GitHub - xayahrainie4793/minimal-elements-of-the-prime-numbers: Minimal elements for the base b representations of the prime numbers > b for the subsequence ordering, for 2 ≤ b ≤ 36

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

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, http://www.numericana.com/answer/primes.htm#euclid, 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.

subsequence ordering	divisibility ordering
the "prime numbers > b" digit strings" in base b	the integers > 1
the set of the minimal elements of the "prime numbers > b" digit strings" in base b under the subsequence ordering (which is exactly the target of this project)	the set of the minimal elements of the integers > 1 under the divisibility ordering (which is exactly the set of the prime numbers)
no common subsequence with length > 1	coprime (no common divisor > 1) (https://en.wikipedia.org/wiki/Coprime_integers, https://primes.utm.edu/glossary/xpage/RelativelyPrime.html, https://www.rieselprime.de/ziki/Coprime, https://mathworld.wolfram.com/RelativelyPrime.html, http://www.numericana.com/answer/primes.htm#coprime)
proper subsequence with length > 1	proper factor (https://en.wikipedia.org/wiki/Proper_factor, https://mathworld.wolfram.com/ProperFactor.html, https://mathworld.wolfram.com/ProperDivisor.html) > 1
longest common subsequence (https://en.wikipedia.org/wiki/Longest_common_subsequence_problem)	greatest common divisor (https://en.wikipedia.org/wiki/Greatest_common_divisor, https://mathworld.wolfram.com/GreatestCommonDivisor.html, http://www.numericana.com/answer/numbers.htm#gcd)
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, http://www.numericana.com/answer/numbers.htm#lcm)
pairwise incomparable strings (no string is a subsequence of another string)	pairwise incomparable numbers (no number divides another number)

This problem is an extension of the original minimal prime problem (https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_17.pdf), https://cs.uwaterloo.ca/~shallit/Papers/br10.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_18.pdf), https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_19.pdf), https://doi.org/10.1080/10586458.2015.1064048 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_20.pdf), https://scholar.colorado.edu/downloads/hh63sw661 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_16.pdf), https://github.com/curtisbright/mepn-data, https://github.com/curtisbright/mepn, https://github.com/RaymondDevillers/primes) to cover Conjectures ‘R Us Sierpinski/Riesel conjectures base b (http://www.noprimeleftbehind.net/crus/, http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-stats.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-top20.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-unproven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/crus-proven.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://primes.utm.edu/bios/page.php?id=1372, https://srbase.my-firewall.org/sr5/, http://www.rechenkraft.net/yoyo/y_status_sieve.php, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian)) with k-values < b, i.e. finding the smallest prime of the form k×bⁿ+1 and k×bⁿ−1 (or proving that such prime does not exist) for all k < b (also to cover dual (http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_1.pdf), https://www.rechenkraft.net/wiki/Five_or_Bust, https://oeis.org/A076336/a076336c.html, http://www.mit.edu/~kenta/three/prime/dual-sierpinski/ezgxggdm/dualsierp-excerpt.txt, http://mit.edu/kenta/www/three/prime/dual-sierpinski/ezgxggdm/dualsierp.txt.gz, https://mersenneforum.org/showpost.php?p=144991&postcount=1, https://mersenneforum.org/showthread.php?t=10761, https://mersenneforum.org/showthread.php?t=6545) Sierpinski/Riesel conjectures base b with k-values < b, i.e. finding the smallest prime of the form bⁿ+k and bⁿ−k (which are the dual forms of k×bⁿ+1 and k×bⁿ−1, respectively) (or proving that such prime does not exist) for all k < b) (also to cover finding the smallest prime of some classic forms (or proving that such prime does not exist), such as bⁿ+2, bⁿ−2, bⁿ+(b−1), bⁿ−(b−1), 2×bⁿ+1, 2×bⁿ−1, (b−1)×bⁿ+1, (b−1)×bⁿ−1, with n ≥ 1, for the same base b (of course, for some bases b the original minimal prime base b problem already covers finding the smallest prime of these forms, e.g. the original minimal prime base b problem covers finding the smallest prime of the form (b−1)×bⁿ+1 if and only if b−1 is not prime, and the original minimal prime base b problem covers finding the smallest prime of the form (b−1)×bⁿ−1 if and only if neither b−1 nor b−2 is prime, but I want the problem covers finding the smallest prime of these forms for all bases b)). The original minimal prime base b problem does not cover Conjectures ‘R Us Sierpinski/Riesel conjectures base b with conjectured k (http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm, https://www.rieselprime.de/Others/CRUS_tab.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt) < b, since in Riesel side, the prime is not minimal prime in original definition if either k−1 or b−1 (or both) is prime, and in Sierpinski side, the prime is not minimal prime in original definition if k is prime (e.g. 25×30³⁴²⁰⁵−1 is not minimal prime in base 30 in original definition, since it is OT₃₄₂₀₅ in base 30, and T (= 29 in decimal) is prime, but it is minimal prime in base 30 if only primes > base are counted), but this extended version of minimal prime base b problem does. (There is someone else who also exclude the single-digit primes, but his research is about substring (https://en.wikipedia.org/wiki/Substring) instead of subsequence, see https://www.mersenneforum.org/showpost.php?p=235383&postcount=42, subsequences can contain consecutive elements which were not consecutive in the original sequence, a subsequence which consists of a consecutive run of elements from the original sequence, such as 234 from 123456, is a substring, substring is a refinement of the subsequence, subsequence is a generalization of substring, substring must be subsequence, but subsequence may not be substring, 514 is a subsequence of 352148, but not a substring of 352148, see the list below of the comparation of "subsequence" and "substring")

subsequence	substring
https://oeis.org/A071062	https://oeis.org/A033274
https://oeis.org/A130448	https://oeis.org/A238334
https://oeis.org/A039995	https://oeis.org/A039997
https://oeis.org/A039994	https://oeis.org/A039996
https://oeis.org/A094535	https://oeis.org/A093301
https://oeis.org/A350508	https://oeis.org/A038103
https://oeis.org/A354113	https://oeis.org/A354114
longest common subsequence problem (https://en.wikipedia.org/wiki/Longest_common_subsequence_problem)	longest common substring problem (https://en.wikipedia.org/wiki/Longest_common_substring_problem)

(we do not research substring in this article, because for substring ordering, "the set of the minimal elements of the base 10 representations of the prime numbers > 10" are very likely to be infinite (and thus almost unsolvable), since all primes of the form 1{0}3 (i.e. https://oeis.org/A159352, for the exponents see https://oeis.org/A049054, also see https://stdkmd.net/nrr/1/10003.htm) or 3{0}1 (i.e. https://oeis.org/A259866, for the exponents see https://oeis.org/A056807, also see https://stdkmd.net/nrr/3/30001.htm) are minimal elements of the base 10 representations of the prime numbers > 10 under the substring ordering, and there is likely infinitely many primes of the form 1{0}3 and infinitely many primes of the form 3{0}1, see https://web.archive.org/web/20100628035147/http://www.math.niu.edu/~rusin/known-math/98/exp_primes and https://mersenneforum.org/showpost.php?p=564786&postcount=3 and https://mersenneforum.org/showpost.php?p=461665&postcount=7)

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, http://www.numericana.com/answer/modular.htm#phi, https://oeis.org/A000010), since b−1 is the number of possible first digit (except 0, all digits can be first digit), and eulerphi(b) is the number of possible last digit (only digits coprime to b can be last digit), by rule of product, there are (b−1)×eulerphi(b) possible (first digit,last digit) combo, and if start with b, then when b is prime, there is an additional possible (first digit,last digit) combo: (1,0), and hence the formula will be (b−1)×eulerphi(b)+1 if b is prime, or (b−1)×eulerphi(b) if b is composite (the fully formula will be (b−1)×eulerphi(b)+isprime(b) or (b−1)×eulerphi(b)+floor((b−eulerphi(b)) / (b−1))), which is more complex, and if start with 1 (i.e. the original minimal prime problem), the formula is much more complex.

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

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

family	smallest allowed n	OEIS sequences for the smallest n such that this form is prime	references
(bⁿ−1)/(b−1)	2	https://oeis.org/A084740 https://oeis.org/A084738 (corresponding primes) https://oeis.org/A246005 (odd b) https://oeis.org/A065854 (prime b) https://oeis.org/A279068 (prime b, corresponding primes) https://oeis.org/A065813 (prime b, n replaced by (n−1)/2) https://oeis.org/A128164 (n = 2 not allowed) https://oeis.org/A285642 (n = 2 not allowed, corresponding primes)	http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German) https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html http://www.primenumbers.net/Henri/us/MersFermus.htm http://www.bitman.name/math/table/379 (in Italian) https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf) https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf)
bⁿ+1	1	https://oeis.org/A079706 https://oeis.org/A084712 (corresponding primes) https://oeis.org/A228101 (log₂ of n) https://oeis.org/A123669 (n = 1 not allowed, corresponding primes)	http://jeppesn.dk/generalized-fermat.html http://www.noprimeleftbehind.net/crus/GFN-primes.htm http://yves.gallot.pagesperso-orange.fr/primes/index.html http://yves.gallot.pagesperso-orange.fr/primes/results.html http://yves.gallot.pagesperso-orange.fr/primes/stat.html
(bⁿ+1)/2 (for odd b)	2		http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt (in German)
2×bⁿ+1	1	https://oeis.org/A119624 https://oeis.org/A253178 (only bases b which have possible primes) https://oeis.org/A098872 (b divisible by 6)	https://mersenneforum.org/showthread.php?t=6918 https://mersenneforum.org/showthread.php?t=19725 (b == 11 mod 12)
2×bⁿ−1	1	https://oeis.org/A119591 https://oeis.org/A098873 (b divisible by 6)	https://mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217
bⁿ+2	1	https://oeis.org/A138066 https://oeis.org/A084713 (corresponding primes) https://oeis.org/A138067 (n = 1 not allowed)
bⁿ−2	2	https://oeis.org/A250200 https://oeis.org/A255707 (n = 1 allowed) https://oeis.org/A084714 (n = 1 allowed, corresponding primes) https://oeis.org/A292201 (prime b, n = 1 allowed)	https://www.primepuzzles.net/puzzles/puzz_887.htm (n = 1 allowed)
3×bⁿ+1	1	https://oeis.org/A098877 (b divisible by 6)
3×bⁿ−1	1	https://oeis.org/A098876 (b divisible by 6)
10×bⁿ+1	1	https://oeis.org/A088782 https://oeis.org/A088622 (corresponding primes)
2×bⁿ+3	1		https://www.primegrid.com/forum_thread.php?id=9538
bⁿ/2+1 (for even b)	2		https://www.primegrid.com/forum_thread.php?id=9538
(b−1)×bⁿ+1	1	https://oeis.org/A305531 https://oeis.org/A087139 (prime b, n replaced by n+1)	https://www.rieselprime.de/ziki/Williams_prime_MP_least https://www.rieselprime.de/ziki/Williams_prime_MP_table https://sites.google.com/view/williams-primes http://www.bitman.name/math/table/477 (in Italian)
(b−1)×bⁿ−1	1	https://oeis.org/A122396 (prime b, n replaced by n+1)	https://harvey563.tripod.com/wills.txt https://www.rieselprime.de/ziki/Williams_prime_MM_least https://www.rieselprime.de/ziki/Williams_prime_MM_table https://sites.google.com/view/williams-primes http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_9.pdf) https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_10.pdf) http://www.bitman.name/math/table/484 (in Italian)
bⁿ+(b−1)	1	https://oeis.org/A076845 https://oeis.org/A076846 (corresponding primes) https://oeis.org/A078178 (n = 1 not allowed) https://oeis.org/A078179 (n = 1 not allowed, corresponding primes)	https://sites.google.com/view/williams-primes
bⁿ−(b−1)	2	https://oeis.org/A113516 https://oeis.org/A343589 (corresponding primes)	https://sites.google.com/view/williams-primes https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html (prime b) http://www.bitman.name/math/table/435 (in Italian) (prime b)
k×bⁿ+1 for all 2 ≤ k ≤ 12	1		https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n https://mersenneforum.org/showthread.php?t=10354
k×bⁿ−1 for all 2 ≤ k ≤ 12	1		https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n https://mersenneforum.org/showthread.php?t=10354

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

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

The primes in forms x{y}, {x}y, xy{x}, {x}yx in base b are near-repdigit primes (https://primes.utm.edu/glossary/xpage/NearRepdigitPrime.html, https://primes.utm.edu/top20/page.php?id=15, https://oeis.org/A164937, https://stdkmd.net/nrr/#factortables_nr, https://stdkmd.net/nrr/records.htm#nrprime, https://stdkmd.net/nrr/records.htm#nrprp) in base b.

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

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

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

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

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

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

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

Proof (https://en.wikipedia.org/wiki/Mathematical_proof, https://mathworld.wolfram.com/Proof.html, https://primes.utm.edu/notes/proofs/): (this proof uses the notation in http://www.wiskundemeisjes.nl/wp-content/uploads/2007/02/minimal.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_12.pdf), i.e. "X ◁ Y" means "X is a subsequence of Y") (below, 𝜆 is the empty string (https://en.wikipedia.org/wiki/Empty_string)) (bold for minimal primes)

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

Case 1: p ends with 1.

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

Case 1.1: p begins with 2.

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

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

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

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

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

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

Case 1.2: p begins with 5.

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

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

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

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

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

Case 1.3: p begins with 8.

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

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

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

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

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

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

Case 1.4: p begins with 9.

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

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

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

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

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

Case 2: p ends with 3.

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

Case 3: p ends with 7.

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

Case 3.1: p begins with 2.

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

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

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

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

Case 3.2: p begins with 5.

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

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

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

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

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

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

Case 3.3: p begins with 7.

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

Case 3.4: p begins with 8.

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

Case 4: p ends with 9.

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

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

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

Case 4.1: p begins with 3.

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

Case 4.2: p begins with 4.

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

Case 4.3: p begins with 6.

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

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

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

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

Case 4.4: p begins with 9.

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

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

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

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

M := ∅
while (L ≠ ∅) do
choose x, a shortest string in L
M := M ∪ {x}
L := L − sup({x})

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

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

Problem: Given strings x, z (may be empty), a digit y, and a base b, does there exist a prime number whose base-b expansion is of the form xy_nz for some n ≥ 0? (If we say "yes", then we should find a such prime (the smallest such prime may be very large, e.g. > 10²⁵⁰⁰⁰, and if so, then we should use 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 xy_nz 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)

Also, there are some examples in decimal (i.e. base b = 10): (references: https://stdkmd.net/nrr/prime/primecount.htm, https://stdkmd.net/nrr/prime/primecount.txt, https://stdkmd.net/nrr/prime/primedifficulty.htm, https://stdkmd.net/nrr/prime/primedifficulty.txt)

x	y	z	answer
7	1	7	Yes! But the smallest such prime is very large, it is 71₁₀₉₀₅7, its algebraic form is (64×10¹⁰⁹⁰⁶+53)/9, its factordb entry is http://factordb.com/index.php?id=1000000000008860930, its primality certificate is http://factordb.com/cert.php?id=1000000000008860930 and https://stdkmd.net/nrr/cert/7/71117_10906.zip
9	4	9	No! All numbers of this form is divisible by some element of {3,7,11,13}, since its algebraic form is (85×10ⁿ⁺¹+41)/9, it is divisible by 3 if n == 0 mod 3, divisible by 7 if n == 5 mod 6, divisible by 11 if n == 0 mod 2, divisible by 13 if n == 1 mod 6
56	1	𝜆 (the empty string (https://en.wikipedia.org/wiki/Empty_string))	Yes! But the smallest such prime is very large, it is 561₁₈₄₇₀, its algebraic form is (505×10¹⁸⁴⁷⁰−1)/9, its factordb entry is http://factordb.com/index.php?id=1100000000301454592, its primality certificate is http://factordb.com/cert.php?id=1100000000301454592 and https://stdkmd.net/nrr/cert/5/56111_18470.zip
38	1	𝜆 (the empty string (https://en.wikipedia.org/wiki/Empty_string))	No! All numbers of this form is either divisible by some element of {3,37} or has a difference-of-two-cubes factorization, since its algebraic form is (343×10ⁿ−1)/9, it is divisible by 3 if n == 1 mod 3, divisible by 37 if n == 2 mod 3, and can be factored to (343×10ⁿ−1)/9 = (7×10^n/3−1) × (49×10^2×n/3+7×10^n/3+1) / 9 if n == 0 mod 3
𝜆 (the empty string (https://en.wikipedia.org/wiki/Empty_string))	2	99	Yes! But the smallest such prime is very large, it is 2₁₉₁₅₁99, its algebraic form is (2×10¹⁹¹⁵³+691)/9, its factordb entry is http://factordb.com/index.php?id=1100000000301493137, its primality certificate is http://factordb.com/cert.php?id=1100000000301493137 and https://stdkmd.net/nrr/cert/2/2w99_19153.zip
𝜆 (the empty string (https://en.wikipedia.org/wiki/Empty_string))	5 (at least one 5)	37	No! All numbers of this form is divisible by some element of {3,7,13,37}, since its algebraic form is (5×10ⁿ⁺²−167)/9, it is divisible by 3 if n == 1 mod 3, divisible by 7 if n == 2 mod 6, divisible by 13 if n == 5 mod 6, divisible by 37 if n == 0 mod 3
44	9 (at least two 9)	𝜆 (the empty string (https://en.wikipedia.org/wiki/Empty_string))	Yes! But the smallest such prime is very large, it is 449₁₁₉₅₉, its algebraic form is 45×10¹¹⁹⁵⁹−1, its factordb entry is http://factordb.com/index.php?id=1100000000291927010, it can be proven prime by the N+1 primality proving (https://primes.utm.edu/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2)

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

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

This conjecture will imply:

There are infinitely many Mersenne primes (i.e. primes of the form 2^p−1 with prime p) (https://en.wikipedia.org/wiki/Mersenne_prime, https://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, https://primes.utm.edu/top20/page.php?id=4, https://www.mersenne.org/, https://www.mersenne.org/primes/, https://www.mersenne.ca/, https://primes.utm.edu/mersenne/) (https://oeis.org/A001348, https://oeis.org/A000668, https://oeis.org/A000043)
There are infinitely many Fermat primes (i.e. primes of the form 2^2ⁿ+1) (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, http://www.prothsearch.com/fermat.html, http://www.fermatsearch.org/) (https://oeis.org/A000215, https://oeis.org/A019434)
There are infinitely many generalized repunit primes (i.e. primes of the form (b^p−1)/(b−1) with prime p) (https://en.wikipedia.org/wiki/Repunit, https://primes.utm.edu/glossary/xpage/Repunit.html, https://primes.utm.edu/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, http://www.bitman.name/math/article/380/231/, http://www.bitman.name/math/table/379, https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf), https://primes.utm.edu/top20/page.php?id=57, https://primes.utm.edu/top20/page.php?id=16, https://oeis.org/A002275) to every base b ≥ 2 which is not a perfect power (i.e. of the form m^r with r > 1) (https://oeis.org/A001597, https://en.wikipedia.org/wiki/Perfect_power, https://mathworld.wolfram.com/PerfectPower.html) (https://oeis.org/A084740, https://oeis.org/A084738, https://oeis.org/A246005, https://oeis.org/A065854, https://oeis.org/A279068, https://oeis.org/A065813, https://oeis.org/A128164, https://oeis.org/A285642)
There are infinitely many generalized Wagstaff primes (i.e. primes of the form (b^p+1)/(b+1) with odd prime p) (https://en.wikipedia.org/wiki/Wagstaff_prime, https://primes.utm.edu/glossary/xpage/WagstaffPrime.html, https://mathworld.wolfram.com/WagstaffPrime.html, https://web.archive.org/web/20211031110623/http://mprime.s3-website.us-west-1.amazonaws.com/wagstaff/, http://www.fermatquotient.com/PrimSerien/GenRepuP.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, http://www.bitman.name/math/table/488, https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_5.pdf), https://primes.utm.edu/top20/page.php?id=67, https://oeis.org/A000979, https://oeis.org/A000978) to every base b ≥ 2 which is neither a perfect odd power (i.e. of the form m^r with odd r > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) nor of the form 4×m⁴ (https://oeis.org/A141046) (https://oeis.org/A084742, https://oeis.org/A084741, https://oeis.org/A126659, https://oeis.org/A065507)
There are infinitely many generalized Fermat primes (i.e. primes of the form b^2ⁿ+1 with even b) (https://primes.utm.edu/glossary/xpage/GeneralizedFermatNumber.html, https://primes.utm.edu/glossary/xpage/GeneralizedFermatPrime.html, https://www.rieselprime.de/ziki/Generalized_Fermat_number, https://mathworld.wolfram.com/GeneralizedFermatNumber.html, http://jeppesn.dk/generalized-fermat.html, http://www.noprimeleftbehind.net/crus/GFN-primes.htm, http://yves.gallot.pagesperso-orange.fr/primes/index.html, http://yves.gallot.pagesperso-orange.fr/primes/results.html, http://yves.gallot.pagesperso-orange.fr/primes/stat.html, http://www.primegrid.com/forum_thread.php?id=3980, http://www.primegrid.com/stats_genefer.php, https://primes.utm.edu/top20/page.php?id=12, http://www.prothsearch.com/fermat.html, http://www.prothsearch.com/GFN06.html, http://www.prothsearch.com/GFN10.html, http://www.prothsearch.com/GFN12.html, http://www.prothsearch.com/GFNfacs.html, http://www.prothsearch.com/GFNsmall.html) to every even base b ≥ 2 which is not a perfect odd power (i.e. of the form m^r with odd r > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) (https://oeis.org/A228101, https://oeis.org/A079706, https://oeis.org/A084712, https://oeis.org/A123669)
There are infinitely many generalized half-Fermat primes (i.e. primes of the form (b^2ⁿ+1)/2 with odd b) (http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt (in German), http://www.prothsearch.com/GFN03.html, http://www.prothsearch.com/GFN05.html, http://www.prothsearch.com/GFN07.html, http://www.prothsearch.com/GFN11.html, http://www.prothsearch.com/GFNfacs.html, http://www.prothsearch.com/GFNsmall.html) to every odd base b ≥ 2 which is not a perfect odd power (i.e. of the form m^r with odd r > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265)
There are infinitely many Williams primes of the first kind (i.e. primes of the form (b−1)×bⁿ−1) (https://harvey563.tripod.com/wills.txt, https://www.rieselprime.de/ziki/Williams_prime_MM_least, https://www.rieselprime.de/ziki/Williams_prime_MM_table, https://sites.google.com/view/williams-primes, http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_9.pdf), https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_10.pdf), http://www.bitman.name/math/table/484 (in Italian)) to every base b ≥ 2 (https://oeis.org/A122396)
There are infinitely many Williams primes of the second kind (i.e. primes of the form (b−1)×bⁿ+1) (https://www.rieselprime.de/ziki/Williams_prime_MP_least, https://www.rieselprime.de/ziki/Williams_prime_MP_table, https://sites.google.com/view/williams-primes, http://www.bitman.name/math/table/477 (in Italian)) to every base b ≥ 2 (https://oeis.org/A305531, https://oeis.org/A087139) (warning: this may be false, (b−1)×bⁿ+1 may be able to be proven to only contain composites by combine of covering congruence and algebraic factorization when b−1 is either a perfect odd power (i.e. of the form m^r with odd r > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) or of the form 4×m⁴ (https://oeis.org/A141046), but the smallest such base b will be very large, however, this is at least true for bases b such that b−1 is neither a perfect odd power (i.e. of the form m^r with odd r > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) nor of the form 4×m⁴ (https://oeis.org/A141046), also at least true for bases b ≤ 10⁶)
There are infinitely many dual Williams primes of the first kind (i.e. primes of the form bⁿ−(b−1)) (https://sites.google.com/view/williams-primes, https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html, http://www.bitman.name/math/table/435 (in Italian)) to every base b ≥ 2 (https://oeis.org/A113516, https://oeis.org/A343589)
There are infinitely many dual Williams primes of the second kind (i.e. primes of the form bⁿ+(b−1)) (https://sites.google.com/view/williams-primes) to every base b ≥ 2 (https://oeis.org/A076845, https://oeis.org/A076846, https://oeis.org/A078178, https://oeis.org/A078179) (warning: this may be false, bⁿ+(b−1) may be able to be proven to only contain composites by combine of covering congruence and algebraic factorization when b−1 is either a perfect odd power (i.e. of the form m^r with odd r > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) or of the form 4×m⁴ (https://oeis.org/A141046), but the smallest such base b will be very large, however, this is at least true for bases b such that b−1 is neither a perfect odd power (i.e. of the form m^r with odd r > 1) (http://mathworld.wolfram.com/OddPower.html, https://oeis.org/A070265) nor of the form 4×m⁴ (https://oeis.org/A141046), also at least true for bases b ≤ 10⁶)
78557 is the smallest Sierpinski number (i.e. odd numbers k such that k×2ⁿ+1 is composite for all n ≥ 1) (http://www.prothsearch.com/sierp.html, http://www.primegrid.com/forum_thread.php?id=1647, http://www.primegrid.com/forum_thread.php?id=972, http://www.primegrid.com/forum_thread.php?id=1750, http://www.primegrid.com/forum_thread.php?id=5758, http://www.primegrid.com/stats_sob_llr.php, http://www.primegrid.com/stats_psp_llr.php, http://www.primegrid.com/stats_esp_llr.php, https://web.archive.org/web/20160405211049/http://www.seventeenorbust.com/, https://en.wikipedia.org/wiki/Sierpi%C5%84ski_number, https://primes.utm.edu/glossary/xpage/SierpinskiNumber.html, https://www.rieselprime.de/ziki/Sierpi%C5%84ski_problem, https://www.rieselprime.de/ziki/Proth_2_Sierpinski, https://mathworld.wolfram.com/SierpinskiNumberoftheSecondKind.html, https://en.wikipedia.org/wiki/Seventeen_or_Bust, https://www.rieselprime.de/ziki/Seventeen_or_Bust, https://web.archive.org/web/20190929190947/https://primes.utm.edu/glossary/xpage/ColbertNumber.html, https://mathworld.wolfram.com/ColbertNumber.html, http://www.numericana.com/answer/primes.htm#sierpinski, http://www.bitman.name/math/article/204 (in Italian), https://oeis.org/A076336, https://oeis.org/A078680, https://oeis.org/A078683, https://oeis.org/A033809, https://oeis.org/A040076, https://oeis.org/A225721, https://oeis.org/A050921, https://oeis.org/A046067, https://oeis.org/A057025, https://oeis.org/A057192, https://oeis.org/A057247)
509203 is the smallest Riesel number (i.e. odd numbers k such that k×2ⁿ−1 is composite for all n ≥ 1) (http://www.prothsearch.com/rieselprob.html, http://www.primegrid.com/forum_thread.php?id=1731, http://www.primegrid.com/stats_trp_llr.php, https://web.archive.org/web/20061021153313/http://stats.rieselsieve.com//queue.php, https://en.wikipedia.org/wiki/Riesel_number, https://primes.utm.edu/glossary/xpage/RieselNumber.html, https://www.rieselprime.de/ziki/Riesel_problem, https://www.rieselprime.de/ziki/Riesel_2_Riesel, https://mathworld.wolfram.com/RieselNumber.html, http://www.bitman.name/math/article/203 (in Italian), https://oeis.org/A076337, https://oeis.org/A101036, https://oeis.org/A050412, https://oeis.org/A052333, https://oeis.org/A108129, https://oeis.org/A040081, https://oeis.org/A038699, https://oeis.org/A046069, https://oeis.org/A057026, https://oeis.org/A128979, https://oeis.org/A101050)
125050976086 is the smallest generalized Sierpinski number to base 3 (i.e. numbers k such that gcd(k+1, 3−1) = 1 and k×3ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159, https://oeis.org/A291437, https://oeis.org/A291438)
63064644938 is the smallest generalized Riesel number to base 3 (i.e. numbers k such that gcd(k−1, 3−1) = 1 and k×3ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://oeis.org/A273987)
66741 is the smallest generalized Sierpinski number to base 4 (i.e. numbers k such that gcd(k+1, 4−1) = 1 and k×4ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159, https://oeis.org/A251057, https://oeis.org/A256002, http://www.rieselprime.de/Related/LiskovetsGallot.htm, https://primepuzzles.net/problems/prob_036.htm)
39939 is the smallest non-square generalized Riesel number to base 4 (i.e. numbers k such that gcd(k−1, 4−1) = 1 and k×4ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://oeis.org/A273987, https://oeis.org/A251757, http://www.rieselprime.de/Related/LiskovetsGallot.htm, https://primepuzzles.net/problems/prob_036.htm)
159986 is the smallest generalized Sierpinski number to base 5 (i.e. numbers k such that gcd(k+1, 5−1) = 1 and k×5ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159, http://www.primegrid.com/forum_thread.php?id=5087, http://www.primegrid.com/stats_sr5_llr.php, https://web.archive.org/web/20131016004333/http://www.sr5.psp-project.de/, https://oeis.org/A345698)
346802 is the smallest generalized Riesel number to base 5 (i.e. numbers k such that gcd(k−1, 5−1) = 1 and k×5ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://oeis.org/A273987, http://www.primegrid.com/forum_thread.php?id=5087, http://www.primegrid.com/stats_sr5_llr.php, https://web.archive.org/web/20131016004333/http://www.sr5.psp-project.de/, https://oeis.org/A345403)
174308 is the smallest generalized Sierpinski number to base 6 (i.e. numbers k such that gcd(k+1, 6−1) = 1 and k×6ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159, https://oeis.org/A244549, https://oeis.org/A250204)
84687 is the smallest generalized Riesel number to base 6 (i.e. numbers k such that gcd(k−1, 6−1) = 1 and k×6ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://oeis.org/A273987, https://oeis.org/A244351, https://oeis.org/A250205)
1112646039348 is the smallest generalized Sierpinski number to base 7 (i.e. numbers k such that gcd(k+1, 7−1) = 1 and k×7ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159)
408034255082 is the smallest generalized Riesel number to base 7 (i.e. numbers k such that gcd(k−1, 7−1) = 1 and k×7ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://oeis.org/A273987)
2344 is the smallest generalized Sierpinski number to base 9 (i.e. numbers k such that gcd(k+1, 9−1) = 1 and k×9ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159)
9175 is the smallest generalized Sierpinski number to base 10 (i.e. numbers k such that gcd(k+1, 10−1) = 1 and k×10ⁿ+1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159, https://oeis.org/A243969)
10176 is the smallest generalized Riesel number to base 10 (i.e. numbers k such that gcd(k−1, 10−1) = 1 and k×10ⁿ−1 is composite for all n ≥ 1) (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://oeis.org/A273987, https://oeis.org/A243974)
78557 is the smallest dual Sierpinski number (i.e. odd numbers k such that 2ⁿ+k is composite for all n ≥ 1) (http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_1.pdf), https://www.rechenkraft.net/wiki/Five_or_Bust, https://oeis.org/A076336/a076336c.html, http://www.mit.edu/~kenta/three/prime/dual-sierpinski/ezgxggdm/dualsierp-excerpt.txt, http://mit.edu/kenta/www/three/prime/dual-sierpinski/ezgxggdm/dualsierp.txt.gz, https://mersenneforum.org/showthread.php?t=10761, https://oeis.org/A067760, https://oeis.org/A123252, https://oeis.org/A094076, https://oeis.org/A139758)
509203 is the smallest dual Riesel number (i.e. odd numbers k such that 2ⁿ−k is composite for all n ≥ 1 such that 2ⁿ > k) (https://mersenneforum.org/showthread.php?t=6545, https://oeis.org/A096502, https://oeis.org/A096822, https://oeis.org/A101462)
125050976086 is the smallest generalized dual Sierpinski number to base 3 (i.e. numbers k such that gcd(k, 3) = 1 and gcd(k+1, 3−1) = 1 and 3ⁿ+k is composite for all n ≥ 1)
63064644938 is the smallest generalized dual Riesel number to base 3 (i.e. numbers k such that gcd(k, 3) = 1 and gcd(k−1, 3−1) = 1 and 3ⁿ−k is composite for all n ≥ 1 such that 3ⁿ > k)
159986 is the smallest generalized dual Sierpinski number to base 5 (i.e. numbers k such that gcd(k, 5) = 1 and gcd(k+1, 5−1) = 1 and 5ⁿ+k is composite for all n ≥ 1)
346802 is the smallest generalized dual Riesel number to base 5 (i.e. numbers k such that gcd(k, 5) = 1 and gcd(k−1, 5−1) = 1 and 5ⁿ−k is composite for all n ≥ 1 such that 5ⁿ > k)
1112646039348 is the smallest generalized dual Sierpinski number to base 7 (i.e. numbers k such that gcd(k, 7) = 1 and gcd(k+1, 7−1) = 1 and 7ⁿ+k is composite for all n ≥ 1)
408034255082 is the smallest generalized dual Riesel number to base 7 (i.e. numbers k such that gcd(k, 7) = 1 and gcd(k−1, 7−1) = 1 and 7ⁿ−k is composite for all n ≥ 1 such that 7ⁿ > k)
201446503145165177 is the smallest reverse Sierpinski base to k = 2 (i.e. bases b such that gcd(2+1, b−1) = 1 and 2×bⁿ+1 is composite for all n ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://mersenneforum.org/showthread.php?t=6918, https://mersenneforum.org/showthread.php?t=19725, https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354, https://oeis.org/A119624, https://oeis.org/A253178, https://oeis.org/A098872)
There are no reverse Riesel bases to k = 2 (i.e. bases b such that gcd(2−1, b−1) = 1 and 2×bⁿ−1 is composite for all n ≥ 1) (https://mersenneforum.org/showthread.php?t=24576, https://www.mersenneforum.org/attachment.php?attachmentid=20976&d=1567314217, https://oeis.org/A119591, https://oeis.org/A098873, https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
There are no reverse Sierpinski bases to k = 3 (i.e. bases b such that gcd(3+1, b−1) = 1 and 3×bⁿ+1 is composite for all n ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354, https://oeis.org/A098877)
There are no reverse Riesel bases to k = 3 (i.e. bases b such that gcd(3−1, b−1) = 1 and 3×bⁿ−1 is composite for all n ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354, https://oeis.org/A098876)
140324348 is the smallest reverse Sierpinski base to k = 5 (i.e. bases b such that gcd(5+1, b−1) = 1 and 5×bⁿ+1 is composite for all n ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
There are no reverse Riesel bases to k = 5 (i.e. bases b such that gcd(5−1, b−1) = 1 and 5×bⁿ−1 is composite for all n ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
There are no reverse Sierpinski bases to k = 7 (i.e. bases b such that gcd(7+1, b−1) = 1 and 7×bⁿ+1 is composite for all n ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
9162668342 is the smallest reverse Riesel base to k = 7 (i.e. bases b such that gcd(7−1, b−1) = 1 and 7×bⁿ−1 is composite for all n ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
177744 is the smallest reverse Sierpinski base to k = 9 (i.e. bases b such that gcd(9+1, b−1) = 1 and 9×bⁿ+1 is composite for all n ≥ 1) (https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1165&context=rhumj (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_2.pdf), https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354)
There are no reverse Riesel bases to k = 9 (i.e. bases b such that gcd(9−1, b−1) = 1 and 9×bⁿ−1 is composite for all n ≥ 1) (https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n, https://mersenneforum.org/showthread.php?t=10354) except the even square bases b and the bases b == 4 mod 10
201446503145165177 is the smallest dual reverse Sierpinski base to k = 2 (i.e. bases b such that gcd(2, b) = 1 and gcd(2+1, b−1) = 1 and bⁿ+2 is composite for all n ≥ 1) (https://oeis.org/A138066, https://oeis.org/A084713, https://oeis.org/A138067)
There are no dual reverse Riesel bases to k = 2 (i.e. bases b such that gcd(2, b) = 1 and gcd(2−1, b−1) = 1 and bⁿ−2 is composite for all n ≥ 1) (https://www.primepuzzles.net/puzzles/puzz_887.htm, https://oeis.org/A255707, https://oeis.org/A084714, https://oeis.org/A250200, https://oeis.org/A292201)
There are no dual reverse Sierpinski bases to k = 3 (i.e. bases b such that gcd(3, b) = 1 and gcd(3+1, b−1) = 1 and bⁿ+3 is composite for all n ≥ 1)
There are no dual reverse Riesel bases to k = 3 (i.e. bases b such that gcd(3, b) = 1 and gcd(3−1, b−1) = 1 and bⁿ−3 is composite for all n ≥ 1)
140324348 is the smallest dual reverse Sierpinski base to k = 5 (i.e. bases b such that gcd(5, b) = 1 and gcd(5+1, b−1) = 1 and bⁿ+5 is composite for all n ≥ 1)
There are no dual reverse Riesel bases to k = 5 (i.e. bases b such that gcd(5, b) = 1 and gcd(5−1, b−1) = 1 and bⁿ−5 is composite for all n ≥ 1)
There are no dual reverse Sierpinski bases to k = 7 (i.e. bases b such that gcd(7, b) = 1 and gcd(7+1, b−1) = 1 and bⁿ+7 is composite for all n ≥ 1)
9162668342 is the smallest dual reverse Riesel base to k = 7 (i.e. bases b such that gcd(7, b) = 1 and gcd(7−1, b−1) = 1 and bⁿ−7 is composite for all n ≥ 1)

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

M := {minimal primes in base b of length 2 or 3}, L := union of all x{Y}z such that x ≠ 0 and gcd(z, b) = 1 and Y is the set of digits y such that xyz has no subsequence in M
While L contains non-simple families: Explore each family of L, and update L. Examine each family of L by: 2.1. Let w be the shortest string in the family. If w has a subsequence in M, then remove the family from L. If w represents a prime, then add w to M and remove the family from L. 2.2. If possible, simplify the family. 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites, and if so then remove the family from L.
Update L, after each split examine the new families as in step 2.

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

We can show that: For the case of covering congruence, the numbers in the family are not equal to any element in S, if n makes the numbers > b, thus these factorizations are nontrivial; for the case of algebraic factorization (if the numbers are factored as F × G / d), both F and G are > d, if n makes the numbers > b, thus these factorizations are nontrivial (the exceptions are the base 9 family {1} and the base 25 family {1} and the base 32 family {1}. For the base 9 family {1}, the algebraic form is (9ⁿ−1)/8 with n ≥ 2, and can be factored to (3ⁿ−1) × (3ⁿ+1) / 8, if n ≥ 3, then both 3ⁿ−1 and 3ⁿ+1 are > 8, thus these factorizations are nontrivial, it only remains to check the case n = 2, but the number with n = 2 is 10 = 2 × 5 is not prime; for the base 25 family {1}, the algebraic form is (25ⁿ−1)/24 with n ≥ 2, and can be factored to (5ⁿ−1) × (5ⁿ+1) / 24, if n ≥ 3, then both 5ⁿ−1 and 5ⁿ+1 are > 24, thus these factorizations are nontrivial, it only remains to check the case n = 2, but the number with n = 2 is 26 = 2 × 13 is not prime; for the base 32 family {1}, the algebraic form is (32ⁿ−1)/31 with n ≥ 2, and can be factored to (2ⁿ−1) × (16ⁿ+8ⁿ+4ⁿ+2ⁿ+1) / 31, if n ≥ 6, then both 2ⁿ−1 and 16ⁿ+8ⁿ+4ⁿ+2ⁿ+1 are > 31, thus these factorizations are nontrivial, it only remains to check the cases n = 2, 3, 4, 5, but the numbers with n = 2, 3, 4, 5 are 33 = 3 × 11, 1057 = 7 × 151, 33825 = 3 × 5² × 11 × 41, 1082401 = 601 × 1801 are not primes); for the case of combine of covering congruence and algebraic factorization (if the numbers are factored as F × G / d), the numbers in the family are not equal to any element in S and both F and G are > d, if n makes the numbers > b, thus these factorizations are nontrivial.

type for proving the	possible bases b	such bases 2 ≤ b ≤ 36
covering congruence with 1 prime	any base b (however, all such families in base b = 2 end with 0 and thus have trailing zeros (https://en.wikipedia.org/wiki/Trailing_zero) and thus not counted)	(2), 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36
covering congruence with 2 primes	b such that b+1 is not a prime power (https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html) (however, the case b = 5 has only four such families: {1}3, {1}4, 3{1}, 4{1}, all of them are covered by the prime 111 (31 in decimal), thus the smallest base b with families which have covering congruence with 2 primes is b = 9)	(5), 9, 11, 13, 14, 17, 19, 20, 21, 23, 25, 27, 29, 32, 33, 34, 35
covering congruence with 3 primes and period 3	b such that omega(b²+b+1) ≥ 3 (where omega is the omega function (https://en.wikipedia.org/wiki/Prime_omega_function, https://oeis.org/A001221), the number of distinct primes dividing n)	16, 25
covering congruence with 3 primes and period 4	b such that b+1 is not a power of 2 (https://en.wikipedia.org/wiki/Power_of_two) and b²+1 is not a prime power (https://en.wikipedia.org/wiki/Prime_power, https://mathworld.wolfram.com/PrimePower.html) (however, bases b = 5, 8, 9, 11, 12, 18 has no such families, base 8 family 6{4}7 is covered by the prime 4₂₂₀7)	(5), (8), (9), (11), (12), 13, 17, (18), 19, 21, 22, 23, 25, 27, 28, 29, 30, 32, 33, 34, 35
algebraic factorization with difference of two squares	b such that b is square (https://en.wikipedia.org/wiki/Square_number, https://www.rieselprime.de/ziki/Square_number, https://mathworld.wolfram.com/SquareNumber.html) (however, base b = 4 has no such families, the family {1} has the prime 11 (5 in decimal))	(4), 9, 16, 25, 36
algebraic factorization with difference of two cubes	b such that b is cube (https://en.wikipedia.org/wiki/Cube_(algebra), https://mathworld.wolfram.com/CubicNumber.html)	8, 27
algebraic factorization with difference of two 5-th powers	b such that b is 5-th power (https://en.wikipedia.org/wiki/Fifth_power_(algebra))	32
algebraic factorization with x⁴+4y⁴	b such that b is 4-th power (https://en.wikipedia.org/wiki/Fourth_power, https://mathworld.wolfram.com/BiquadraticNumber.html)	16
combine of covering congruence with 1 prime and algebraic factorization with difference of two squares	b such that b is not square (https://en.wikipedia.org/wiki/Square_number, https://www.rieselprime.de/ziki/Square_number, https://mathworld.wolfram.com/SquareNumber.html) and b+1 has a prime factor p == 1 mod 4	12, 14, 19, 24, 28, 29, 33, 34

(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)

(only list the families which can be proven to be not covered by any prime > b (this would include all such families of the form x{0}y and all such families of the form x{y} (unless y = 1) and all such families of the form {x}y (unless x = 1)), the only exceptions are the base 21 family B0{H}6H (which is covered by the smallest prime in the family B{H}6H in base 21 (if such prime exists)) and the base 8 family 6{4}7 (which is covered by the prime 4₂₂₀7 in base 8))

(this table is sorted by: single trivial prime factor (2{0}1 in base b = 10 through D{6}R in base b = 28) → covering congruence with 2 primes ({1}5 in base b = 9 through {X}5 in base b = 34) → covering congruence with ≥ 3 primes (6{4}7 in base b = 8 through {G}L in base b = 32) → algebraic factorization ({1} in base b = 9 through 9{S}IJ in base b = 36) → combine of covering congruence and algebraic factorization (8{D} in base b = 14 through {X}P in base b = 34))

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 (if n = 0 already makes the number > b, then start with n = 0))
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	7ⁿ⁺¹+1 (n ≥ 0)	always divisible by 2	http://factordb.com/index.php?query=7%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}3	7ⁿ⁺¹+3 (n ≥ 0)	always divisible by 2	http://factordb.com/index.php?query=7%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
7	1{0}5	7ⁿ⁺¹+5 (n ≥ 0)	always divisible by 2 (in fact, always divisible by 6)	http://factordb.com/index.php?query=7%5E%28n%2B1%29%2B5&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	3{0}1	3×7ⁿ⁺¹+1 (n ≥ 0)	always divisible by 2	http://factordb.com/index.php?query=3*7%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	3{0}5	3×7ⁿ⁺¹+5 (n ≥ 0)	always divisible by 2	http://factordb.com/index.php?query=3*7%5E%28n%2B1%29%2B5&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	5{0}1	5×7ⁿ⁺¹+1 (n ≥ 0)	always divisible by 2 (in fact, always divisible by 6)	http://factordb.com/index.php?query=5*7%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	5{0}3	5×7ⁿ⁺¹+3 (n ≥ 0)	always divisible by 2	http://factordb.com/index.php?query=5*7%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
7	1{0}2	7ⁿ⁺¹+2 (n ≥ 0)	always divisible by 3	http://factordb.com/index.php?query=7%5E%28n%2B1%29%2B2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
7	2{0}1	2×7ⁿ⁺¹+1 (n ≥ 0)	always divisible by 3	http://factordb.com/index.php?query=2*7%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	4{0}5	4×7ⁿ⁺¹+5 (n ≥ 0)	always divisible by 3	http://factordb.com/index.php?query=4*7%5E%28n%2B1%29%2B5&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	5{0}4	5×7ⁿ⁺¹+4 (n ≥ 0)	always divisible by 3	http://factordb.com/index.php?query=5*7%5E%28n%2B1%29%2B4&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{0}1	7^m+n+2+7ⁿ⁺¹+1 (m+n ≥ 0)	always divisible by 3 (non-simple family)	–
7	1{0}3{0}5	7^m+n+2+3×7ⁿ⁺¹+5 (m+n ≥ 0)	always divisible by 3 (non-simple family)	–
7	1{0}5{0}3	7^m+n+2+5×7ⁿ⁺¹+3 (m+n ≥ 0)	always divisible by 3 (non-simple family)	–
7	3{0}1{0}5	3×7^m+n+2+7ⁿ⁺¹+5 (m+n ≥ 0)	always divisible by 3 (non-simple family)	–
7	3{0}5{0}1	3×7^m+n+2+5×7ⁿ⁺¹+1 (m+n ≥ 0)	always divisible by 3 (non-simple family)	–
7	5{0}1{0}3	5×7^m+n+2+7ⁿ⁺¹+3 (m+n ≥ 0)	always divisible by 3 (non-simple family)	–
7	5{0}3{0}1	5×7^m+n+2+3×7ⁿ⁺¹+1 (m+n ≥ 0)	always divisible by 3 (non-simple family)	–
7	1{0}1{0}1{0}1	7^r+m+n+3+7^m+n+2+7ⁿ⁺¹+1 (r+m+n ≥ 0)	always divisible by 2 (non-simple family)	–
7	1{0}1{0}2	7^m+n+2+7ⁿ⁺¹+2 (m+n ≥ 0)	always divisible by 2 (non-simple family)	–
7	1{0}2{0}1	7^m+n+2+2×7ⁿ⁺¹+1 (m+n ≥ 0)	always divisible by 2 (non-simple family)	–
7	2{0}1{0}1	2×7^m+n+2+7ⁿ⁺¹+1 (m+n ≥ 0)	always divisible by 2 (non-simple family)	–
7	4{0}5{0}5	4×7^m+n+2+5×7ⁿ⁺¹+5 (m+n ≥ 0)	always divisible by 2 (non-simple family)	–
7	5{0}4{0}5	5×7^m+n+2+4×7ⁿ⁺¹+5 (m+n ≥ 0)	always divisible by 2 (non-simple family)	–
7	5{0}5{0}4	5×7^m+n+2+5×7ⁿ⁺¹+4 (m+n ≥ 0)	always divisible by 2 (non-simple family)	–
8	2{0}5	2×8ⁿ⁺¹+5 (n ≥ 0)	always divisible by 7	http://factordb.com/index.php?query=2*8%5E%28n%2B1%29%2B5&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
8	4{0}3	4×8ⁿ⁺¹+3 (n ≥ 0)	always divisible by 7	http://factordb.com/index.php?query=4*8%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
8	6{0}1	6×8ⁿ⁺¹+1 (n ≥ 0)	always divisible by 7	http://factordb.com/index.php?query=6*8%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
8	44{0}3	36×8ⁿ⁺¹+3 (n ≥ 0)	always divisible by 3	http://factordb.com/index.php?query=36*8%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
8	6{0}11	6×8ⁿ⁺²+9 (n ≥ 0)	always divisible by 3	http://factordb.com/index.php?query=6*8%5E%28n%2B2%29%2B9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
9	{7}62	(7×9ⁿ⁺²−119)/8 (n ≥ 0)	always divisible by 7	http://factordb.com/index.php?query=%287*9%5E%28n%2B2%29-119%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
11	2{5}3	(5×11ⁿ⁺¹−5)/2 (n ≥ 0)	always divisible by 5 (in fact, always divisible by 25)	http://factordb.com/index.php?query=%285*11%5E%28n%2B1%29-5%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11	3{5}2	(7×11ⁿ⁺¹−7)/2 (n ≥ 0)	always divisible by 5 (in fact, always divisible by 35)	http://factordb.com/index.php?query=%287*11%5E%28n%2B1%29-7%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11	3{5}9	(7×11ⁿ⁺¹+7)/2 (n ≥ 0)	always divisible by 7	http://factordb.com/index.php?query=%287*11%5E%28n%2B1%29%2B7%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11	A{5}2	(21×11ⁿ⁺¹−7)/2 (n ≥ 0)	always divisible by 7	http://factordb.com/index.php?query=%2821*11%5E%28n%2B1%29-7%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11	A{5}9	(21×11ⁿ⁺¹+7)/2 (n ≥ 0)	always divisible by 7	http://factordb.com/index.php?query=%2821*11%5E%28n%2B1%29%2B7%29%2F2&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
12	A{0}21	10×12ⁿ⁺²+25	always divisible by 5	http://factordb.com/index.php?query=10*12%5E%28n%2B2%29%2B25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3{6}A	(7×13ⁿ⁺¹+7)/2 (n ≥ 0)	always divisible by 7	http://factordb.com/index.php?query=%287*13%5E%28n%2B1%29%2B7%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	5{A}C	(35×13ⁿ⁺¹+7)/6 (n ≥ 0)	always divisible by 7	http://factordb.com/index.php?query=%2835*13%5E%28n%2B1%29%2B7%29%2F6&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	A{6}3	(21×13ⁿ⁺¹−7)/2 (n ≥ 0)	always divisible by 7	http://factordb.com/index.php?query=%2821*13%5E%28n%2B1%29-7%29%2F2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	C{A}5	(77×13ⁿ⁺¹−35)/6 (n ≥ 0)	always divisible by 7	http://factordb.com/index.php?query=%2877*13%5E%28n%2B1%29-35%29%2F6&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
14	8{6}9	(110×14ⁿ⁺¹+33)/13 (n ≥ 0)	always divisible by 11	http://factordb.com/index.php?query=%28110*14%5E%28n%2B1%29%2B33%29%2F13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
14	40{4}9	(732×14ⁿ⁺¹+61)/13 (n ≥ 0)	always divisible by 61	http://factordb.com/index.php?query=%28732*14%5E%28n%2B1%29%2B61%29%2F13&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
15	9{6}8	(66×15ⁿ⁺¹+11)/7 (n ≥ 0)	always divisible by 11	http://factordb.com/index.php?query=%2866*15%5E%28n%2B1%29%2B11%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	2{C}3	(14×16ⁿ⁺¹−49)/5 (n ≥ 0)	always divisible by 7	http://factordb.com/index.php?query=%2814*16%5E%28n%2B1%29-49%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	A{4}1	(154×16ⁿ⁺¹−49)/15 (n ≥ 0)	always divisible by 7	http://factordb.com/index.php?query=%28154*16%5E%28n%2B1%29-49%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	8{C}F	(44×16ⁿ⁺¹+11)/5 (n ≥ 0)	always divisible by 11	http://factordb.com/index.php?query=%2844*16%5E%28n%2B1%29%2B11%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	8{A}F	(26×16ⁿ⁺¹+13)/3 (n ≥ 0)	always divisible by 13	http://factordb.com/index.php?query=%2826*16%5E%28n%2B1%29%2B13%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	A{6}9	(52×16ⁿ⁺¹+13)/5 (n ≥ 0)	always divisible by 13	http://factordb.com/index.php?query=%2852*16%5E%28n%2B1%29%2B13%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
21	B0{H}6H	(4637×21ⁿ⁺²−4637)/20 (n ≥ 0)	always divisible by 4637	http://factordb.com/index.php?query=%284637*21%5E%28n%2B2%29-4637%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	4{O}9	(44×28ⁿ⁺¹−143)/9 (n ≥ 0)	always divisible by 11	http://factordb.com/index.php?query=%2844*28%5E%28n%2B1%29-143%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	N{6}R	(209×28ⁿ⁺¹+187)/9 (n ≥ 0)	always divisible by 11	http://factordb.com/index.php?query=%28209*28%5E%28n%2B1%29%2B187%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	D{6}R	(119×28ⁿ⁺¹+187)/9 (n ≥ 0)	always divisible by 17	http://factordb.com/index.php?query=%28119*28%5E%28n%2B1%29%2B187%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
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}	(9^m+n+1+40×9ⁿ−1)/8 (m+n ≥ 1)	always divisible by some element of {2,5} (non-simple family) divisible by 2 if m+n is even, divisible by 5 if m+n is odd	–
9	{7}2{7}	(7×9^m+n+1−40×9ⁿ−7)/8 (m+n ≥ 1)	always divisible by some element of {2,5} (non-simple family) divisible by 2 if m+n is even, divisible by 5 if m+n is odd	–
9	5{0}{7}	(40×9^m+n+7×9ⁿ−7)/8 (m+n ≥ 1)	always divisible by some element of {2,5} (non-simple family) divisible by 2 if n is odd, divisible by 5 if n is even	–
9	{3}{0}5	(3×9^m+n+1−3×9ⁿ⁺¹+40)/8 (m+n ≥ 1)	always divisible by some element of {2,5} (non-simple family) divisible by 2 if m is odd, divisible by 5 if m is even	–
9	{7}{0}5	(7×9^m+n+1−7×9ⁿ⁺¹+40)/8 (m+n ≥ 1)	always divisible by some element of {2,5} (non-simple family) divisible by 2 if m is odd, divisible by 5 if m 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	(11ⁿ⁺¹+5)/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%2B5%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}9	(11ⁿ⁺¹+7)/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%2B7%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}9	(7×11ⁿ⁺¹+13)/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%2B13%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}A	(7×11ⁿ⁺¹+23)/10 (n ≥ 1)	always divisible by some element of {2,3} divisible by 2 if the length is odd, divisible by 3 if the length is even	http://factordb.com/index.php?query=%287*11%5E%28n%2B1%29%2B23%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	3{0}{5}	(6×11^m+n+11ⁿ−1)/2	always divisible by some element of {2,3} (non-simple family) divisible by 2 if n is odd, divisible by 3 if n is even	–
11	{5}{0}3	(11^m+n+1−11ⁿ⁺¹+6)/2	always divisible by some element of {2,3} (non-simple family) divisible by 2 if m is odd, divisible by 3 if m is even	–
14	4{0}1	4×14ⁿ⁺¹+1 (n ≥ 0)	always divisible by some element of {3,5} divisible by 3 if n is even, divisible by 5 if n is odd	http://factordb.com/index.php?query=4*14%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
14	B{0}1	11×14ⁿ⁺¹+1 (n ≥ 0)	always divisible by some element of {3,5} divisible by 3 if n is odd, divisible by 5 if n is even	http://factordb.com/index.php?query=11*14%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
14	3{D}	4×14ⁿ−1 (n ≥ 1)	always divisible by some element of {3,5} divisible by 3 if n is even, divisible by 5 if n is odd	http://factordb.com/index.php?query=4*14%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
14	A{D}	11×14ⁿ−1 (n ≥ 1)	always divisible by some element of {3,5} divisible by 3 if n is odd, divisible by 5 if n is even	http://factordb.com/index.php?query=11*14%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
14	1{0}B	14ⁿ⁺¹+11 (n ≥ 0)	always divisible by some element of {3,5} divisible by 3 if n is odd, divisible by 5 if n is even	http://factordb.com/index.php?query=14%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
14	{D}3	14ⁿ⁺¹−11 (n ≥ 1)	always divisible by some element of {3,5} divisible by 3 if n is even, divisible by 5 if n is odd	http://factordb.com/index.php?query=14%5E%28n%2B1%29-11&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
14	{4}9	(4×14ⁿ⁺¹+61)/13 (n ≥ 1)	always divisible by some element of {3,5} divisible by 3 if n is even, divisible by 5 if n is odd	http://factordb.com/index.php?query=%284*14%5E%28n%2B1%29%2B61%29%2F13&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
14	{8}5	(8×14ⁿ⁺¹−47)/13 (n ≥ 1)	always divisible by some element of {3,5} divisible by 3 if n is odd, divisible by 5 if n is even	http://factordb.com/index.php?query=%288*14%5E%28n%2B1%29-47%29%2F13&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
20	8{0}1	8×20ⁿ⁺¹+1 (n ≥ 0)	always divisible by some element of {3,7} divisible by 3 if n is odd, divisible by 7 if n is even	http://factordb.com/index.php?query=8*20%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
20	D{0}1	13×20ⁿ⁺¹+1 (n ≥ 0)	always divisible by some element of {3,7} divisible by 3 if n is even, divisible by 7 if n is odd	http://factordb.com/index.php?query=13*20%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
20	7{J}	8×20ⁿ−1 (n ≥ 1)	always divisible by some element of {3,7} divisible by 3 if n is odd, divisible by 7 if n is even	http://factordb.com/index.php?query=8*20%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
20	C{J}	13×20ⁿ−1 (n ≥ 1)	always divisible by some element of {3,7} divisible by 3 if n is even, divisible by 7 if n is odd	http://factordb.com/index.php?query=13*20%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
20	1{0}D	20ⁿ⁺¹+13 (n ≥ 0)	always divisible by some element of {3,7} divisible by 3 if n is even, divisible by 7 if n is odd	http://factordb.com/index.php?query=20%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
20	{J}7	20ⁿ⁺¹−13 (n ≥ 1)	always divisible by some element of {3,7} divisible by 3 if n is odd, divisible by 7 if n is even	http://factordb.com/index.php?query=20%5E%28n%2B1%29-13&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
25	D{1}	(313×25ⁿ−1)/24 (n ≥ 1)	always divisible by some element of {2,13} divisible by 2 if n is odd, divisible by 13 if n is even	http://factordb.com/index.php?query=%28313*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
25	E{1}	(337×25ⁿ−1)/24 (n ≥ 1)	always divisible by some element of {2,13} divisible by 2 if n is even, divisible by 13 if n is odd	http://factordb.com/index.php?query=%28337*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
25	1E{1}	(937×25ⁿ−1)/24 (n ≥ 0)	always divisible by some element of {2,13} divisible by 2 if n is odd, divisible by 13 if n is even	http://factordb.com/index.php?query=%28937*25%5En-1%29%2F24&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
25	1F{1}	(961×25ⁿ−1)/24 (n ≥ 0)	always divisible by some element of {2,13} divisible by 2 if n is even, divisible by 13 if n is odd	http://factordb.com/index.php?query=%28961*25%5En-1%29%2F24&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
32	A{0}1	10×32ⁿ⁺¹+1 (n ≥ 0)	always divisible by some element of {3,11} divisible by 3 if n is even, divisible by 11 if n is odd	http://factordb.com/index.php?query=10*32%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
32	N{0}1	23×32ⁿ⁺¹+1 (n ≥ 0)	always divisible by some element of {3,11} divisible by 3 if n is odd, divisible by 11 if n is even	http://factordb.com/index.php?query=23*32%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
32	9{V}	10×32ⁿ−1 (n ≥ 1)	always divisible by some element of {3,11} divisible by 3 if n is even, divisible by 11 if n is odd	http://factordb.com/index.php?query=10*32%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
32	M{V}	23×32ⁿ−1 (n ≥ 1)	always divisible by some element of {3,11} divisible by 3 if n is odd, divisible by 11 if n is even	http://factordb.com/index.php?query=23*32%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
32	1{0}N	32ⁿ⁺¹+23 (n ≥ 0)	always divisible by some element of {3,11} divisible by 3 if n is odd, divisible by 11 if n is even	http://factordb.com/index.php?query=32%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
32	{V}9	32ⁿ⁺¹−23 (n ≥ 1)	always divisible by some element of {3,11} divisible by 3 if n is even, divisible by 11 if n is odd	http://factordb.com/index.php?query=32%5E%28n%2B1%29-23&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
34	6{0}1	6×34ⁿ⁺¹+1 (n ≥ 0)	always divisible by some element of {5,7} divisible by 5 if n is even, divisible by 7 if n is odd	http://factordb.com/index.php?query=6*34%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
34	5{X}	6×34ⁿ−1 (n ≥ 1)	always divisible by some element of {5,7} divisible by 5 if n is even, divisible by 7 if n is odd	http://factordb.com/index.php?query=6*34%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
34	S{X}	29×34ⁿ−1 (n ≥ 1)	always divisible by some element of {5,7} divisible by 5 if n is odd, divisible by 7 if n is even	http://factordb.com/index.php?query=29*34%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
34	{X}5	34ⁿ⁺¹−29 (n ≥ 1)	always divisible by some element of {5,7} divisible by 5 if n is even, divisible by 7 if n is odd	http://factordb.com/index.php?query=34%5E%28n%2B1%29-29&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
8	6{4}7	(46×8ⁿ⁺¹+17)/7 (n ≥ 0)	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)	http://factordb.com/index.php?query=%2846*8%5E%28n%2B1%29%2B17%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	95{0}3	122×13ⁿ⁺¹+3 (n ≥ 0)	always divisible by some element of {5,7,17} divisible by 7 if n is even, divisible by 5 if n == 3 mod 4, divisible by 17 if n == 1 mod 4	http://factordb.com/index.php?query=122*13%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
13	3{0}95	3×13ⁿ⁺²+122 (n ≥ 0)	always divisible by some element of {5,7,17} divisible by 7 if n is odd, divisible by 5 if n == 2 mod 4, divisible by 17 if n == 0 mod 4	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
16	{4}D	(4×16ⁿ⁺¹+131)/15 (n ≥ 1)	always divisible by some element of {3,7,13} divisible by 3 if n == 2 mod 3, divisible by 7 if n == 1 mod 3, divisible by 13 if n == 0 mod 3	http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29%2B131%29%2F15&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	{8}F	(8×16ⁿ⁺¹+97)/15 (n ≥ 1)	always divisible by some element of {3,7,13} divisible by 3 if n == 0 mod 3, divisible by 7 if n == 2 mod 3, divisible by 13 if n == 1 mod 3	http://factordb.com/index.php?query=%288*16%5E%28n%2B1%29%2B97%29%2F15&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17	7F{0}D	134×17ⁿ⁺¹+13 (n ≥ 0)	always divisible by some element of {3,5,29} divisible by 3 if n is odd, divisible by 5 if n == 2 mod 4, divisible by 29 if n == 0 mod 4	http://factordb.com/index.php?query=134*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
17	D{0}7F	13×17ⁿ⁺²+134 (n ≥ 0)	always divisible by some element of {3,5,29} divisible by 3 if n is even, divisible by 5 if n == 3 mod 4, divisible by 29 if n == 1 mod 4	http://factordb.com/index.php?query=13*17%5E%28n%2B2%29%2B134&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
21	{7}D	(7×21ⁿ⁺¹+113)/20 (n ≥ 1)	always divisible by some element of {2,13,17} divisible by 2 if n is odd, divisible by 13 if n == 0 mod 4, divisible by 17 if n == 2 mod 4	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
23	7L{0}1	182×23ⁿ⁺¹+1 (n ≥ 0)	always divisible by some element of {3,5,53} divisible by 3 if n is odd, divisible by 5 if n == 2 mod 4, divisible by 53 if n == 0 mod 4	http://factordb.com/index.php?query=182*23%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
23	1{0}7L	23ⁿ⁺²+182 (n ≥ 0)	always divisible by some element of {3,5,53} divisible by 3 if n is even, divisible by 5 if n == 3 mod 4, divisible by 53 if n == 1 mod 4	http://factordb.com/index.php?query=23%5E%28n%2B2%29%2B182&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
23	{D}GA	(13×23ⁿ⁺²+1439)/22 (n ≥ 0)	always divisible by some element of {2,5,7,37,79} divisible by 2 if n is even, divisible by 5 if n == 1 mod 4, divisible by 7 if n == 0 mod 3, divisible by 37 if n == 7 mod 12, divisible by 79 if n == 2 mod 3	http://factordb.com/index.php?query=%2813*23%5E%28n%2B2%29%2B1439%29%2F22&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
23	L{5}L	(467×23ⁿ⁺¹+347)/22 (n ≥ 0)	always divisible by some element of {2,5,7,13,37} divisible by 2 if n is even, divisible by 5 if n == 1 mod 4, divisible by 7 if n == 0 mod 3, divisible by 13 if n == 1 mod 6, divisible by 37 if n == 11 mod 12	http://factordb.com/index.php?query=%28467*23%5E%28n%2B1%29%2B347%29%2F22&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
25	9{N}	(239×25ⁿ−23)/24 (n ≥ 1)	always divisible by some element of {3,7,31} divisible by 3 if n == 0 mod 3, divisible by 7 if n == 2 mod 3, divisible by 31 if n == 1 mod 3	http://factordb.com/index.php?query=%28239*25%5En-23%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
25	{N}E	(23×25ⁿ⁺¹−239)/24 (n ≥ 1)	always divisible by some element of {3,7,31} divisible by 3 if n == 2 mod 3, divisible by 7 if n == 0 mod 3, divisible by 31 if n == 1 mod 3	http://factordb.com/index.php?query=%2823*25%5E%28n%2B1%29-239%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
25	27{1}	(1369×25ⁿ−1)/24 (n ≥ 0)	always divisible by some element of {3,7,31} divisible by 3 if n == 0 mod 3, divisible by 7 if n == 2 mod 3, divisible by 31 if n == 1 mod 3	http://factordb.com/index.php?query=%281369*25%5En-1%29%2F24&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
27	JP{0}1	538×27ⁿ⁺¹+1 (n ≥ 0)	always divisible by some element of {5,7,73} divisible by 7 if n is odd, divisible by 5 if n == 2 mod 4, divisible by 73 if n == 0 mod 4	http://factordb.com/index.php?query=538*27%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
27	1{0}JP	27ⁿ⁺²+538 (n ≥ 0)	always divisible by some element of {5,7,73} divisible by 7 if n is even, divisible by 5 if n == 3 mod 4, divisible by 73 if n == 1 mod 4	http://factordb.com/index.php?query=27%5E%28n%2B2%29%2B538&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
27	J{0}2	19×27ⁿ⁺¹+2 (n ≥ 0)	always divisible by some element of {5,7,73} divisible by 7 if n is odd, divisible by 5 if n == 0 mod 4, divisible by 73 if n == 2 mod 4	http://factordb.com/index.php?query=19*27%5E%28n%2B1%29%2B2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
29	{2}{5}	(2×29^m+n+3×29ⁿ−5)/28	always divisible by some element of {2,3,5} (non-simple family) divisible by 2 if n is even, divisible by 3 if m and n are both even or both odd, divisible by 5 if m is even	–
29	{5}{2}	(5×29^m+n−3×29ⁿ−2)/28	always divisible by some element of {2,3,5} (non-simple family) divisible by 2 if m is even, divisible by 3 if m and n are both even or both odd, divisible by 5 if n is even	–
29	{M}{P}	(22×29^m+n+3×29ⁿ−25)/28	always divisible by some element of {2,3,5} (non-simple family) divisible by 2 if n is even, divisible by 3 if m and n are both even or both odd, divisible by 5 if m is even	–
29	{P}{M}	(25×29^m+n−3×29ⁿ−22)/28	always divisible by some element of {2,3,5} (non-simple family) divisible by 2 if m is even, divisible by 3 if m and n are both even or both odd, divisible by 5 if n is even	–
30	A{0}9J	10×30ⁿ⁺²+289 (n ≥ 0)	always divisible by some element of {7,13,19,31} divisible by 7 if n == 0 mod 3, divisible by 13 if n == 4 mod 6, divisible by 19 if n == 2 mod 3, divisible by 31 if n is odd	http://factordb.com/index.php?query=10*30%5E%28n%2B2%29%2B289&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
31	O{5}	(145×31ⁿ−1)/6 (n ≥ 1)	always divisible by some element of {2,3,7,19} divisible by 2 if n is even, divisible by 3 if n == 0 mod 3, divisible by 7 if n == 1 mod 6, divisible by 19 if n == 5 mod 6	http://factordb.com/index.php?query=%28145*31%5En-1%29%2F6&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
32	8{0}V	8×32ⁿ⁺¹+31 (n ≥ 0)	always divisible by some element of {3,5,41} divisible by 3 if n is odd, divisible by 5 if n == 2 mod 4, divisible by 41 if n == 0 mod 4	http://factordb.com/index.php?query=8*32%5E%28n%2B1%29%2B31&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
32	{G}L	(16×32ⁿ⁺¹+139)/31 (n ≥ 1)	always divisible by some element of {3,5,41} divisible by 3 if n is even, divisible by 5 if n == 3 mod 4, divisible by 41 if n == 1 mod 4	http://factordb.com/index.php?query=%2816*32%5E%28n%2B1%29%2B139%29%2F31&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}	(9ⁿ−1)/8 (n ≥ 2)	difference-of-two-squares factorization (9ⁿ−1)/8 = (3ⁿ−1) × (3ⁿ+1) / 8	http://factordb.com/index.php?query=%289%5En-1%29%2F8&use=n&n=2&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
8	1{0}1	8ⁿ⁺¹+1 (n ≥ 0)	sum-of-two-cubes factorization 8ⁿ⁺¹+1 = (2ⁿ⁺¹+1) × (4ⁿ⁺¹−2ⁿ⁺¹+1)	http://factordb.com/index.php?query=8%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
9	3{1}	(25×9ⁿ−1)/8 (n ≥ 1)	difference-of-two-squares factorization (25×9ⁿ−1)/8 = (5×3ⁿ−1) × (5×3ⁿ+1) / 8	http://factordb.com/index.php?query=%2825*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	3{8}	4×9ⁿ−1 (n ≥ 1)	difference-of-two-squares factorization 4×9ⁿ−1 = (2×3ⁿ−1) × (2×3ⁿ+1)	http://factordb.com/index.php?query=4*9%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
9	{8}5	9ⁿ⁺¹−4 (n ≥ 1)	difference-of-two-squares factorization 9ⁿ⁺¹−4 = (3ⁿ⁺¹−2) × (3ⁿ⁺¹+2)	http://factordb.com/index.php?query=9%5E%28n%2B1%29-4&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}35	4×9ⁿ⁺²−49 (n ≥ 0)	difference-of-two-squares factorization 4×9ⁿ⁺²−49 = (2×3ⁿ⁺²−7) × (2×3ⁿ⁺²+7)	http://factordb.com/index.php?query=4*9%5E%28n%2B2%29-49&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	8{F}	9×16ⁿ−1 (n ≥ 1)	difference-of-two-squares factorization 9×16ⁿ−1 = (3×4ⁿ−1) × (3×4ⁿ+1)	http://factordb.com/index.php?query=9*16%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	{F}7	16ⁿ⁺¹−9 (n ≥ 1)	difference-of-two-squares factorization 16ⁿ⁺¹−9 = (4ⁿ⁺¹−3) × (4ⁿ⁺¹+3)	http://factordb.com/index.php?query=16%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
16	{4}1	(4×16ⁿ⁺¹−49)/15 (n ≥ 1)	difference-of-two-squares factorization (4×16ⁿ⁺¹−49)/15 = (2×4ⁿ⁺¹−7) × (2×4ⁿ⁺¹+7) / 15	http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29-49%29%2F15&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	B{4}1	(169×16ⁿ⁺¹−49)/15 (n ≥ 0)	difference-of-two-squares factorization (169×16ⁿ⁺¹−49)/15 = (13×4ⁿ⁺¹−7) × (13×4ⁿ⁺¹+7) / 15	http://factordb.com/index.php?query=%28169*16%5E%28n%2B1%29-49%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	1{5}	(4×16ⁿ−1)/3 (n ≥ 1)	difference-of-two-squares factorization (4×16ⁿ−1)/3 = (2×4ⁿ−1) × (2×4ⁿ+1) / 3	http://factordb.com/index.php?query=%284*16%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	8{5}	(25×16ⁿ−1)/3 (n ≥ 1)	difference-of-two-squares factorization (25×16ⁿ−1)/3 = (5×4ⁿ−1) × (5×4ⁿ+1) / 3	http://factordb.com/index.php?query=%2825*16%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	10{5}	(49×16ⁿ−1)/3 (n ≥ 1)	difference-of-two-squares factorization (49×16ⁿ−1)/3 = (7×4ⁿ−1) × (7×4ⁿ+1) / 3	http://factordb.com/index.php?query=%2849*16%5En-1%29%2F3&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	A1{5}	(484×16ⁿ−1)/3 (n ≥ 0)	difference-of-two-squares factorization (484×16ⁿ−1)/3 = (22×4ⁿ−1) × (22×4ⁿ+1) / 3	http://factordb.com/index.php?query=%28484*16%5En-1%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	7{3}	(36×16ⁿ−1)/5 (n ≥ 1)	difference-of-two-squares factorization (36×16ⁿ−1)/5 = (6×4ⁿ−1) × (6×4ⁿ+1) / 5	http://factordb.com/index.php?query=%2836*16%5En-1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	3{F}AF	4×16ⁿ⁺²−81 (n ≥ 0)	difference-of-two-squares factorization 4×16ⁿ⁺²−81 = (2×4ⁿ⁺²−9) × (2×4ⁿ⁺²+9)	http://factordb.com/index.php?query=4*16%5E%28n%2B2%29-81&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	30{F}AF	49×16ⁿ⁺²−81 (n ≥ 0)	difference-of-two-squares factorization 49×16ⁿ⁺²−81 = (7×4ⁿ⁺²−9) × (7×4ⁿ⁺²+9)	http://factordb.com/index.php?query=49*16%5E%28n%2B2%29-81&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	3{F}A0F	4×16ⁿ⁺³−1521 (n ≥ 0)	difference-of-two-squares factorization 4×16ⁿ⁺³−1521 = (2×4ⁿ⁺³−39) × (2×4ⁿ⁺³+39)	http://factordb.com/index.php?query=4*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
16	30{F}A0F	49×16ⁿ⁺³−1521 (n ≥ 0)	difference-of-two-squares factorization 49×16ⁿ⁺³−1521 = (7×4ⁿ⁺³−39) × (7×4ⁿ⁺³+39)	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
16	{3}23	(16ⁿ⁺²−81)/5 (n ≥ 0)	difference-of-two-squares factorization (16ⁿ⁺²−81)/5 = (4ⁿ⁺²−9) × (4ⁿ⁺²+9) / 5 (in fact, difference-of-4th-powers factorization) (16ⁿ⁺²−81)/5 = (2ⁿ⁺²−3) × (2ⁿ⁺²+3) × (4ⁿ⁺²+9) / 5	http://factordb.com/index.php?query=%2816%5E%28n%2B2%29-81%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	{5}45	(16ⁿ⁺²−49)/3 (n ≥ 0)	difference-of-two-squares factorization (16ⁿ⁺²−49)/3 = (4ⁿ⁺²−7) × (4ⁿ⁺²+7) / 3	http://factordb.com/index.php?query=%2816%5E%28n%2B2%29-49%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	{C}B	(4×16ⁿ⁺¹−9)/5 (n ≥ 1)	difference-of-two-squares factorization (4×16ⁿ⁺¹−9)/5 = (2×4ⁿ⁺¹−3) × (2×4ⁿ⁺¹+3) / 5	http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29-9%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	{C}D	(4×16ⁿ⁺¹+1)/5 (n ≥ 1)	Aurifeuillian factorization of x⁴+4×y⁴ (4×16ⁿ⁺¹+1)/5 = (2×4ⁿ⁺¹−2×2ⁿ⁺¹+1) × (2×4ⁿ⁺¹+2×2ⁿ⁺¹+1) / 5	http://factordb.com/index.php?query=%284*16%5E%28n%2B1%29%2B1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	{C}DD	(4×16ⁿ⁺²+81)/5 (n ≥ 0)	Aurifeuillian factorization of x⁴+4×y⁴ (4×16ⁿ⁺²+81)/5 = (2×4ⁿ⁺²−6×2ⁿ⁺²+9) × (2×4ⁿ⁺²+6×2ⁿ⁺²+9) / 5	http://factordb.com/index.php?query=%284*16%5E%28n%2B2%29%2B81%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
25	{1}	(25ⁿ−1)/24 (n ≥ 2)	difference-of-two-squares factorization (25ⁿ−1)/24 = (5ⁿ−1) × (5ⁿ+1) / 24	http://factordb.com/index.php?query=%2825%5En-1%29%2F24&use=n&n=2&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
25	2{1}	(49×25ⁿ−1)/24 (n ≥ 1)	difference-of-two-squares factorization (49×25ⁿ−1)/24 = (7×5ⁿ−1) × (7×5ⁿ+1) / 24	http://factordb.com/index.php?query=%2849*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
25	5{1}	(121×25ⁿ−1)/24 (n ≥ 1)	difference-of-two-squares factorization (121×25ⁿ−1)/24 = (11×5ⁿ−1) × (11×5ⁿ+1) / 24	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
25	7{1}	(169×25ⁿ−1)/24 (n ≥ 1)	difference-of-two-squares factorization (169×25ⁿ−1)/24 = (13×5ⁿ−1) × (13×5ⁿ+1) / 24	http://factordb.com/index.php?query=%28169*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
25	C{1}	(289×25ⁿ−1)/24 (n ≥ 1)	difference-of-two-squares factorization (289×25ⁿ−1)/24 = (17×5ⁿ−1) × (17×5ⁿ+1) / 24	http://factordb.com/index.php?query=%28289*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
25	F{1}	(361×25ⁿ−1)/24 (n ≥ 1)	difference-of-two-squares factorization (361×25ⁿ−1)/24 = (19×5ⁿ−1) × (19×5ⁿ+1) / 24	http://factordb.com/index.php?query=%28361*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
25	M{1}	(529×25ⁿ−1)/24 (n ≥ 1)	difference-of-two-squares factorization (529×25ⁿ−1)/24 = (23×5ⁿ−1) × (23×5ⁿ+1) / 24	http://factordb.com/index.php?query=%28529*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
25	7C{1}	(4489×25ⁿ−1)/24 (n ≥ 0)	difference-of-two-squares factorization (4489×25ⁿ−1)/24 = (67×5ⁿ−1) × (67×5ⁿ+1) / 24	http://factordb.com/index.php?query=%284489*25%5En-1%29%2F24&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
25	D5{1}	(7921×25ⁿ−1)/24 (n ≥ 0)	difference-of-two-squares factorization (7921×25ⁿ−1)/24 = (89×5ⁿ−1) × (89×5ⁿ+1) / 24	http://factordb.com/index.php?query=%287921*25%5En-1%29%2F24&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
25	1{3}	(9×25ⁿ−1)/8 (n ≥ 1)	difference-of-two-squares factorization (9×25ⁿ−1)/8 = (3×5ⁿ−1) × (3×5ⁿ+1) / 8	http://factordb.com/index.php?query=%289*25%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
25	1{8}	(4×25ⁿ−1)/3 (n ≥ 1)	difference-of-two-squares factorization (4×25ⁿ−1)/3 = (2×5ⁿ−1) × (2×5ⁿ+1) / 3	http://factordb.com/index.php?query=%284*25%5En-1%29%2F3&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
25	5{8}	(16×25ⁿ−1)/3 (n ≥ 1)	difference-of-two-squares factorization (16×25ⁿ−1)/3 = (4×5ⁿ−1) × (4×5ⁿ+1) / 3	http://factordb.com/index.php?query=%2816*25%5En-1%29%2F3&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
25	A{3}	(81×25ⁿ−1)/8 (n ≥ 1)	difference-of-two-squares factorization (81×25ⁿ−1)/8 = (9×5ⁿ−1) × (9×5ⁿ+1) / 8	http://factordb.com/index.php?query=%2881*25%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
25	L{8}	(64×25ⁿ−1)/3 (n ≥ 1)	difference-of-two-squares factorization (64×25ⁿ−1)/3 = (8×5ⁿ−1) × (8×5ⁿ+1) / 3	http://factordb.com/index.php?query=%2864*25%5En-1%29%2F3&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
25	{3}2	(25ⁿ⁺¹−9)/8 (n ≥ 1)	difference-of-two-squares factorization (25ⁿ⁺¹−9)/8 = (5ⁿ⁺¹−3) × (5ⁿ⁺¹+3) / 8	http://factordb.com/index.php?query=%2825%5E%28n%2B1%29-9%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
25	{8}3	(25ⁿ⁺¹−16)/3 (n ≥ 1)	difference-of-two-squares factorization (25ⁿ⁺¹−16)/3 = (5ⁿ⁺¹−4) × (5ⁿ⁺¹+4) / 3	http://factordb.com/index.php?query=%2825%5E%28n%2B1%29-16%29%2F3&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
25	{8}7	(25ⁿ⁺¹−4)/3 (n ≥ 1)	difference-of-two-squares factorization (25ⁿ⁺¹−4)/3 = (5ⁿ⁺¹−2) × (5ⁿ⁺¹+2) / 3	http://factordb.com/index.php?query=%2825%5E%28n%2B1%29-4%29%2F3&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
25	{3}2I	(25ⁿ⁺²−81)/8 (n ≥ 0)	difference-of-two-squares factorization (25ⁿ⁺²−81)/8 = (5ⁿ⁺²−9) × (5ⁿ⁺²+9) / 8	http://factordb.com/index.php?query=%2825%5E%28n%2B2%29-81%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
25	{8}5I	(25ⁿ⁺²−196)/3 (n ≥ 0)	difference-of-two-squares factorization (25ⁿ⁺²−196)/3 = (5ⁿ⁺²−14) × (5ⁿ⁺²+14) / 3	http://factordb.com/index.php?query=%2825%5E%28n%2B2%29-196%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
25	{8}7C	(25ⁿ⁺²−64)/3 (n ≥ 0)	difference-of-two-squares factorization (25ⁿ⁺²−64)/3 = (5ⁿ⁺²−8) × (5ⁿ⁺²+8) / 3	http://factordb.com/index.php?query=%2825%5E%28n%2B2%29-64%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
27	8{0}1	8×27ⁿ⁺¹+1 (n ≥ 0)	sum-of-two-cubes factorization 8×27ⁿ⁺¹+1 = (2×3ⁿ⁺¹+1) × (4×9ⁿ⁺¹−2×3ⁿ⁺¹+1)	http://factordb.com/index.php?query=8*27%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
27	1{0}8	27ⁿ⁺¹+8 (n ≥ 0)	sum-of-two-cubes factorization 27ⁿ⁺¹+8 = (3ⁿ⁺¹+2) × (9ⁿ⁺¹−2×3ⁿ⁺¹+4)	http://factordb.com/index.php?query=27%5E%28n%2B1%29%2B8&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
27	{D}E	(27ⁿ⁺¹+1)/2 (n ≥ 1)	sum-of-two-cubes factorization (27ⁿ⁺¹+1)/2 = (3ⁿ⁺¹+1) × (9ⁿ⁺¹−3ⁿ⁺¹+1) / 2	http://factordb.com/index.php?query=%2827%5E%28n%2B1%29%2B1%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
27	7{Q}	8×27ⁿ−1 (n ≥ 1)	difference-of-two-cubes factorization 8×27ⁿ−1 = (2×3ⁿ−1) × (4×9ⁿ+2×3ⁿ+1)	http://factordb.com/index.php?query=8*27%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
27	{Q}J	27ⁿ⁺¹−8 (n ≥ 1)	difference-of-two-cubes factorization 27ⁿ⁺¹−8 = (3ⁿ⁺¹−2) × (9ⁿ⁺¹+2×3ⁿ⁺¹+4)	http://factordb.com/index.php?query=27%5E%28n%2B1%29-8&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
27	9{G}	(125×27ⁿ−8)/13 (n ≥ 1)	difference-of-two-cubes factorization (125×27ⁿ−8)/13 = (5×3ⁿ−2) × (25×9ⁿ+10×3ⁿ+4) / 13	http://factordb.com/index.php?query=%28125*27%5En-8%29%2F13&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
32	1{0}1	32ⁿ⁺¹+1 (n ≥ 0)	sum-of-two-5th-powers factorization 32ⁿ⁺¹+1 = (2ⁿ⁺¹+1) × (16ⁿ⁺¹−8ⁿ⁺¹+4ⁿ⁺¹−2ⁿ⁺¹+1)	http://factordb.com/index.php?query=32%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
32	{1}	(32ⁿ−1)/31 (n ≥ 2)	difference-of-two-5th-powers factorization (32ⁿ−1)/31 = (2ⁿ−1) × (16ⁿ+8ⁿ+4ⁿ+2ⁿ+1) / 31	http://factordb.com/index.php?query=%2832%5En-1%29%2F31&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	3{7}	(16×36ⁿ−1)/5 (n ≥ 1)	difference-of-two-squares factorization (16×36ⁿ−1)/5 = (4×6ⁿ−1) × (4×6ⁿ+1) / 5	http://factordb.com/index.php?query=%2816*36%5En-1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	3{Z}	4×36ⁿ−1 (n ≥ 1)	difference-of-two-squares factorization 4×36ⁿ−1 = (2×6ⁿ−1) × (2×6ⁿ+1)	http://factordb.com/index.php?query=4*36%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	8{Z}	9×36ⁿ−1 (n ≥ 1)	difference-of-two-squares factorization 9×36ⁿ−1 = (3×6ⁿ−1) × (3×6ⁿ+1)	http://factordb.com/index.php?query=9*36%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	O{Z}	25×36ⁿ−1 (n ≥ 1)	difference-of-two-squares factorization 25×36ⁿ−1 = (5×6ⁿ−1) × (5×6ⁿ+1)	http://factordb.com/index.php?query=25*36%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	{Z}B	36ⁿ⁺¹−25 (n ≥ 1)	difference-of-two-squares factorization 36ⁿ⁺¹−25 = (6ⁿ⁺¹−5) × (6ⁿ⁺¹+5)	http://factordb.com/index.php?query=36%5E%28n%2B1%29-25&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
36	8{Z}B	9×36ⁿ⁺¹−25 (n ≥ 0)	difference-of-two-squares factorization 9×36ⁿ⁺¹−25 = (3×6ⁿ⁺¹−5) × (3×6ⁿ⁺¹+5)	http://factordb.com/index.php?query=9*36%5E%28n%2B1%29-25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	F{Z}B	16×36ⁿ⁺¹−25 (n ≥ 0)	difference-of-two-squares factorization 16×36ⁿ⁺¹−25 = (4×6ⁿ⁺¹−5) × (4×6ⁿ⁺¹+5)	http://factordb.com/index.php?query=16*36%5E%28n%2B1%29-25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	{Z}RZ	36ⁿ⁺²−289 (n ≥ 0)	difference-of-two-squares factorization 36ⁿ⁺²−289 = (6ⁿ⁺²−17) × (6ⁿ⁺²+17)	http://factordb.com/index.php?query=36%5E%28n%2B2%29-289&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	O{Z}RZ	25×36ⁿ⁺²−289 (n ≥ 0)	difference-of-two-squares factorization 25×36ⁿ⁺²−289 = (5×6ⁿ⁺²−17) × (5×6ⁿ⁺²+17)	http://factordb.com/index.php?query=25*36%5E%28n%2B2%29-289&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	O{5}	(169×36ⁿ−1)/7 (n ≥ 1)	difference-of-two-squares factorization (169×36ⁿ−1)/7 = (13×6ⁿ−1) × (13×6ⁿ+1) / 7	http://factordb.com/index.php?query=%28169*36%5En-1%29%2F7&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
36	O{7}	(121×36ⁿ−1)/5 (n ≥ 1)	difference-of-two-squares factorization (121×36ⁿ−1)/5 = (11×6ⁿ−1) × (11×6ⁿ+1) / 5	http://factordb.com/index.php?query=%28121*36%5En-1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	{9}1	(9×36ⁿ⁺¹−289)/35 (n ≥ 1)	difference-of-two-squares factorization (9×36ⁿ⁺¹−289)/35 = (3×6ⁿ⁺¹−17) × (3×6ⁿ⁺¹+17) / 35	http://factordb.com/index.php?query=%289*36%5E%28n%2B1%29-289%29%2F35&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
36	T{9}1	(1024×36ⁿ⁺¹−289)/35 (n ≥ 0)	difference-of-two-squares factorization (1024×36ⁿ⁺¹−289)/35 = (32×6ⁿ⁺¹−17) × (32×6ⁿ⁺¹+17) / 35	http://factordb.com/index.php?query=%281024*36%5E%28n%2B1%29-289%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	{G}D	(16×36ⁿ⁺¹−121)/35 (n ≥ 1)	difference-of-two-squares factorization (16×36ⁿ⁺¹−121)/35 = (4×6ⁿ⁺¹−11) × (4×6ⁿ⁺¹+11) / 35	http://factordb.com/index.php?query=%2816*36%5E%28n%2B1%29-121%29%2F35&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
36	{G}8D	(16×36ⁿ⁺²−10201)/35 (n ≥ 0)	difference-of-two-squares factorization (16×36ⁿ⁺²−10201)/35 = (4×6ⁿ⁺²−101) × (4×6ⁿ⁺²+101) / 35	http://factordb.com/index.php?query=%2816*36%5E%28n%2B2%29-10201%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	R{G}D	(961×36ⁿ⁺¹−121)/35 (n ≥ 0)	difference-of-two-squares factorization (961×36ⁿ⁺¹−121)/35 = (31×6ⁿ⁺¹−11) × (31×6ⁿ⁺¹+11) / 35	http://factordb.com/index.php?query=%28961*36%5E%28n%2B2%29-121%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	3{G}8D	(121×36ⁿ⁺²−10201)/35 (n ≥ 0)	difference-of-two-squares factorization (121×36ⁿ⁺²−10201)/35 = (11×6ⁿ⁺²−101) × (11×6ⁿ⁺²+101) / 35	http://factordb.com/index.php?query=%28121*36%5E%28n%2B2%29-10201%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	R{G}8D	(961×36ⁿ⁺²−10201)/35 (n ≥ 0)	difference-of-two-squares factorization (961×36ⁿ⁺²−10201)/35 = (31×6ⁿ⁺²−101) × (31×6ⁿ⁺²+101) / 35	http://factordb.com/index.php?query=%28961*36%5E%28n%2B2%29-10201%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	{K}H	(4×36ⁿ⁺¹−25)/7 (n ≥ 1)	difference-of-two-squares factorization (4×36ⁿ⁺¹−25)/7 = (2×6ⁿ⁺¹−5) × (2×6ⁿ⁺¹+5) / 7	http://factordb.com/index.php?query=%284*36%5E%28n%2B1%29-25%29%2F7&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
36	B{K}H	(81×36ⁿ⁺¹−25)/7 (n ≥ 0)	difference-of-two-squares factorization (81×36ⁿ⁺¹−25)/7 = (9×6ⁿ⁺¹−5) × (9×6ⁿ⁺¹+5) / 7	http://factordb.com/index.php?query=%2881*36%5E%28n%2B1%29-25%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	3{K}IH	(25×36ⁿ⁺²−529)/7 (n ≥ 0)	difference-of-two-squares factorization (25×36ⁿ⁺²−529)/7 = (5×6ⁿ⁺²−23) × (5×6ⁿ⁺²+23) / 7	http://factordb.com/index.php?query=%2825*36%5E%28n%2B2%29-529%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	B{K}IH	(81×36ⁿ⁺²−529)/7 (n ≥ 0)	difference-of-two-squares factorization (81×36ⁿ⁺²−529)/7 = (9×6ⁿ⁺²−23) × (9×6ⁿ⁺²+23) / 7	http://factordb.com/index.php?query=%2881*36%5E%28n%2B2%29-529%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	{S}J	(4×36ⁿ⁺¹−49)/5 (n ≥ 1)	difference-of-two-squares factorization (4×36ⁿ⁺¹−49)/5 = (2×6ⁿ⁺¹−7) × (2×6ⁿ⁺¹+7) / 5	http://factordb.com/index.php?query=%284*36%5E%28n%2B1%29-49%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	{S}IJ	(4×36ⁿ⁺²−1849)/5 (n ≥ 0)	difference-of-two-squares factorization (4×36ⁿ⁺²−1849)/5 = (2×6ⁿ⁺²−43) × (2×6ⁿ⁺²+43) / 5	http://factordb.com/index.php?query=%284*36%5E%28n%2B2%29-1849%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	1{S}J	(9×36ⁿ⁺¹−49)/5 (n ≥ 0)	difference-of-two-squares factorization (9×36ⁿ⁺¹−49)/5 = (3×6ⁿ⁺¹−7) × (3×6ⁿ⁺¹+7) / 5	http://factordb.com/index.php?query=%289*36%5E%28n%2B1%29-49%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	C{S}J	(64×36ⁿ⁺¹−49)/5 (n ≥ 0)	difference-of-two-squares factorization (64×36ⁿ⁺¹−49)/5 = (8×6ⁿ⁺¹−7) × (8×6ⁿ⁺¹+7) / 5	http://factordb.com/index.php?query=%2864*36%5E%28n%2B1%29-49%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	X{S}J	(169×36ⁿ⁺¹−49)/5 (n ≥ 0)	difference-of-two-squares factorization (169×36ⁿ⁺¹−49)/5 = (13×6ⁿ⁺¹−7) × (13×6ⁿ⁺¹+7) / 5	http://factordb.com/index.php?query=%28169*36%5E%28n%2B1%29-49%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	1{S}GJ	(9×36ⁿ⁺²−2209)/5 (n ≥ 0)	difference-of-two-squares factorization (9×36ⁿ⁺²−2209)/5 = (3×6ⁿ⁺²−47) × (3×6ⁿ⁺²+47) / 5	http://factordb.com/index.php?query=%289*36%5E%28n%2B2%29-2209%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	9{S}GJ	(49×36ⁿ⁺²−2209)/5 (n ≥ 0)	difference-of-two-squares factorization (49×36ⁿ⁺²−2209)/5 = (7×6ⁿ⁺²−47) × (7×6ⁿ⁺²+47) / 5	http://factordb.com/index.php?query=%2849*36%5E%28n%2B2%29-2209%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	C{S}GJ	(64×36ⁿ⁺²−2209)/5 (n ≥ 0)	difference-of-two-squares factorization (64×36ⁿ⁺²−2209)/5 = (8×6ⁿ⁺²−47) × (8×6ⁿ⁺²+47) / 5	http://factordb.com/index.php?query=%2864*36%5E%28n%2B2%29-2209%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	X{S}GJ	(169×36ⁿ⁺²−2209)/5 (n ≥ 0)	difference-of-two-squares factorization (169×36ⁿ⁺²−2209)/5 = (13×6ⁿ⁺²−47) × (13×6ⁿ⁺²+47) / 5	http://factordb.com/index.php?query=%28169*36%5E%28n%2B2%29-2209%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	1{S}IJ	(9×36ⁿ⁺²−1849)/5 (n ≥ 0)	difference-of-two-squares factorization (9×36ⁿ⁺²−1849)/5 = (3×6ⁿ⁺²−43) × (3×6ⁿ⁺²+43) / 5	http://factordb.com/index.php?query=%289*36%5E%28n%2B2%29-1849%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	9{S}IJ	(49×36ⁿ⁺²−1849)/5 (n ≥ 0)	difference-of-two-squares factorization (49×36ⁿ⁺²−1849)/5 = (7×6ⁿ⁺²−43) × (7×6ⁿ⁺²+43) / 5	http://factordb.com/index.php?query=%2849*36%5E%28n%2B2%29-1849%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
14	8{D}	9×14ⁿ−1 (n ≥ 1)	combine of factor 5 and difference-of-two-squares factorization divisible by 5 if n is odd, 9×14ⁿ−1 = (3×14^n/2−1) × (3×14^n/2+1) if n is even	http://factordb.com/index.php?query=9*14%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
12	{B}9B	12ⁿ⁺²−25 (n ≥ 0)	combine of factor 13 and difference-of-two-squares factorization divisible by 13 if n is odd, 12ⁿ⁺²−25 = (12^(n+2)/2−5) × (12^(n+2)/2+5) if n is even	http://factordb.com/index.php?query=12%5E%28n%2B2%29-25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
14	{D}5	14ⁿ⁺¹−9 (n ≥ 1)	combine of factor 5 and difference-of-two-squares factorization divisible by 5 if n is even, 14ⁿ⁺¹−9 = (14^(n+1)/2−3) × (14^(n+1)/2+3) if n is odd	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
17	1{9}	(25×17ⁿ−9)/16 (n ≥ 1)	combine of factor 2 and difference-of-two-squares factorization divisible by 2 if n is odd, (25×17ⁿ−9)/16 = (5×17^n/2−3) × (5×17^n/2+3) / 16 if n is even	http://factordb.com/index.php?query=%2825*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
17	7{9}	(121×17ⁿ−9)/16 (n ≥ 1)	combine of factor 2 and difference-of-two-squares factorization divisible by 2 if n is odd, (121×17ⁿ−9)/16 = (11×17^n/2−3) × (11×17^n/2+3) / 16 if n is even	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
17	{9}2	(9×17ⁿ⁺¹−121)/16 (n ≥ 1)	combine of factor 2 and difference-of-two-squares factorization divisible by 2 if n is even, (9×17ⁿ⁺¹−121)/16 = (3×17^(n+1)/2−11) × (3×17^(n+1)/2+11) / 16 if n is odd	http://factordb.com/index.php?query=%289*17%5E%28n%2B1%29-121%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
17	{9}8	(9×17ⁿ⁺¹−25)/16 (n ≥ 1)	combine of factor 2 and difference-of-two-squares factorization divisible by 2 if n is even, (9×17ⁿ⁺¹−25)/16 = (3×17^(n+1)/2−5) × (3×17^(n+1)/2+5) / 16 if n is odd	http://factordb.com/index.php?query=%289*17%5E%28n%2B1%29-25%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
19	1{6}	(4×19ⁿ−1)/3 (n ≥ 1)	combine of factor 5 and difference-of-two-squares factorization divisible by 5 if n is odd, (4×19ⁿ−1)/3 = (2×19^n/2−1) × (2×19^n/2+1) / 3 if n is even	http://factordb.com/index.php?query=%284*19%5En-1%29%2F3&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
19	{6}5	(19ⁿ⁺¹−4)/3 (n ≥ 1)	combine of factor 5 and difference-of-two-squares factorization divisible by 5 if n is even, (19ⁿ⁺¹−4)/3 = (19^(n+1)/2−2) × (19^(n+1)/2+2) / 3 if n is odd	http://factordb.com/index.php?query=%2819%5E%28n%2B1%29-4%29%2F3&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
19	7{2}	(64×19ⁿ−1)/9 (n ≥ 1)	combine of factor 5 and difference-of-two-squares factorization divisible by 5 if n is odd, (64×19ⁿ−1)/9 = (8×19^n/2−1) × (8×19^n/2+1) / 9 if n is even	http://factordb.com/index.php?query=%2864*19%5En-1%29%2F9&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
19	89{6}	(484×19ⁿ−1)/3 (n ≥ 0)	combine of factor 5 and difference-of-two-squares factorization divisible by 5 if n is odd, (484×19ⁿ−1)/3 = (22×19^n/2−1) × (22×19^n/2+1) / 3 if n is even	http://factordb.com/index.php?query=%28484*19%5En-1%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
24	3{N}	4×24ⁿ−1 (n ≥ 1)	combine of factor 5 and difference-of-two-squares factorization divisible by 5 if n is odd, 4×24ⁿ−1 = (2×24^n/2−1) × (2×24^n/2+1) if n is even	http://factordb.com/index.php?query=4*24%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
24	5{N}	6×24ⁿ−1 (n ≥ 1)	combine of factor 5 and difference-of-two-squares factorization divisible by 5 if n is even, 6×24ⁿ−1 = (12×24^(n−1)/2−1) × (12×24^(n−1)/2+1) if n is odd	http://factordb.com/index.php?query=6*24%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
24	8{N}	9×24ⁿ−1 (n ≥ 1)	combine of factor 5 and difference-of-two-squares factorization divisible by 5 if n is odd, 9×24ⁿ−1 = (3×24^n/2−1) × (3×24^n/2+1) if n is even	http://factordb.com/index.php?query=9*24%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
24	{6}1	(6×24ⁿ⁺¹−121)/23 (n ≥ 1)	combine of factor 5 and difference-of-two-squares factorization divisible by 5 if n is odd, (6×24ⁿ⁺¹−121)/23 = (12×24^n/2−11) × (12×24^n/2+11) / 23 if n is even	http://factordb.com/index.php?query=%286*24%5E%28n%2B1%29-121%29%2F23&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
24	{N}LN	24ⁿ⁺²−49 (n ≥ 0)	combine of factor 5 and difference-of-two-squares factorization divisible by 5 if n is odd, 24ⁿ⁺²−49 = (24^(n+2)/2−7) × (24^(n+2)/2+7) if n is even	http://factordb.com/index.php?query=24%5E%28n%2B2%29-49&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
33	F{W}	16×33ⁿ−1 (n ≥ 1)	combine of factor 17 and difference-of-two-squares factorization divisible by 17 if n is odd, 16×33ⁿ−1 = (4×33^n/2−1) × (4×33^n/2+1) if n is even	http://factordb.com/index.php?query=16*33%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
33	{W}H	33ⁿ⁺¹−16 (n ≥ 1)	combine of factor 17 and difference-of-two-squares factorization divisible by 17 if n is even, 33ⁿ⁺¹−16 = (33^(n+1)/2−4) × (33^(n+1)/2+4) if n is odd	http://factordb.com/index.php?query=33%5E%28n%2B1%29-16&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
33	3{P}	(121×33ⁿ−25)/32 (n ≥ 1)	combine of factor 2 and difference-of-two-squares factorization divisible by 2 if n is odd, (121×33ⁿ−25)/32 = (11×33^n/2−5) × (11×33^n/2+5) / 32 if n is even	http://factordb.com/index.php?query=%28121*33%5En-25%29%2F32&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
33	D{P}	(441×33ⁿ−25)/32 (n ≥ 1)	combine of factor 2 and difference-of-two-squares factorization divisible by 2 if n is odd, (441×33ⁿ−25)/32 = (21×33^n/2−5) × (21×33^n/2+5) / 32 if n is even	http://factordb.com/index.php?query=%28441*33%5En-25%29%2F32&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
33	{9}4	(9×33ⁿ⁺¹−169)/32 (n ≥ 1)	combine of factor 2 and difference-of-two-squares factorization divisible by 2 if n is even, (9×33ⁿ⁺¹−169)/32 = (3×33^(n+1)/2−13) × (3×33^(n+1)/2+13) / 32 if n is odd	http://factordb.com/index.php?query=%289*33%5E%28n%2B1%29-169%29%2F32&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
34	1{B}	(4×34ⁿ−1)/3 (n ≥ 1)	combine of factor 5 and difference-of-two-squares factorization divisible by 5 if n is odd, (4×34ⁿ−1)/3 = (2×34^n/2−1) × (2×34^n/2+1) / 3 if n is even	http://factordb.com/index.php?query=%284*34%5En-1%29%2F3&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
34	8{X}	9×34ⁿ−1 (n ≥ 1)	combine of factor 5 and difference-of-two-squares factorization divisible by 5 if n is odd, 9×34ⁿ−1 = (3×34^n/2−1) × (3×34^n/2+1) if n is even	http://factordb.com/index.php?query=9*34%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
34	{X}P	34ⁿ⁺¹−9 (n ≥ 1)	combine of factor 5 and difference-of-two-squares factorization divisible by 5 if n is even, 34ⁿ⁺¹−9 = (34^(n+1)/2−3) × (34^(n+1)/2+3) if n is odd	http://factordb.com/index.php?query=34%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

Also families which contain only one very small prime > b: (this is because: for the case of covering congruence, an element in S is indeed in the family and > b, to make the factorizations be trivial; for the case of algebraic factorization (if the numbers are factored as F × G / d), F (or G) is equal to d and G (or F) is prime > b, to make the factorizations be trivial; for the case of combine of covering congruence and algebraic factorization (if the numbers are factored as F × G / d), an element in S is indeed in the family and > b or/and F (or G) is equal to d and G (or F) is prime > b, to make the factorizations be trivial)

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)	the only prime > b in this family	this prime > b written in decimal	why this family contains only this prime > 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 (if n = 0 already makes the number > b, then start with n = 0))
9	2{7}5	(23×9ⁿ⁺¹−23)/8 (n ≥ 0)	25	23	always divisible by 23	http://factordb.com/index.php?query=%2823*9%5E%28n%2B1%29-23%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	5{7}2	(47×9ⁿ⁺¹−47)/8 (n ≥ 0)	52	47	always divisible by 47	http://factordb.com/index.php?query=%2847*9%5E%28n%2B1%29-47%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
11	3{7}4	(37×11ⁿ⁺¹−37)/10 (n ≥ 0)	34	37	always divisible by 37	http://factordb.com/index.php?query=%2837*11%5E%28n%2B1%29-37%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11	4{7}3	(47×11ⁿ⁺¹−47)/10 (n ≥ 0)	43	47	always divisible by 47	http://factordb.com/index.php?query=%2847*11%5E%28n%2B1%29-47%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
27	2{0}J	2×27ⁿ⁺¹+19 (n ≥ 0)	2J	73	always divisible by some element of {5,7,73} divisible by 7 if n is odd, divisible by 5 if n == 2 mod 4, divisible by 73 if n == 0 mod 4	http://factordb.com/index.php?query=2*27%5E%28n%2B1%29%2B19&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	{1}	(4ⁿ−1)/3 (n ≥ 2)	11	5	difference-of-two-squares factorization but 11 is prime, and 11 is the only prime > b in this family (4ⁿ−1)/3 = (2ⁿ−1) × (2ⁿ+1) / 3	http://factordb.com/index.php?query=%284%5En-1%29%2F3&use=n&n=2&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
8	{1}	(8ⁿ−1)/7 (n ≥ 2)	111	73	difference-of-two-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	http://factordb.com/index.php?query=%288%5En-1%29%2F7&use=n&n=2&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	{1}	(16ⁿ−1)/15 (n ≥ 2)	11	17	difference-of-two-squares factorization but 11 is prime, and 11 is the only prime > b in this family (16ⁿ−1)/15 = (4ⁿ−1) × (4ⁿ+1) / 15 (in fact, difference-of-4th-powers factorization) (16ⁿ−1)/15 = (2ⁿ−1) × (2ⁿ+1) × (4ⁿ+1) / 15	http://factordb.com/index.php?query=%2816%5En-1%29%2F15&use=n&n=2&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
27	{1}	(27ⁿ−1)/26 (n ≥ 2)	111	757	difference-of-two-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	http://factordb.com/index.php?query=%2827%5En-1%29%2F26&use=n&n=2&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
27	{G}7	(8×27ⁿ⁺¹−125)/13 (n ≥ 1)	G7	439	difference-of-two-cubes factorization but G7 is prime, and G7 is the only prime > b in this family (8×27ⁿ⁺¹−125)/13 = (2×3ⁿ⁺¹−5) × (4×9ⁿ⁺¹+10×3ⁿ⁺¹+25) / 13	http://factordb.com/index.php?query=%288*27%5E%28n%2B1%29-125%29%2F13&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
36	{1}	(36ⁿ−1)/35 (n ≥ 2)	11	37	difference-of-two-squares factorization but 11 is prime, and 11 is the only prime > b in this family (36ⁿ−1)/35 = (6ⁿ−1) × (6ⁿ+1) / 35	http://factordb.com/index.php?query=%2836%5En-1%29%2F35&use=n&n=2&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

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.

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)/d when d > 1 (however, of course, the numbers (a×bⁿ+c)/d with |c| ≠ 1 and/or d <> 1 and/or a > bⁿ can only be PRP (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, https://stdkmd.net/nrr/records.htm#probableprimenumbers, https://stdkmd.net/nrr/repunit/prpfactors.htm, http://factordb.com/listtype.php?t=1) tested; the numbers a×2ⁿ±1 (with a < 2ⁿ) are the fastest to test, a×2ⁿ+1 numbers are tested using the Proth algorithm (https://en.wikipedia.org/wiki/Proth%27s_theorem, https://www.rieselprime.de/ziki/Proth%27s_theorem, https://mathworld.wolfram.com/ProthsTheorem.html, http://www.numericana.com/answer/primes.htm#proth), a×2ⁿ−1 numbers are tested using the Lucas-Lehmer-Riesel algorithm (https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer%E2%80%93Riesel_test); the numbers a×bⁿ±1 (with b > 2, a < bⁿ) can also be definitely prime (https://en.wikipedia.org/wiki/Provable_prime, https://stdkmd.net/nrr/records.htm#primenumbers, http://factordb.com/listtype.php?t=4) tested, a×bⁿ+1 numbers are tested using the N−1 Pocklington algorithm (https://primes.utm.edu/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1), a×bⁿ−1 numbers are tested using the N+1 Morrison algorithm (https://primes.utm.edu/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2)), 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 ≤ 6 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/, https://www.rieselprime.de/ziki/Factoring_Database) 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, https://stdkmd.net/nrr/records.htm#probableprimenumbers, https://stdkmd.net/nrr/repunit/prpfactors.htm, 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/)).

Some of the left families may cover another left family, e.g. the base 19 left family 5{H}5 covers another base 19 left family 5{H}05, and if the smallest prime in family 5{H}5 in base 19 has length n, and the family 5{H}05 in base 19 has no prime with length ≤ n, then family 5{H}05 in base 19 can be removed from the unsolved families for base 19, however, if the smallest prime in family 5{H}5 in base 19 has length n, but the family 5{H}05 in base 19 is not tested to length n or more, then family 5{H}05 in base 19 should not be removed from the unsolved families for base 19, since a number in family 5{H}05 covers the prime in family 5{H}5 with length n if and only if the length of this number is ≥ n+1; besides, the base 19 left family FH0{H} covers another base 19 left family FHHH0{H}, and if the smallest prime in family FH0{H} in base 19 has length n, and the family FHHH0{H} in base 19 has no prime with length ≤ n+1, then family FHHH0{H} in base 19 can be removed from the unsolved families for base 19, however, if the smallest prime in family FH0{H} in base 19 has length n, but the family FHHH0{H} in base 19 is not tested to length n+1 or more, then family FHHH0{H} in base 19 should not be removed from the unsolved families for base 19, since a number in family FHHH0{H} covers the prime in family FH0{H} with length n if and only if the length of this number is ≥ n+2; besides, the base 21 left family {9}D covers another base 21 left family F{9}D, and if the smallest prime in family {9}D in base 21 has length n, and the family F{9}D in base 21 has no prime with length ≤ n, then family F{9}D in base 21 can be removed from the unsolved families for base 21, however, if the smallest prime in family {9}D in base 21 has length n, but the family F{9}D in base 21 is not tested to length n or more, then family F{9}D in base 21 should not be removed from the unsolved families for base 21, since a number in family F{9}D covers the prime in family {9}D with length n if and only if the length of this number is ≥ n+1 (if a family has no primes, then we say "the smallest prime in this family has length ∞ (https://en.wikipedia.org/wiki/Infinity, https://primes.utm.edu/glossary/xpage/Infinite.html, https://mathworld.wolfram.com/Infinity.html) (instead of 0 or −1)", see http://gladhoboexpress.blogspot.com/2019/05/prime-sandwiches-made-with-one-derbread.html and http://chesswanks.com/seq/a306861.txt (for the OEIS sequence https://oeis.org/A306861) and http://chesswanks.com/seq/a269254.txt (for the OEIS sequence https://oeis.org/A269254) (since this is more convenient, e.g. the n of the smallest prime in the base 13 family A3_nA, this family has been searched to n = 358000 with no prime or probable prime found, we can use ">358000" for the n of the smallest prime in the base 13 family A3_nA (while for the n of the smallest prime in the base 13 family 95_n, it is 197420), ">358000" includes infinity (since infinity is > 358000) but does not includes 0 or −1, it is still possible that there is no prime in the base 13 family A3_nA, although by the heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://primes.utm.edu/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, http://www.utm.edu/~caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf)) above, this is very impossible, also "the smallest n ≥ 1 such that (a×bⁿ+c)/gcd(a+c,b−1) is prime" should be the infimum (https://en.wikipedia.org/wiki/Infimum, https://mathworld.wolfram.com/Infimum.html) of the set S of the numbers n ≥ 1 such that (a×bⁿ+c)/gcd(a+c,b−1) is prime, and if there is no n ≥ 1 such that (a×bⁿ+c)/gcd(a+c,b−1) is prime, then this set S is the empty set (https://en.wikipedia.org/wiki/Empty_set, https://mathworld.wolfram.com/EmptySet.html), and by the definition of "inf", the infimum of the empty set is ∞), ∞ is > any finite number, e.g. "the smallest n ≥ 1 such that k×2ⁿ+1 is prime" is ∞ for k = 78557, 157114, 271129, 271577, 314228, 322523, 327739, 482719, ..., while it is 31172165 for k = 10223 and 13018586 for k = 19249, another example is "the smallest n such that (bⁿ−1)/(b−1) is prime" is ∞ for b = 9, 25, 32, 49, 64, 81, 121, 125, 144, 169, 216, 225, 243, 289, 324, 343, 361, 441, 484, 529, 625, 729, 784, 841, 900, 961, 1000, ..., while it is 62903 for b = 691 and 41189 for b = 693).

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 ≤ 6 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 as well as the length of these unproven probable primes 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)

b	index of this minimal prime in base b (assuming the primality of all probable primes in base b)	base-b form of this unproven probable prime	algebraic ((a×bⁿ+c)/gcd(a+c,b−1)) form of this unproven probable prime (which is a minimal prime assuming its primality)	factordb entry of this unproven probable prime (which is a minimal prime assuming its primality)	Primo input file of this unproven probable prime (which is a minimal prime assuming its primality)	factorization of the numbers in corresponding family (n is the number of digits in the "{}", start with the smallest n making the number > b (if n = 0 already makes the number > b, then start with n = 0))
11	1068	57₆₂₆₆₈	(57×11⁶²⁶⁶⁸−7)/10	http://factordb.com/index.php?id=1100000003573679860	http://factordb.com/cert.php?id=1100000003573679860&inputfile	http://factordb.com/index.php?query=%2857*11%5En-7%29%2F10&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3194	C5₂₃₇₅₅C	(149×13²³⁷⁵⁶+79)/12	http://factordb.com/index.php?id=1100000003590647776	http://factordb.com/cert.php?id=1100000003590647776&inputfile	http://factordb.com/index.php?query=%28149*13%5E%28n%2B1%29%2B79%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3195	80₃₂₀₁₇111	8×13³²⁰²⁰+183	http://factordb.com/index.php?id=1100000000490878060	http://factordb.com/cert.php?id=1100000000490878060&inputfile	http://factordb.com/index.php?query=8*13%5E%28n%2B3%29%2B183&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3196	95₁₉₇₄₂₀	(113×13¹⁹⁷⁴²⁰−5)/12	http://factordb.com/index.php?id=1100000003943359311	(no Primo input file, since this unproven probable prime is too large (> 10¹⁴⁹⁹⁹⁹) to be PRP-tested in factordb, and factordb does not have Primo input file for numbers with status (http://factordb.com/status.html, http://factordb.com/distribution.php) "U" (i.e. in http://factordb.com/listtype.php?t=2), factordb only has Primo input file for numbers with status "PRP" (i.e. in http://factordb.com/listtype.php?t=1))	http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	2345	DB₃₂₂₃₄	(206×16³²²³⁴−11)/15	http://factordb.com/index.php?id=1100000002383583629	http://factordb.com/cert.php?id=1100000002383583629&inputfile	http://factordb.com/index.php?query=%28206*16%5En-11%29%2F15&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	2346	4₇₂₇₈₅DD	(4×16⁷²⁷⁸⁷+2291)/15	http://factordb.com/index.php?id=1100000003615909841	http://factordb.com/cert.php?id=1100000003615909841&inputfile	http://factordb.com/index.php?query=%284*16%5E%28n%2B2%29%2B2291%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	2347	3₁₁₆₁₃₇AF	(16¹¹⁶¹³⁹+619)/5	http://factordb.com/index.php?id=1100000003851731988	http://factordb.com/cert.php?id=1100000003851731988&inputfile	http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
22	8003	BK₂₂₀₀₁5	(251×22²²⁰⁰²−335)/21	http://factordb.com/index.php?id=1100000003594696838	http://factordb.com/cert.php?id=1100000003594696838&inputfile	http://factordb.com/index.php?query=%28251*22%5E%28n%2B1%29-335%29%2F21&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25250	5₁₉₃₉₁6F	(26¹⁹³⁹³+179)/5	http://factordb.com/index.php?id=1100000003850151202	http://factordb.com/cert.php?id=1100000003850151202&inputfile	http://factordb.com/index.php?query=%2826%5E%28n%2B2%29%2B179%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25251	7₂₀₂₇₉OL	(7×26²⁰²⁸¹+11393)/25	http://factordb.com/index.php?id=1100000003892628605	http://factordb.com/cert.php?id=1100000003892628605&inputfile	http://factordb.com/index.php?query=%287*26%5E%28n%2B2%29%2B11393%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25252	LD0₂₀₉₇₅7	559×26²⁰⁹⁷⁶+7	http://factordb.com/index.php?id=1100000003892628658	http://factordb.com/cert.php?id=1100000003892628658&inputfile	http://factordb.com/index.php?query=559*26%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25253	6K₂₃₃₀₀5	(34×26²³³⁰¹−79)/5	http://factordb.com/index.php?id=1100000003892628745	http://factordb.com/cert.php?id=1100000003892628745&inputfile	http://factordb.com/index.php?query=%2834*26%5E%28n%2B1%29-79%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25254	J0₄₄₃₀₃KCB	19×26⁴⁴³⁰⁶+13843	http://factordb.com/index.php?id=1100000003968156595	http://factordb.com/cert.php?id=1100000003968156595&inputfile	http://factordb.com/index.php?query=19*26%5E%28n%2B3%29%2B13843&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25255	M0₆₁₁₈₆2BB	22×26⁶¹¹⁸⁹+1649	http://factordb.com/index.php?id=1100000003968169875	http://factordb.com/cert.php?id=1100000003968169875&inputfile	http://factordb.com/index.php?query=22*26%5E%28n%2B3%29%2B1649&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25526	N6₂₄₀₅₁LR	(209×28²⁴⁰⁵³+3967)/9	http://factordb.com/index.php?id=1100000003879667576	http://factordb.com/cert.php?id=1100000003879667576&inputfile	http://factordb.com/index.php?query=%28209*28%5E%28n%2B2%29%2B3967%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25527	5OA₃₁₂₃₈F	(4438×28³¹²³⁹+125)/27	http://factordb.com/index.php?id=1100000003880455200	http://factordb.com/cert.php?id=1100000003880455200&inputfile	http://factordb.com/index.php?query=%284438*28%5E%28n%2B1%29%2B125%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25528	O4O₉₄₅₃₅9	(6092×28⁹⁴⁵³⁶−143)/9	http://factordb.com/index.php?id=1100000000808118231	http://factordb.com/cert.php?id=1100000000808118231&inputfile	http://factordb.com/index.php?query=%286092*28%5E%28n%2B1%29-143%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
30	2618	I0₂₄₆₀₈D	18×30²⁴⁶⁰⁹+13	http://factordb.com/index.php?id=1100000003593967511	http://factordb.com/cert.php?id=1100000003593967511&inputfile	http://factordb.com/index.php?query=18*30%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35284	7K₂₆₅₆₇Z	(53×36²⁶⁵⁶⁸+101)/7	http://factordb.com/index.php?id=1100000003896952461	http://factordb.com/cert.php?id=1100000003896952461&inputfile	http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35285	S0₇₅₀₀₇8H	28×36⁷⁵⁰⁰⁹+305	http://factordb.com/index.php?id=1100000004020085177	http://factordb.com/cert.php?id=1100000004020085177&inputfile	http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35286	P₈₁₉₉₃SZ	(5×36⁸¹⁹⁹⁵+821)/7	http://factordb.com/index.php?id=1100000002394962083	http://factordb.com/cert.php?id=1100000002394962083&inputfile	http://factordb.com/index.php?query=%285*36%5E%28n%2B2%29%2B821%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show

All these numbers are strong probable primes (https://en.wikipedia.org/wiki/Strong_pseudoprime, https://primes.utm.edu/glossary/xpage/StrongPRP.html, https://mathworld.wolfram.com/StrongPseudoprime.html) to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61 (see https://oeis.org/A014233 and https://oeis.org/A141768 and https://oeis.org/A001262 and https://oeis.org/A074773 and http://ntheory.org/data/spsps.txt), 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 http://ntheory.org/data/slpsps-baillie.txt), and trial factored (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, http://www.numericana.com/answer/factoring.htm#trial) to 10¹⁶ (i.e. all these numbers are 10¹⁶-rough numbers (https://en.wikipedia.org/wiki/Rough_number, https://mathworld.wolfram.com/RoughNumber.html)), thus, all these numbers are Baillie–PSW probable primes (https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test, https://mathworld.wolfram.com/Baillie-PSWPrimalityTest.html), and no composites < 2⁶⁴ pass the Baillie–PSW probable prime test (see http://ntheory.org/pseudoprimes.html and https://faculty.lynchburg.edu/~nicely/misc/bpsw.html), thus if one of these numbers is in fact composite, it will be a pseudoprime to the Baillie–PSW probable prime test, which currently no single example is known!

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 ≤ 6 unsolved families) and the factorization of the numbers in these families:

(you can calculate "equivalent searching limit of length in decimal" by: "current searching limit of length of this family" × log(b), where log is the common logarithm (https://en.wikipedia.org/wiki/Common_logarithm, https://mathworld.wolfram.com/CommonLogarithm.html), i.e. the logarithm with base 10)

b	base-b form of the unsolved family	algebraic ((a×bⁿ+c)/gcd(a+c,b−1)) form of the unsolved family	current searching limit of length of this family	factorization of the numbers in this family (n is the number of digits in the "{}", start with the smallest n making the number > b (if n = 0 already makes the number > b, then start with n = 0))
13	A{3}A	(41×13ⁿ⁺¹+27)/4	358000	http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	85{M}B	(5347×26ⁿ⁺¹−297)/25	100000	http://factordb.com/index.php?query=%285347*26%5E%28n%2B1%29-297%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	{A}6F	(2×26ⁿ⁺²−497)/5	100000	http://factordb.com/index.php?query=%282*26%5E%28n%2B2%29-497%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	{H}MH	(17×26ⁿ⁺²+3233)/25	100000	http://factordb.com/index.php?query=%2817*26%5E%28n%2B2%29%2B3233%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	{I}GL	(18×26ⁿ⁺²−1243)/25	100000	http://factordb.com/index.php?query=%2818*26%5E%28n%2B2%29-1243%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	O{A}F	(658×28ⁿ⁺¹+125)/27	543203	http://factordb.com/index.php?query=(658*28^(n%2B1)%2B125)/27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	B{0}EUV	11×36ⁿ⁺³+19255	100000	http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	HM{0}N	634×36ⁿ⁺¹+23	100000	http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	N{0}YYN	23×36ⁿ⁺³+45311	100000	http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	O{L}Z	(123×36ⁿ⁺¹+67)/5	100000	http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show

The large proven primes (> 10²⁹⁹) for bases b = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 (the solved or near-solved bases, i.e. the bases b with ≤ 6 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 as well as the length of these primes 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)

b	index of this minimal prime in base b	base-b form of this minimal prime	algebraic ((a×bⁿ+c)/gcd(a+c,b−1)) form of this minimal prime	factordb entry of this minimal prime	primality certificate for this minimal prime	factorization of the numbers in corresponding family (n is the number of digits in the "{}", start with the smallest n making the number > b (if n = 0 already makes the number > b, then start with n = 0))
9	149	76₃₂₉2	(31×9³³⁰−19)/4	http://factordb.com/index.php?id=1100000002359003642	http://factordb.com/cert.php?id=1100000002359003642	http://factordb.com/index.php?query=%2831*9%5E%28n%2B1%29-19%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
9	150	27₆₈₆07	(23×9⁶⁸⁸−511)/8	http://factordb.com/index.php?id=1100000002495467486	http://factordb.com/cert.php?id=1100000002495467486	http://factordb.com/index.php?query=%2823*9%5E%28n%2B2%29-511%29%2F8&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
9	151	30₁₁₅₈11	3×9¹¹⁶⁰+10	http://factordb.com/index.php?id=1100000002376318423	http://factordb.com/cert.php?id=1100000002376318423	http://factordb.com/index.php?query=3*9%5E%28n%2B2%29%2B10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11	1065	A₇₁₃58	11⁷¹⁵−58	http://factordb.com/index.php?id=1100000003576826487	http://factordb.com/cert.php?id=1100000003576826487	http://factordb.com/index.php?query=11%5E%28n%2B2%29-58&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11	1066	7₇₅₉44	(7×11⁷⁶¹−367)/10	http://factordb.com/index.php?id=1100000002505568840	http://factordb.com/cert.php?id=1100000002505568840	http://factordb.com/index.php?query=%287*11%5E%28n%2B2%29-367%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
11	1067	557₁₀₁₁	(607×11¹⁰¹¹−7)/10	http://factordb.com/index.php?id=1100000002361376522	http://factordb.com/cert.php?id=1100000002361376522	http://factordb.com/index.php?query=%28607*11%5En-7%29%2F10&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3165	50₂₇₀44	5×13²⁷²+56	http://factordb.com/index.php?id=1100000002632397005	http://factordb.com/cert.php?id=1100000002632397005	http://factordb.com/index.php?query=5*13%5E%28n%2B2%29%2B56&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3166	9₂₇₁095	(3×13²⁷⁴−6103)/4	http://factordb.com/index.php?id=1100000003590431654	http://factordb.com/cert.php?id=1100000003590431654	http://factordb.com/index.php?query=%283*13%5E%28n%2B3%29-6103%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3167	10₂₈₆7771	13²⁹⁰+16654	http://factordb.com/index.php?id=1100000003590431633	http://factordb.com/cert.php?id=1100000003590431633	http://factordb.com/index.php?query=13%5E%28n%2B4%29%2B16654&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3168	9₃₀₈1	(3×13³⁰⁹−35)/4	http://factordb.com/index.php?id=1100000000840126705	proven prime by N−1 test (https://primes.utm.edu/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since N−1 is 39/4×(13³⁰⁸−1), thus factor N−1 is equivalent to factor 13³⁰⁸−1, and for the factorization of 13³⁰⁸−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN=	http://factordb.com/index.php?query=%283*13%5E%28n%2B1%29-35%29%2F4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3169	B₃₄₁C4	(11×13³⁴³+61)/12	http://factordb.com/index.php?id=1100000003590431618	http://factordb.com/cert.php?id=1100000003590431618	http://factordb.com/index.php?query=%2811*13%5E%28n%2B2%29%2B61%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3170	8B₃₄₃	(107×13³⁴³−11)/12	http://factordb.com/index.php?id=1100000002321018736	http://factordb.com/cert.php?id=1100000002321018736	http://factordb.com/index.php?query=%28107*13%5En-11%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3171	710₃₇₁111	92×13³⁷⁴+183	http://factordb.com/index.php?id=1100000003590431609	http://factordb.com/cert.php?id=1100000003590431609	http://factordb.com/index.php?query=92*13%5E%28n%2B3%29%2B183&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3172	75₃₇₅7	(89×13³⁷⁶+19)/12	http://factordb.com/index.php?id=1100000003590431596	http://factordb.com/cert.php?id=1100000003590431596	http://factordb.com/index.php?query=%2889*13%5E%28n%2B1%29%2B19%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3173	9B0₃₉₁9	128×13³⁹²+9	http://factordb.com/index.php?id=1100000002632396790	http://factordb.com/cert.php?id=1100000002632396790	http://factordb.com/index.php?query=128*13%5E%28n%2B1%29%2B9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3174	7B0B₃₉₇	(15923×13³⁹⁷−11)/12	http://factordb.com/index.php?id=1100000003590431574	http://factordb.com/cert.php?id=1100000003590431574	http://factordb.com/index.php?query=%2815923*13%5En-11%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3175	10₄₁₄93	13⁴¹⁶+120	http://factordb.com/index.php?id=1100000002523249240	http://factordb.com/cert.php?id=1100000002523249240	http://factordb.com/index.php?query=13%5E%28n%2B2%29%2B120&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3176	81010₄₁₅1	17746×13⁴¹⁶+1	http://factordb.com/index.php?id=1100000003590431555	proven prime by N−1 test (https://primes.utm.edu/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since N−1 is trivially fully factored	http://factordb.com/index.php?query=17746*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3177	8110₄₃₅1	1366×13⁴³⁶+1	http://factordb.com/index.php?id=1100000002373259109	proven prime by N−1 test (https://primes.utm.edu/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since N−1 is trivially fully factored	http://factordb.com/index.php?query=1366*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3178	B7₄₈₆	(139×13⁴⁸⁶−7)/12	http://factordb.com/index.php?id=1100000002321015892	http://factordb.com/cert.php?id=1100000002321015892	http://factordb.com/index.php?query=%28139*13%5En-7%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3179	B₅₆₃C	(11×13⁵⁶⁴+1)/12	http://factordb.com/index.php?id=1100000000000217927	proven prime by N−1 test (https://primes.utm.edu/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since N−1 is 11/12×(13⁵⁶⁴−1), thus factor N−1 is equivalent to factor 13⁵⁶⁴−1, and for the factorization of 13⁵⁶⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN=	http://factordb.com/index.php?query=%2811*13%5E%28n%2B1%29%2B1%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3180	1B₅₇₆	(23×13⁵⁷⁶−11)/12	http://factordb.com/index.php?id=1100000002321021456	proven prime by N−1 test (https://primes.utm.edu/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since N−1 is 23/12×(13⁵⁷⁶−1), thus factor N−1 is equivalent to factor 13⁵⁷⁶−1, and for the factorization of 13⁵⁷⁶−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN=	http://factordb.com/index.php?query=%2823*13%5En-11%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3181	80₆₉₃87	8×13⁶⁹⁵+111	http://factordb.com/index.php?id=1100000002615636527	http://factordb.com/cert.php?id=1100000002615636527	http://factordb.com/index.php?query=8*13%5E%28n%2B2%29%2B111&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3182	CC5₇₁₃	(2021×13⁷¹³−5)/12	http://factordb.com/index.php?id=1100000002615627353	http://factordb.com/cert.php?id=1100000002615627353	http://factordb.com/index.php?query=%282021*13%5En-5%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3183	B₈₃₄74	(11×13⁸³⁶−719)/12	http://factordb.com/index.php?id=1100000003590430871	http://factordb.com/cert.php?id=1100000003590430871	http://factordb.com/index.php?query=%2811*13%5E%28n%2B2%29-719%29%2F12&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3184	9₉₆₈B	(3×13⁹⁶⁹+5)/4	http://factordb.com/index.php?id=1100000000258566244	http://factordb.com/cert.php?id=1100000000258566244	http://factordb.com/index.php?query=%283*13%5E%28n%2B1%29%2B5%29%2F4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3185	10₁₂₉₅181	13¹²⁹⁸+274	http://factordb.com/index.php?id=1100000002615445013	http://factordb.com/cert.php?id=1100000002615445013	http://factordb.com/index.php?query=13%5E%28n%2B3%29%2B274&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3186	9₁₃₆₂5	(3×13¹³⁶³−19)/4	http://factordb.com/index.php?id=1100000002321017776	http://factordb.com/cert.php?id=1100000002321017776	http://factordb.com/index.php?query=%283*13%5E%28n%2B1%29-19%29%2F4&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3187	7₁₅₀₄1	(7×13¹⁵⁰⁵−79)/12	http://factordb.com/index.php?id=1100000002320890755	http://factordb.com/cert.php?id=1100000002320890755	http://factordb.com/index.php?query=%287*13%5E%28n%2B1%29-79%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3188	930₁₅₅₁1	120×13¹⁵⁵²+1	http://factordb.com/index.php?id=1100000000765961452	proven prime by N−1 test (https://primes.utm.edu/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since N−1 is trivially fully factored	http://factordb.com/index.php?query=120*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3189	720₂₂₉₇2	93×13²²⁹⁸+2	http://factordb.com/index.php?id=1100000002632396910	http://factordb.com/cert.php?id=1100000002632396910	http://factordb.com/index.php?query=93*13%5E%28n%2B1%29%2B2&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3190	1770₂₇₀₃17	267×13²⁷⁰⁵+20	http://factordb.com/index.php?id=1100000003590430825	http://factordb.com/cert.php?id=1100000003590430825	http://factordb.com/index.php?query=267*13%5E%28n%2B2%29%2B20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3191	390₆₂₆₆1	48×13⁶²⁶⁷+1	http://factordb.com/index.php?id=1100000000765961441	proven prime by N−1 test (https://primes.utm.edu/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since N−1 is trivially fully factored	http://factordb.com/index.php?query=48*13%5E%28n%2B1%29%2B1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3192	B0₆₅₄₀BBA	11×13⁶⁵⁴³+2012	http://factordb.com/index.php?id=1100000002616382906	http://factordb.com/cert.php?id=1100000002616382906	http://factordb.com/index.php?query=11*13%5E%28n%2B3%29%2B2012&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
13	3193	C₁₀₆₃₁92	13¹⁰⁶³³−50	http://factordb.com/index.php?id=1100000003590493750	http://factordb.com/cert.php?id=1100000003590493750	http://factordb.com/index.php?query=13%5E%28n%2B2%29-50&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
14	649	34D₇₀₈	47×14⁷⁰⁸−1	http://factordb.com/index.php?id=1100000001540144903	proven prime by N+1 test (https://primes.utm.edu/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2), since N+1 is trivially fully factored	http://factordb.com/index.php?query=47*14%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
14	650	4D₁₉₆₉₈	5×14¹⁹⁶⁹⁸−1	http://factordb.com/index.php?id=1100000000884560233	proven prime by N+1 test (https://primes.utm.edu/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2), since N+1 is trivially fully factored	http://factordb.com/index.php?query=5*14%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	2328	880₂₄₆7	136×16²⁴⁷+7	http://factordb.com/index.php?id=1100000002468140199	proven prime by N−1 test (https://primes.utm.edu/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), N−1 is 2³×3×7×13×25703261×(289-digit prime)	http://factordb.com/index.php?query=136*16%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	2329	D4₂₆₃D	(199×16²⁶⁴+131)/15	http://factordb.com/index.php?id=1100000002468170238	http://factordb.com/cert.php?id=1100000002468170238	http://factordb.com/index.php?query=%28199*16%5E%28n%2B1%29%2B131%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	2330	E0₂₆₁4DD	14×16²⁶⁴+1245	http://factordb.com/index.php?id=1100000003588388352	http://factordb.com/cert.php?id=1100000003588388352	http://factordb.com/index.php?query=14*16%5E%28n%2B3%29%2B1245&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	2331	8C0₂₉₀ED	140×16²⁹²+237	http://factordb.com/index.php?id=1100000003588388307	http://factordb.com/cert.php?id=1100000003588388307	http://factordb.com/index.php?query=140*16%5E%28n%2B2%29%2B237&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	2332	DA₃₀₅5	(41×16³⁰⁶−17)/3	http://factordb.com/index.php?id=1100000003588388284	http://factordb.com/cert.php?id=1100000003588388284	http://factordb.com/index.php?query=%2841*16%5E%28n%2B1%29-17%29%2F3&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	2333	CE80₄₂₂D	3304×16⁴²³+13	http://factordb.com/index.php?id=1100000003588388257	http://factordb.com/cert.php?id=1100000003588388257	http://factordb.com/index.php?query=3304*16%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	2334	5F₅₄₄6F	6×16⁵⁴⁶−145	http://factordb.com/index.php?id=1100000002604723967	http://factordb.com/cert.php?id=1100000002604723967	http://factordb.com/index.php?query=6*16%5E%28n%2B2%29-145&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	2335	88F₅₄₅	137×16⁵⁴⁵−1	http://factordb.com/index.php?id=1100000000413679658	proven prime by N+1 test (https://primes.utm.edu/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2), since N+1 is trivially fully factored	http://factordb.com/index.php?query=137*16%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	2336	BE0₇₉₂BB	190×16⁷⁹⁴+187	http://factordb.com/index.php?id=1100000003588387938	http://factordb.com/cert.php?id=1100000003588387938	http://factordb.com/index.php?query=190*16%5E%28n%2B2%29%2B187&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	2337	D9₁₀₅₂	(68×16¹⁰⁵²−3)/5	http://factordb.com/index.php?id=1100000002321036020	http://factordb.com/cert.php?id=1100000002321036020	http://factordb.com/index.php?query=%2868*16%5En-3%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	2338	FAF₁₀₆₂45	251×16¹⁰⁶⁴−187	http://factordb.com/index.php?id=1100000003588387610	http://factordb.com/cert.php?id=1100000003588387610	http://factordb.com/index.php?query=251*16%5E%28n%2B2%29-187&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	2339	F8₁₅₁₇F	(233×16¹⁵¹⁸+97)/15	http://factordb.com/index.php?id=1100000000633744824	http://factordb.com/cert.php?id=1100000000633744824	http://factordb.com/index.php?query=%28233*16%5E%28n%2B1%29%2B97%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	2340	20₁₇₁₃321	2×16¹⁷¹⁶+801	http://factordb.com/index.php?id=1100000003588386735	http://factordb.com/cert.php?id=1100000003588386735	http://factordb.com/index.php?query=2*16%5E%28n%2B3%29%2B801&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	2341	300F₁₉₆₀AF	769×16¹⁹⁶²−81	http://factordb.com/index.php?id=1100000003588368750	http://factordb.com/cert.php?id=1100000003588368750	http://factordb.com/index.php?query=769*16%5E%28n%2B2%29-81&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	2342	90₃₅₄₂91	9×16³⁵⁴⁴+145	http://factordb.com/index.php?id=1100000000633424191	http://factordb.com/cert.php?id=1100000000633424191	http://factordb.com/index.php?query=9*16%5E%28n%2B2%29%2B145&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	2343	5BC₃₇₀₀D	(459×16³⁷⁰¹+1)/5	http://factordb.com/index.php?id=1100000000993764322	http://factordb.com/cert.php?id=1100000000993764322	http://factordb.com/index.php?query=%28459*16%5E%28n%2B1%29%2B1%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
16	2344	D0B₁₇₈₀₄	(3131×16¹⁷⁸⁰⁴−11)/15	http://factordb.com/index.php?id=1100000003589278511	http://factordb.com/cert.php?id=1100000003589278511	http://factordb.com/index.php?query=%283131*16%5En-11%29%2F15&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
18	547	80₂₉₈B	8×18²⁹⁹+11	http://factordb.com/index.php?id=1100000002355574745	http://factordb.com/cert.php?id=1100000002355574745	http://factordb.com/index.php?query=8*18%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
18	548	H₇₆₆FH	18⁷⁶⁸−37	http://factordb.com/index.php?id=1100000003590430490	http://factordb.com/cert.php?id=1100000003590430490	http://factordb.com/index.php?query=18%5E%28n%2B2%29-37&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
18	549	C0₆₂₆₈C5	12×18⁶²⁷⁰+221	http://factordb.com/index.php?id=1100000003590442437	http://factordb.com/cert.php?id=1100000003590442437	http://factordb.com/index.php?query=12*18%5E%28n%2B2%29%2B221&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
20	3301	H₂₄₇A0H	(17×20²⁵⁰−59677)/19	http://factordb.com/index.php?id=1100000003590502619	http://factordb.com/cert.php?id=1100000003590502619	http://factordb.com/index.php?query=%2817*20%5E%28n%2B3%29-59677%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
20	3302	7₂₄₉A7	(7×20²⁵¹+1133)/19	http://factordb.com/index.php?id=1100000003590502602	http://factordb.com/cert.php?id=1100000003590502602	http://factordb.com/index.php?query=%287*20%5E%28n%2B2%29%2B1133%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
20	3303	J7₂₇₀	(368×20²⁷⁰−7)/19	http://factordb.com/index.php?id=1100000002325395462	http://factordb.com/cert.php?id=1100000002325395462	http://factordb.com/index.php?query=%28368*20%5En-7%29%2F19&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
20	3304	J₃₃₀CCC7	20³³⁴−58953	http://factordb.com/index.php?id=1100000003590502572	http://factordb.com/cert.php?id=1100000003590502572	http://factordb.com/index.php?query=20%5E%28n%2B4%29-58953&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
20	3305	40₃₈₇404B	4×20³⁹¹+32091	http://factordb.com/index.php?id=1100000003590502563	http://factordb.com/cert.php?id=1100000003590502563	http://factordb.com/index.php?query=4*20%5E%28n%2B4%29%2B32091&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
20	3306	EC0₄₂₉7	292×20⁴³⁰+7	http://factordb.com/index.php?id=1100000002633348702	http://factordb.com/cert.php?id=1100000002633348702	http://factordb.com/index.php?query=292*20%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
20	3307	G₄₄₇99	(16×20⁴⁴⁹−2809)/19	http://factordb.com/index.php?id=1100000000840126753	http://factordb.com/cert.php?id=1100000000840126753	http://factordb.com/index.php?query=%2816*20%5E%28n%2B2%29-2809%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
20	3308	3A₅₂₇3	(67×20⁵²⁸−143)/19	http://factordb.com/index.php?id=1100000003590502531	http://factordb.com/cert.php?id=1100000003590502531	http://factordb.com/index.php?query=%2867*20%5E%28n%2B1%29-143%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
20	3309	E₅₆₆C7	(14×20⁵⁶⁸−907)/19	http://factordb.com/index.php?id=1100000003590502516	http://factordb.com/cert.php?id=1100000003590502516	http://factordb.com/index.php?query=%2814*20%5E%28n%2B2%29-907%29%2F19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
20	3310	JCJ₆₂₉	393×20⁶²⁹−1	http://factordb.com/index.php?id=1100000001559454258	proven prime by N+1 test (https://primes.utm.edu/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2), since N+1 is trivially fully factored	http://factordb.com/index.php?query=393*20%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
20	3311	J₆₅₅05J	20⁶⁵⁸−7881	http://factordb.com/index.php?id=1100000003590502490	http://factordb.com/cert.php?id=1100000003590502490	http://factordb.com/index.php?query=20%5E%28n%2B3%29-7881&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
20	3312	50₁₁₆₃AJ	5×20¹¹⁶⁵+219	http://factordb.com/index.php?id=1100000003590502412	http://factordb.com/cert.php?id=1100000003590502412	http://factordb.com/index.php?query=5*20%5E%28n%2B2%29%2B219&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
20	3313	CD₂₄₄₉	(241×20²⁴⁴⁹−13)/19	http://factordb.com/index.php?id=1100000002325393915	http://factordb.com/cert.php?id=1100000002325393915	http://factordb.com/index.php?query=%28241*20%5En-13%29%2F19&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
20	3314	G0₆₂₆₉D	16×20⁶²⁷⁰+13	http://factordb.com/index.php?id=1100000003590539457	http://factordb.com/cert.php?id=1100000003590539457	http://factordb.com/index.php?query=16*20%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
22	7984	I7G0₂₅₄H	8882×22²⁵⁵+17	http://factordb.com/index.php?id=1100000003591372788	http://factordb.com/cert.php?id=1100000003591372788	http://factordb.com/index.php?query=8882*22%5E%28n%2B1%29%2B17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
22	7985	D0₂₅₅5EEF	13×22²⁵⁹+60339	http://factordb.com/index.php?id=1100000003591371932	http://factordb.com/cert.php?id=1100000003591371932	http://factordb.com/index.php?query=13*22%5E%28n%2B4%29%2B60339&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
22	7986	IK₃₂₂F	(398×22³²³−125)/21	http://factordb.com/index.php?id=1100000000840384145	http://factordb.com/cert.php?id=1100000000840384145	http://factordb.com/index.php?query=%28398*22%5E%28n%2B1%29-125%29%2F21&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
22	7987	C0₃₄₀G9	12×22³⁴²+361	http://factordb.com/index.php?id=1100000000840384159	http://factordb.com/cert.php?id=1100000000840384159	http://factordb.com/index.php?query=12*22%5E%28n%2B2%29%2B361&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
22	7988	77E₃₄₈K7	(485×22³⁵⁰+373)/3	http://factordb.com/index.php?id=1100000003591369779	http://factordb.com/cert.php?id=1100000003591369779	http://factordb.com/index.php?query=%28485*22%5E%28n%2B2%29%2B373%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
22	7989	J₃₇₉KJ	(19×22³⁸¹+443)/21	http://factordb.com/index.php?id=1100000003591369027	http://factordb.com/cert.php?id=1100000003591369027	http://factordb.com/index.php?query=%2819*22%5E%28n%2B2%29%2B443%29%2F21&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
22	7990	J₃₈₈EJ	(19×22³⁹⁰−2329)/21	http://factordb.com/index.php?id=1100000003591367729	http://factordb.com/cert.php?id=1100000003591367729	http://factordb.com/index.php?query=%2819*22%5E%28n%2B2%29-2329%29%2F21&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
22	7991	DJ₄₀₀	(292×22⁴⁰⁰−19)/21	http://factordb.com/index.php?id=1100000002325880110	http://factordb.com/cert.php?id=1100000002325880110	http://factordb.com/index.php?query=%28292*22%5En-19%29%2F21&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
22	7992	E₄₀₄K7	(2×22⁴⁰⁶+373)/3	http://factordb.com/index.php?id=1100000003591366298	http://factordb.com/cert.php?id=1100000003591366298	http://factordb.com/index.php?query=%282*22%5E%28n%2B2%29%2B373%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
22	7993	66F₄₅₃B3	(971×22⁴⁵⁵−705)/7	http://factordb.com/index.php?id=1100000003591365809	http://factordb.com/cert.php?id=1100000003591365809	http://factordb.com/index.php?query=%28971*22%5E%28n%2B2%29-705%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
22	7994	L0₄₅₄B63	21×22⁴⁵⁷+5459	http://factordb.com/index.php?id=1100000003591365331	http://factordb.com/cert.php?id=1100000003591365331	http://factordb.com/index.php?query=21*22%5E%28n%2B3%29%2B5459&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
22	7995	L₄₈₃G3	22⁴⁸⁵−129	http://factordb.com/index.php?id=1100000003591364730	http://factordb.com/cert.php?id=1100000003591364730	http://factordb.com/index.php?query=22%5E%28n%2B2%29-129&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
22	7996	E60₄₉₆L	314×22⁴⁹⁷+21	http://factordb.com/index.php?id=1100000000632703239	http://factordb.com/cert.php?id=1100000000632703239	http://factordb.com/index.php?query=314*22%5E%28n%2B1%29%2B21&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
22	7997	I₆₂₆AF	(6×22⁶²⁸−1259)/7	http://factordb.com/index.php?id=1100000000632724334	http://factordb.com/cert.php?id=1100000000632724334	http://factordb.com/index.php?query=%286*22%5E%28n%2B2%29-1259%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
22	7998	K0₇₆₀EC1	20×22⁷⁶³+7041	http://factordb.com/index.php?id=1100000000632724415	http://factordb.com/cert.php?id=1100000000632724415	http://factordb.com/index.php?query=20*22%5E%28n%2B3%29%2B7041&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
22	7999	J0₇₆₇IGGJ	19×22⁷⁷¹+199779	http://factordb.com/index.php?id=1100000003591362567	http://factordb.com/cert.php?id=1100000003591362567	http://factordb.com/index.php?query=19*22%5E%28n%2B4%29%2B199779&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
22	8000	7₉₅₉K7	(22⁹⁶¹+857)/3	http://factordb.com/index.php?id=1100000003591361817	http://factordb.com/cert.php?id=1100000003591361817	http://factordb.com/index.php?query=%2822%5E%28n%2B2%29%2B857%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
22	8001	L₂₃₈₅KE7	22²³⁸⁸−653	http://factordb.com/index.php?id=1100000003591360774	http://factordb.com/cert.php?id=1100000003591360774	http://factordb.com/index.php?query=22%5E%28n%2B3%29-653&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
22	8002	7₃₈₁₅2L	(22³⁸¹⁷−289)/3	http://factordb.com/index.php?id=1100000003591359839	http://factordb.com/cert.php?id=1100000003591359839	http://factordb.com/index.php?query=%2822%5E%28n%2B2%29-289%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
24	3400	I0₂₄₁I5	18×24²⁴³+437	http://factordb.com/index.php?id=1100000002633360037	http://factordb.com/cert.php?id=1100000002633360037	http://factordb.com/index.php?query=18*24%5E%28n%2B2%29%2B437&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
24	3401	D0₂₅₉KKD	13×24²⁶²+12013	http://factordb.com/index.php?id=1100000003593270725	http://factordb.com/cert.php?id=1100000003593270725	http://factordb.com/index.php?query=13*24%5E%28n%2B3%29%2B12013&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
24	3402	C7₂₉₈	(283×24²⁹⁸−7)/23	http://factordb.com/index.php?id=1100000002326181235	http://factordb.com/cert.php?id=1100000002326181235	http://factordb.com/index.php?query=%28283*24%5En-7%29%2F23&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
24	3403	20₃₁₃7	2×24³¹⁴+7	http://factordb.com/index.php?id=1100000002355610241	http://factordb.com/cert.php?id=1100000002355610241	http://factordb.com/index.php?query=2*24%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
24	3404	BC0₃₃₁B	276×24³³²+11	http://factordb.com/index.php?id=1100000002633359842	http://factordb.com/cert.php?id=1100000002633359842	http://factordb.com/index.php?query=276*24%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
24	3405	N₂₆₄₄LLN	24²⁶⁴⁷−1201	http://factordb.com/index.php?id=1100000003593270089	http://factordb.com/cert.php?id=1100000003593270089	http://factordb.com/index.php?query=24%5E%28n%2B3%29-1201&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
24	3406	D₂₆₉₈LD	(13×24²⁷⁰⁰+4403)/23	http://factordb.com/index.php?id=1100000003593269876	http://factordb.com/cert.php?id=1100000003593269876	http://factordb.com/index.php?query=%2813*24%5E%28n%2B2%29%2B4403%29%2F23&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
24	3407	A0₂₉₅₁8ID	10×24²⁹⁵⁴+5053	http://factordb.com/index.php?id=1100000003593269654	http://factordb.com/cert.php?id=1100000003593269654	http://factordb.com/index.php?query=10*24%5E%28n%2B3%29%2B5053&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
24	3408	88N₅₉₅₁	201×24⁵⁹⁵¹−1	http://factordb.com/index.php?id=1100000003593275880	proven prime by N+1 test (https://primes.utm.edu/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2), since N+1 is trivially fully factored	http://factordb.com/index.php?query=201*24%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
24	3409	N00N₈₁₂₉LN	13249×24⁸¹³¹−49	http://factordb.com/index.php?id=1100000003593391606	http://factordb.com/cert.php?id=1100000003593391606	http://factordb.com/index.php?query=13249*24%5E%28n%2B2%29-49&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25174	OL0₂₁₄M9	645×26²¹⁶+581	http://factordb.com/index.php?id=1100000000840631576	proven prime by N−1 test (https://primes.utm.edu/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), N−1 is 2²×5²×7×223×42849349×(296-digit prime)	http://factordb.com/index.php?query=645*26%5E%28n%2B2%29%2B581&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25175	1A₂₁₉P	(7×26²²⁰+73)/5	http://factordb.com/index.php?id=1100000000840631595	http://factordb.com/cert.php?id=1100000000840631595	http://factordb.com/index.php?query=%287*26%5E%28n%2B1%29%2B73%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25176	A₂₂₃DP	(2×26²²⁵+463)/5	http://factordb.com/index.php?id=1100000003850155262	http://factordb.com/cert.php?id=1100000003850155262	http://factordb.com/index.php?query=%282*26%5E%28n%2B2%29%2B463%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25177	6J₂₂₅	(169×26²²⁵−19)/25	http://factordb.com/index.php?id=1100000002328050895	http://factordb.com/cert.php?id=1100000002328050895	http://factordb.com/index.php?query=%28169*26%5En-19%29%2F25&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
26	25178	O₂₂₈5	(24×26²²⁹−499)/25	http://factordb.com/index.php?id=1100000002328059255	http://factordb.com/cert.php?id=1100000002328059255	http://factordb.com/index.php?query=%2824*26%5E%28n%2B1%29-499%29%2F25&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
26	25179	K0₂₃₀K0IP	20×26²³⁴+352013	http://factordb.com/index.php?id=1100000000840631669	http://factordb.com/cert.php?id=1100000000840631669	http://factordb.com/index.php?query=20*26%5E%28n%2B4%29%2B352013&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25180	B0₂₃₆OB	11×26²³⁸+635	http://factordb.com/index.php?id=1100000002634136234	http://factordb.com/cert.php?id=1100000002634136234	http://factordb.com/index.php?query=11*26%5E%28n%2B2%29%2B635&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25181	11G0₂₃₉9	718×26²⁴⁰+9	http://factordb.com/index.php?id=1100000000840631687	http://factordb.com/cert.php?id=1100000000840631687	http://factordb.com/index.php?query=718*26%5E%28n%2B1%29%2B9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25182	K0₂₄₁E5	20×26²⁴³+369	http://factordb.com/index.php?id=1100000002634136479	http://factordb.com/cert.php?id=1100000002634136479	http://factordb.com/index.php?query=20*26%5E%28n%2B2%29%2B369&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25183	J₂₄₃0L	(19×26²⁴⁵−12319)/25	http://factordb.com/index.php?id=1100000003850155263	http://factordb.com/cert.php?id=1100000003850155263	http://factordb.com/index.php?query=%2819*26%5E%28n%2B2%29-12319%29%2F25&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
26	25184	B₂₅₁I9	(11×26²⁵³+4489)/25	http://factordb.com/index.php?id=1100000003850155264	http://factordb.com/cert.php?id=1100000003850155264	http://factordb.com/index.php?query=%2811*26%5E%28n%2B2%29%2B4489%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25185	F₂₅₀0PCF	(3×26²⁵⁴−1284793)/5	http://factordb.com/index.php?id=1100000000840631708	http://factordb.com/cert.php?id=1100000000840631708	http://factordb.com/index.php?query=%283*26%5E%28n%2B4%29-1284793%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25186	4E7₂₆₂	(2957×26²⁶²−7)/25	http://factordb.com/index.php?id=1100000003850155265	http://factordb.com/cert.php?id=1100000003850155265	http://factordb.com/index.php?query=%282957*26%5En-7%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25187	E7₂₆₄OL	(357×26²⁶⁶+11393)/25	http://factordb.com/index.php?id=1100000003850155266	http://factordb.com/cert.php?id=1100000003850155266	http://factordb.com/index.php?query=%28357*26%5E%28n%2B2%29%2B11393%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25188	EIL₂₆₇	(9571×26²⁶⁷−21)/25	http://factordb.com/index.php?id=1100000000840631801	http://factordb.com/cert.php?id=1100000000840631801	http://factordb.com/index.php?query=%289571*26%5En-21%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25189	6₂₆₈4F	(6×26²⁷⁰−1081)/25	http://factordb.com/index.php?id=1100000000840631976	http://factordb.com/cert.php?id=1100000000840631976	http://factordb.com/index.php?query=%286*26%5E%28n%2B2%29-1081%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25190	D020₂₇₃H	8790×26²⁷⁴+17	http://factordb.com/index.php?id=1100000003850155267	http://factordb.com/cert.php?id=1100000003850155267	http://factordb.com/index.php?query=8790*26%5E%28n%2B1%29%2B17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25191	B₂₉₁KB	(11×26²⁹³+5839)/25	http://factordb.com/index.php?id=1100000003850155268	http://factordb.com/cert.php?id=1100000003850155268	http://factordb.com/index.php?query=%2811*26%5E%28n%2B2%29%2B5839%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25192	5₂₉₃O5	(26²⁹⁵+2469)/5	http://factordb.com/index.php?id=1100000003850155269	http://factordb.com/cert.php?id=1100000003850155269	http://factordb.com/index.php?query=%2826%5E%28n%2B2%29%2B2469%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25193	D7₃₀₀	(332×26³⁰⁰−7)/25	http://factordb.com/index.php?id=1100000002328053362	http://factordb.com/cert.php?id=1100000002328053362	http://factordb.com/index.php?query=%28332*26%5En-7%29%2F25&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
26	25194	E₃₀₅IL	(14×26³⁰⁷+2761)/25	http://factordb.com/index.php?id=1100000000840632032	http://factordb.com/cert.php?id=1100000000840632032	http://factordb.com/index.php?query=%2814*26%5E%28n%2B2%29%2B2761%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25195	PO5₃₁₂	(3371×26³¹²−1)/5	http://factordb.com/index.php?id=1100000003850155270	http://factordb.com/cert.php?id=1100000003850155270	http://factordb.com/index.php?query=%283371*26%5En-1%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25196	47₃₁₄	(107×26³¹⁴−7)/25	http://factordb.com/index.php?id=1100000002328050727	http://factordb.com/cert.php?id=1100000002328050727	http://factordb.com/index.php?query=%28107*26%5En-7%29%2F25&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
26	25197	A₃₃₅60F	(2×26³³⁸−14797)/5	http://factordb.com/index.php?id=1100000000840632163	http://factordb.com/cert.php?id=1100000000840632163	http://factordb.com/index.php?query=%282*26%5E%28n%2B3%29-14797%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25198	O5K5₃₄₁	(81871×26³⁴¹−1)/5	http://factordb.com/index.php?id=1100000003850155271	http://factordb.com/cert.php?id=1100000003850155271	http://factordb.com/index.php?query=%2887871*26%5En-1%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25199	9K₃₄₃AP	(49×26³⁴⁵−1279)/5	http://factordb.com/index.php?id=1100000000840632228	proven prime by N−1 test (https://primes.utm.edu/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since N−1 is 1274/5×(26³⁴⁴−1), thus factor N−1 is equivalent to factor 26³⁴⁴−1, and for the factorization of 26³⁴⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&Exp=344&c0=-&EN=	http://factordb.com/index.php?query=%2849*26%5E%28n%2B2%29-1279%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25200	8₃₅₄1	(8×26³⁵⁵−183)/25	http://factordb.com/index.php?id=1100000000840632517	proven prime by N−1 test (https://primes.utm.edu/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since N−1 is 208/25×(26³⁵⁴−1), thus factor N−1 is equivalent to factor 26³⁵⁴−1, and for the factorization of 26³⁵⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&Exp=354&c0=-&EN=	http://factordb.com/index.php?query=%288*26%5E%28n%2B1%29-183%29%2F25&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
26	25201	L0₃₅₆66K9	21×26³⁶⁰+110041	http://factordb.com/index.php?id=1100000000840632748	http://factordb.com/cert.php?id=1100000000840632748	http://factordb.com/index.php?query=21*26%5E%28n%2B4%29%2B110041&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25202	K0₃₅₈KIP	20×26³⁶¹+14013	http://factordb.com/index.php?id=1100000000840632880	http://factordb.com/cert.php?id=1100000000840632880	http://factordb.com/index.php?query=20*26%5E%28n%2B3%29%2B14013&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25203	J0₃₆₀A0P	19×26³⁶³+6785	http://factordb.com/index.php?id=1100000003850155272	http://factordb.com/cert.php?id=1100000003850155272	http://factordb.com/index.php?query=19*26%5E%28n%2B3%29%2B6785&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25204	OK6₃₇₆9	(16106×26³⁷⁷+69)/25	http://factordb.com/index.php?id=1100000000840633320	http://factordb.com/cert.php?id=1100000000840633320	http://factordb.com/index.php?query=%2816106*26%5E%28n%2B1%29%2B69%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25205	J₄₀₆7	(19×26⁴⁰⁷−319)/25	http://factordb.com/index.php?id=1100000002328055467	http://factordb.com/cert.php?id=1100000002328055467	http://factordb.com/index.php?query=%2819*26%5E%28n%2B1%29-319%29%2F25&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
26	25206	9B₄₃₇	(236×26⁴³⁷−11)/25	http://factordb.com/index.php?id=1100000002328051905	http://factordb.com/cert.php?id=1100000002328051905	http://factordb.com/index.php?query=%28236*26%5En-11%29%2F25&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
26	25207	3₄₄₂GL	(3×26⁴⁴⁴+8897)/25	http://factordb.com/index.php?id=1100000003850155273	http://factordb.com/cert.php?id=1100000003850155273	http://factordb.com/index.php?query=%283*26%5E%28n%2B2%29%2B8897%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25208	1M₄₉₁P	(47×26⁴⁹²+53)/25	http://factordb.com/index.php?id=1100000000840633390	http://factordb.com/cert.php?id=1100000000840633390	http://factordb.com/index.php?query=%2847*26%5E%28n%2B1%29%2B53%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25209	40₅₀₉GL	4×26⁵¹¹+437	http://factordb.com/index.php?id=1100000000840633483	http://factordb.com/cert.php?id=1100000000840633483	http://factordb.com/index.php?query=4*26%5E%28n%2B2%29%2B437&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25210	BFA₅₁₁5	(1507×26⁵¹²−27)/5	http://factordb.com/index.php?id=1100000003850155274	http://factordb.com/cert.php?id=1100000003850155274	http://factordb.com/index.php?query=%281507*26%5E%28n%2B1%29-27%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25211	LK₅₁₈5	(109×26⁵¹⁹−79)/5	http://factordb.com/index.php?id=1100000003850155276	http://factordb.com/cert.php?id=1100000003850155276	http://factordb.com/index.php?query=%28109*26%5E%28n%2B1%29-79%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25212	MI₅₄₃3	(568×26⁵⁴⁴−393)/25	http://factordb.com/index.php?id=1100000003850155277	http://factordb.com/cert.php?id=1100000003850155277	http://factordb.com/index.php?query=%28568*26%5E%28n%2B1%29-393%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25213	E4E₅₅₉7	(9214×26⁵⁶⁰−189)/25	http://factordb.com/index.php?id=1100000003850155278	http://factordb.com/cert.php?id=1100000003850155278	http://factordb.com/index.php?query=%289214*26%5E%28n%2B1%29-189%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25214	80₅₇₇C7	8×26⁵⁷⁹+319	http://factordb.com/index.php?id=1100000002634136160	http://factordb.com/cert.php?id=1100000002634136160	http://factordb.com/index.php?query=8*26%5E%28n%2B2%29%2B319&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25215	9E0₆₁₉B	248×26⁶²⁰+11	http://factordb.com/index.php?id=1100000002634136193	http://factordb.com/cert.php?id=1100000002634136193	http://factordb.com/index.php?query=248*26%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25216	G60₆₁₈KJ	422×26⁶²⁰+539	http://factordb.com/index.php?id=1100000003850155283	http://factordb.com/cert.php?id=1100000003850155283	http://factordb.com/index.php?query=422*26%5E%28n%2B2%29%2B539&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25217	OO0₆₂₀D3	648×26⁶²²+341	http://factordb.com/index.php?id=1100000003850155285	http://factordb.com/cert.php?id=1100000003850155285	http://factordb.com/index.php?query=648*26%5E%28n%2B2%29%2B341&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25218	K0₆₅₃IP	20×26⁶⁵⁵+493	http://factordb.com/index.php?id=1100000000840633594	http://factordb.com/cert.php?id=1100000000840633594	http://factordb.com/index.php?query=20*26%5E%28n%2B2%29%2B493&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25219	J00J₆₉₈L	(321119×26⁶⁹⁹+31)/25	http://factordb.com/index.php?id=1100000003850155288	http://factordb.com/cert.php?id=1100000003850155288	http://factordb.com/index.php?query=%28321119*26%5E%28n%2B1%29%2B31%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25220	B0₇₇₂90J	11×26⁷⁷⁵+6103	http://factordb.com/index.php?id=1100000003850155290	http://factordb.com/cert.php?id=1100000003850155290	http://factordb.com/index.php?query=11*26%5E%28n%2B3%29%2B6103&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25221	J₈₄₄B	(19×26⁸⁴⁵−219)/25	http://factordb.com/index.php?id=1100000002328055693	http://factordb.com/cert.php?id=1100000002328055693	http://factordb.com/index.php?query=%2819*26%5E%28n%2B1%29-219%29%2F25&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
26	25222	H₈₅₅M0H	(17×26⁸⁵⁸+73433)/25	http://factordb.com/index.php?id=1100000003850155291	http://factordb.com/cert.php?id=1100000003850155291	http://factordb.com/index.php?query=%2817*26%5E%28n%2B3%29%2B73433%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25223	J₈₆₁OOL	(19×26⁸⁶⁴+87781)/25	http://factordb.com/index.php?id=1100000003850155296	http://factordb.com/cert.php?id=1100000003850155296	http://factordb.com/index.php?query=%2819*26%5E%28n%2B3%29%2B87781%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25224	B0₉₇₉H	11×26⁹⁸⁰+17	http://factordb.com/index.php?id=1100000002355639467	http://factordb.com/cert.php?id=1100000002355639467	http://factordb.com/index.php?query=11*26%5E%28n%2B1%29%2B17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25225	L0₉₉₁4000J	21×26⁹⁹⁶+1827923	http://factordb.com/index.php?id=1100000003850155301	http://factordb.com/cert.php?id=1100000003850155301	http://factordb.com/index.php?query=21*26%5E%28n%2B5%29%2B1827923&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25226	E0₁₁₀₁K2B	14×26¹¹⁰⁴+13583	http://factordb.com/index.php?id=1100000003850155305	http://factordb.com/cert.php?id=1100000003850155305	http://factordb.com/index.php?query=14*26%5E%28n%2B3%29%2B13583&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25227	G₁₁₀₅OO9	(16×26¹¹⁰⁸+140209)/25	http://factordb.com/index.php?id=1100000000840633717	http://factordb.com/cert.php?id=1100000000840633717	http://factordb.com/index.php?query=%2816*26%5E%28n%2B3%29%2B140209%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25228	MC0₁₁₀₉N	584×26¹¹¹⁰+23	http://factordb.com/index.php?id=1100000002634136576	http://factordb.com/cert.php?id=1100000002634136576	http://factordb.com/index.php?query=584*26%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25229	G₁₁₅₉9	(16×26¹¹⁶⁰−191)/25	http://factordb.com/index.php?id=1100000000840633844	http://factordb.com/cert.php?id=1100000000840633844	http://factordb.com/index.php?query=%2816*26%5E%28n%2B1%29-191%29%2F25&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
26	25230	A₁₂₉₅06F	(2×26¹²⁹⁸−34297)/5	http://factordb.com/index.php?id=1100000000840633998	http://factordb.com/cert.php?id=1100000000840633998	http://factordb.com/index.php?query=%282*26%5E%28n%2B3%29-34297%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25231	KIA₁₂₉₈F	(2692×26¹²⁹⁹+23)/5	http://factordb.com/index.php?id=1100000000840634108	http://factordb.com/cert.php?id=1100000000840634108	http://factordb.com/index.php?query=%282692*26%5E%28n%2B1%29%2B23%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25232	L7₁₃₁₉OL	(532×26¹³²¹+11393)/25	http://factordb.com/index.php?id=1100000003850155311	http://factordb.com/cert.php?id=1100000003850155311	http://factordb.com/index.php?query=%28532*26%5E%28n%2B2%29%2B11393%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25233	J₁₃₆₅L	(19×26¹³⁶⁶+31)/25	http://factordb.com/index.php?id=1100000002328055922	http://factordb.com/cert.php?id=1100000002328055922	http://factordb.com/index.php?query=%2819*26%5E%28n%2B1%29%2B31%29%2F25&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
26	25234	2₁₄₉₈H	(2×26¹⁴⁹⁹+373)/25	http://factordb.com/index.php?id=1100000002328050300	http://factordb.com/cert.php?id=1100000002328050300	http://factordb.com/index.php?query=%282*26%5E%28n%2B1%29%2B373%29%2F25&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
26	25235	O5₁₅₀₉	(121×26¹⁵⁰⁹−1)/5	http://factordb.com/index.php?id=1100000000894500022	http://factordb.com/cert.php?id=1100000000894500022	http://factordb.com/index.php?query=%28121*26%5En-1%29%2F5&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25236	DM₁₅₁₉P	(347×26¹⁵²⁰+53)/25	http://factordb.com/index.php?id=1100000003850155312	http://factordb.com/cert.php?id=1100000003850155312	http://factordb.com/index.php?query=%28347*26%5E%28n%2B1%29%2B53%29%2F25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25237	J0₁₅₂₃P	19×26¹⁵²⁴+25	http://factordb.com/index.php?id=1100000002355640604	http://factordb.com/cert.php?id=1100000002355640604	http://factordb.com/index.php?query=19*26%5E%28n%2B1%29%2B25&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25238	F₁₅₆₉PCF	(3×26¹⁵⁷²+33407)/5	http://factordb.com/index.php?id=1100000000840634210	http://factordb.com/cert.php?id=1100000000840634210	http://factordb.com/index.php?query=%283*26%5E%28n%2B3%29%2B33407%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25239	N0₁₆₄₇NEN	23×26¹⁶⁵⁰+15935	http://factordb.com/index.php?id=1100000003850155313	http://factordb.com/cert.php?id=1100000003850155313	http://factordb.com/index.php?query=23*26%5E%28n%2B3%29%2B15935&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25240	5₁₈₈₅4P	(26¹⁸⁸⁷−31)/5	http://factordb.com/index.php?id=1100000003850155314	http://factordb.com/cert.php?id=1100000003850155314	http://factordb.com/index.php?query=%2826%5E%28n%2B2%29-31%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25241	6K₂₅₅₆A5	(34×26²⁵⁵⁸−1379)/5	http://factordb.com/index.php?id=1100000003850155315	http://factordb.com/cert.php?id=1100000003850155315	http://factordb.com/index.php?query=%2834*26%5E%28n%2B2%29-1379%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25242	70₂₆₁₃CN	7×26²⁶¹⁵+335	http://factordb.com/index.php?id=1100000002634136105	http://factordb.com/cert.php?id=1100000002634136105	http://factordb.com/index.php?query=7*26%5E%28n%2B2%29%2B335&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25243	E0₂₆₇₃H	14×26²⁶⁷⁴+17	http://factordb.com/index.php?id=1100000002355640062	http://factordb.com/cert.php?id=1100000002355640062	http://factordb.com/index.php?query=14*26%5E%28n%2B1%29%2B17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25244	G60₂₇₄₀J	422×26²⁷⁴¹+19	http://factordb.com/index.php?id=1100000002634136363	http://factordb.com/cert.php?id=1100000002634136363	http://factordb.com/index.php?query=422*26%5E%28n%2B1%29%2B19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25245	B₂₉₁₅9	(11×26²⁹¹⁶−61)/25	http://factordb.com/index.php?id=1100000002328052611	http://factordb.com/cert.php?id=1100000002328052611	http://factordb.com/index.php?query=%2811*26%5E%28n%2B1%29-61%29%2F25&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
26	25246	J₄₂₂₂P	(19×26⁴²²³+131)/25	http://factordb.com/index.php?id=1100000002328056865	http://factordb.com/cert.php?id=1100000002328056865	http://factordb.com/index.php?query=%2819*26%5E%28n%2B1%29%2B131%29%2F25&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
26	25247	K0₄₃₆₄I5	20×26⁴³⁶⁶+473	http://factordb.com/index.php?id=1100000002634136508	http://factordb.com/cert.php?id=1100000002634136508	http://factordb.com/index.php?query=20*26%5E%28n%2B2%29%2B473&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
26	25248	M₈₇₇₂P	(22×26⁸⁷⁷³+53)/25	http://factordb.com/index.php?id=1100000000758011195	http://factordb.com/cert.php?id=1100000000758011195	http://factordb.com/index.php?query=%2822*26%5E%28n%2B1%29%2B53%29%2F25&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
26	25249	9GDK₁₅₉₂₀P	(32569×26¹⁵⁹²¹+21)/5	http://factordb.com/index.php?id=1100000003850155316	http://factordb.com/cert.php?id=1100000003850155316	http://factordb.com/index.php?query=%2832569*26%5E%28n%2B1%29%2B21%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25485	JN₂₀₆	(536×28²⁰⁶−23)/27	http://factordb.com/index.php?id=1100000002611724435	proven prime by N−1 test (https://primes.utm.edu/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), N−1 is 2×1061×1171×74311×(289-digit prime)	http://factordb.com/index.php?query=%28536*28%5En-23%29%2F27&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
28	25486	3₂₁₁M9	(28²¹³+4841)/9	http://factordb.com/index.php?id=1100000003850161936	http://factordb.com/cert.php?id=1100000003850161936	http://factordb.com/index.php?query=%2828%5E%28n%2B2%29%2B4841%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25487	HD0₂₁₃D	489×28²¹⁴+13	http://factordb.com/index.php?id=1100000003850161937	http://factordb.com/cert.php?id=1100000003850161937	http://factordb.com/index.php?query=489*28%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25488	64O₂₁₇9	(1556×28²¹⁸−143)/9	http://factordb.com/index.php?id=1100000000840840215	http://factordb.com/cert.php?id=1100000000840840215	http://factordb.com/index.php?query=%281556*28%5E%28n%2B1%29-143%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25489	G0₂₁₇A0N	16×28²²⁰+7863	http://factordb.com/index.php?id=1100000003850161938	http://factordb.com/cert.php?id=1100000003850161938	http://factordb.com/index.php?query=16*28%5E%28n%2B3%29%2B7863&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25490	55OA₂₂₆F	(110278×28²²⁷+125)/27	http://factordb.com/index.php?id=1100000003850161939	http://factordb.com/cert.php?id=1100000003850161939	http://factordb.com/index.php?query=%28110278*28%5E%28n%2B1%29%2B125%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25491	L0₂₂₉Q3	21×28²³¹+731	http://factordb.com/index.php?id=1100000003850161940	http://factordb.com/cert.php?id=1100000003850161940	http://factordb.com/index.php?query=21*28%5E%28n%2B2%29%2B731&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25492	B0₂₃₁7ID	11×28²³⁴+6005	http://factordb.com/index.php?id=1100000003850161941	http://factordb.com/cert.php?id=1100000003850161941	http://factordb.com/index.php?query=11*28%5E%28n%2B3%29%2B6005&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25493	PM₂₃₃B	(697×28²³⁴−319)/27	http://factordb.com/index.php?id=1100000003850161942	http://factordb.com/cert.php?id=1100000003850161942	http://factordb.com/index.php?query=%28697*28%5E%28n%2B1%29-319%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25494	K0₂₃₈OF	20×28²⁴⁰+687	http://factordb.com/index.php?id=1100000000840840142	http://factordb.com/cert.php?id=1100000000840840142	http://factordb.com/index.php?query=20*28%5E%28n%2B2%29%2B687&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25495	I₂₆₂E3	(2×28²⁶⁴−383)/3	http://factordb.com/index.php?id=1100000003850161943	http://factordb.com/cert.php?id=1100000003850161943	http://factordb.com/index.php?query=%282*28%5E%28n%2B2%29-383%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
28	25496	C5A₂₇₃F	(9217×28²⁷⁴+125)/27	http://factordb.com/index.php?id=1100000003850161944	http://factordb.com/cert.php?id=1100000003850161944	http://factordb.com/index.php?query=%289217*28%5E%28n%2B1%29%2B125%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25497	J0₂₇₆IMB	19×28²⁷⁹+14739	http://factordb.com/index.php?id=1100000003850161945	http://factordb.com/cert.php?id=1100000003850161945	http://factordb.com/index.php?query=19*28%5E%28n%2B3%29%2B14739&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25498	F0₂₈₂QAP	15×28²⁸⁵+20689	http://factordb.com/index.php?id=1100000000840840006	http://factordb.com/cert.php?id=1100000000840840006	http://factordb.com/index.php?query=15*28%5E%28n%2B3%29%2B20689&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25499	M0₂₉₆KKN	22×28²⁹⁹+16263	http://factordb.com/index.php?id=1100000003850161946	http://factordb.com/cert.php?id=1100000003850161946	http://factordb.com/index.php?query=22*28%5E%28n%2B3%29%2B16263&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25500	C₃₁₀43	(4×28³¹²−2101)/9	http://factordb.com/index.php?id=1100000003850161947	http://factordb.com/cert.php?id=1100000003850161947	http://factordb.com/index.php?query=%284*28%5E%28n%2B2%29-2101%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25501	RN₃₁₉	(752×28³¹⁹−23)/27	http://factordb.com/index.php?id=1100000002611723967	http://factordb.com/cert.php?id=1100000002611723967	http://factordb.com/index.php?query=%28752*28%5En-23%29%2F27&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
28	25502	CA₃₂₀F	(334×28³²¹+125)/27	http://factordb.com/index.php?id=1100000000840839995	http://factordb.com/cert.php?id=1100000000840839995	http://factordb.com/index.php?query=%28334*28%5E%28n%2B1%29%2B125%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25503	D6₃₂₆LR	(119×28³²⁸+3967)/9	http://factordb.com/index.php?id=1100000003850161948	http://factordb.com/cert.php?id=1100000003850161948	http://factordb.com/index.php?query=%28119*28%5E%28n%2B2%29%2B3967%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25504	B₃₅₀AB	(11×28³⁵²−767)/27	http://factordb.com/index.php?id=1100000003850161949	http://factordb.com/cert.php?id=1100000003850161949	http://factordb.com/index.php?query=%2811*28%5E%28n%2B2%29-767%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25505	GA0₃₅₅N	458×28³⁵⁶+23	http://factordb.com/index.php?id=1100000003850161950	http://factordb.com/cert.php?id=1100000003850161950	http://factordb.com/index.php?query=458*28%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25506	A0₃₅₆P7P	10×28³⁵⁹+19821	http://factordb.com/index.php?id=1100000003850161951	http://factordb.com/cert.php?id=1100000003850161951	http://factordb.com/index.php?query=10*28%5E%28n%2B3%29%2B19821&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25507	J₃₆₃H	(19×28³⁶⁴−73)/27	http://factordb.com/index.php?id=1100000002611724460	http://factordb.com/cert.php?id=1100000002611724460	http://factordb.com/index.php?query=%2819*28%5E%28n%2B1%29-73%29%2F27&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
28	25508	4B₃₈₁	(119×28³⁸¹−11)/27	http://factordb.com/index.php?id=1100000002611724588	http://factordb.com/cert.php?id=1100000002611724588	http://factordb.com/index.php?query=%28119*28%5En-11%29%2F27&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
28	25509	EB0₄₀₅1	403×28⁴⁰⁶+1	http://factordb.com/index.php?id=1100000001534442374	proven prime by N−1 test (https://primes.utm.edu/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since N−1 is trivially fully factored	http://factordb.com/index.php?query=403*28%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
28	25510	AN₄₆₁	(293×28⁴⁶¹−23)/27	http://factordb.com/index.php?id=1100000002611724556	http://factordb.com/cert.php?id=1100000002611724556	http://factordb.com/index.php?query=%28293*28%5En-23%29%2F27&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
28	25511	4O₆₁₄09	(44×28⁶¹⁶−6191)/9	http://factordb.com/index.php?id=1100000000840839989	http://factordb.com/cert.php?id=1100000000840839989	http://factordb.com/index.php?query=%2844*28%5E%28n%2B2%29-6191%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25512	2D₆₄₁	(67×28⁶⁴¹−13)/27	http://factordb.com/index.php?id=1100000002611725341	http://factordb.com/cert.php?id=1100000002611725341	http://factordb.com/index.php?query=%2867*28%5En-13%29%2F27&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
28	25513	70₇₄₈M5	7×28⁷⁵⁰+621	http://factordb.com/index.php?id=1100000003850161956	http://factordb.com/cert.php?id=1100000003850161956	http://factordb.com/index.php?query=7*28%5E%28n%2B2%29%2B621&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25514	4A0₈₀₄B	122×28⁸⁰⁵+11	http://factordb.com/index.php?id=1100000003850161957	http://factordb.com/cert.php?id=1100000003850161957	http://factordb.com/index.php?query=122*28%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25515	LK₉₂₅F	(587×28⁹²⁶−155)/27	http://factordb.com/index.php?id=1100000000840839978	http://factordb.com/cert.php?id=1100000000840839978	http://factordb.com/index.php?query=%28587*28%5E%28n%2B1%29-155%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25516	J0₁₀₇₁AC5	19×28¹⁰⁷⁴+8181	http://factordb.com/index.php?id=1100000003850161959	http://factordb.com/cert.php?id=1100000003850161959	http://factordb.com/index.php?query=19*28%5E%28n%2B3%29%2B8181&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25517	J0₁₂₅₂J5	19×28¹²⁵⁴+537	http://factordb.com/index.php?id=1100000003850161963	http://factordb.com/cert.php?id=1100000003850161963	http://factordb.com/index.php?query=19*28%5E%28n%2B2%29%2B537&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25518	5₁₃₀₄6F	(5×28¹³⁰⁶+1021)/27	http://factordb.com/index.php?id=1100000003850161964	http://factordb.com/cert.php?id=1100000003850161964	http://factordb.com/index.php?query=%285*28%5E%28n%2B2%29%2B1021%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25519	5₁₃₃₂P8P	(5×28¹³³⁵+426163)/27	http://factordb.com/index.php?id=1100000003850161965	http://factordb.com/cert.php?id=1100000003850161965	http://factordb.com/index.php?query=%285*28%5E%28n%2B3%29%2B426163%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25520	5I₁₃₇₀F	(17×28¹³⁷¹−11)/3	http://factordb.com/index.php?id=1100000003850161972	http://factordb.com/cert.php?id=1100000003850161972	http://factordb.com/index.php?query=%2817*28%5E%28n%2B1%29-11%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
28	25521	A₁₄₂₃6F	(10×28¹⁴²⁵−2899)/27	http://factordb.com/index.php?id=1100000000840839947	http://factordb.com/cert.php?id=1100000000840839947	http://factordb.com/index.php?query=%2810*28%5E%28n%2B2%29-2899%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25522	G0₁₈₉₉AN	16×28¹⁹⁰¹+303	http://factordb.com/index.php?id=1100000003850161973	http://factordb.com/cert.php?id=1100000003850161973	http://factordb.com/index.php?query=16*28%5E%28n%2B2%29%2B303&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25523	5₃₇₄₆8P	(5×28³⁷⁴⁸+2803)/27	http://factordb.com/index.php?id=1100000003850161974	http://factordb.com/cert.php?id=1100000003850161974	http://factordb.com/index.php?query=%285*28%5E%28n%2B2%29%2B2803%29%2F27&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25524	QO₄₂₃₉69	(242×28⁴²⁴¹−4679)/9	http://factordb.com/index.php?id=1100000000840839934	http://factordb.com/cert.php?id=1100000000840839934	http://factordb.com/index.php?query=%28242*28%5E%28n%2B2%29-4679%29%2F9&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
28	25525	D0₅₂₆₇77D	13×28⁵²⁷⁰+5697	http://factordb.com/index.php?id=1100000003850151420	http://factordb.com/cert.php?id=1100000003850151420	http://factordb.com/index.php?query=13*28%5E%28n%2B3%29%2B5697&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
30	2613	AN₂₀₆	(313×30²⁰⁶−23)/29	http://factordb.com/index.php?id=1100000002327651073	http://factordb.com/cert.php?id=1100000002327651073	http://factordb.com/index.php?query=%28313*30%5En-23%29%2F29&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
30	2614	M₂₄₁QB	(22×30²⁴³+3139)/29	http://factordb.com/index.php?id=1100000003593408295	http://factordb.com/cert.php?id=1100000003593408295	http://factordb.com/index.php?query=%2822*30%5E%28n%2B2%29%2B3139%29%2F29&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
30	2615	M0₅₄₇SS7	22×30⁵⁵⁰+26047	http://factordb.com/index.php?id=1100000003593407988	http://factordb.com/cert.php?id=1100000003593407988	http://factordb.com/index.php?query=22*30%5E%28n%2B3%29%2B26047&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
30	2616	C0₁₀₂₂1	12×30¹⁰²³+1	http://factordb.com/index.php?id=1100000000785448736	proven prime by N−1 test (https://primes.utm.edu/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1), since N−1 is trivially fully factored	http://factordb.com/index.php?query=12*30%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
30	2617	5₄₈₈₂J	(5×30⁴⁸⁸³+401)/29	http://factordb.com/index.php?id=1100000002327649423	http://factordb.com/cert.php?id=1100000002327649423	http://factordb.com/index.php?query=%285*30%5E%28n%2B1%29%2B401%29%2F29&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
30	2619	OT₃₄₂₀₅	25×30³⁴²⁰⁵−1	http://factordb.com/index.php?id=1100000000800812865	proven prime by N+1 test (https://primes.utm.edu/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2), since N+1 is trivially fully factored	http://factordb.com/index.php?query=25*30%5En-1&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35212	Q₁₉₅77	(26×36¹⁹⁷−24631)/35	http://factordb.com/index.php?id=1100000003807362350	http://factordb.com/cert.php?id=1100000003807362350	http://factordb.com/index.php?query=%2826*36%5E%28n%2B2%29-24631%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35213	W0₁₉₉ND	32×36²⁰¹+841	http://factordb.com/index.php?id=1100000002634136732	http://factordb.com/cert.php?id=1100000002634136732	http://factordb.com/index.php?query=32*36%5E%28n%2B2%29%2B841&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35214	G0₂₀₄YT	16×36²⁰⁶+1253	http://factordb.com/index.php?id=1100000002634137789	http://factordb.com/cert.php?id=1100000002634137789	http://factordb.com/index.php?query=16*36%5E%28n%2B2%29%2B1253&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35215	RHY₂₂₃H	(34649×36²²⁴−629)/35	http://factordb.com/index.php?id=1100000003807362353	http://factordb.com/cert.php?id=1100000003807362353	http://factordb.com/index.php?query=%2834649*36%5E%28n%2B1%29-629%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35216	T0₂₂₃ST	29×36²²⁵+1037	http://factordb.com/index.php?id=1100000002634136882	http://factordb.com/cert.php?id=1100000002634136882	http://factordb.com/index.php?query=29*36%5E%28n%2B2%29%2B1037&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35217	J0₂₂₄U6V	19×36²²⁷+39127	http://factordb.com/index.php?id=1100000003807362355	http://factordb.com/cert.php?id=1100000003807362355	http://factordb.com/index.php?query=19*36%5E%28n%2B3%29%2B39127&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35218	BE0₂₃₅IV	410×36²³⁷+679	http://factordb.com/index.php?id=1100000003807362356	http://factordb.com/cert.php?id=1100000003807362356	http://factordb.com/index.php?query=410*36%5E%28n%2B2%29%2B679&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35219	E0₂₃₆KY1	14×36²³⁹+27145	http://factordb.com/index.php?id=1100000000840634520	http://factordb.com/cert.php?id=1100000000840634520	http://factordb.com/index.php?query=14*36%5E%28n%2B3%29%2B27145&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35220	JXV0₂₄₄B	25843×36²⁴⁵+11	http://factordb.com/index.php?id=1100000003807362357	http://factordb.com/cert.php?id=1100000003807362357	http://factordb.com/index.php?query=25843*36%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35221	5Q₂₄₉7	(201×36²⁵⁰−691)/35	http://factordb.com/index.php?id=1100000003807362359	http://factordb.com/cert.php?id=1100000003807362359	http://factordb.com/index.php?query=%28201*36%5E%28n%2B1%29-691%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35222	N30₂₅₃H	831×36²⁵⁴+17	http://factordb.com/index.php?id=1100000002634137359	http://factordb.com/cert.php?id=1100000002634137359	http://factordb.com/index.php?query=831*36%5E%28n%2B1%29%2B17&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35223	Y₂₆₁AH	(34×36²⁶³−30869)/35	http://factordb.com/index.php?id=1100000003807362360	http://factordb.com/cert.php?id=1100000003807362360	http://factordb.com/index.php?query=%2834*36%5E%28n%2B2%29-30869%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35224	90₂₇₇VV	9×36²⁷⁹+1147	http://factordb.com/index.php?id=1100000002634138388	http://factordb.com/cert.php?id=1100000002634138388	http://factordb.com/index.php?query=9*36%5E%28n%2B2%29%2B1147&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35225	J0₂₈₁VB	19×36²⁸³+1127	http://factordb.com/index.php?id=1100000002634137683	http://factordb.com/cert.php?id=1100000002634137683	http://factordb.com/index.php?query=19*36%5E%28n%2B2%29%2B1127&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35226	J0₂₈₁WV	19×36²⁸³+1183	http://factordb.com/index.php?id=1100000002634137660	http://factordb.com/cert.php?id=1100000002634137660	http://factordb.com/index.php?query=19*36%5E%28n%2B2%29%2B1183&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35227	DE0₂₈₁61	482×36²⁸³+217	http://factordb.com/index.php?id=1100000003807362361	http://factordb.com/cert.php?id=1100000003807362361	http://factordb.com/index.php?query=482*36%5E%28n%2B2%29%2B217&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35228	9H₂₉₇	(332×36²⁹⁷−17)/35	http://factordb.com/index.php?id=1100000002332535884	http://factordb.com/cert.php?id=1100000002332535884	http://factordb.com/index.php?query=%28332*36%5En-17%29%2F35&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
36	35229	M70₂₉₇FD	799×36²⁹⁹+553	http://factordb.com/index.php?id=1100000003807362363	http://factordb.com/cert.php?id=1100000003807362363	http://factordb.com/index.php?query=799*36%5E%28n%2B2%29%2B553&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35230	9X₃₀₁B	(348×36³⁰²−803)/35	http://factordb.com/index.php?id=1100000003807362364	http://factordb.com/cert.php?id=1100000003807362364	http://factordb.com/index.php?query=%28348*36%5E%28n%2B1%29-803%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35231	XE0₃₂₅7	1202×36³²⁶+7	http://factordb.com/index.php?id=1100000002634136674	http://factordb.com/cert.php?id=1100000002634136674	http://factordb.com/index.php?query=1202*36%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
36	35232	KP₃₃₀SZ	(145×36³³²+821)/7	http://factordb.com/index.php?id=1100000000840634515	http://factordb.com/cert.php?id=1100000000840634515	http://factordb.com/index.php?query=%28145*36%5E%28n%2B2%29%2B821%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35233	5₃₄₇QP	(36³⁴⁹+5431)/7	http://factordb.com/index.php?id=1100000003807362365	http://factordb.com/cert.php?id=1100000003807362365	http://factordb.com/index.php?query=%2836%5E%28n%2B2%29%2B5431%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35234	E0₃₄₉6U1	14×36³⁵²+8857	http://factordb.com/index.php?id=1100000000840634509	http://factordb.com/cert.php?id=1100000000840634509	http://factordb.com/index.php?query=14*36%5E%28n%2B3%29%2B8857&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35235	K0₃₆₇E6T	20×36³⁷⁰+18389	http://factordb.com/index.php?id=1100000003807362367	http://factordb.com/cert.php?id=1100000003807362367	http://factordb.com/index.php?query=20*36%5E%28n%2B3%29%2B18389&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35236	U0₃₇₀WP	30×36³⁷²+1177	http://factordb.com/index.php?id=1100000000840634503	http://factordb.com/cert.php?id=1100000000840634503	http://factordb.com/index.php?query=30*36%5E%28n%2B2%29%2B1177&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35237	P8Z₃₉₀	909×36³⁹⁰−1	http://factordb.com/index.php?id=1100000000764100228	proven prime by N+1 test (https://primes.utm.edu/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2), since N+1 is trivially fully factored	http://factordb.com/index.php?query=909*36%5En-1&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35238	90₃₉₇4B	9×36³⁹⁹+155	http://factordb.com/index.php?id=1100000002634138490	http://factordb.com/cert.php?id=1100000002634138490	http://factordb.com/index.php?query=9*36%5E%28n%2B2%29%2B155&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35239	50₄₀₅WW5	5×36⁴⁰⁸+42629	http://factordb.com/index.php?id=1100000003807362369	http://factordb.com/cert.php?id=1100000003807362369	http://factordb.com/index.php?query=5*36%5E%28n%2B3%29%2B42629&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35240	V0₄₀₅EE4B	31×36⁴⁰⁹+671483	http://factordb.com/index.php?id=1100000003807362370	http://factordb.com/cert.php?id=1100000003807362370	http://factordb.com/index.php?query=31*36%5E%28n%2B4%29%2B671483&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35241	TTR0₄₃₄T	38655×36⁴³⁵+29	http://factordb.com/index.php?id=1100000003807362372	http://factordb.com/cert.php?id=1100000003807362372	http://factordb.com/index.php?query=38655*36%5E%28n%2B1%29%2B29&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35242	LK₄₄₃Z	(151×36⁴⁴⁴+101)/7	http://factordb.com/index.php?id=1100000000840634496	http://factordb.com/cert.php?id=1100000000840634496	http://factordb.com/index.php?query=%28151*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35243	Q₄₅₄7	(26×36⁴⁵⁵−691)/35	http://factordb.com/index.php?id=1100000002332534290	http://factordb.com/cert.php?id=1100000002332534290	http://factordb.com/index.php?query=%2826*36%5E%28n%2B1%29-691%29%2F35&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
36	35244	R0₄₅₈7	27×36⁴⁵⁹+7	http://factordb.com/index.php?id=1100000002356257765	http://factordb.com/cert.php?id=1100000002356257765	http://factordb.com/index.php?query=27*36%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
36	35245	50₄₆₀KGZ	5×36⁴⁶³+26531	http://factordb.com/index.php?id=1100000003807362374	http://factordb.com/cert.php?id=1100000003807362374	http://factordb.com/index.php?query=5*36%5E%28n%2B3%29%2B26531&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35246	K0₄₇₄OY1	20×36⁴⁷⁷+32329	http://factordb.com/index.php?id=1100000000840634488	http://factordb.com/cert.php?id=1100000000840634488	http://factordb.com/index.php?query=20*36%5E%28n%2B3%29%2B32329&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35247	B₄₇₈KB	(11×36⁴⁸⁰+11329)/35	http://factordb.com/index.php?id=1100000003807362381	http://factordb.com/cert.php?id=1100000003807362381	http://factordb.com/index.php?query=%2811*36%5E%28n%2B2%29%2B11329%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35248	WY₅₀₇H	(1154×36⁵⁰⁸−629)/35	http://factordb.com/index.php?id=1100000003807362386	http://factordb.com/cert.php?id=1100000003807362386	http://factordb.com/index.php?query=%281154*36%5E%28n%2B1%29-629%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35249	G0₅₁₀USJ	16×36⁵¹³+39907	http://factordb.com/index.php?id=1100000003807362389	http://factordb.com/cert.php?id=1100000003807362389	http://factordb.com/index.php?query=16*36%5E%28n%2B3%29%2B39907&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35250	Z₅₂₇EX7	36⁵³⁰−27317	http://factordb.com/index.php?id=1100000003807362391	http://factordb.com/cert.php?id=1100000003807362391	http://factordb.com/index.php?query=36%5E%28n%2B3%29-27317&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35251	EY0₅₃₄A1	538×36⁵³⁶+361	http://factordb.com/index.php?id=1100000000840634482	http://factordb.com/cert.php?id=1100000000840634482	http://factordb.com/index.php?query=538*36%5E%28n%2B2%29%2B361&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35252	Z0₅₆₃995	35×36⁵⁶⁶+11993	http://factordb.com/index.php?id=1100000003807362394	http://factordb.com/cert.php?id=1100000003807362394	http://factordb.com/index.php?query=35*36%5E%28n%2B3%29%2B11993&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35253	F₅₉₀95	(3×36⁵⁹²−1585)/7	http://factordb.com/index.php?id=1100000003807362398	http://factordb.com/cert.php?id=1100000003807362398	http://factordb.com/index.php?query=%283*36%5E%28n%2B2%29-1585%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35254	990₅₉₁B	333×36⁵⁹²+11	http://factordb.com/index.php?id=1100000002634138415	http://factordb.com/cert.php?id=1100000002634138415	http://factordb.com/index.php?query=333*36%5E%28n%2B1%29%2B11&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35255	J₆₇₅T	(19×36⁶⁷⁶+331)/35	http://factordb.com/index.php?id=1100000002332534943	http://factordb.com/cert.php?id=1100000002332534943	http://factordb.com/index.php?query=%2819*36%5E%28n%2B1%29%2B331%29%2F35&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
36	35256	FZ₇₀₈OEB	16×36⁷¹¹−15037	http://factordb.com/index.php?id=1100000003807362403	http://factordb.com/cert.php?id=1100000003807362403	http://factordb.com/index.php?query=16*36%5E%28n%2B3%29-15037&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35257	EX₇₃₂B	(523×36⁷³³−803)/35	http://factordb.com/index.php?id=1100000003807362408	http://factordb.com/cert.php?id=1100000003807362408	http://factordb.com/index.php?query=%28523*36%5E%28n%2B1%29-803%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35258	M₇₆₄8B	(22×36⁷⁶⁶−18047)/35	http://factordb.com/index.php?id=1100000003807362414	http://factordb.com/cert.php?id=1100000003807362414	http://factordb.com/index.php?query=%2822*36%5E%28n%2B2%29-18047%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35259	3EG₇₇₇D	(4286×36⁷⁷⁸−121)/35	http://factordb.com/index.php?id=1100000003807362419	http://factordb.com/cert.php?id=1100000003807362419	http://factordb.com/index.php?query=%284286*36%5E%28n%2B1%29-121%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35260	W₉₆₂7	(32×36⁹⁶³−907)/35	http://factordb.com/index.php?id=1100000002332533447	http://factordb.com/cert.php?id=1100000002332533447	http://factordb.com/index.php?query=%2832*36%5E%28n%2B1%29-907%29%2F35&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
36	35261	G₉₇₉88D	(16×36⁹⁸²−373081)/35	http://factordb.com/index.php?id=1100000003807362435	http://factordb.com/cert.php?id=1100000003807362435	http://factordb.com/index.php?query=%2816*36%5E%28n%2B3%29-373081%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35262	70₁₀₅₀XQN	7×36¹⁰⁵³+43727	http://factordb.com/index.php?id=1100000003807362444	http://factordb.com/cert.php?id=1100000003807362444	http://factordb.com/index.php?query=7*36%5E%28n%2B3%29%2B43727&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35263	EB0₁₀₈₃UV	515×36¹⁰⁸⁵+1111	http://factordb.com/index.php?id=1100000003807362457	http://factordb.com/cert.php?id=1100000003807362457	http://factordb.com/index.php?query=515*36%5E%28n%2B2%29%2B1111&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35264	F02R0₁₂₃₇D	699939×36¹²³⁸+13	http://factordb.com/index.php?id=1100000003807362472	http://factordb.com/cert.php?id=1100000003807362472	http://factordb.com/index.php?query=699939*36%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35265	50₁₃₁₃WMN	5×36¹³¹⁶+42287	http://factordb.com/index.php?id=1100000003807362473	http://factordb.com/cert.php?id=1100000003807362473	http://factordb.com/index.php?query=5*36%5E%28n%2B3%29%2B42287&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35266	V0₁₃₂₈444B	31×36¹³³²+191963	http://factordb.com/index.php?id=1100000003807362474	http://factordb.com/cert.php?id=1100000003807362474	http://factordb.com/index.php?query=31*36%5E%28n%2B4%29%2B191963&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35267	SI0₁₇₁₂KH	1026×36¹⁷¹⁴+737	http://factordb.com/index.php?id=1100000003807362475	http://factordb.com/cert.php?id=1100000003807362475	http://factordb.com/index.php?query=1026*36%5E%28n%2B2%29%2B737&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35268	Z0₁₇₁₄387	35×36¹⁷¹⁷+4183	http://factordb.com/index.php?id=1100000003807362477	http://factordb.com/cert.php?id=1100000003807362477	http://factordb.com/index.php?query=35*36%5E%28n%2B3%29%2B4183&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35269	5₁₉₃₆3Z	(36¹⁹³⁸−295)/7	http://factordb.com/index.php?id=1100000003807362478	http://factordb.com/cert.php?id=1100000003807362478	http://factordb.com/index.php?query=%2836%5E%28n%2B2%29-295%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35270	40₂₄₇₈RV	4×36²⁴⁸⁰+1003	http://factordb.com/index.php?id=1100000002634138559	http://factordb.com/cert.php?id=1100000002634138559	http://factordb.com/index.php?query=4*36%5E%28n%2B2%29%2B1003&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35271	IS0₂₆₈₄A0H	676×36²⁶⁸⁷+12977	http://factordb.com/index.php?id=1100000003807362479	http://factordb.com/cert.php?id=1100000003807362479	http://factordb.com/index.php?query=676*36%5E%28n%2B3%29%2B12977&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35272	5Z₂₈₅₉95	6×36²⁸⁶¹−967	http://factordb.com/index.php?id=1100000003807362480	http://factordb.com/cert.php?id=1100000003807362480	http://factordb.com/index.php?query=6*36%5E%28n%2B2%29-967&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35273	Q₂₉₄₂2D	(26×36²⁹⁴⁴−30721)/35	http://factordb.com/index.php?id=1100000003807362481	http://factordb.com/cert.php?id=1100000003807362481	http://factordb.com/index.php?query=%2826*36%5E%28n%2B2%29-30721%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35274	D0₃₀₄₇6E01	13×36³⁰⁵¹+298081	http://factordb.com/index.php?id=1100000003807362482	http://factordb.com/cert.php?id=1100000003807362482	http://factordb.com/index.php?query=13*36%5E%28n%2B4%29%2B298081&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35275	CNS₃₄₂₄J	(2279×36³⁴²⁵−49)/5	http://factordb.com/index.php?id=1100000003807362483	http://factordb.com/cert.php?id=1100000003807362483	http://factordb.com/index.php?query=%282279*36%5E%28n%2B1%29-49%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35276	4₃₉₂₅V	(4×36³⁹²⁶+941)/35	http://factordb.com/index.php?id=1100000002332536659	http://factordb.com/cert.php?id=1100000002332536659	http://factordb.com/index.php?query=%284*36%5E%28n%2B1%29%2B941%29%2F35&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
36	35277	OZ₃₉₃₂AZ	25×36³⁹³⁴−901	http://factordb.com/index.php?id=1100000000840634476	http://factordb.com/cert.php?id=1100000000840634476	http://factordb.com/index.php?query=25*36%5E%28n%2B2%29-901&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35278	RY₄₅₆₂H	(979×36⁴⁵⁶³−629)/35	http://factordb.com/index.php?id=1100000003807362485	http://factordb.com/cert.php?id=1100000003807362485	http://factordb.com/index.php?query=%28979*36%5E%28n%2B1%29-629%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35279	T09₄₆₁₈1	(36549×36⁴⁶¹⁹−289)/35	http://factordb.com/index.php?id=1100000003807362486	http://factordb.com/cert.php?id=1100000003807362486	http://factordb.com/index.php?query=%2836549*36%5E%28n%2B1%29-289%29%2F35&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35280	FZ₅₇₇₇3P	16×36⁵⁷⁷⁹−1163	http://factordb.com/index.php?id=1100000003807362487	http://factordb.com/cert.php?id=1100000003807362487	http://factordb.com/index.php?query=16*36%5E%28n%2B2%29-1163&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35281	EO0₆₁₇₇V	528×36⁶¹⁷⁸+31	http://factordb.com/index.php?id=1100000003807362488	http://factordb.com/cert.php?id=1100000003807362488	http://factordb.com/index.php?query=528*36%5E%28n%2B1%29%2B31&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35282	VL0₇₂₅₈J	1137×36⁷²⁵⁹+19	http://factordb.com/index.php?id=1100000003807362489	http://factordb.com/cert.php?id=1100000003807362489	http://factordb.com/index.php?query=1137*36%5E%28n%2B1%29%2B19&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show
36	35283	J₁₀₁₁₇LJ	(19×36¹⁰¹¹⁹+2501)/35	http://factordb.com/index.php?id=1100000003807362491	http://factordb.com/cert.php?id=1100000003807362491	http://factordb.com/index.php?query=%2819*36%5E%28n%2B2%29%2B2501%29%2F35&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

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 (https://primes.utm.edu/prove/prove2_3.html, https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test, https://en.wikipedia.org/wiki/Strong_pseudoprime, https://primes.utm.edu/glossary/xpage/StrongPRP.html, https://www.rieselprime.de/ziki/Miller-Rabin_pseudoprimality_test, https://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html, https://mathworld.wolfram.com/StrongPseudoprime.html, http://www.numericana.com/answer/pseudo.htm#rabin, http://www.numericana.com/answer/pseudo.htm#strong, http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm, http://ntheory.org/data/spsps.txt, https://sites.google.com/view/strong-pseudoprime, https://sites.google.com/view/bases-strong-pseudoprime, https://oeis.org/A001262, https://oeis.org/A020229, https://oeis.org/A072276, https://oeis.org/A014233, https://oeis.org/A181782, https://oeis.org/A071294, https://oeis.org/A141768, https://oeis.org/A195328, https://oeis.org/A329759, https://oeis.org/A298756) to all prime bases p < 64 and has passed the Baillie–PSW primality test (https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test, https://mathworld.wolfram.com/Baillie-PSWPrimalityTest.html) and has trial factored (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, http://www.numericana.com/answer/factoring.htm#trial) 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, this candidate for minimal prime base 11 has 65263 decimal digits, while the top record ordinary prime (https://primes.utm.edu/glossary/xpage/OrdinaryPrime.html) (i.e. neither N−1 (https://primes.utm.edu/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1) nor N+1 (https://primes.utm.edu/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2) can be ≥ 1/4 factored (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm)) has 57125 decimal digits (the entry of this prime in top definitely primes is https://primes.utm.edu/primes/page.php?id=134371), see https://primes.utm.edu/top20/page.php?id=27, and 65263 > 57125) (you can click the "show" in the factordb page to see these primes (or probable primes) written in base 10 and base b as well as the length of these primes (or probable primes) 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 primes > 10²⁹⁹ (for the factorization of N−1 and N+1 for the primes < 10²⁹⁹, you can just enter "-1" or "+1" after the prime in the searching box, then click "Factorize!"))

b	number of minimal primes (or probable prime, which is a minimal prime assuming its primality) base b	base-b form of the top 10 known minimal prime (or probable prime, which is a minimal prime assuming its primality) base b (write "d_n" if there are 5 or more (n) consecutive same digits d)	length of the top 10 known minimal prime (or probable prime, which is a minimal prime assuming its primality) base b	algebraic ((a×bⁿ+c)/gcd(a+c,b−1)) form of the top 10 known minimal prime (or probable prime, which is a minimal prime assuming its primality) base b	factordb entry of this minimal prime (or probable prime, which is a minimal prime assuming its primality)	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	http://factordb.com/index.php?id=3	0	–
3	3	111 21 12	3 2 2	13 7 5	http://factordb.com/index.php?id=13 http://factordb.com/index.php?id=7 http://factordb.com/index.php?id=5	0	–
4	5	221 31 23 13 11	3 2 2 2 2	41 13 11 7 5	http://factordb.com/index.php?id=41 http://factordb.com/index.php?id=13 http://factordb.com/index.php?id=11 http://factordb.com/index.php?id=7 http://factordb.com/index.php?id=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	http://factordb.com/index.php?id=1100000000034686071 http://factordb.com/index.php?id=9391 http://factordb.com/index.php?id=3121 http://factordb.com/index.php?id=2341 http://factordb.com/index.php?id=2251 http://factordb.com/index.php?id=1951 http://factordb.com/index.php?id=1249 http://factordb.com/index.php?id=653 http://factordb.com/index.php?id=401 http://factordb.com/index.php?id=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	http://factordb.com/index.php?id=5209 http://factordb.com/index.php?id=1033 http://factordb.com/index.php?id=1009 http://factordb.com/index.php?id=31 http://factordb.com/index.php?id=29 http://factordb.com/index.php?id=23 http://factordb.com/index.php?id=19 http://factordb.com/index.php?id=17 http://factordb.com/index.php?id=13 http://factordb.com/index.php?id=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	http://factordb.com/index.php?id=116315256993601 http://factordb.com/index.php?id=207532837 http://factordb.com/index.php?id=2882377 http://factordb.com/index.php?id=134471 http://factordb.com/index.php?id=91631 http://factordb.com/index.php?id=62819 http://factordb.com/index.php?id=50459 http://factordb.com/index.php?id=28813 http://factordb.com/index.php?id=16871 http://factordb.com/index.php?id=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	http://factordb.com/index.php?id=1100000000416605822 http://factordb.com/index.php?id=25131694349141 http://factordb.com/index.php?id=549755813881 http://factordb.com/index.php?id=8589035809 http://factordb.com/index.php?id=127027489 http://factordb.com/index.php?id=76695841 http://factordb.com/index.php?id=11983381 http://factordb.com/index.php?id=1495381 http://factordb.com/index.php?id=1474901 http://factordb.com/index.php?id=1198399	0	–
9	151	30₁₁₅₈11 27₆₈₆07 76₃₂₉2 561₃₆ 10₂₅57 30₂₀51 8₁₉335 727₁₅07 51₁₃61 10₁₁507	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	http://factordb.com/index.php?id=1100000002376318423 http://factordb.com/index.php?id=1100000002495467486 http://factordb.com/index.php?id=1100000002359003642 http://factordb.com/index.php?id=1100000001554010824 http://factordb.com/index.php?id=1100000002512830927 http://factordb.com/index.php?id=1100000000032261811 http://factordb.com/index.php?id=1100000002495736583 http://factordb.com/index.php?id=1100000003446800389 http://factordb.com/index.php?id=1055192051985121 http://factordb.com/index.php?id=22876792455373	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	http://factordb.com/index.php?id=1100000000204142046 http://factordb.com/index.php?id=555555555551 http://factordb.com/index.php?id=80555551 http://factordb.com/index.php?id=66600049 http://factordb.com/index.php?id=66000049 http://factordb.com/index.php?id=60000049 http://factordb.com/index.php?id=22000001 http://factordb.com/index.php?id=5200007 http://factordb.com/index.php?id=946669 http://factordb.com/index.php?id=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	http://factordb.com/index.php?id=1100000003573679860 http://factordb.com/index.php?id=1100000002361376522 http://factordb.com/index.php?id=1100000002505568840 http://factordb.com/index.php?id=1100000003576826487 http://factordb.com/index.php?id=1100000003576826769 http://factordb.com/index.php?id=1100000002518512744 http://factordb.com/index.php?id=1100000002391585327 http://factordb.com/index.php?id=1100000002632393378 http://factordb.com/index.php?id=1100000002391531300 http://factordb.com/index.php?id=1100000003576826781	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	http://factordb.com/index.php?id=1100000002375054575 http://factordb.com/index.php?id=1100000002354113100 http://factordb.com/index.php?id=5159780039 http://factordb.com/index.php?id=388177921 http://factordb.com/index.php?id=32847239 http://factordb.com/index.php?id=32555521 http://factordb.com/index.php?id=2985817 http://factordb.com/index.php?id=2488397 http://factordb.com/index.php?id=1097113 http://factordb.com/index.php?id=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	http://factordb.com/index.php?id=1100000003943359311 http://factordb.com/index.php?id=1100000000490878060 http://factordb.com/index.php?id=1100000003590647776 http://factordb.com/index.php?id=1100000003590493750 http://factordb.com/index.php?id=1100000002616382906 http://factordb.com/index.php?id=1100000000765961441 http://factordb.com/index.php?id=1100000003590430825 http://factordb.com/index.php?id=1100000002632396910 http://factordb.com/index.php?id=1100000000765961452 http://factordb.com/index.php?id=1100000002320890755	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	http://factordb.com/index.php?id=1100000000884560233 http://factordb.com/index.php?id=1100000001540144903 http://factordb.com/index.php?id=1100000003575856650 http://factordb.com/index.php?id=1100000002321014379 http://factordb.com/index.php?id=1100000000823937973 http://factordb.com/index.php?id=1100000002631073246 http://factordb.com/index.php?id=1100000002384401372 http://factordb.com/index.php?id=1100000002631077787 http://factordb.com/index.php?id=1100000000840126683 http://factordb.com/index.php?id=1100000002321038522	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	http://factordb.com/index.php?id=1100000002454891840 http://factordb.com/index.php?id=1100000002454900849 http://factordb.com/index.php?id=1100000000823937997 http://factordb.com/index.php?id=1100000003588407143 http://factordb.com/index.php?id=1100000003588407386 http://factordb.com/index.php?id=1100000002632398579 http://factordb.com/index.php?id=1100000002321033312 http://factordb.com/index.php?id=1100000002391199877 http://factordb.com/index.php?id=1100000003588407806 http://factordb.com/index.php?id=1100000000851967288	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	http://factordb.com/index.php?id=1100000003851731988 http://factordb.com/index.php?id=1100000003615909841 http://factordb.com/index.php?id=1100000002383583629 http://factordb.com/index.php?id=1100000003589278511 http://factordb.com/index.php?id=1100000000993764322 http://factordb.com/index.php?id=1100000000633424191 http://factordb.com/index.php?id=1100000003588368750 http://factordb.com/index.php?id=1100000003588386735 http://factordb.com/index.php?id=1100000000633744824 http://factordb.com/index.php?id=1100000003588387610	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	http://factordb.com/index.php?id=1100000003993647842 http://factordb.com/index.php?id=1100000000765961389 http://factordb.com/index.php?id=1100000003883765450 http://factordb.com/index.php?id=1100000003848346668 http://factordb.com/index.php?id=1100000003838755382 http://factordb.com/index.php?id=1100000003815568647 http://factordb.com/index.php?id=1100000003802992457 http://factordb.com/index.php?id=1100000003782940761 http://factordb.com/index.php?id=1100000003782940760 http://factordb.com/index.php?id=1100000001100010542	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	http://factordb.com/index.php?id=1100000003590442437 http://factordb.com/index.php?id=1100000003590430490 http://factordb.com/index.php?id=1100000002355574745 http://factordb.com/index.php?id=1100000002632837015 http://factordb.com/index.php?id=1100000002321052894 http://factordb.com/index.php?id=1100000000819230161 http://factordb.com/index.php?id=1100000002631240657 http://factordb.com/index.php?id=1100000003590430474 http://factordb.com/index.php?id=1100000003590430470 http://factordb.com/index.php?id=1100000003590430465	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	http://factordb.com/index.php?id=1100000004163040839 http://factordb.com/index.php?id=1100000003998413751 http://factordb.com/index.php?id=1100000000808118332 http://factordb.com/index.php?id=1100000003949188041 http://factordb.com/index.php?id=1100000003949189035 http://factordb.com/index.php?id=1100000000808118328 http://factordb.com/index.php?id=1100000003968090004 http://factordb.com/index.php?id=1100000003949189435 http://factordb.com/index.php?id=1100000003968087931 http://factordb.com/index.php?id=1100000003949187975	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	http://factordb.com/index.php?id=1100000003590539457 http://factordb.com/index.php?id=1100000002325393915 http://factordb.com/index.php?id=1100000003590502412 http://factordb.com/index.php?id=1100000003590502490 http://factordb.com/index.php?id=1100000001559454258 http://factordb.com/index.php?id=1100000003590502516 http://factordb.com/index.php?id=1100000003590502531 http://factordb.com/index.php?id=1100000000840126753 http://factordb.com/index.php?id=1100000002633348702 http://factordb.com/index.php?id=1100000003590502563	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	http://factordb.com/index.php?id=1100000000808118331 http://factordb.com/index.php?id=1100000003996110311 http://factordb.com/index.php?id=1100000003996110479 http://factordb.com/index.php?id=1100000003996110718 http://factordb.com/index.php?id=1100000003996110944 http://factordb.com/index.php?id=1100000003996111130 http://factordb.com/index.php?id=1100000003996112263 http://factordb.com/index.php?id=1100000003996112521 http://factordb.com/index.php?id=1100000003996112710 http://factordb.com/index.php?id=1100000000777265872	12	100000
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	http://factordb.com/index.php?id=1100000003594696838 http://factordb.com/index.php?id=1100000003591359839 http://factordb.com/index.php?id=1100000003591360774 http://factordb.com/index.php?id=1100000003591361817 http://factordb.com/index.php?id=1100000003591362567 http://factordb.com/index.php?id=1100000000632724415 http://factordb.com/index.php?id=1100000000632724334 http://factordb.com/index.php?id=1100000000632703239 http://factordb.com/index.php?id=1100000003591364730 http://factordb.com/index.php?id=1100000003591365331	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	http://factordb.com/index.php?id=1100000004149262767 http://factordb.com/index.php?id=1100000004149263445 http://factordb.com/index.php?id=1100000004149263708 http://factordb.com/index.php?id=1100000004149263936 http://factordb.com/index.php?id=1100000004149264140 http://factordb.com/index.php?id=1100000004149264399 http://factordb.com/index.php?id=1100000004149264460 http://factordb.com/index.php?id=1100000004149265024 http://factordb.com/index.php?id=1100000004149265030 http://factordb.com/index.php?id=1100000004149265063	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	http://factordb.com/index.php?id=1100000003593391606 http://factordb.com/index.php?id=1100000003593275880 http://factordb.com/index.php?id=1100000003593269654 http://factordb.com/index.php?id=1100000003593269876 http://factordb.com/index.php?id=1100000003593270089 http://factordb.com/index.php?id=1100000002633359842 http://factordb.com/index.php?id=1100000002355610241 http://factordb.com/index.php?id=1100000002326181235 http://factordb.com/index.php?id=1100000003593270725 http://factordb.com/index.php?id=1100000002633360037	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	http://factordb.com/index.php?id=1100000004141587423 http://factordb.com/index.php?id=1100000004141587957 http://factordb.com/index.php?id=1100000000819229846 http://factordb.com/index.php?id=1100000004141588107 http://factordb.com/index.php?id=1100000004141588234 http://factordb.com/index.php?id=1100000004141590563 http://factordb.com/index.php?id=1100000004141590642 http://factordb.com/index.php?id=1100000004141590737 http://factordb.com/index.php?id=1100000004141590789 http://factordb.com/index.php?id=1100000004141590830	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	http://factordb.com/index.php?id=1100000003968169875 http://factordb.com/index.php?id=1100000003968156595 http://factordb.com/index.php?id=1100000003892628745 http://factordb.com/index.php?id=1100000003892628658 http://factordb.com/index.php?id=1100000003892628605 http://factordb.com/index.php?id=1100000003850151202 http://factordb.com/index.php?id=1100000003850155316 http://factordb.com/index.php?id=1100000000758011195 http://factordb.com/index.php?id=1100000002634136508 http://factordb.com/index.php?id=1100000002328056865	4	100000
27	102850~102896	O44L₆₆₀₁₆7 NJ0₆₄₃₆₉H ME₄₉₆₄₀9G PH0₄₇₈₉₀1 QF₄₇₁₆₅AF5 J0₄₀₇₉₁PD 510₃₉₁₆₄I07 NGN0₃₆₃₂₉N 153F₃₅₈₃₁5 L₃₅₅₆₄GLG	66020 64372 49643 47893 47169 40794 39169 36333 35835 35567	(457829×27⁶⁶⁰¹⁷−385)/26 640×27⁶⁴³⁷⁰+17 (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	http://factordb.com/index.php?id=1100000004374140861 http://factordb.com/index.php?id=1100000004374138999 http://factordb.com/index.php?id=1100000000819229859 http://factordb.com/index.php?id=1100000004102754118 http://factordb.com/index.php?id=1100000004102755880 http://factordb.com/index.php?id=1100000004102758254 http://factordb.com/index.php?id=1100000004102875088 http://factordb.com/index.php?id=1100000004103372866 http://factordb.com/index.php?id=1100000004103376142 http://factordb.com/index.php?id=1100000000819229833	46	75000
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	http://factordb.com/index.php?id=1100000000808118231 http://factordb.com/index.php?id=1100000003880455200 http://factordb.com/index.php?id=1100000003879667576 http://factordb.com/index.php?id=1100000003850151420 http://factordb.com/index.php?id=1100000000840839934 http://factordb.com/index.php?id=1100000003850161974 http://factordb.com/index.php?id=1100000003850161973 http://factordb.com/index.php?id=1100000000840839947 http://factordb.com/index.php?id=1100000003850161972 http://factordb.com/index.php?id=1100000003850161965	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	http://factordb.com/index.php?id=1100000000800812865 http://factordb.com/index.php?id=1100000003593967511 http://factordb.com/index.php?id=1100000002327649423 http://factordb.com/index.php?id=1100000000785448736 http://factordb.com/index.php?id=1100000003593407988 http://factordb.com/index.php?id=1100000003593408295 http://factordb.com/index.php?id=1100000002327651073 http://factordb.com/index.php?id=1100000002356282476 http://factordb.com/index.php?id=1100000003593409109 http://factordb.com/index.php?id=1100000003593409165	0	–
31
32	168868~169008	H₄₈₈₈₂4H L₄₆₉₄₂S6L L₄₅₉₄₂A99 S60₄₅₄₀₃L 9A₄₂₂₂₅L 70₄₁₇₈₃UT 6₄₁₄₇₆AOF K₄₀₁₁₇U9 PI₃₉₂₅₇3 D₃₈₇₉₇OD	48884 46945 45945 45406 42227 41786 41479 40119 39259 38799	(17×32⁴⁸⁸⁸⁴−12913)/31 (21×32⁴⁶⁹⁴⁵+207307)/31 (21×32⁴⁵⁹⁴⁵−361481)/31 902×32⁴⁵⁴⁰⁴+21 (289×32⁴²²²⁶+331)/31 7×32⁴¹⁷⁸⁵+989 (6×32⁴¹⁴⁷⁹+145105)/31 (20×32⁴⁰¹¹⁹+9559)/31 (793×32³⁹²⁵⁸−483)/31 (13×32³⁸⁷⁹⁹+10899)/31	http://factordb.com/index.php?id=1100000004230594707 http://factordb.com/index.php?id=1100000004230594958 http://factordb.com/index.php?id=1100000004230596384 http://factordb.com/index.php?id=1100000004230596933 http://factordb.com/index.php?id=1100000004230597870 http://factordb.com/index.php?id=1100000004230601701 http://factordb.com/index.php?id=1100000004230609228 http://factordb.com/index.php?id=1100000004230610293 http://factordb.com/index.php?id=1100000004230611223 http://factordb.com/index.php?id=1100000004230611188	140	50000
33	279994~280093	FL₄₈₄₀₇7 FUD₄₇₅₄₆ 70J0₄₅₅₉₀P L7₄₃₀₅₂IS U0₄₁₆₀₄BMW 50V₄₀₉₆₂ B0₄₀₂₅₅7N H7₃₉₃₀₃BS F0R0₃₈₅₅₀T WF₃₈₂₅₁W	48409 47548 45594 43055 41608 40964 40258 39306 38554 38253	(501×33⁴⁸⁴⁰⁸−469)/32 (16813×33⁴⁷⁵⁴⁶−13)/32 7642×33⁴⁵⁵⁹¹+25 (679×33⁴³⁰⁵⁴+12281)/32 30×33⁴¹⁶⁰⁷+12737 (5311×33⁴⁰⁹⁶²−31)/32 11×33⁴⁰²⁵⁷+254 (551×33³⁹³⁰⁵+4889)/32 16362×33³⁸⁵⁵¹+29 (1039×33³⁸²⁵²+529)/32	http://factordb.com/index.php?id=1100000004378726800 http://factordb.com/index.php?id=1100000004378727308 http://factordb.com/index.php?id=1100000004378727577 http://factordb.com/index.php?id=1100000004378727887 http://factordb.com/index.php?id=1100000004378728175 http://factordb.com/index.php?id=1100000004378738121 http://factordb.com/index.php?id=1100000004378738229 http://factordb.com/index.php?id=1100000004378738467 http://factordb.com/index.php?id=1100000004378738500 http://factordb.com/index.php?id=1100000004378738574	99	50000
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	http://factordb.com/index.php?id=1100000004125629992 http://factordb.com/index.php?id=1100000004125644307 http://factordb.com/index.php?id=1100000004125646708 http://factordb.com/index.php?id=1100000004125649832 http://factordb.com/index.php?id=1100000004125652107 http://factordb.com/index.php?id=1100000004125653362 http://factordb.com/index.php?id=1100000004125708803 http://factordb.com/index.php?id=1100000004125741107 http://factordb.com/index.php?id=1100000004125751526 http://factordb.com/index.php?id=1100000004125755574	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	http://factordb.com/index.php?id=1100000002394962083 http://factordb.com/index.php?id=1100000004020085177 http://factordb.com/index.php?id=1100000003896952461 http://factordb.com/index.php?id=1100000003807362491 http://factordb.com/index.php?id=1100000003807362489 http://factordb.com/index.php?id=1100000003807362488 http://factordb.com/index.php?id=1100000003807362487 http://factordb.com/index.php?id=1100000003807362486 http://factordb.com/index.php?id=1100000003807362485 http://factordb.com/index.php?id=1100000000840634476	4	100000

Links for top (probable) primes: (also pages for the largest known prime: https://en.wikipedia.org/wiki/Largest_known_prime_number, https://en.wikipedia.org/wiki/List_of_largest_known_primes_and_probable_primes, http://www.numericana.com/answer/primes.htm#history, and related pages: https://en.wikipedia.org/wiki/Megaprime, https://primes.utm.edu/glossary/xpage/TitanicPrime.html, https://primes.utm.edu/glossary/xpage/GiganticPrime.html, https://primes.utm.edu/glossary/xpage/Megaprime.html, https://www.rieselprime.de/ziki/Titanic_prime, https://www.rieselprime.de/ziki/Gigantic_prime, https://www.rieselprime.de/ziki/Megaprime, https://www.rieselprime.de/ziki/Gigaprime, https://mathworld.wolfram.com/TitanicPrime.html, https://mathworld.wolfram.com/GiganticPrime.html)

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, html version)
https://primes.utm.edu/primes/search.php?OnList=all&Number=1000000 (all numbers in the list of top definitely primes, text version)
https://primes.utm.edu/primes/ (index page of top definitely primes)
https://primes.utm.edu/primes/download.php (download page of top definitely primes)
https://primes.utm.edu/primes/status.php?hours=72 (recently 3 days found top definitely primes)
https://primes.utm.edu/primes/status.php?hours=1000 (recently found top definitely primes)
https://primes.utm.edu/primes/status.php?hours=0 (top definitely primes which are in process or awaiting verification)
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)

and the search result page for the (probable) primes of special forms: (note: a large prime of the form (a×bⁿ+c)/d with small a, b, c, d and large n can be easily proven prime if and only if c = ±1 and d = 1)

Definitely primes (i.e. c = ±1 and d = 1):

Probable primes (i.e. c ≠ ±1 and/or d ≠ 1):

Home page of Proth Primes Search (search of primes of the form k×2ⁿ+1 with odd k): http://www.prothsearch.com/

Home page of Riesel Prime Search (search of primes of the form k×2ⁿ−1 with odd k): https://web.archive.org/web/20210817181915/http://www.15k.org/

References of minimal primes (https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics), https://primes.utm.edu/glossary/xpage/MinimalPrime.html) (the original definition, i.e. prime > base (b) is not required):

http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_11.pdf) (base 10)
https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_17.pdf) (bases 2 to 30)
https://cs.uwaterloo.ca/~shallit/Papers/br10.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_18.pdf) (bases 2 to 30)
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) (bases 2 to 30)
https://doi.org/10.1080/10586458.2015.1064048 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_20.pdf) (bases 2 to 30)
https://scholar.colorado.edu/downloads/hh63sw661 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_16.pdf) (bases 2 to 10) (warning: the datas for bases 8 and 10 have errors, the data for base 8 misses the prime 6101 and the data for base 10 misses the primes 9001 and 9049 and instead wrongly includes the primes 90001, 90469, and 9000049, and the correct values of S_m for bases 8 and 10 are 15 and 26 (instead of 14 and 27), respectively)
https://github.com/curtisbright/mepn-data (bases 2 to 30)
https://github.com/curtisbright/mepn (bases 2 to 30)
https://github.com/RaymondDevillers/primes (bases 28 to 50)
http://recursed.blogspot.com/2006/12/prime-game.html (base 10)
https://inzitan.blogspot.com/2007/07/prime-game.html (in Spain) (base 10)
http://www.pourlascience.fr/ewb_pages/a/article-nombres-premiers-inevitables-et-pyramidaux-24978.php (in French) (base 10)
http://villemin.gerard.free.fr/aNombre/TYPMULTI/PremInev.htm (base 10)
https://schoolbag.info/mathematics/numbers/66.html (base 10)
https://www.microsiervos.com/archivo/ciencia/2-3-5-7-11.html (in Spain) (base 10)
https://math.stackexchange.com/questions/58292/how-to-check-if-an-integer-has-a-prime-number-in-it (base 10)
https://www.metafilter.com/62794/3-is-an-odd-prime-5-is-an-odd-prime-7-is-an-odd-prime-9-is-a-very-odd-prime (base 10)
https://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1165031124 (base 10)
https://www.cristal.univ-lille.fr/profil/jdelahay/pls:2002:094.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_100.pdf) (bases 2 to 10) (warning: the data for base 8 has error, the data misses the primes 444641 and 444444441)
https://logs.esolangs.org/freenode-esoteric/2011-02-04.html (bases 2 to 10) (warning: the data for base 8 has error, the data misses the prime 111 and instead wrongly includes the primes 1101, 101111, 600111, 1000011, 1000111, 4411111, 64111111, 601111111, 41111111111111111, and possibly 6111111111111111111111 if the author of this article continues to search)
http://www.bitman.name/math/article/730 (in Italian) (bases 2 to 20)
http://www.bitman.name/math/table/497 (in Italian) (bases 2 to 16)
http://www.bitman.name/math/table/498 (in Italian) (base 17)
http://www.bitman.name/math/table/499 (in Italian) (base 18)
http://www.bitman.name/math/table/500 (in Italian) (base 19)
http://www.bitman.name/math/table/501 (in Italian) (base 20)
https://www.primepuzzles.net/puzzles/puzz_178.htm (base 10)
https://oeis.org/A071062 (base 10)

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, https://www.numbersaplenty.com/set/truncatable_prime/), 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://rosettacode.org/wiki/Find_largest_left_truncatable_prime_in_a_given_base (bases 3 to 17)
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)
http://www.primerecords.dk/left-truncatable.htm (base 10)
http://rosettacode.org/wiki/Truncatable_primes (base 10)
https://www.primepuzzles.net/puzzles/puzz_002.htm (base 10)
https://web.archive.org/web/20041204160717/http://www.wschnei.de/digit-related-numbers/circular-primes.html (base 10)
http://www.bitman.name/math/article/1155 (in Italian) (bases 2 to 20)
http://www.bitman.name/math/table/524 (in Italian) (bases 2 to 20)
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, https://www.numbersaplenty.com/set/truncatable_prime/), 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)
http://rosettacode.org/wiki/Truncatable_primes (base 10)
https://www.primepuzzles.net/puzzles/puzz_002.htm (base 10)
https://web.archive.org/web/20041204160717/http://www.wschnei.de/digit-related-numbers/circular-primes.html (base 10)
http://www.bitman.name/math/article/1155 (in Italian) (bases 2 to 20)
http://www.bitman.name/math/table/525 (in Italian) (bases 2 to 20)
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:

https://www.primepuzzles.net/puzzles/puzz_178.htm
https://github.com/curtisbright/mepn-data/blob/master/data/primes1mod4/minimal.10.txt
https://www.primepuzzles.net/puzzles/Minimal%20Primes%204k+1,%204k-1,%20pu%20178.doc (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/doc_1.doc) (warning: the data has many errors, the data wrongly including many primes which are not minimal primes, including the largest "minimal 4k+1 prime" in the list: 9₆₃₀493, this prime is not a minimal 4k+1 prime since 9949 is also a prime == 1 mod 4, and 9949 is a subsequence of 9₆₃₀493, there are 146 (instead of 173) minimal 4k+1 primes and 113 (instead of 138) minimal 4k−1 primes, and the largest minimal 4k+1 prime is 8₇₇33 = (8*10⁷⁹−503)/9 instead of 9₆₃₀493 = 10⁶³³−507)
https://oeis.org/A111055

Primes == 3 mod 4:

https://www.primepuzzles.net/puzzles/puzz_178.htm
https://github.com/curtisbright/mepn-data/blob/master/data/primes3mod4/minimal.10.txt
https://www.primepuzzles.net/puzzles/Minimal%20Primes%204k+1,%204k-1,%20pu%20178.doc (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/doc_1.doc) (warning: the data has many errors, the data wrongly including many primes which are not minimal primes, including the largest "minimal 4k+1 prime" in the list: 9₆₃₀493, this prime is not a minimal 4k+1 prime since 9949 is also a prime == 1 mod 4, and 9949 is a subsequence of 9₆₃₀493, there are 146 (instead of 173) minimal 4k+1 primes and 113 (instead of 138) minimal 4k−1 primes, and the largest minimal 4k+1 prime is 8₇₇33 = (8*10⁷⁹−503)/9 instead of 9₆₃₀493 = 10⁶³³−507)
https://oeis.org/A111056 (warning: the b-file does not include the prime 2₁₉₁₅₁99)

Palindromic primes:

https://www.primepuzzles.net/puzzles/puzz_178.htm
https://oeis.org/A114835 (warning: the b-file does not include the probable prime 994₃₄₀₁₉99)

Composites:

Squares:

Powers of 2:

Multiples of 3:

https://oeis.org/A071073

Multiples of 4:

https://oeis.org/A071072

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?")
http://www.math.com/students/calculators/source/prime-number.htm
https://www.calculatorsoup.com/calculators/math/prime-number-calculator.php
https://onlinemathtools.com/test-prime-number
https://www.bigprimes.net/primalitytest
https://www.archimedes-lab.org/primOmatic.html
http://www.sonic.net/~undoc/java/PrimeCalc.html
http://www.primzahlen.de/primzahltests/testverfahren.htm (in German)
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, in fact, links 2, 6, 10, 11, 12, 13 can also calculate the next prime above N, besides, links 2, 6 can also calculate the previous prime below 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
https://www.archimedes-lab.org/primOmatic.html
http://www.proftnj.com/calcprem.htm (in French) (use the box "Factoriser un nombre" and click "Factoriser")
http://factordb.com/ (online factor database)
http://myfactorcollection.mooo.com:8090/dbio.html (online factor database for numbers of the form bⁿ±1)
https://web.archive.org/web/20120722020628/http://homes.cerias.purdue.edu/~ssw/cun/prime.php (online factor database for numbers of the form bⁿ±1 for 2 ≤ b ≤ 12)
https://web.archive.org/web/20120330032919/http://homes.cerias.purdue.edu/~ssw/cun/clientold.html (online factor database for numbers of the form bⁿ±1 for 2 ≤ b ≤ 12)

Base converters:

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: (see https://primes.utm.edu/notes/faq/LongestList.html and https://primes.utm.edu/notes/faq/x_digit_primes.html, although we can quickly determine whether a number < 10³⁰⁰ is prime or not, but there is no list of all primes < 10³⁰⁰, since the space which we need is larger than the number of elementary particles (https://en.wikipedia.org/wiki/Elementary_particle) (i.e. quarks (https://en.wikipedia.org/wiki/Quark) and electrons (https://en.wikipedia.org/wiki/Electron)) in the universe (https://en.wikipedia.org/wiki/Universe))

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/b000040.txt
https://oeis.org/A000040/a000040.txt
https://oeis.org/A000040/b000040_1.txt
https://oeis.org/A000040/a000040_1B.7z
http://www.prime-numbers.org/
http://prime-numbers.org/sample.zip
https://metanumbers.com/prime-numbers
https://www.calculatorsoup.com/calculators/math/prime-numbers.php
https://www2.cs.arizona.edu/icon/oddsends/primes.htm
https://www.numbersaplenty.com/set/prime_number/more.php
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 (warning: this site incorrectly includes 1 as a prime and misses the primes 3229 and 3329)
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
http://www.primzahlen.de/primzahltests/2-100003.htm (in German)
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))
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
https://sites.google.com/view/prime-factorization-of-integer
https://web.archive.org/web/20060210182347/http://bearnol.is-a-geek.com/Panfur%20Project/ (warning: this site does not factor the composite numbers 15, 51, 85, 91, 255, 435, 451, 561, 595, 679, 703, 771, 1105, 1261, 1285, 1351, 1387, ...)
http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/
http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/?infinity
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)
http://factordb.com/index.php?query=n&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show (from factordb)

Lists of small integers in various bases:

https://en.wikipedia.org/wiki/Table_of_bases

Also, programs related to this research: (some of these programs can also be downloaded in http://www.fermatsearch.org/download.php or https://www.mersenne.org/download/freeware.php or https://download.mersenne.ca/) (some of these programs need to use GMP (https://gmplib.org/))

Primality (or probable primality) testing (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://primes.utm.edu/prove/index.html) programs (https://www.rieselprime.de/ziki/Primality_testing_program):

Sieving (https://www.rieselprime.de/ziki/Sieving, https://www.rieselprime.de/ziki/Sieving_a_range_of_sequences, https://mathworld.wolfram.com/Sieve.html, http://www.rechenkraft.net/yoyo/y_status_sieve.php) programs (https://www.rieselprime.de/ziki/Sieving_program):

Integer factoring (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) programs (https://www.rieselprime.de/ziki/Factoring_program):

For 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

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