33n+5) / 32 |
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33 | {9}4 | combine of factor 2 and difference-of-squares factorization even length is divisible by 2, odd length has factorization (9×332×n−169)/32 = (3×33n−13) × (3×33n+13) / 32 |
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34 | 1{B} | combine of factor 5 and difference-of-squares factorization even length is divisible by 5, odd length has factorization (4×342×n−1)/3 = (2×34n−1) × (2×34n+1) / 3 |
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34 | 8{X} | combine of factor 5 and difference-of-squares factorization even length is divisible by 5, odd length has factorization 9×342×n−1 = (3×34n−1) × (3×34n+1) |
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34 | {X}P | combine of factor 5 and difference-of-squares factorization odd length is divisible by 5, even length has factorization 342×n−9 = (34n−3) × (34n+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 (4n−1)/3 = (2n−1) × (2n+1) / 3 |
8 | {1} | difference-of-cubes factorization but 111 is prime, and 111 is the only prime > b in this family (8n−1)/7 = (2n−1) × (4n+2n+1) / 7 |
16 | {1} | difference-of-squares factorization but 11 is prime, and 11 is the only prime > b in this family (16n−1)/15 = (4n−1) × (4n+1) / 15 |
27 | {1} | difference-of-cubes factorization but 111 is prime, and 111 is the only prime > b in this family (27n−1)/26 = (3n−1) × (9n+3n+1) / 26 |
27 | {G}7 | difference-of-cubes factorization but G7 is prime, and G7 is the only prime > b in this family (8×27n−125)/13 = (2×3n−5) × (4×9n+10×3n+25) / 13 |
36 | {1} | difference-of-squares factorization but 11 is prime, and 11 is the only prime > b in this family (36n−1)/35 = (6n−1) × (6n+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 Nn−1 can be ≥ 1/3 factored for a small n. If either N−1 or N+1 (or both) can be ≥ 1/2 factored, then we can use the Pocklington N−1 primality test (https://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 Nn−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×bn+c)/gcd(a+c,b−1) for some fixed a, b, c such that a ≥ 1, b ≥ 2 (b is the base), c ≠ 0, gcd(a,c) = 1, gcd(b,c) = 1. 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×bn+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×bn+c has algebraic factors (e.g. difference-of-two-squares factorization (https://en.wikipedia.org/wiki/Difference_of_two_squares) and sum/difference-of-two-nth-powers factorization with odd n > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html) and Aurifeuillean factorization (https://en.wikipedia.org/wiki/Aurifeuillean_factorization, https://www.rieselprime.de/ziki/Aurifeuillian_factor, https://mathworld.wolfram.com/AurifeuilleanFactorization.html, http://myfactorcollection.mooo.com:8090/source/cyclo.cpp, http://myfactorcollection.mooo.com:8090/LCD_2_199, http://myfactorcollection.mooo.com:8090/LCD_2_998) of x4+4y4), 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×bn+c)/gcd(a+c,b−1), the case of "Mersenne number" in https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/srsieve_1.1.4/algebraic.c is the case which a is rational power of b, c = −1 and the case which a is rational power of b, c = 1, gcd(a+c,b−1) ≥ 3, and the case of "GFN" in https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/srsieve_1.1.4/algebraic.c is the case which a is rational power of b, c = 1, gcd(a+c,b−1) is either 1 or 2)). Since this program cannot handle the general case (a×bn+c)/gcd(a+c,b−1) when gcd(a+c,b−1) > 1 we only used it to sieve the sequence a×bn+c for primes p not dividing gcd(a+c,b−1), and initialized the list of candidates to not include n for which there is some prime p dividing gcd(a+c,b−1) for which p dividing (a×bn+c)/gcd(a+c,b−1). The program had to be modified slightly to remove a check which would prevent it from running in the case when a, b, and c were all odd (since then 2 divides a×bn+c, but 2 may not divide (a×bn+c)/gcd(a+c,b−1)) (see https://github.com/curtisbright/mepn-data/commit/1b55b353f46c707bbe52897573914128b3303960).
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×bn+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)
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)
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)
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 "dn" if there are 5 or more (n) consecutive same digits d) | length of the top 10 known minimal prime base b | algebraic ((a×bn+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 9643 |
0 | – |
8 | 75 | 42207 51325 7121 7777461 7471 481 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 51612A 5012657 1012551 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 AAA0001 BBBAA1 A00065 44AAA1 BBBB1 |
42 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 248821 |
0 | – |
13 | 3196~3197 | 95197420 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 34D708 8D14185 886B 408349 8C793 1879B 6B772B 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 DB36272 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 J233046J 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 I626AF 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 BF034431D 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 M8772P 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 NGN036329N 153F358315 L35564GLG PN033401J 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 G01899AN 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 C010221 M0547SS7 M241QB AN206 50164B J153QJ 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 R17333CA MF1690135 CT167940H 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 NA0417331 K038239J4J 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 VL07258J EO06177V FZ57773P T0946181 RY4562H 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×bn±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:
- http://primerecords.dk/left-truncatable.txt (base 10)
- http://chesswanks.com/num/LTPs/ (bases 3 to 120)
- 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)
- https://oeis.org/A103443 (largest left-truncatable prime in base b)
- https://oeis.org/A103463 (length of the largest left-truncatable prime in base b)
- 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:
- http://primerecords.dk/right-truncatable.txt (base 10)
- http://fatphil.org/maths/rtp/rtp.html (bases 3 to 90)
- 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)
- https://oeis.org/A023107 (largest right-truncatable prime in base b)
- https://oeis.org/A103483 (length of the largest right-truncatable prime in base b)
- 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:
Primes == 3 mod 4:
Palindromic primes:
Composites:
Squares:
Powers of 2:
- https://oeis.org/A071071/a071071.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_13.pdf)
- https://oeis.org/A071071
Multiples of 3:
Multiples of 4:
Other sets:
- 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)
- 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:
- https://primes.utm.edu/curios/includes/primetest.php
- https://www.numberempire.com/primenumbers.php
- http://www.numbertheory.org/php/lucas.html
- http://www.javascripter.net/faq/numberisprime.htm
- http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm
- http://www.javascripter.net/math/calculators/100digitbigintcalculator.htm (just type x and click "prime?")
- https://www.bigprimes.net/primalitytest
- https://www.archimedes-lab.org/primOmatic.html
- http://www.sonic.net/~undoc/java/PrimeCalc.html
- http://www.proftnj.com/calcprem.htm (in French) (use the box "Rechercher si un nombre est premier" and click "Rechercher")
- https://primes.utm.edu/nthprime/ (calculate the nth prime)
- http://factordb.com/nextprime.php (calculate the next (probable) prime above N)
Integer factorizers:
- https://www.numberempire.com/numberfactorizer.php
- https://www.alpertron.com.ar/ECM.HTM
- http://www.javascripter.net/math/calculators/primefactorscalculator.htm
- https://betaprojects.com/calculators/prime_factors.html
- https://www.emathhelp.net/calculators/pre-algebra/prime-factorization-calculator/
- http://www.numbertheory.org/php/factor.html
- https://primefan.tripod.com/Factorer.html
- https://www.calculatorsoup.com/calculators/math/prime-factors.php
- https://www.calculator.net/prime-factorization-calculator.html
- http://www.se16.info/js/factor.htm
- http://math.fau.edu/Richman/mla/factor-f.htm
- http://www.rsok.com/~jrm/factor.html
- http://www.brennen.net/primes/FactorApplet.html (need run with Java)
- https://web.archive.org/web/20161004191531/http://britton.disted.camosun.bc.ca/jbprimefactor.htm
- http://wims.unice.fr/~wims/en_tool~algebra~factor.en.html
- http://www.analyzemath.com/Calculators_3/prime_factors.html
- http://www.proftnj.com/calcprem.htm (in French) (use the box "Factoriser un nombre" and click "Factoriser")
Base converters:
- https://baseconvert.com/
- https://www.calculand.com/unit-converter/zahlen.php
- https://www.dcode.fr/base-n-convert
- https://www.cut-the-knot.org/Curriculum/Algorithms/BaseConversion.shtml
- http://www.tonymarston.net/php-mysql/converter.php
- http://math.fau.edu/Richman/mla/convert.htm
- https://web.archive.org/web/20190629223750/http://thedevtoolkit.com/tools/base_conversion
- http://www.kwuntung.net/hkunit/base/base.php (in Chinese)
- https://linesegment.web.fc2.com/application/math/numbers/RadixConversion.html (in Japanese)
Expression generators:
- https://stdkmd.net/nrr/exprgen.htm (only support base 10 forms)
- 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:
- https://primes.utm.edu/lists/small/1000.txt
- https://primes.utm.edu/lists/small/10000.txt
- https://primes.utm.edu/lists/small/100000.txt
- https://primes.utm.edu/lists/small/millions/
- https://oeis.org/A000040/a000040.txt
- https://oeis.org/A000040/b000040_1.txt
- https://oeis.org/A000040/a000040_1B.7z
- https://metanumbers.com/prime-numbers
- https://www.calculatorsoup.com/calculators/math/prime-numbers.php
- https://www2.cs.arizona.edu/icon/oddsends/primes.htm
- https://cdn1.byjus.com/wp-content/uploads/2021/10/Prime-Numbers-from-1-to-1000.png
- http://noe-education.org/D11102.php (in French)
- https://web.archive.org/web/20060513054350/http://www.walter-fendt.de/m14i/primes_i.htm (in Italian)
- https://primefan.tripod.com/500Primes1.html
- https://www.gutenberg.org/files/65/65.txt
- http://www.primos.mat.br/indexen.html
- https://www.walter-fendt.de/html5/men/primenumbers_en.htm
- http://www.rsok.com/~jrm/printprimes.html
- http://www.numbertheory.org/php/prime_generator.html
- https://jocelyn.quizz.chat/np/cache/index.html (in French)
- http://www.sosmath.com/tables/prime/prime.html
- https://www.bigprimes.net/archive/prime
- https://web.archive.org/web/20201130071856/http://www.mathematical.com/primelist1to100kk.html
- https://web.archive.org/web/20191118082053/http://www.tsm-resources.com/alists/prim.html
- https://web.archive.org/web/20090917191047/http://planetmath.org/encyclopedia/FirstThousandPositivePrimeNumbers.html
- 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)
- https://en.wikipedia.org/wiki/List_of_prime_numbers#The_first_1000_prime_numbers
Lists of factorizations of small integers:
- http://primefan.tripod.com/500factored.html
- http://www.sosmath.com/tables/factor/factor.html
- http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/
- https://oeis.org/A027750/a027750.txt (all (prime or composite or unit) factors of N)
- http://factorzone.tripod.com/factors.htm (all (prime or composite or unit) factors of N)
- http://functions.wolfram.com/NumberTheoryFunctions/Divisors/03/02 (all (prime or composite or unit) factors of N)
- https://en.wikipedia.org/wiki/Table_of_prime_factors
- https://en.wikipedia.org/wiki/Table_of_divisors (all (prime or composite or unit) factors of N)
Lists of small integers in various 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):
- 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)
- 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)
- 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)
- 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):
- 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)
- 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):
- 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)
- MSIEVE (https://sourceforge.net/projects/msieve, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/msieve153_win64)
- GGNFS (http://sourceforge.net/projects/ggnfs, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/GGNFS)
- 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