(In progess to add bases b = 17 and b = 21)

Unfortunately, for every base b, there are infinitely many strong pseudoprimes, see https://www.ams.org/journals/mcom/1980-35-151/S0025-5718-1980-0572872-7/S0025-5718-1980-0572872-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_45.pdf), even more worse, for any given finite set of bases, there are infinitely strong pseudoprimes to these bases simultaneously, i.e. no finite set of bases is sufficient for all composite numbers, Alford, Granville, and Pomerance have shown that there exist infinitely many composite numbers n whose smallest compositeness witness is at least (ln(n))^{1/(3×ln(ln(ln(n))))}, see https://math.dartmouth.edu/~carlp/PDF/reliable.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_121.pdf), however, there are no "strong Carmichael numbers" (i.e. numbers that are strong pseudoprimes to all bases coprime to them), and given a random base, the probability that a number is a strong pseudoprime to that base is less than 1/4, and if the generalized Riemann hypothesis (https://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis, https://mathworld.wolfram.com/GeneralizedRiemannHypothesis.html, https://t5k.org/notes/rh.html) is true, then every composite number n has smallest compositeness witness less than 2×(ln(n))², also, when the number n to be tested is small, trying all bases b < 2×(ln(n))² is not necessary, as much smaller sets of potential witnesses are known to suffice. For example: (also see https://oeis.org/A014233 for the smallest composite number which is strong pseudoprime to all of the first n prime bases)

test bases b	the smallest composite number which is strong pseudoprime to all these bases b	prime factorization
2	2047	23 × 89
3	121	11²
4	341	11 × 31
5	781	11 × 71
6	217	7 × 31
7	25	5²
8	9	3²
9	91	7 × 13
10	9	3²
11	133	7 × 19
12	91	7 × 13
13	85	5 × 17
14	15	3 × 5
15	1687	7 × 241
16	15	3 × 5
17	9	3²
18	25	5²
19	9	3²
20	21	3 × 7
21	221	13 × 17
22	21	3 × 7
23	169	13²
24	25	5²
25	217	7 × 31
26	9	3²
27	121	11²
28	9	3²
29	15	3 × 5
30	49	7²
31	15	3 × 5
32	25	5²
33	545	5 × 109
34	33	3 × 11
35	9	3²
36	35	5 × 7
95	1891	31 × 61
240	1991	11 × 181
385	1891	31 × 61
777	1541	23 × 67
933	1387	19 × 73
1320	4097	17 × 241
2, 3	1373653	829 × 1657
31, 73	9080191	2131 × 4261
2, 3, 5	25326001	2251 × 11251
350, 3958281543	170584961	7541 × 22621
2, 3, 5, 7	3215031751	151 × 751 × 28351
2, 7, 61	4759123141	48781 × 97561
2, 379215, 457083754	75792980677	137653 × 550609
2, 13, 23, 1662803	1122004669633	611557 × 1834669
2, 3, 5, 7, 11	2152302898747	6763 × 10627 × 29947
2, 3, 5, 7, 11, 13	3474749660383	1303 × 16927 × 157543
2, 1215, 34862, 574237825	21652684502221	3290341 × 6580681
2, 3, 5, 7, 11, 13, 17	341550071728321	10670053 × 32010157
2, 3, 5, 7, 11, 13, 17, 19, 23	3825123056546413051	149491 × 747451 × 34233211
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37	318665857834031151167461	399165290221 × 798330580441
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41	3317044064679887385961981	1287836182261 × 2575672364521
all primes p ≤ n	> e^{(n/2)^1/2} (under the generalized Riemann hypothesis)	–

These numbers are semiprimes (https://en.wikipedia.org/wiki/Semiprime, https://t5k.org/glossary/xpage/Semiprime.html, https://mathworld.wolfram.com/Semiprime.html, https://www.numbersaplenty.com/set/semiprime/, https://oeis.org/A001358) of the form (m+1)×(2×m+1) or (m+1)×(3×m+1), (m+1)×(2×m+1) is Fermat pseudoprime to base b (assuming b is coprime to the number) if and only if b is quadratic residue (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod the prime 2×m+1, thus 1/2 of the bases coprime to the number; (m+1)×(3×m+1) is Fermat pseudoprime to base b (assuming b is coprime to the number) if and only if b is cubic residue (https://en.wikipedia.org/wiki/Cubic_residue, https://mathworld.wolfram.com/CubicResidue.html) mod the prime 3×m+1, thus 1/3 of the bases coprime to the number. https://oeis.org/A129521 is a subsequence of https://oeis.org/A191311 (i.e. all numbers of the form p×(2×p−1) with p and 2×p−1 both primes are half-Carmichael numbers (i.e. composite numbers c such that b^c−1 == 1 mod c for half of the bases b coprime to c, the Carmichael numbers are the composite numbers c such that b^c−1 == 1 mod c for all bases b coprime to c)) (and the numbers in https://oeis.org/A191311 but not in https://oeis.org/A129521 are the numbers in https://oeis.org/A191592, the smallest such number (except the trivial number 4, which is the only one such even number, the only one such non-squarefree number, and the only one such semiprime, note that all Carmichael numbers are odd, squarefree, and have at least 3 prime factors) is 11305 = 5 × 7 × 17 × 19). e.g. 15₁₅ (base 16) is Fermat pseudoprime to base b (assuming b is coprime to the number) if and only if b is cubic residue (https://en.wikipedia.org/wiki/Cubic_residue, https://mathworld.wolfram.com/CubicResidue.html) mod the prime 2³¹−1, thus 1/3 of the bases coprime to the number; 15₃₀ (base 16) is Fermat pseudoprime to base b (assuming b is coprime to the number) if and only if b is cubic residue (https://en.wikipedia.org/wiki/Cubic_residue, https://mathworld.wolfram.com/CubicResidue.html) mod the prime 2⁶¹−1, thus 1/3 of the bases coprime to the number; 15₆₃ (base 16) is Fermat pseudoprime to base b (assuming b is coprime to the number) if and only if b is cubic residue (https://en.wikipedia.org/wiki/Cubic_residue, https://mathworld.wolfram.com/CubicResidue.html) mod the prime 2¹²⁷−1, thus 1/3 of the bases coprime to the number; 85₆ (base 16) is Fermat pseudoprime to base b (assuming b is coprime to the number) if and only if b is cubic residue (https://en.wikipedia.org/wiki/Cubic_residue, https://mathworld.wolfram.com/CubicResidue.html) mod the prime 5×2¹²−1, thus 1/3 of the bases coprime to the number; 85₃₆ (base 16) is Fermat pseudoprime to base b (assuming b is coprime to the number) if and only if b is cubic residue (https://en.wikipedia.org/wiki/Cubic_residue, https://mathworld.wolfram.com/CubicResidue.html) mod the prime 5×2⁷²−1, thus 1/3 of the bases coprime to the number; 1₁₃ (base 9) is Fermat pseudoprime to base b (assuming b is coprime to the number) if and only if b is quadratic residue (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod the prime (3¹³−1)/2, thus 1/2 of the bases coprime to the number.

number	factordb entry	bases 2 ≤ b ≤ 64 such that this number is Fermat pseudoprime (called "Fermat liars")	count
15₁₅ in base b = 16 (see https://web.archive.org/web/19991117032157/http://ourworld.compuserve.com/homepages/hlifchitz/MersFermus.htm (which is an erroneous version of http://www.primenumbers.net/Henri/us/MersFermus.htm))	http://factordb.com/index.php?id=1000000000043569055	2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 29, 32, 36, 41, 48, 54, 58, 59, 64	20 (31.75%)
15₃₀ in base b = 16 (see https://web.archive.org/web/19991117032157/http://ourworld.compuserve.com/homepages/hlifchitz/MersFermus.htm (which is an erroneous version of http://www.primenumbers.net/Henri/us/MersFermus.htm))	http://factordb.com/index.php?id=1000000000043569085	2, 3, 4, 6, 8, 9, 11, 12, 13, 16, 18, 19, 22, 24, 26, 27, 31, 32, 33, 35, 36, 38, 39, 44, 48, 52, 53, 54, 57, 61, 62, 64	32 (50.79%)
15₆₃ in base b = 16 (see https://web.archive.org/web/19991117032157/http://ourworld.compuserve.com/homepages/hlifchitz/MersFermus.htm (which is an erroneous version of http://www.primenumbers.net/Henri/us/MersFermus.htm) and 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))	http://factordb.com/index.php?id=1000000000043569151	2, 3, 4, 6, 8, 9, 11, 12, 13, 16, 17, 18, 19, 22, 24, 26, 27, 29, 31, 32, 33, 34, 36, 37, 38, 39, 41, 44, 47, 48, 51, 52, 54, 57, 58, 59, 61, 62, 64	39 (61.90%)
85₆ in base b = 16	http://factordb.com/index.php?id=139810133	5, 7, 8, 12, 13, 17, 18, 25, 27, 35, 38, 40, 44, 46, 49, 56, 57, 59, 60, 64	20 (31.75%)
85₃₆ in base b = 16	http://factordb.com/index.php?id=1100000000348829387	3, 5, 8, 9, 13, 15, 17, 22, 24, 25, 27, 28, 29, 39, 40, 41, 45, 46, 47, 51, 53, 62, 64	23 (36.51%)
1₁₃ in base b = 9 (see https://web.archive.org/web/19991117032157/http://ourworld.compuserve.com/homepages/hlifchitz/MersFermus.htm (which is an erroneous version of http://www.primenumbers.net/Henri/us/MersFermus.htm))	http://factordb.com/index.php?id=317733228541	2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 32, 35, 36, 37, 38, 39, 40, 41, 42, 45, 46, 48, 49, 50, 52, 54, 56, 57, 58, 59, 60, 61, 63, 64	50 (79.37%)
58₉ in base b = 25	http://factordb.com/index.php?id=20345052083333	2, 4, 8, 13, 16, 26, 27, 31, 32, 43, 45, 47, 52, 54, 62, 63, 64	17 (26.98%)
I0901 in base b = 26 (see https://github.com/curtisbright/mepn-data/commit/7565d197d7b438b437871bf71614a6f8914397f7)	http://factordb.com/index.php?id=8231653	2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 18, 21, 22, 23, 24, 25, 26, 27, 28, 32, 33, 36, 39, 41, 42, 43, 44, 46, 48, 49, 50, 52, 53, 54, 56, 59, 63, 64	40 (63.49%)
28462346×3⁷+1 (see https://www.mersenneforum.org/showpost.php?p=137730&postcount=1)	http://factordb.com/index.php?id=62247150703	3, 4, 7, 9, 10, 11, 12, 13, 16, 17, 19, 21, 25, 27, 28, 29, 30, 33, 36, 39, 40, 41, 44, 46, 47, 48, 49, 51, 52, 53, 57, 59, 61, 62, 63, 64	36 (57.14%)

This list is the time complexity (https://en.wikipedia.org/wiki/Time_complexity) for the primality testing (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/prove3.html, https://t5k.org/prove/prove4.html) methods and the 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) methods:

method (primality testing (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/prove3.html, https://t5k.org/prove/prove4.html) or 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))	time complexity (https://en.wikipedia.org/wiki/Time_complexity) (where n is the number to be primality tested (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html, https://t5k.org/prove/prove3.html, https://t5k.org/prove/prove4.html) or integer 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), p is the prime factor to be found, O is the big O notation (https://en.wikipedia.org/wiki/Big_O_notation, https://t5k.org/glossary/xpage/BigOh.html, https://mathworld.wolfram.com/Big-ONotation.html), o is the little o notation (https://en.wikipedia.org/wiki/Big_O_notation#Little-o_notation, https://t5k.org/glossary/xpage/LittleOh.html, https://mathworld.wolfram.com/Little-ONotation.html), L is the L-notation (https://en.wikipedia.org/wiki/L-notation), 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), log is the natural logarithm (https://en.wikipedia.org/wiki/Natural_logarithm, https://t5k.org/glossary/xpage/Log.html, https://mathworld.wolfram.com/NaturalLogarithm.html))	usage for numbers	record number	factordb entry of the record number	number of decimal digits of the record number	reference of the record number
Fermat probable primality testing (https://t5k.org/prove/prove2_2.html, https://en.wikipedia.org/wiki/Fermat_primality_test, https://en.wikipedia.org/wiki/Fermat_pseudoprime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Fermat_pseudoprimality_test, https://mathworld.wolfram.com/FermatPseudoprime.html, https://www.numbersaplenty.com/set/Poulet_number/, http://www.numericana.com/answer/pseudo.htm#pseudoprime, http://ntheory.org/data/psps.txt, https://sites.google.com/view/fermat-pseudoprime, https://sites.google.com/view/bases-fermat-pseudoprime, https://oeis.org/A001567, https://oeis.org/A005935, https://oeis.org/A005936, https://oeis.org/A005938, https://oeis.org/A052155, https://oeis.org/A083737, https://oeis.org/A083739, https://oeis.org/A083876, https://oeis.org/A271221, https://oeis.org/A348258, https://oeis.org/A181780, https://oeis.org/A063994, https://oeis.org/A194946, https://oeis.org/A195327, https://oeis.org/A002997, https://oeis.org/A191311, https://oeis.org/A090086, https://oeis.org/A007535)	O(log(n))	any number, but only probabilistic (https://en.wikipedia.org/wiki/Probabilistic_algorithm) primality testing	(10^8177207−1)/9 (the largest known unproven probable prime (https://en.wikipedia.org/wiki/Probable_prime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Probable_prime, https://mathworld.wolfram.com/ProbablePrime.html, http://www.primenumbers.net/prptop/prptop.php, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#probableprimenumbers, https://stdkmd.net/nrr/repunit/prpfactors.htm, https://www.mersenne.ca/prp.php?show=1, https://www.alfredreichlg.de/10w7/prp.html, http://factordb.com/listtype.php?t=1))	http://factordb.com/index.php?id=1100000002579528773&open=prime	8177207	http://www.primenumbers.net/prptop/prptop.php
Miller–Rabin probable primality testing (https://t5k.org/prove/prove2_3.html, https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test, https://en.wikipedia.org/wiki/Strong_pseudoprime, https://t5k.org/glossary/xpage/MillersTest.html, https://t5k.org/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://faculty.lynchburg.edu/~nicely/misc/mpzspsp.html, http://factordb.com/prooffailed.php, 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/A020231, https://oeis.org/A020233, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773, https://oeis.org/A014233, https://oeis.org/A006945, https://oeis.org/A089825, https://oeis.org/A181782, https://oeis.org/A071294, https://oeis.org/A141768, https://oeis.org/A195328, https://oeis.org/A329759, https://oeis.org/A298756)	O(log(n))	any number, but only probabilistic (https://en.wikipedia.org/wiki/Probabilistic_algorithm) primality testing	(10^8177207−1)/9 (the largest known unproven probable prime (https://en.wikipedia.org/wiki/Probable_prime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Probable_prime, https://mathworld.wolfram.com/ProbablePrime.html, http://www.primenumbers.net/prptop/prptop.php, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#probableprimenumbers, https://stdkmd.net/nrr/repunit/prpfactors.htm, https://www.mersenne.ca/prp.php?show=1, https://www.alfredreichlg.de/10w7/prp.html, http://factordb.com/listtype.php?t=1))	http://factordb.com/index.php?id=1100000002579528773&open=prime	8177207	http://www.primenumbers.net/prptop/prptop.php
Lucas probable primality testing (https://en.wikipedia.org/wiki/Lucas_pseudoprime, https://mathworld.wolfram.com/LucasPseudoprime.html, https://www.mathpages.com/home/kmath127/kmath127.htm, http://ntheory.org/data/lpsps-baillie.txt, https://oeis.org/A217120)	O(log(n))	any number, but only probabilistic (https://en.wikipedia.org/wiki/Probabilistic_algorithm) primality testing	(10^8177207−1)/9 (the largest known unproven probable prime (https://en.wikipedia.org/wiki/Probable_prime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Probable_prime, https://mathworld.wolfram.com/ProbablePrime.html, http://www.primenumbers.net/prptop/prptop.php, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#probableprimenumbers, https://stdkmd.net/nrr/repunit/prpfactors.htm, https://www.mersenne.ca/prp.php?show=1, https://www.alfredreichlg.de/10w7/prp.html, http://factordb.com/listtype.php?t=1))	http://factordb.com/index.php?id=1100000002579528773&open=prime	8177207	http://www.primenumbers.net/prptop/prptop.php
strong Lucas probable primality testing (https://en.wikipedia.org/wiki/Strong_Lucas_pseudoprime, https://mathworld.wolfram.com/StrongLucasPseudoprime.html, http://ntheory.org/data/slpsps-baillie.txt, https://oeis.org/A217255)	O(log(n))	any number, but only probabilistic (https://en.wikipedia.org/wiki/Probabilistic_algorithm) primality testing	(10^8177207−1)/9 (the largest known unproven probable prime (https://en.wikipedia.org/wiki/Probable_prime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Probable_prime, https://mathworld.wolfram.com/ProbablePrime.html, http://www.primenumbers.net/prptop/prptop.php, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#probableprimenumbers, https://stdkmd.net/nrr/repunit/prpfactors.htm, https://www.mersenne.ca/prp.php?show=1, https://www.alfredreichlg.de/10w7/prp.html, http://factordb.com/listtype.php?t=1))	http://factordb.com/index.php?id=1100000002579528773&open=prime	8177207	http://www.primenumbers.net/prptop/prptop.php
Baillie–PSW primality testing (https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test, https://mathworld.wolfram.com/Baillie-PSWPrimalityTest.html, http://pseudoprime.com/dopo.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_318.pdf))	O(log(n))	any number, but only probabilistic (https://en.wikipedia.org/wiki/Probabilistic_algorithm) primality testing	(10^8177207−1)/9 (the largest known unproven probable prime (https://en.wikipedia.org/wiki/Probable_prime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Probable_prime, https://mathworld.wolfram.com/ProbablePrime.html, http://www.primenumbers.net/prptop/prptop.php, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#probableprimenumbers, https://stdkmd.net/nrr/repunit/prpfactors.htm, https://www.mersenne.ca/prp.php?show=1, https://www.alfredreichlg.de/10w7/prp.html, http://factordb.com/listtype.php?t=1))	http://factordb.com/index.php?id=1100000002579528773&open=prime	8177207	http://www.primenumbers.net/prptop/prptop.php
Pépin primality testing (https://en.wikipedia.org/wiki/P%C3%A9pin%27s_test, https://t5k.org/glossary/xpage/PepinsTest.html, https://www.rieselprime.de/ziki/P%C3%A9pin%27s_test, https://mathworld.wolfram.com/PepinsTest.html)	O(log(n))	Fermat numbers (https://en.wikipedia.org/wiki/Fermat_number, https://t5k.org/glossary/xpage/FermatNumber.html, https://www.rieselprime.de/ziki/Fermat_number, https://mathworld.wolfram.com/FermatNumber.html, https://mathworld.wolfram.com/FermatPrime.html, https://pzktupel.de/Primetables/TableFermat.php)	65537 (the largest known Fermat prime (https://en.wikipedia.org/wiki/Fermat_number, https://t5k.org/glossary/xpage/FermatNumber.html, https://www.rieselprime.de/ziki/Fermat_number, https://mathworld.wolfram.com/FermatNumber.html, https://mathworld.wolfram.com/FermatPrime.html, https://pzktupel.de/Primetables/TableFermat.php, http://www.prothsearch.com/fermat.html, https://t5k.org/top20/page.php?id=8, https://t5k.org/primes/search.php?Comment=Divides&OnList=all&Number=1000000&Style=HTML, http://www.fermatsearch.org/, https://64ordle.au/fermat/, http://www.fermatsearch.org/factors/faclist.php, http://www.fermatsearch.org/factors/composite.php))	http://factordb.com/index.php?id=65537&open=prime	5	http://www.fermatsearch.org/
Lucas–Lehmer primality testing (https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test, https://www.rieselprime.de/ziki/Lucas-Lehmer_test, https://mathworld.wolfram.com/Lucas-LehmerTest.html, https://t5k.org/notes/proofs/LucasLehmer.html, http://www.numericana.com/answer/primes.htm#lucas-lehmer)	O(log(n))	Mersenne numbers (https://en.wikipedia.org/wiki/Mersenne_prime, https://t5k.org/glossary/xpage/MersenneNumber.html, https://t5k.org/glossary/xpage/Mersennes.html, https://www.rieselprime.de/ziki/Mersenne_number, https://www.rieselprime.de/ziki/Mersenne_prime, https://mathworld.wolfram.com/MersenneNumber.html, https://mathworld.wolfram.com/MersennePrime.html, https://pzktupel.de/Primetables/TableMersenne.php)	2^82589933−1 (the largest known Mersenne prime (https://en.wikipedia.org/wiki/Mersenne_prime, https://t5k.org/glossary/xpage/MersenneNumber.html, https://t5k.org/glossary/xpage/Mersennes.html, https://www.rieselprime.de/ziki/Mersenne_number, https://www.rieselprime.de/ziki/Mersenne_prime, https://mathworld.wolfram.com/MersenneNumber.html, https://mathworld.wolfram.com/MersennePrime.html, https://pzktupel.de/Primetables/TableMersenne.php, https://t5k.org/top20/page.php?id=4, https://t5k.org/primes/search.php?Comment=Mersenne%20[[:digit:]]&OnList=all&Number=1000000&Style=HTML, https://www.mersenne.org/, https://www.mersenne.ca/, https://www.mersenne.org/primes/, https://www.mersenne.ca/prime.php, https://t5k.org/mersenne/))	http://www.factordb.com/index.php?id=1100000001257221107&open=prime	24862048	http://www.mersenne.org/primes/
Proth primality testing (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)	O(log(n))	numbers of the form k×2ⁿ+1 with k odd and k < 2ⁿ, such numbers are called "Proth numbers" (https://en.wikipedia.org/wiki/Proth_prime, https://t5k.org/glossary/xpage/ProthPrime.html, https://www.rieselprime.de/ziki/Proth_prime, https://mathworld.wolfram.com/ProthNumber.html, http://www.prothsearch.com/frequencies.html, http://www.prothsearch.com/history.html, https://www.rieselprime.de/Data/PStatistics.htm, https://www.rieselprime.de/Data/PRanges50.htm, https://www.rieselprime.de/Data/PRanges300.htm, https://www.rieselprime.de/Data/PRanges1200.htm, http://irvinemclean.com/maths/pfaq6.htm, https://www.numbersaplenty.com/set/Proth_number/, https://web.archive.org/web/20230706041914/https://pzktupel.de/Primetables/TableProthTOP10KK.php, https://web.archive.org/web/20231030081449/https://pzktupel.de/Primetables/ProthK.txt, https://pzktupel.de/Primetables/TableProthTOP10KS.php, https://pzktupel.de/Primetables/ProthS.txt, https://t5k.org/top20/page.php?id=66, https://www.primegrid.com/forum_thread.php?id=2665, https://www.primegrid.com/stats_pps_llr.php, https://www.primegrid.com/stats_ppse_llr.php, https://www.primegrid.com/stats_mega_llr.php, https://www.primegrid.com/primes/primes.php?project=PPS&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=PPSE&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=MEG&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, http://boincvm.proxyma.ru:30080/test4vm/public/pps_dc_status.php, https://www.rieselprime.de/ziki/PrimeGrid_Proth_Prime_Search, https://www.rieselprime.de/ziki/PrimeGrid_Proth_Prime_Search_Extended, https://www.rieselprime.de/ziki/PrimeGrid_Proth_Mega_Prime_Search), and such primes are called "Proth primes"	10223×2^31172165+1 (the largest known Proth prime (https://en.wikipedia.org/wiki/Proth_prime, https://t5k.org/glossary/xpage/ProthPrime.html, https://www.rieselprime.de/ziki/Proth_prime, https://mathworld.wolfram.com/ProthNumber.html, http://www.prothsearch.com/frequencies.html, http://www.prothsearch.com/history.html, https://www.rieselprime.de/Data/PStatistics.htm, https://www.rieselprime.de/Data/PRanges50.htm, https://www.rieselprime.de/Data/PRanges300.htm, https://www.rieselprime.de/Data/PRanges1200.htm, http://irvinemclean.com/maths/pfaq6.htm, https://www.numbersaplenty.com/set/Proth_number/, https://web.archive.org/web/20230706041914/https://pzktupel.de/Primetables/TableProthTOP10KK.php, https://web.archive.org/web/20231030081449/https://pzktupel.de/Primetables/ProthK.txt, https://pzktupel.de/Primetables/TableProthTOP10KS.php, https://pzktupel.de/Primetables/ProthS.txt, https://t5k.org/top20/page.php?id=66, https://www.primegrid.com/forum_thread.php?id=2665, https://www.primegrid.com/stats_pps_llr.php, https://www.primegrid.com/stats_ppse_llr.php, https://www.primegrid.com/stats_mega_llr.php, https://www.primegrid.com/primes/primes.php?project=PPS&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=PPSE&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=MEG&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, http://boincvm.proxyma.ru:30080/test4vm/public/pps_dc_status.php, https://www.rieselprime.de/ziki/PrimeGrid_Proth_Prime_Search, https://www.rieselprime.de/ziki/PrimeGrid_Proth_Prime_Search_Extended, https://www.rieselprime.de/ziki/PrimeGrid_Proth_Mega_Prime_Search))	http://factordb.com/index.php?id=1100000000881266827&open=prime	9383761	https://t5k.org/primes/lists/all.txt
Lucas–Lehmer–Riesel primality testing (https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer%E2%80%93Riesel_test)	O(log(n))	numbers of the form k×2ⁿ−1 with k odd and k < 2ⁿ, such numbers are called "Proth numbers of the second kind" (https://www.rieselprime.de/ziki/Riesel_prime, https://www.rieselprime.de/Data/Statistics.htm, http://irvinemclean.com/maths/pfaq6.htm, https://web.archive.org/web/20230628151418/https://pzktupel.de/Primetables/TableRieselTOP10KK.php, https://web.archive.org/web/20231030081316/https://pzktupel.de/Primetables/RieselK.txt, https://pzktupel.de/Primetables/TableRieselTOP10KS.php, https://pzktupel.de/Primetables/RieselS.txt, http://www.noprimeleftbehind.net/stats/index.php?content=prime_list, http://www.noprimeleftbehind.net/prpnet/, http://www.noprimeleftbehind.net:9000/all.html, http://www.noprimeleftbehind.net:4000/all.html, http://www.noprimeleftbehind.net:2000/all.html, http://www.noprimeleftbehind.net:1468/all.html, http://www.noprimeleftbehind.net:1400/all.html, https://www.rieselprime.de/ziki/NPLB_Drive_17, https://www.rieselprime.de/ziki/NPLB_Drive_18, https://www.rieselprime.de/ziki/NPLB_Drive_19, https://www.rieselprime.de/ziki/NPLB_Drive_High-n), and such primes are called "Proth primes of the second kind"	3×2^20928756−1 (the largest known Proth prime of the second kind (https://www.rieselprime.de/ziki/Riesel_prime, https://www.rieselprime.de/Data/Statistics.htm, http://irvinemclean.com/maths/pfaq6.htm, https://web.archive.org/web/20230628151418/https://pzktupel.de/Primetables/TableRieselTOP10KK.php, https://web.archive.org/web/20231030081316/https://pzktupel.de/Primetables/RieselK.txt, https://pzktupel.de/Primetables/TableRieselTOP10KS.php, https://pzktupel.de/Primetables/RieselS.txt, http://www.noprimeleftbehind.net/stats/index.php?content=prime_list, http://www.noprimeleftbehind.net/prpnet/, http://www.noprimeleftbehind.net:9000/all.html, http://www.noprimeleftbehind.net:4000/all.html, http://www.noprimeleftbehind.net:2000/all.html, http://www.noprimeleftbehind.net:1468/all.html, http://www.noprimeleftbehind.net:1400/all.html, https://www.rieselprime.de/ziki/NPLB_Drive_17, https://www.rieselprime.de/ziki/NPLB_Drive_18, https://www.rieselprime.de/ziki/NPLB_Drive_19, https://www.rieselprime.de/ziki/NPLB_Drive_High-n))	http://factordb.com/index.php?id=1100000004660727594&open=prime	6300184	https://t5k.org/primes/lists/all.txt
Pocklington (N−1) primality testing (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, 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)	O(log(n))	numbers n such that n−1 can be ≥ 1/2 factored (i.e. the product of the known prime factors of n−1 is ≥ the square root (https://en.wikipedia.org/wiki/Square_root, https://www.rieselprime.de/ziki/Square_root, https://mathworld.wolfram.com/SquareRoot.html) of it)	1963736^1048576+1 (the largest known generalized Proth prime (https://en.wikipedia.org/wiki/Proth_prime, https://t5k.org/glossary/xpage/ProthPrime.html, https://www.rieselprime.de/ziki/Proth_prime, https://mathworld.wolfram.com/ProthNumber.html, http://www.prothsearch.com/frequencies.html, http://www.prothsearch.com/history.html, https://www.rieselprime.de/Data/PStatistics.htm, https://www.rieselprime.de/Data/PRanges50.htm, https://www.rieselprime.de/Data/PRanges300.htm, https://www.rieselprime.de/Data/PRanges1200.htm, http://irvinemclean.com/maths/pfaq6.htm, https://www.numbersaplenty.com/set/Proth_number/, https://web.archive.org/web/20230706041914/https://pzktupel.de/Primetables/TableProthTOP10KK.php, https://web.archive.org/web/20231030081449/https://pzktupel.de/Primetables/ProthK.txt, https://pzktupel.de/Primetables/TableProthTOP10KS.php, https://pzktupel.de/Primetables/ProthS.txt, https://pzktupel.de/Primetables/TableProthGen.php, https://pzktupel.de/Primetables/TableProthGen.txt, https://sites.google.com/view/proth-primes, https://t5k.org/primes/search_proth.php, https://t5k.org/top20/page.php?id=66, https://www.primegrid.com/forum_thread.php?id=2665, https://www.primegrid.com/stats_pps_llr.php, https://www.primegrid.com/stats_ppse_llr.php, https://www.primegrid.com/stats_mega_llr.php, https://www.primegrid.com/primes/primes.php?project=PPS&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=PPSE&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=MEG&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, http://boincvm.proxyma.ru:30080/test4vm/public/pps_dc_status.php, https://www.rieselprime.de/ziki/PrimeGrid_Proth_Prime_Search, https://www.rieselprime.de/ziki/PrimeGrid_Proth_Prime_Search_Extended, https://www.rieselprime.de/ziki/PrimeGrid_Proth_Mega_Prime_Search) with base b > 2)	http://factordb.com/index.php?id=1100000003905175851&open=prime	6598776	https://t5k.org/primes/lists/all.txt
Morrison (N+1) primality testing (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2)	O(log(n))	numbers n such that n+1 can be ≥ 1/2 factored (i.e. the product of the known prime factors of n+1 is ≥ the square root (https://en.wikipedia.org/wiki/Square_root, https://www.rieselprime.de/ziki/Square_root, https://mathworld.wolfram.com/SquareRoot.html) of it)	3622×5^7558139−1 (the largest known generalized Proth prime of the second kind (https://www.rieselprime.de/ziki/Riesel_prime, https://www.rieselprime.de/Data/Statistics.htm, http://irvinemclean.com/maths/pfaq6.htm, https://web.archive.org/web/20230628151418/https://pzktupel.de/Primetables/TableRieselTOP10KK.php, https://web.archive.org/web/20231030081316/https://pzktupel.de/Primetables/RieselK.txt, https://pzktupel.de/Primetables/TableRieselTOP10KS.php, https://pzktupel.de/Primetables/RieselS.txt, https://pzktupel.de/Primetables/TableRieselGen.php, https://pzktupel.de/Primetables/TableRieselGen.txt, https://sites.google.com/view/proth-primes, http://www.noprimeleftbehind.net/stats/index.php?content=prime_list, https://t5k.org/primes/search_proth.php, http://www.noprimeleftbehind.net/prpnet/, http://www.noprimeleftbehind.net:9000/all.html, http://www.noprimeleftbehind.net:4000/all.html, http://www.noprimeleftbehind.net:2000/all.html, http://www.noprimeleftbehind.net:1468/all.html, http://www.noprimeleftbehind.net:1400/all.html, https://www.rieselprime.de/ziki/NPLB_Drive_17, https://www.rieselprime.de/ziki/NPLB_Drive_18, https://www.rieselprime.de/ziki/NPLB_Drive_19, https://www.rieselprime.de/ziki/NPLB_Drive_High-n) with base b > 2)	http://factordb.com/index.php?id=1100000002920355417&open=prime	5282917	https://t5k.org/primes/lists/all.txt
elliptic curve (ECPP) primality testing (https://t5k.org/prove/prove4_2.html, https://en.wikipedia.org/wiki/Elliptic_curve_primality, https://t5k.org/glossary/xpage/ECPP.html, https://mathworld.wolfram.com/EllipticCurvePrimalityProving.html, http://irvinemclean.com/maths/pfaq7.htm, https://t5k.org/top20/page.php?id=27, https://t5k.org/primes/search.php?Comment=ECPP&OnList=all&Number=1000000&Style=HTML, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/~morain/Prgms/ecpp.english.html, https://www.multiprecision.org/cm/ecpp.html, https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1199989-X/S0025-5718-1993-1199989-X.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_256.pdf), https://arxiv.org/pdf/2404.05506.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_428.pdf))	O(log(n)^4+ε) for some ε > 0	any number	(10⁸⁶⁴⁵³−1)/9 (the largest known proven ordinary prime (https://t5k.org/glossary/xpage/OrdinaryPrime.html))	http://factordb.com/index.php?id=1100000000046752372&open=prime	86453	https://t5k.org/top20/page.php?id=27
Adleman–Pomerance–Rumely (APR) primality testing (https://en.wikipedia.org/wiki/Adleman%E2%80%93Pomerance%E2%80%93Rumely_primality_test, https://www.rieselprime.de/ziki/Adleman%E2%80%93Pomerance%E2%80%93Rumely_primality_test, https://mathworld.wolfram.com/Adleman-Pomerance-RumelyPrimalityTest.html, https://t5k.org/prove/prove4_1.html, https://t5k.org/primes/search.php?Comment=APR-CL%20assisted&OnList=all&Number=1000000&Style=HTML)	O(log(n)^{log(log(log(n)))})	any number	(none)	(none)	(none)	(none)
Agrawal–Kayal–Saxena (AKS) primality testing (https://en.wikipedia.org/wiki/AKS_primality_test, https://mathworld.wolfram.com/AKSPrimalityTest.html, https://t5k.org/prove/prove4_3.html, http://www.numericana.com/answer/primes.htm#aks, http://cr.yp.to/papers/aks-20030125-retypeset20220327.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_70.pdf), http://www.cse.iitk.ac.in/users/manindra/algebra/primality_v6.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_231.pdf))	O(log(n)¹²)	any number	(none)	(none)	(none)	(none)
special number field sieve (SNFS) integer factoring (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (sections "Special number field sieve by size of number factored:" and "Special number field sieve by SNFS difficulty:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=snfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#smallpolynomial, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/)	e^{(1+o(1))×(32/9×log(n))^1/3×log(log(n))^2/3} = L_n(1/3, (32/9)^1/3)	numbers of special forms, e.g. numbers of the form a×bⁿ±c with small a, b, c and large n	2¹¹⁹³−1	http://factordb.com/index.php?id=1000000000000001193&open=ecm	360 (SNFS difficulty)	https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (sections "Special number field sieve by size of number factored:" and "Special number field sieve by SNFS difficulty:")
general number field sieve (GNFS) integer factoring (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "General number field sieve by size of number factored:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=gnfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/)	e^{((64/9)^1/3+o(1))×log(n)^1/3×log(log(n))^2/3} = L_n(1/3, (64/9)^1/3)	any number	the RSA number (https://en.wikipedia.org/wiki/RSA_numbers, https://t5k.org/glossary/xpage/RSAExample.html, https://mathworld.wolfram.com/RSANumber.html, http://www.ontko.com/pub/rayo/primes/rsa_fact.html, http://www.loria.fr/~zimmerma/records/rsa.html, https://web.archive.org/web/20061209135708/http://www.rsasecurity.com/rsalabs/node.asp?id=2093, https://web.archive.org/web/20130521030319/https://www.rsa.com/rsalabs/challenges/factoring/challengenumbers.txt) RSA-250 (see https://www.mersenneforum.org/showthread.php?t=25312 for more information of this number)	http://factordb.com/index.php?id=1100000000104374168&open=ecm	250 (GNFS difficulty)	https://en.wikipedia.org/wiki/Integer_factorization_records#Numbers_of_a_general_form
trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172)	O(p)	any prime factor	9999999999999937 (the largest prime < 10¹⁶)	http://factordb.com/index.php?id=9999999999999937&open=ecm	16	(none)
Pollard (P−1) integer factoring (https://en.wikipedia.org/wiki/Pollard%27s_p_%E2%88%92_1_algorithm, https://www.rieselprime.de/ziki/P-1_factorization_method, https://mathworld.wolfram.com/Pollardp-1FactorizationMethod.html, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "Pollard p-1:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=p-1&maxrows=10000, http://www.numericana.com/answer/factoring.htm#p-1, http://factordb.com/listecm.php?c=2, http://www.loria.fr/~zimmerma/records/Pminus1.html, https://web.archive.org/web/20021015212913/http://www.users.globalnet.co.uk/~aads/Pminus1.html, https://web.archive.org/web/20231002022529/https://colin.barker.pagesperso-orange.fr/lpa/big_pm1.htm, https://www.mersenne.org/report_pminus1/, https://www.mersenne.ca/userfactors/pm1/1, https://www.mersenne.ca/smooth.php, https://www.mersenne.ca/p1missed.php, https://www.mersenne.ca/prob.php)	O(log(p))	prime factors p such that p−1 is smooth (https://en.wikipedia.org/wiki/Smooth_number, https://mathworld.wolfram.com/SmoothNumber.html, https://oeis.org/A003586, https://oeis.org/A051037, https://oeis.org/A002473, https://oeis.org/A051038, https://oeis.org/A080197, https://oeis.org/A080681, https://oeis.org/A080682, https://oeis.org/A080683) (i.e. the greatest prime factor (http://mathworld.wolfram.com/GreatestPrimeFactor.html, https://oeis.org/A006530) of p−1 is small)	prime factor of 960¹¹⁹−1	http://factordb.com/index.php?id=1100000000216738427&open=ecm	66	http://www.loria.fr/~zimmerma/records/Pminus1.html
Williams (P+1) integer factoring (https://en.wikipedia.org/wiki/Williams%27s_p_%2B_1_algorithm, https://www.rieselprime.de/ziki/P%2B1_factorization_method, https://mathworld.wolfram.com/WilliamspPlus1FactorizationMethod.html, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "p+1:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=p%2b1&maxrows=10000, http://www.numericana.com/answer/factoring.htm#p+1, http://factordb.com/listecm.php?c=3, http://www.loria.fr/~zimmerma/records/Pplus1.html, https://www.mersenne.org/report_pplus1/, https://www.mersenne.ca/userfactors/pp1/1, https://www.mersenne.ca/pplus1.php)	O(log(p))	prime factors p such that p+1 is smooth (https://en.wikipedia.org/wiki/Smooth_number, https://mathworld.wolfram.com/SmoothNumber.html, https://oeis.org/A003586, https://oeis.org/A051037, https://oeis.org/A002473, https://oeis.org/A051038, https://oeis.org/A080197, https://oeis.org/A080681, https://oeis.org/A080682, https://oeis.org/A080683) (i.e. the greatest prime factor (http://mathworld.wolfram.com/GreatestPrimeFactor.html, https://oeis.org/A006530) of p+1 is small)	prime factor of the Lucas number (https://en.wikipedia.org/wiki/Lucas_number, https://t5k.org/glossary/xpage/LucasNumber.html, https://mathworld.wolfram.com/LucasNumber.html, https://www.numbersaplenty.com/set/Lucas_number/, https://t5k.org/top20/page.php?id=48, https://t5k.org/primes/search.php?Comment=^Lucas%20number&OnList=all&Number=1000000&Style=HTML, https://pzktupel.de/Primetables/TableLucas.php, https://oeis.org/A000032, https://oeis.org/A000204, https://oeis.org/A005479, https://oeis.org/A001606) L₂₃₆₆	http://factordb.com/index.php?id=1100000000216896584&open=ecm	60	http://www.loria.fr/~zimmerma/records/Pplus1.html
Lenstra elliptic-curve (ECM) integer factoring (https://en.wikipedia.org/wiki/Lenstra_elliptic-curve_factorization, https://www.rieselprime.de/ziki/Elliptic_curve_method, https://mathworld.wolfram.com/EllipticCurveFactorizationMethod.html, http://www.numericana.com/answer/factoring.htm#ecm, http://factordb.com/listecm.php?c=1, http://factordb.com/listecm.php?c=4, http://www.loria.fr/~zimmerma/ecmnet/, http://www.loria.fr/~zimmerma/records/ecmnet.html, http://www.loria.fr/~zimmerma/records/factor.html, http://www.loria.fr/~zimmerma/records/top50.html, http://www.loria.fr/~zimmerma/records/ecm/params.html, https://oeis.org/wiki/OEIS_sequences_needing_factors#ECM_efforts, https://stdkmd.net/nrr/records.htm#largefactorecm, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "Elliptic curve method:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=ecm&maxrows=10000, http://maths-people.anu.edu.au/~brent/factors.html, http://maths-people.anu.edu.au/~brent/ftp/champs.txt, https://www.alpertron.com.ar/ECM.HTM, https://www.alpertron.com.ar/ECMREC.HTM, https://homes.cerias.purdue.edu/~ssw/cun/press/tech.html, https://homes.cerias.purdue.edu/~ssw/cun/press/nontech.html, http://www.mersenne.org/report_ECM/, https://www.mersenne.ca/userfactors/ecm/1, https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, https://kurtbeschorner.de/ecm-efforts.htm, http://www.rechenkraft.net/yoyo//y_factors_ecm.php, http://www.rechenkraft.net/yoyo/y_status_ecm.php, http://www.wraithx.net/math/ecmprobs/ecmprobs.html, http://www.loria.fr/~zimmerma/papers/ecm-entry.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_460.pdf))	e^{(2^1/2+o(1))×log(n)^1/2×log(log(n))^1/2} = L_p(1/2, 2^1/2)	any prime factor	prime factor of 7³³⁷+1	http://factordb.com/index.php?id=1100000000632146801&open=ecm	83	http://www.loria.fr/~zimmerma/records/top50.html

Thus, for these numbers, we usually use the general purpose tests (https://t5k.org/prove/prove4.html), which do not require factorization of N±1 and can be used for ordinary primes (https://t5k.org/glossary/xpage/OrdinaryPrime.html).

Primes p such that p−1 has many factors	Primes p such that p+1 has many factors
Fermat probable primality test (https://t5k.org/prove/prove2_2.html, https://en.wikipedia.org/wiki/Fermat_primality_test, https://en.wikipedia.org/wiki/Fermat_pseudoprime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Fermat_pseudoprimality_test, https://mathworld.wolfram.com/FermatPseudoprime.html, https://www.numbersaplenty.com/set/Poulet_number/, http://www.numericana.com/answer/pseudo.htm#pseudoprime, http://ntheory.org/data/psps.txt, https://sites.google.com/view/fermat-pseudoprime, https://sites.google.com/view/bases-fermat-pseudoprime, https://oeis.org/A001567, https://oeis.org/A005935, https://oeis.org/A005936, https://oeis.org/A005938, https://oeis.org/A052155, https://oeis.org/A083737, https://oeis.org/A083739, https://oeis.org/A083876, https://oeis.org/A271221, https://oeis.org/A348258, https://oeis.org/A181780, https://oeis.org/A063994, https://oeis.org/A194946, https://oeis.org/A195327, https://oeis.org/A002997, https://oeis.org/A191311, https://oeis.org/A090086, https://oeis.org/A007535)	Lucas probable primality test (https://en.wikipedia.org/wiki/Lucas_pseudoprime, https://mathworld.wolfram.com/LucasPseudoprime.html, https://www.mathpages.com/home/kmath127/kmath127.htm, http://ntheory.org/data/lpsps-baillie.txt, https://oeis.org/A217120)
Miller–Rabin probable primality test (https://t5k.org/prove/prove2_3.html, https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test, https://en.wikipedia.org/wiki/Strong_pseudoprime, https://t5k.org/glossary/xpage/MillersTest.html, https://t5k.org/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://faculty.lynchburg.edu/~nicely/misc/mpzspsp.html, http://factordb.com/prooffailed.php, 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/A020231, https://oeis.org/A020233, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773, https://oeis.org/A014233, https://oeis.org/A006945, https://oeis.org/A089825, https://oeis.org/A181782, https://oeis.org/A071294, https://oeis.org/A141768, https://oeis.org/A195328, https://oeis.org/A329759, https://oeis.org/A298756)	Strong Lucas probable primality test (https://en.wikipedia.org/wiki/Strong_Lucas_pseudoprime, https://mathworld.wolfram.com/StrongLucasPseudoprime.html, http://ntheory.org/data/slpsps-baillie.txt, https://oeis.org/A217255)
Carmichael number (https://oeis.org/A002997, https://en.wikipedia.org/wiki/Carmichael_number, https://t5k.org/glossary/xpage/CarmichaelNumber.html, https://mathworld.wolfram.com/CarmichaelNumber.html, https://www.numbersaplenty.com/set/Carmichael_number/, http://www.numericana.com/answer/modular.htm#carmichael, http://www.s369624816.websitehome.co.uk/rgep/carpsp.html, http://www.s369624816.websitehome.co.uk/rgep/cartable.html)	Lucas–Carmichael number (https://oeis.org/A006972, https://en.wikipedia.org/wiki/Lucas%E2%80%93Carmichael_number)
Euler's totient function (https://oeis.org/A000010, https://en.wikipedia.org/wiki/Euler%27s_totient_function, https://t5k.org/glossary/xpage/EulersPhi.html, https://mathworld.wolfram.com/TotientFunction.html, http://www.numericana.com/answer/modular.htm#phi), its range is https://oeis.org/A002202, and the even numbers not in its range are https://oeis.org/A005277	Dedekind psi function (https://oeis.org/A001615, https://en.wikipedia.org/wiki/Dedekind_psi_function, https://mathworld.wolfram.com/DedekindFunction.html), its range is https://oeis.org/A203444, and the even numbers not in its range are https://oeis.org/A307055
Pépin primality test (https://en.wikipedia.org/wiki/P%C3%A9pin%27s_test, https://t5k.org/glossary/xpage/PepinsTest.html, https://www.rieselprime.de/ziki/P%C3%A9pin%27s_test, https://mathworld.wolfram.com/PepinsTest.html) for Fermat numbers (https://en.wikipedia.org/wiki/Fermat_number, https://t5k.org/glossary/xpage/FermatNumber.html, https://www.rieselprime.de/ziki/Fermat_number, https://mathworld.wolfram.com/FermatNumber.html, https://mathworld.wolfram.com/FermatPrime.html, https://pzktupel.de/Primetables/TableFermat.php), i.e. numbers of the form 2ⁿ+1 (https://oeis.org/A000051), if 2ⁿ+1 is prime, then n must be power of 2, such numbers are https://oeis.org/A000215, and such primes are https://oeis.org/A019434	Lucas–Lehmer primality test (https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test, https://www.rieselprime.de/ziki/Lucas-Lehmer_test, https://mathworld.wolfram.com/Lucas-LehmerTest.html, https://t5k.org/notes/proofs/LucasLehmer.html, http://www.numericana.com/answer/primes.htm#lucas-lehmer) for Mersenne numbers (https://en.wikipedia.org/wiki/Mersenne_prime, https://t5k.org/glossary/xpage/MersenneNumber.html, https://t5k.org/glossary/xpage/Mersennes.html, https://www.rieselprime.de/ziki/Mersenne_number, https://www.rieselprime.de/ziki/Mersenne_prime, https://mathworld.wolfram.com/MersenneNumber.html, https://mathworld.wolfram.com/MersennePrime.html, https://pzktupel.de/Primetables/TableMersenne.php), i.e. numbers of the form 2ⁿ−1 (https://oeis.org/A000225), if 2ⁿ−1 is prime, then n must be prime, such numbers are https://oeis.org/A001348, and such primes are https://oeis.org/A000668
Pépin primality test numbers: https://oeis.org/A060377	Lucas–Lehmer primality test numbers: https://oeis.org/A003010
Residues of Pépin primality test for Fermat numbers: https://oeis.org/A152153	Residues of Lucas–Lehmer primality test for Mersenne numbers: https://oeis.org/A095847
Possible bases for Pépin primality test for Fermat numbers (the original base for Pépin primality test is 3): https://oeis.org/A129802	Possible starting values for Lucas–Lehmer primality test for Mersenne numbers (the original starting value for Lucas–Lehmer primality test is 4): https://oeis.org/A018844
Proth primality test (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) for numbers of the form k×2ⁿ+1 with k odd and k < 2ⁿ, such numbers are called "Proth numbers" (https://en.wikipedia.org/wiki/Proth_prime, https://t5k.org/glossary/xpage/ProthPrime.html, https://www.rieselprime.de/ziki/Proth_prime, https://mathworld.wolfram.com/ProthNumber.html, http://www.prothsearch.com/frequencies.html, http://www.prothsearch.com/history.html, https://www.rieselprime.de/Data/PStatistics.htm, https://www.rieselprime.de/Data/PRanges50.htm, https://www.rieselprime.de/Data/PRanges300.htm, https://www.rieselprime.de/Data/PRanges1200.htm, http://irvinemclean.com/maths/pfaq6.htm, https://www.numbersaplenty.com/set/Proth_number/, https://web.archive.org/web/20230706041914/https://pzktupel.de/Primetables/TableProthTOP10KK.php, https://web.archive.org/web/20231030081449/https://pzktupel.de/Primetables/ProthK.txt, https://pzktupel.de/Primetables/TableProthTOP10KS.php, https://pzktupel.de/Primetables/ProthS.txt, https://t5k.org/top20/page.php?id=66, https://www.primegrid.com/forum_thread.php?id=2665, https://www.primegrid.com/stats_pps_llr.php, https://www.primegrid.com/stats_ppse_llr.php, https://www.primegrid.com/stats_mega_llr.php, https://www.primegrid.com/primes/primes.php?project=PPS&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=PPSE&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=MEG&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, http://boincvm.proxyma.ru:30080/test4vm/public/pps_dc_status.php, https://www.rieselprime.de/ziki/PrimeGrid_Proth_Prime_Search, https://www.rieselprime.de/ziki/PrimeGrid_Proth_Prime_Search_Extended, https://www.rieselprime.de/ziki/PrimeGrid_Proth_Mega_Prime_Search), and such primes are called "Proth primes", such numbers are https://oeis.org/A080075, and such primes are https://oeis.org/A080076	Lucas–Lehmer–Riesel primality test (https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer%E2%80%93Riesel_test) for numbers of the form k×2ⁿ−1 with k odd and k < 2ⁿ, such numbers are called "Proth numbers of the second kind" (https://www.rieselprime.de/ziki/Riesel_prime, https://www.rieselprime.de/Data/Statistics.htm, http://irvinemclean.com/maths/pfaq6.htm, https://web.archive.org/web/20230628151418/https://pzktupel.de/Primetables/TableRieselTOP10KK.php, https://web.archive.org/web/20231030081316/https://pzktupel.de/Primetables/RieselK.txt, https://pzktupel.de/Primetables/TableRieselTOP10KS.php, https://pzktupel.de/Primetables/RieselS.txt, http://www.noprimeleftbehind.net/stats/index.php?content=prime_list, http://www.noprimeleftbehind.net/prpnet/, http://www.noprimeleftbehind.net:9000/all.html, http://www.noprimeleftbehind.net:4000/all.html, http://www.noprimeleftbehind.net:2000/all.html, http://www.noprimeleftbehind.net:1468/all.html, http://www.noprimeleftbehind.net:1400/all.html, https://www.rieselprime.de/ziki/NPLB_Drive_17, https://www.rieselprime.de/ziki/NPLB_Drive_18, https://www.rieselprime.de/ziki/NPLB_Drive_19, https://www.rieselprime.de/ziki/NPLB_Drive_High-n), and such primes are called "Proth primes of the second kind", such numbers are https://oeis.org/A112714, and such primes are https://oeis.org/A112715
Pocklington N−1 primality test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, 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) for numbers N such that N−1 can be ≥ 1/2 factored (i.e. the product of the known prime factors of N−1 is ≥ the square root (https://en.wikipedia.org/wiki/Square_root, https://www.rieselprime.de/ziki/Square_root, https://mathworld.wolfram.com/SquareRoot.html) of it)	Morrison N+1 primality test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) for numbers N such that N+1 can be ≥ 1/2 factored (i.e. the product of the known prime factors of N+1 is ≥ the square root (https://en.wikipedia.org/wiki/Square_root, https://www.rieselprime.de/ziki/Square_root, https://mathworld.wolfram.com/SquareRoot.html) of it)
List of primes of the form k×2ⁿ+1 with odd k (http://www.prothsearch.com/riesel1.html, http://www.prothsearch.com/riesel1a.html, http://www.prothsearch.com/riesel1b.html, http://www.prothsearch.com/riesel1c.html, http://www.prothsearch.com/Proth-k-1200-I.txt, http://www.prothsearch.com/Proth-k-1200-II.txt, http://www.prothsearch.com/Proth-k-1200-III.txt, http://www.prothsearch.com/Proth-k-10000-I.txt, http://www.prothsearch.com/Proth-k-10000-II.txt, http://www.prothsearch.com/Proth-k-10000-III.txt, http://www.prothsearch.com/Proth-k-100000-I.txt, http://www.prothsearch.com/Proth-k-100000-II.txt, http://www.prothsearch.com/Proth-k-mega.txt, http://irvinemclean.com/maths/sierpin.htm, http://irvinemclean.com/maths/sierpin2.htm, http://irvinemclean.com/maths/sierpin3.htm, http://irvinemclean.com/maths/sierhigh.txt, https://www.rieselprime.de/Data/P00001.htm, https://www.rieselprime.de/Data/P00300.htm, https://www.rieselprime.de/Data/P02000.htm, https://www.rieselprime.de/Data/P04000.htm, https://www.rieselprime.de/Data/P06000.htm, https://www.rieselprime.de/Data/P08000.htm, https://www.rieselprime.de/ziki/Proth_2_1-300, https://www.rieselprime.de/ziki/Proth_2_300-2000, https://www.rieselprime.de/ziki/Proth_2_2000-4000, https://www.rieselprime.de/ziki/Proth_2_4000-6000, https://www.rieselprime.de/ziki/Proth_2_6000-8000, https://www.rieselprime.de/ziki/Proth_2_8000-10000, https://www.rieselprime.de/ziki/Proth_2_10e4-10e5, https://www.rieselprime.de/ziki/Proth_2_10e5-10e6, https://www.rieselprime.de/ziki/Proth_2_10e6-10e7, https://www.rieselprime.de/ziki/Proth_2_10e7-10e8, https://www.rieselprime.de/ziki/Proth_2_10e8-10e9, https://www.rieselprime.de/ziki/Proth_2_10e9-10e10, https://www.rieselprime.de/ziki/Proth_2_10e10-infinity, https://harvey563.tripod.com/proths.txt, https://sites.google.com/view/proth-primes, https://www.primegrid.com/forum_thread.php?id=7945, https://www.primegrid.com/forum_thread.php?id=2665, https://www.primegrid.com/stats_321_llr.php, https://www.primegrid.com/stats_pps_llr.php, https://www.primegrid.com/stats_ppse_llr.php, https://www.primegrid.com/stats_mega_llr.php, https://www.primegrid.com/primes/primes.php?project=321&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=27&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=121&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=PPS&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=PPSE&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=MEG&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://web.archive.org/web/20231223043356/https://www.mersenneforum.org/321search/index.html, https://web.archive.org/web/20110601231527/http://www.bodang.com/12121/, https://web.archive.org/web/20100518081012/http://www.bodang.com/12121/27k/, https://oeis.org/A002253, https://oeis.org/A002254, https://oeis.org/A032353, https://oeis.org/A002256, https://oeis.org/A002261, https://oeis.org/A032356, https://oeis.org/A002258, https://oeis.org/A002259, https://oeis.org/A032359, https://oeis.org/A032360, https://oeis.org/A032361, https://oeis.org/A032362, https://oeis.org/A032363, https://oeis.org/A032364, https://oeis.org/A032365, https://oeis.org/A032366, https://oeis.org/A032367, https://oeis.org/A032368, https://oeis.org/A002269, https://oeis.org/A032370, https://oeis.org/A032371, https://oeis.org/A032372, https://oeis.org/A032373, https://oeis.org/A032374, https://oeis.org/A032375, https://oeis.org/A032376, https://oeis.org/A032377, https://oeis.org/A002274, https://oeis.org/A032379, https://oeis.org/A032380, https://oeis.org/A032381, https://oeis.org/A032382, https://oeis.org/A032383, https://oeis.org/A032384, https://oeis.org/A032385, https://oeis.org/A032386, https://oeis.org/A032387, https://oeis.org/A032388, https://oeis.org/A032389, https://oeis.org/A032390, https://oeis.org/A032391, https://oeis.org/A032392, https://oeis.org/A032393, https://oeis.org/A032394, https://oeis.org/A032395, https://oeis.org/A032396, https://oeis.org/A032397, https://oeis.org/A032398, https://oeis.org/A032399)	List of primes of the form k×2ⁿ−1 with odd k (http://www.prothsearch.com/riesel2.html, https://web.archive.org/web/20210715115849/http://www.15k.org/riesellist.html, https://web.archive.org/web/20210715164816/http://www.15k.org/Summary00300.htm, https://web.archive.org/web/20210715164824/http://www.15k.org/Summary02000.htm, https://web.archive.org/web/20210715164823/http://www.15k.org/Summary04000.htm, https://web.archive.org/web/20210715164815/http://www.15k.org/Summary06000.htm, https://web.archive.org/web/20210715164832/http://www.15k.org/Summary08000.htm, https://web.archive.org/web/20091027083904/http://www.geocities.com/primes_r_us/riesel/prime300.html, https://web.archive.org/web/20091027083901/http://www.geocities.com/primes_r_us/riesel/prime400.html, https://web.archive.org/web/20091027083903/http://www.geocities.com/primes_r_us/riesel/prime500.html, https://web.archive.org/web/20091027083858/http://www.geocities.com/primes_r_us/riesel/prime600.html, https://web.archive.org/web/20091027083905/http://www.geocities.com/primes_r_us/riesel/prime700.html, https://web.archive.org/web/20091027083900/http://www.geocities.com/primes_r_us/riesel/prime800.html, https://web.archive.org/web/20091027083906/http://www.geocities.com/primes_r_us/riesel/prime900.html, http://www.noprimeleftbehind.net/gary/primes-kx2n-1-001.htm, http://www.noprimeleftbehind.net/gary/Rieselprimes-ranges.htm, https://www.rieselprime.de/Data/00001.htm, https://www.rieselprime.de/Data/00300.htm, https://www.rieselprime.de/Data/02000.htm, https://www.rieselprime.de/Data/04000.htm, https://www.rieselprime.de/Data/06000.htm, https://www.rieselprime.de/Data/08000.htm, https://www.rieselprime.de/Data/10e04.htm, https://www.rieselprime.de/Data/10e05.htm, https://www.rieselprime.de/Data/10e06.htm, https://www.rieselprime.de/Data/10e07.htm, https://www.rieselprime.de/Data/10e08.htm, https://www.rieselprime.de/Data/10e09.htm, https://www.rieselprime.de/Data/10e10.htm, https://www.rieselprime.de/Data/10e04a.txt, https://www.rieselprime.de/ziki/Riesel_2_1-300, https://www.rieselprime.de/ziki/Riesel_2_300-2000, https://www.rieselprime.de/ziki/Riesel_2_2000-4000, https://www.rieselprime.de/ziki/Riesel_2_4000-6000, https://www.rieselprime.de/ziki/Riesel_2_6000-8000, https://www.rieselprime.de/ziki/Riesel_2_8000-10000, https://www.rieselprime.de/ziki/Riesel_2_10e4-10e5, https://www.rieselprime.de/ziki/Riesel_2_10e5-10e6, https://www.rieselprime.de/ziki/Riesel_2_10e6-10e7, https://www.rieselprime.de/ziki/Riesel_2_10e7-10e8, https://www.rieselprime.de/ziki/Riesel_2_10e8-10e9, https://www.rieselprime.de/ziki/Riesel_2_10e9-10e10, https://www.rieselprime.de/ziki/Riesel_2_10e10-infinity, https://harvey563.tripod.com/reisels.txt, https://sites.google.com/view/proth-primes, https://www.primegrid.com/forum_thread.php?id=7945, https://www.primegrid.com/stats_321_llr.php, https://www.primegrid.com/primes/primes.php?project=321&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=27&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=121&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://web.archive.org/web/20231223043356/https://www.mersenneforum.org/321search/index.html, https://web.archive.org/web/20110601231527/http://www.bodang.com/12121/, https://web.archive.org/web/20100518081012/http://www.bodang.com/12121/27k/, https://oeis.org/A002235, https://oeis.org/A001770, https://oeis.org/A001771, https://oeis.org/A002236, https://oeis.org/A001772, https://oeis.org/A001773, https://oeis.org/A002237, https://oeis.org/A001774, https://oeis.org/A001775, https://oeis.org/A002238, https://oeis.org/A050537, https://oeis.org/A050538, https://oeis.org/A050539, https://oeis.org/A050540, https://oeis.org/A050541, https://oeis.org/A002240, https://oeis.org/A050543, https://oeis.org/A050544, https://oeis.org/A050545, https://oeis.org/A050546, https://oeis.org/A050547, https://oeis.org/A002242, https://oeis.org/A050549, https://oeis.org/A050550, https://oeis.org/A050551, https://oeis.org/A050552, https://oeis.org/A050553, https://oeis.org/A050554, https://oeis.org/A050555, https://oeis.org/A050556, https://oeis.org/A050557, https://oeis.org/A050558, https://oeis.org/A050559, https://oeis.org/A050560, https://oeis.org/A050561, https://oeis.org/A050562, https://oeis.org/A050563, https://oeis.org/A050564, https://oeis.org/A050565, https://oeis.org/A050566, https://oeis.org/A050567, https://oeis.org/A050568, https://oeis.org/A050569, https://oeis.org/A050570, https://oeis.org/A050571, https://oeis.org/A050572, https://oeis.org/A050573, https://oeis.org/A050574, https://oeis.org/A050575)
List of primes of the form k×10ⁿ+1 with k neither divisible by 10 nor == 2 mod 3 (https://stdkmd.net/nrr/prime/prime_p1.htm, https://stdkmd.net/nrr/prime/prime_p1.txt, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#k10np1, https://oeis.org/A056807, https://oeis.org/A056806, https://oeis.org/A056805, https://oeis.org/A056804, https://oeis.org/A056797, https://oeis.org/A294396, https://oeis.org/A289051, https://oeis.org/A295325, https://oeis.org/A273002, https://oeis.org/A282456, https://oeis.org/A267420, https://oeis.org/A109397, https://oeis.org/A171612, https://oeis.org/A293824, https://oeis.org/A271107, https://oeis.org/A282280, https://oeis.org/A276118, https://oeis.org/A267865, https://oeis.org/A004203, https://oeis.org/A109800, https://oeis.org/A271361, https://oeis.org/A109503, https://oeis.org/A293001, https://oeis.org/A109749, https://oeis.org/A109713 (unfortunately, currently there are no OEIS sequences for k = 21, 24, 27, 31, 34, 36, 37, 43, 45, 46, 48, 51, 52, 57, 58, 61, 64, 67, 72, 73, 75, 76, 78, 79, 81, 82, 84, 85, 87, 91, 93, 94, 96, 97))	List of primes of the form k×10ⁿ−1 with k neither divisible by 10 nor == 1 mod 3 (http://www.noprimeleftbehind.net/gary/primes-kx10n-1.htm, https://stdkmd.net/nrr/prime/prime_m1.htm, https://stdkmd.net/nrr/prime/prime_m1.txt, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#k10nm1, https://oeis.org/A002957, https://oeis.org/A056703, https://oeis.org/A056712, https://oeis.org/A056716, https://oeis.org/A056721, https://oeis.org/A056725, https://oeis.org/A111391, https://oeis.org/A257036, https://oeis.org/A257037, https://oeis.org/A257038, https://oeis.org/A257039, https://oeis.org/A257040, https://oeis.org/A257041 (unfortunately, currently there are no OEIS sequences for 12 ≤ k ≤ 89))
Sierpinski problem (http://www.prothsearch.com/sierp.html, https://www.primegrid.com/forum_thread.php?id=1647, https://www.primegrid.com/forum_thread.php?id=972, https://www.primegrid.com/forum_thread.php?id=1750, https://www.primegrid.com/forum_thread.php?id=5758, https://www.primegrid.com/stats_sob_llr.php, https://www.primegrid.com/stats_psp_llr.php, https://www.primegrid.com/stats_esp_llr.php, https://www.primegrid.com/primes/primes.php?project=SOB&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=PSP&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=ESP&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://web.archive.org/web/20160405211049/http://www.seventeenorbust.com/, https://en.wikipedia.org/wiki/Sierpi%C5%84ski_number, https://t5k.org/glossary/xpage/SierpinskiNumber.html, https://www.rieselprime.de/ziki/Sierpi%C5%84ski_number, https://www.rieselprime.de/ziki/Sierpi%C5%84ski_problem, https://www.rieselprime.de/ziki/Proth_2_Sierpinski, https://www.rieselprime.de/ziki/Proth_2_Count-0, https://mathworld.wolfram.com/SierpinskiNumberoftheSecondKind.html, https://en.wikipedia.org/wiki/Seventeen_or_Bust, https://www.rieselprime.de/ziki/Seventeen_or_Bust, https://web.archive.org/web/20190929190947/https://primes.utm.edu/glossary/xpage/ColbertNumber.html, https://mathworld.wolfram.com/ColbertNumber.html, http://www.numericana.com/answer/primes.htm#sierpinski, http://www.bitman.name/math/article/204 (in Italian), https://www.ams.org/journals/mcom/1983-40-161/S0025-5718-1983-0679453-8/S0025-5718-1983-0679453-8.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_40.pdf), https://www.fq.math.ca/Scanned/33-3/izotov.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_46.pdf), http://www.digizeitschriften.de/download/PPN378850199_0015/PPN378850199_0015___log24.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_213.pdf), https://www.ams.org/journals/mcom/1981-37-155/S0025-5718-1981-0616376-2/S0025-5718-1981-0616376-2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_214.pdf), https://www.ams.org/journals/mcom/1983-41-164/S0025-5718-1983-0717710-7/S0025-5718-1983-0717710-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_215.pdf), http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://oeis.org/A076336), finding and proving the smallest odd k such that k×2ⁿ+1 is composite for all n ≥ 1, the smallest such k is conjectured to be 78557, such k are called Sierpinski numbers	Riesel problem (http://www.prothsearch.com/rieselprob.html, https://www.primegrid.com/forum_thread.php?id=1731, https://www.primegrid.com/stats_trp_llr.php, https://www.primegrid.com/primes/primes.php?project=TRP&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://web.archive.org/web/20061021145019/http://rieselsieve.com/, https://web.archive.org/web/20061021153313/http://stats.rieselsieve.com//queue.php, https://en.wikipedia.org/wiki/Riesel_number, https://t5k.org/glossary/xpage/RieselNumber.html, https://www.rieselprime.de/ziki/Riesel_number, https://www.rieselprime.de/ziki/Riesel_problem_1, https://www.rieselprime.de/ziki/Riesel_problem_2, https://www.rieselprime.de/ziki/Riesel_problem_3, https://www.rieselprime.de/ziki/Riesel_problem_4, https://www.rieselprime.de/ziki/Riesel_2_Riesel, https://www.rieselprime.de/ziki/Riesel_2_Count-0, https://mathworld.wolfram.com/RieselNumber.html, https://en.wikipedia.org/wiki/Riesel_Sieve, https://www.rieselprime.de/ziki/Riesel_Sieve, https://www.rieselprime.de/ziki/PrimeGrid_The_Riesel_Problem, http://www.bitman.name/math/article/203 (in Italian), http://www.iakovlev.org/zip/riesel2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_8.pdf), https://doi.org/10.1016/j.jnt.2008.02.004 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_47.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL16/Ismailescu/ismailescu3.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_51.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL18/Baczkowski/bacz2.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_61.pdf), https://arxiv.org/pdf/1110.4671.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_62.pdf), https://oeis.org/A076337, https://oeis.org/A101036), finding and proving the smallest odd k such that k×2ⁿ−1 is composite for all n ≥ 1, the smallest such k is conjectured to be 509203, such k are called Riesel numbers
Smallest n such that k×2ⁿ+1 is prime (https://oeis.org/A078680, https://oeis.org/A078683 (corresponding primes), https://oeis.org/A033809 (odd k), https://oeis.org/A040076 (n = 0 allowed), https://oeis.org/A225721 (n = 0 allowed, k replaced by k+1), https://oeis.org/A050921 (n = 0 allowed, corresponding primes), https://oeis.org/A046067 (odd k, n = 0 allowed), https://oeis.org/A057025 (odd k, n = 0 allowed, corresponding primes), https://oeis.org/A057192 (prime k, n = 0 allowed), https://oeis.org/A057247 (prime k, corresponding primes), https://oeis.org/A046068 (odd k, n = 0 allowed, second n))	Smallest n such that k×2ⁿ−1 is prime (https://oeis.org/A050412 (k replaced by k−1), https://oeis.org/A052333 (corresponding primes, k replaced by k−1), https://oeis.org/A108129 (odd k), https://oeis.org/A040081 (n = 0 allowed), https://oeis.org/A038699 (n = 0 allowed, corresponding primes), https://oeis.org/A046069 (odd k, n = 0 allowed), https://oeis.org/A057026 (odd k, n = 0 allowed, corresponding primes), https://oeis.org/A128979 (prime k), https://oeis.org/A101050 (prime k, n = 0 allowed), https://oeis.org/A257495 (k such that k−1 is prime, k replaced by k−1), https://oeis.org/A046070 (odd k, n = 0 allowed, second n))
Real Sierpinski problem (http://boincvm.proxyma.ru:30080/test4vm/public/pps_snob_problem.php, https://www.primegrid.com/forum_thread.php?id=9107, https://www.rieselprime.de/ziki/PrimeGrid_New_Sierpi%C5%84ski_Problem), finding and proving the smallest odd k such that k×2ⁿ+1 is composite for all n such that 2ⁿ > k (i.e. all "Proth numbers" of the form k×2ⁿ+1 are composite, i.e. there is no "Proth prime" of the form k×2ⁿ+1), the smallest such k is conjectured to be 78557	Real Riesel problem (http://www.noprimeleftbehind.net/crus/Real-Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Real-Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/Real-Riesel-conjecture-base2-remain.htm, http://www.noprimeleftbehind.net/crus/all-ks-real-riesel-base2.zip, https://www.rieselprime.de/ziki/Riesel_problem_real_1, https://www.mersenneforum.org/showthread.php?t=29051), finding and proving the smallest odd k such that k×2ⁿ−1 is composite for all n such that 2ⁿ > k (i.e. all "Proth numbers of the second kind" of the form k×2ⁿ−1 are composite, i.e. there is no "Proth prime of the second kind" of the form k×2ⁿ−1), the smallest such k is conjectured to be 509203
Dual Sierpinski problem (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://www.mersenneforum.org/showthread.php?t=10761), finding and proving the smallest odd k such that 2ⁿ+k is composite for all n ≥ 1, the smallest such k is conjectured to be 78557	Dual Riesel problem (https://www.mersenneforum.org/showthread.php?t=6545), finding and proving the smallest odd k such that \|2ⁿ−k\| is composite for all n ≥ 1, the smallest such k is conjectured to be 509203
List of primes of the form 2ⁿ+k with odd k (https://sites.google.com/view/proth-primes, https://oeis.org/A057732, https://oeis.org/A059242, https://oeis.org/A057195, https://oeis.org/A057196, https://oeis.org/A102633, https://oeis.org/A102634, https://oeis.org/A057197, https://oeis.org/A057200, https://oeis.org/A057221, https://oeis.org/A057201, https://oeis.org/A057203, https://oeis.org/A157006, https://oeis.org/A157007, https://oeis.org/A156982, https://oeis.org/A247952, https://oeis.org/A247953, https://oeis.org/A220077 (unfortunately, currently there are no OEIS sequences for 37 ≤ k ≤ 99))	List of primes of the form 2ⁿ−k with odd k (https://sites.google.com/view/proth-primes, https://oeis.org/A050414, https://oeis.org/A059608, https://oeis.org/A059609, https://oeis.org/A059610, https://oeis.org/A096817, https://oeis.org/A096818, https://oeis.org/A059612, https://oeis.org/A059611, https://oeis.org/A096819, https://oeis.org/A096820, https://oeis.org/A057202 (allow negative primes), https://oeis.org/A057220 (allow negative primes), https://oeis.org/A356826 (unfortunately, currently there are no OEIS sequences for 25 ≤ k ≤ 99 except k = 29, also for k = 23 only "allow negative primes" version exists in OEIS))
List of primes of the form 10ⁿ+k with k coprime to 10 and not == 2 mod 3 (https://stdkmd.net/nrr/prime/prime_pk.htm, https://stdkmd.net/nrr/prime/prime_pk.txt, https://oeis.org/A049054, https://oeis.org/A088274, https://oeis.org/A088275, https://oeis.org/A095688, https://oeis.org/A108052, https://oeis.org/A108050, https://oeis.org/A108312, https://oeis.org/A107083, https://oeis.org/A107084, https://oeis.org/A135109, https://oeis.org/A135108, https://oeis.org/A108049, https://oeis.org/A108054, https://oeis.org/A135118, https://oeis.org/A135119, https://oeis.org/A135116, https://oeis.org/A135115, https://oeis.org/A135113, https://oeis.org/A135114, https://oeis.org/A135132, https://oeis.org/A135131, https://oeis.org/A137848, https://oeis.org/A135117, https://oeis.org/A110918, https://oeis.org/A135112, https://oeis.org/A135107, https://oeis.org/A110980)	List of primes of the form 10ⁿ−k with k coprime to 10 and not == 1 mod 3 (https://stdkmd.net/nrr/prime/prime_mk.htm, https://stdkmd.net/nrr/prime/prime_mk.txt, https://oeis.org/A089675, https://oeis.org/A095714, https://oeis.org/A092767, https://oeis.org/A108326, https://oeis.org/A108327, https://oeis.org/A108328, https://oeis.org/A108329, https://oeis.org/A108330, https://oeis.org/A108364, https://oeis.org/A108365, https://oeis.org/A178406, https://oeis.org/A178175, https://oeis.org/A178429, https://oeis.org/A178430, https://oeis.org/A108493, https://oeis.org/A108506, https://oeis.org/A178433, https://oeis.org/A177866, https://oeis.org/A178434, https://oeis.org/A178436, https://oeis.org/A178437, https://oeis.org/A178438, https://oeis.org/A108331, https://oeis.org/A108332, https://oeis.org/A178531, https://oeis.org/A178439)
Smallest n such that 2ⁿ+k is prime for odd k (https://oeis.org/A067760, https://oeis.org/A123252 (corresponding primes), https://oeis.org/A094076 (prime k, n = 0 allowed), https://oeis.org/A139758 (prime k, n = 0 allowed, corresponding primes))	Smallest n such that 2ⁿ−k is prime for odd k (https://oeis.org/A096502, https://oeis.org/A096822 (corresponding primes), https://oeis.org/A101462 (prime k), https://oeis.org/A252168 (allow negative primes), https://oeis.org/A276417 (only negative primes), https://oeis.org/A188903 (only negative primes, n replaced by 2ⁿ))
Generalized Sierpinski problems to bases b > 2 (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, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://web.archive.org/web/20231011144408/https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian), https://oeis.org/A123159), finding and proving the smallest k such that k×bⁿ+1 is composite for all n ≥ 1	Generalized Riesel problems to bases b > 2 (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, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, http://www.bitman.name/math/article/2005 (in Italian), https://oeis.org/A273987), finding and proving the smallest k such that k×bⁿ−1 is composite for all n ≥ 1
Sophie Germain prime (https://en.wikipedia.org/wiki/Sophie_Germain_prime, https://t5k.org/glossary/xpage/SophieGermainPrime.html, https://www.rieselprime.de/ziki/Sophie_Germain_prime, https://mathworld.wolfram.com/SophieGermainPrime.html, https://www.numbersaplenty.com/set/Sophie_Germain_prime/, https://oeis.org/A005384, https://t5k.org/top20/page.php?id=2, https://t5k.org/primes/search.php?Comment=Sophie%20Germain&OnList=all&Number=1000000&Style=HTML, https://www.primegrid.com/forum_thread.php?id=1450, https://www.primegrid.com/stats_sgs_llr.php, https://www.primegrid.com/primes/primes.php?project=SGS&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.rieselprime.de/Related/RieselTwinSG.htm, https://www.primepuzzles.net/problems/prob_049.htm, https://www.rieselprime.de/ziki/PrimeGrid_Sophie_Germain_Search)	Sophie Germain prime of the second kind (https://oeis.org/A005382, https://harvey563.tripod.com/cunninghams.txt)
Safe prime (https://en.wikipedia.org/wiki/Safe_prime, https://t5k.org/glossary/xpage/SafePrime.html, https://oeis.org/A005385, https://www.rieselprime.de/Related/RieselTwinSG.htm, https://www.primepuzzles.net/problems/prob_049.htm)	Safe prime of the second kind (https://oeis.org/A005383, https://harvey563.tripod.com/cunninghams.txt)
Cunningham chain of the first kind (https://en.wikipedia.org/wiki/Cunningham_chain, https://t5k.org/glossary/xpage/CunninghamChain.html, https://mathworld.wolfram.com/CunninghamChain.html, https://t5k.org/top20/page.php?id=19, https://t5k.org/primes/search.php?Comment=Cunningham%20chain&OnList=all&Number=1000000&Style=HTML, https://www.pzktupel.de/JensKruseAndersen/CC.php, http://www.primerecords.dk/Cunningham_Chain_records.htm, https://oeis.org/A005602, https://oeis.org/A057331)	Cunningham chain of the second kind (https://en.wikipedia.org/wiki/Cunningham_chain, https://t5k.org/glossary/xpage/CunninghamChain.html, https://mathworld.wolfram.com/CunninghamChain.html, https://t5k.org/top20/page.php?id=20, https://t5k.org/primes/search.php?Comment=Cunningham%20chain&OnList=all&Number=1000000&Style=HTML, https://www.pzktupel.de/JensKruseAndersen/CC.php, http://www.primerecords.dk/Cunningham_Chain_records.htm, https://oeis.org/A005603, https://oeis.org/A057330)
Pierpont prime (class 1− prime) (https://en.wikipedia.org/wiki/Pierpont_prime, https://t5k.org/glossary/xpage/PierpontPrime.html, https://mathworld.wolfram.com/PierpontPrime.html, https://www.numbersaplenty.com/set/Pierpont_prime/, https://oeis.org/A005109)	Pierpont prime of the second kind (class 1+ prime) (https://oeis.org/A005105)
Pollard P−1 integer factorization method (https://en.wikipedia.org/wiki/Pollard%27s_p_%E2%88%92_1_algorithm, https://www.rieselprime.de/ziki/P-1_factorization_method, https://mathworld.wolfram.com/Pollardp-1FactorizationMethod.html, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "Pollard p-1:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=p-1&maxrows=10000, http://www.numericana.com/answer/factoring.htm#p-1, http://factordb.com/listecm.php?c=2, http://www.loria.fr/~zimmerma/records/Pminus1.html, https://web.archive.org/web/20021015212913/http://www.users.globalnet.co.uk/~aads/Pminus1.html, https://web.archive.org/web/20231002022529/https://colin.barker.pagesperso-orange.fr/lpa/big_pm1.htm, https://www.mersenne.org/report_pminus1/, https://www.mersenne.ca/userfactors/pm1/1, https://www.mersenne.ca/smooth.php, https://www.mersenne.ca/p1missed.php, https://www.mersenne.ca/prob.php)	Williams P+1 integer factorization method (https://en.wikipedia.org/wiki/Williams%27s_p_%2B_1_algorithm, https://www.rieselprime.de/ziki/P%2B1_factorization_method, https://mathworld.wolfram.com/WilliamspPlus1FactorizationMethod.html, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "p+1:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=p%2b1&maxrows=10000, http://www.numericana.com/answer/factoring.htm#p+1, http://factordb.com/listecm.php?c=3, http://www.loria.fr/~zimmerma/records/Pplus1.html, https://www.mersenne.org/report_pplus1/, https://www.mersenne.ca/userfactors/pp1/1, https://www.mersenne.ca/pplus1.php)

(also all known large definitely twin prime pairs (https://en.wikipedia.org/wiki/Twin_prime, https://t5k.org/glossary/xpage/TwinPrime.html, https://www.rieselprime.de/ziki/Twin_prime, https://mathworld.wolfram.com/TwinPrimes.html, https://www.numbersaplenty.com/set/twin_primes/, https://oeis.org/A001097, https://oeis.org/A077800, https://oeis.org/A001359, https://oeis.org/A006512, https://t5k.org/lists/small/1ktwins.txt, https://t5k.org/lists/small/10ktwins.txt, https://t5k.org/lists/small/100ktwins.txt, https://t5k.org/top20/page.php?id=1, https://t5k.org/primes/search.php?Comment=Twin&OnList=all&Number=1000000&Style=HTML, https://pzktupel.de/ktuplets.php#largest2, https://pzktupel.de/KTHIST/kt002.php, https://pzktupel.de/counting/PI_02.php, https://pzktupel.de/RecordGaps/GAP02.php, https://pzktupel.de/RecordGaps/GAP02FO.html, http://www.fermatquotient.com/PrimLuecken/ZwillingsRekordLuecken.txt, http://www.fermatquotient.com/PrimLuecken/ZwillingsErstDiff.txt, https://web.archive.org/web/20090220132900/https://www.primegrid.com/stats_tps_llr.php, https://www.primegrid.com/primes/primes.php?project=TPS&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, http://www.noprimeleftbehind.net/tps/, http://www.noprimeleftbehind.net:12000/all.html, http://www.noprimeleftbehind.net:13000/all.html, https://www.rieselprime.de/Related/RieselTwinSG.htm, https://www.rieselprime.de/RPS/Efforts/9ks.htm, https://www.rieselprime.de/Related/FirstKTwin.htm, http://www.noprimeleftbehind.net/gary/twins100K.htm, http://www.noprimeleftbehind.net/gary/twins1M.htm, https://www.primepuzzles.net/problems/prob_049.htm, https://stdkmd.net/nrr/prime/prime_tw.htm, https://stdkmd.net/nrr/prime/prime_tw.txt, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#qrtwin, https://pzktupel.de/SMPDF/SM02.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_445.pdf)) (p,q) are a prime p such that p+1 has many factors and a prime q such that q−1 has many factors, in fact, p+1 = q−1)

All these numbers are strong probable primes (https://t5k.org/prove/prove2_3.html, https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test, https://en.wikipedia.org/wiki/Strong_pseudoprime, https://t5k.org/glossary/xpage/MillersTest.html, https://t5k.org/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://faculty.lynchburg.edu/~nicely/misc/mpzspsp.html, http://factordb.com/prooffailed.php, 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/A020231, https://oeis.org/A020233, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773, https://oeis.org/A014233, https://oeis.org/A006945, https://oeis.org/A089825, 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 ≤ 64 (i.e. 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/psps.txt), and strong Lucas probable primes (https://en.wikipedia.org/wiki/Strong_Lucas_pseudoprime, 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://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) to 10¹⁶ (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, http://pseudoprime.com/dopo.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_318.pdf))), and since all of them are > 10²⁵⁰⁰⁰, thus the property that they are in fact composite is less than 10⁻²⁰⁰⁰ (but still not zero!), see https://t5k.org/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) (bases 11, 13, 16, 22, 30 have no unsolved families but have unproven probable primes, thus, we can 99.999999...999999% (with >2000 9's) say that the minimal sets for bases 11, 13, 16, 22, 30 are these five sets, but we still cannot 100% say this)

A reference of many types of pseudoprimes: http://ntheory.org/pseudoprimes.html; also references of data for all base 2 Fermat pseudoprimes < 2⁶⁴: http://www.cecm.sfu.ca/Pseudoprimes/index-2-to-64.html and https://web.archive.org/web/20220921163920/http://www.janfeitsma.nl/math/psp2/index; also examples of strong pseudoprimes to many bases: https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1192971-8/S0025-5718-1993-1192971-8.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_41.pdf), https://arxiv.org/pdf/1207.0063.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_42.pdf), https://arxiv.org/pdf/1509.00864.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_43.pdf), https://www.sciencedirect.com/science/article/pii/S0747717185710425?via%3Dihub (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_44.pdf), https://t5k.org/curios/page.php?number_id=4265, https://t5k.org/prove/prove2_3.html, https://faculty.lynchburg.edu/~nicely/misc/mpzspsp.html, http://factordb.com/prooffailed.php, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773

If 57₆₂₆₆₈ (base 11) is in fact composite, then 5{7} will be an additional unsolved family for base 11.
If C5₂₃₇₅₅C (base 13) is in fact composite, then C{5}C will be an additional unsolved family for base 13.
If 80₃₂₀₁₇111 (base 13) is in fact composite, then 8{0}111 will be an additional unsolved family for base 13.
If 95₁₉₇₄₂₀ (base 13) is in fact composite, then 9{5} will be an additional unsolved family for base 13.
If A3₅₉₂₁₉₇A (base 13) is in fact composite, then A{3}A will be an additional unsolved family for base 13.
If DB₃₂₂₃₄ (base 16) is in fact composite, then D{B} will be an additional unsolved family for base 16.
If 4₇₂₇₈₅DD (base 16) is in fact composite, then {4}DD will be an additional unsolved family for base 16.
If 3₁₁₆₁₃₇AF (base 16) is in fact composite, then {3}AF will be an additional unsolved family for base 16.
If BK₂₂₀₀₁5 (base 22) is in fact composite, then B{K}5 will be an additional unsolved family for base 22.
If 5₁₉₃₉₁6F (base 26) is in fact composite, then {5}6F will be an additional unsolved family for base 26.
If 7₂₀₂₇₉OL (base 26) is in fact composite, then {7}OL will be an additional unsolved family for base 26.
If LD0₂₀₉₇₅7 (base 26) is in fact composite, then LD{0}7 will be an additional unsolved family for base 26.
If 6K₂₃₃₀₀5 (base 26) is in fact composite, then 6{K}5 will be an additional unsolved family for base 26.
If J0₄₄₃₀₃KCB (base 26) is in fact composite, then J{0}KCB will be an additional unsolved family for base 26.
If M0₆₁₁₈₆2BB (base 26) is in fact composite, then M{0}2BB will be an additional unsolved family for base 26.
If N6₂₄₀₅₁LR (base 28) is in fact composite, then N{6}LR will be an additional unsolved family for base 28.
If 5OA₃₁₂₃₈F (base 28) is in fact composite, then 5O{A}F will be an additional unsolved family for base 28 (unless a prime in the family O{A}F is known, since the family 5O{A}F is covered by the unsolved family O{A}F (if the prime in the family 5O{A}F has longer length than the prime in the family O{A}F) (i.e. the family O{A}F is a subfamily of the family 5O{A}F), and there is still a possibility that the smallest prime in the family 5O{A}F has longer length than the smallest prime in the family O{A}F, thus we still cannot definitely say that there are 25529 minimal primes in base 28 even if we assume the heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, https://web.archive.org/web/20230911032453/https://www.utm.edu/staff/caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf), https://arxiv.org/pdf/2103.04483.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_113.pdf)) that all unsolved families have a prime, this is reasonable, references: https://t5k.org/mersenne/heuristic.html, https://t5k.org/notes/faq/NextMersenne.html, https://t5k.org/glossary/xpage/Repunit.html, https://web.archive.org/web/20100628035147/http://www.math.niu.edu/~rusin/known-math/98/exp_primes, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906, https://mathoverflow.net/questions/268918/density-of-primes-in-sequences-of-the-form-anb, https://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Mathematics/2023_September_25#Are_there_infinitely_many_primes_of_the_form_1000%E2%80%A60007.2C_333%E2%80%A63331.2C_7111%E2%80%A6111.2C_or_3444%E2%80%A64447_in_base_10?, https://arxiv.org/pdf/2307.07894.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_203.pdf), https://web.archive.org/web/20231002020455/http://yves.gallot.pagesperso-orange.fr/papers/weight.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_216.pdf), https://www.mersenneforum.org/showthread.php?t=12327, https://oeis.org/A234285 (the comment by Farideh Firoozbakht, although this comment is not true, there is no prime for s = 509203 and s = −78557, s = 509203 has a covering set of {3, 5, 7, 13, 17, 241}, and s = −78557 has a covering set of {3, 5, 7, 13, 19, 37, 73}), https://www.mersenneforum.org/showpost.php?p=564786&postcount=3, https://www.mersenneforum.org/showpost.php?p=461665&postcount=7, https://www.mersenneforum.org/showpost.php?p=344985&postcount=293, https://www.mersenneforum.org/showpost.php?p=625978&postcount=1027, also the graphs https://t5k.org/gifs/lg_lg_Mn.gif (for the family {1} in base b = 2) and https://t5k.org/gifs/repunits.gif (for the family {1} in base b = 10) and https://www.mersenneforum.org/attachment.php?attachmentid=4010&d=1642088235 (for the family 2{0}1 in base b = 3), we can only definitely say that there are either 25528 or 25529 minimal primes in base 28 if we assume the heuristic argument (https://en.wikipedia.org/wiki/Heuristic_argument, https://t5k.org/glossary/xpage/Heuristic.html, https://mathworld.wolfram.com/Heuristic.html, https://web.archive.org/web/20230911032453/https://www.utm.edu/staff/caldwell/preprints/Heuristics.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_112.pdf), https://arxiv.org/pdf/2103.04483.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_113.pdf)) that all unsolved families have a prime).
If O4O₉₄₅₃₅9 (base 28) is in fact composite, then O4{O}9 will be an additional unsolved family for base 28.
If I0₂₄₆₀₈D (base 30) is in fact composite, then I{0}D will be an additional unsolved family for base 30.
If 7K₂₆₅₆₇Z (base 36) is in fact composite, then 7{K}Z will be an additional unsolved family for base 36.
If S0₇₅₀₀₇8H (base 36) is in fact composite, then S{0}8H will be an additional unsolved family for base 36.
If P₈₁₉₉₃SZ (base 36) is in fact composite, then {P}SZ will be an additional unsolved family for base 36.

For the files in this page:

File "unprovenPRP b": The unproven probable primes which are minimal primes in base b (assuming their primality), expressed as base b strings with x(y^n)z format (which in the README files it is denoted as "xy_nz") (y^n means n copies of y, e.g. 34(5^6)78 is 3455555578, this n is written in decimal)
File "unprovenPRPalgebraic b": The algebraic ((a×bⁿ+c)/gcd(a+c,b−1)) form of the unproven probable primes which are minimal primes in base b (assuming their primality)
File "primoinput b n": The input file for Primo for the nth minimal prime in base b (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309) (local copy from factordb (http://factordb.com/, https://www.rieselprime.de/ziki/Factoring_Database)), they should be renamed to ".in" files before inputting to Primo, e.g. file "primoinput11_1068" is the input file for Primo for the 1068th minimal prime in base 11, i.e. the input file for Primo for the probable prime 57₆₂₆₆₈ in base 11, which equals the probable prime (57×11⁶²⁶⁶⁸−7)/10 (for more input files for Primo in factordb, see http://factordb.com/primobatch.php and http://factordb.com/primobatch.php?digits=300&files=10&parts=1&start=Generate+zip and http://factordb.com/primobatch.php?digits=300&files=32000&parts=32&start=Generate+zip). (Note: There is no input file for Primo for the probable prime 95₁₉₇₄₂₀ in base 13 (which equals the probable prime (113×13¹⁹⁷⁴²⁰−5)/12) and A3₅₉₂₁₉₇A in base 13 (which equals the probable prime (41×13⁵⁹²¹⁹⁸+27)/4) since these two probable primes are too large (>10¹⁹⁹⁹⁹⁹) to be PRP-tested in factordb, and factordb does not have input file for Primo 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 input file for Primo for numbers with status "PRP" (i.e. in http://factordb.com/listtype.php?t=1))

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