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unproven-probable-primes

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

Primality of these probable primes (https://en.wikipedia.org/wiki/Probable_prime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Probable_prime, https://mathworld.wolfram.com/ProbablePrime.html, http://www.primenumbers.net/prptop/prptop.php, https://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) in the minimal sets for bases b = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36 (the "easy" bases (bases b with ≤ 150 minimal primes > 10299 (base b = 26 has 82 known minimal (probable) primes > 10299 and 4 unsolved families, base b = 36 has 75 known minimal (probable) primes > 10299 and 4 unsolved families, base b = 17 has 99 known minimal (probable) primes > 10299 and 18 unsolved families, base b = 21 has 80 known minimal (probable) primes > 10299 and 12 unsolved families, base b = 19 has 201 known minimal (probable) primes > 10299 and 23 unsolved families))) are not certificated yet (technically, probable (https://en.wikipedia.org/wiki/Probabilistic_algorithm) primality tests (https://t5k.org/prove/prove2.html) such as Miller–Rabin 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, https://miller-rabin.appspot.com/, http://www.pi-e.de/Miller-Rabin-Pseudoprimzahlen.htm (in German), 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/A020230, https://oeis.org/A020231, https://oeis.org/A020232, https://oeis.org/A020233, https://oeis.org/A020234, https://oeis.org/A020235, https://oeis.org/A020236, https://oeis.org/A020237, https://oeis.org/A020238, https://oeis.org/A020239, https://oeis.org/A020240, https://oeis.org/A020241, https://oeis.org/A020242, https://oeis.org/A020243, https://oeis.org/A020244, https://oeis.org/A020245, https://oeis.org/A020246, https://oeis.org/A020247, https://oeis.org/A020248, https://oeis.org/A020249, https://oeis.org/A020250, https://oeis.org/A020251, https://oeis.org/A020252, https://oeis.org/A020253, https://oeis.org/A020254, https://oeis.org/A020255, https://oeis.org/A020256, https://oeis.org/A020257, https://oeis.org/A020258, https://oeis.org/A020259, https://oeis.org/A020260, https://oeis.org/A020261, https://oeis.org/A020262, 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) and Baillie–PSW primality test (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)) are used for these numbers, the Baillie–PSW primality test is the combine of the Miller–Rabin 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, https://miller-rabin.appspot.com/, http://www.pi-e.de/Miller-Rabin-Pseudoprimzahlen.htm (in German), 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/A020230, https://oeis.org/A020231, https://oeis.org/A020232, https://oeis.org/A020233, https://oeis.org/A020234, https://oeis.org/A020235, https://oeis.org/A020236, https://oeis.org/A020237, https://oeis.org/A020238, https://oeis.org/A020239, https://oeis.org/A020240, https://oeis.org/A020241, https://oeis.org/A020242, https://oeis.org/A020243, https://oeis.org/A020244, https://oeis.org/A020245, https://oeis.org/A020246, https://oeis.org/A020247, https://oeis.org/A020248, https://oeis.org/A020249, https://oeis.org/A020250, https://oeis.org/A020251, https://oeis.org/A020252, https://oeis.org/A020253, https://oeis.org/A020254, https://oeis.org/A020255, https://oeis.org/A020256, https://oeis.org/A020257, https://oeis.org/A020258, https://oeis.org/A020259, https://oeis.org/A020260, https://oeis.org/A020261, https://oeis.org/A020262, 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) with base b = 2 and the strong Lucas primality test (https://en.wikipedia.org/wiki/Strong_Lucas_pseudoprime, https://mathworld.wolfram.com/StrongLucasPseudoprime.html, http://ntheory.org/data/slpsps-baillie.txt, http://www.hoegge.dk/lucasselfridgeprps.txt, https://oeis.org/A217255) with parameters P = 1 and Q = (1−D)/4, where D is the first number in the sequence 5, −7, 9, −11, 13, −15, 17, −19, ... such that the Jacobi symbol (https://en.wikipedia.org/wiki/Jacobi_symbol, https://t5k.org/glossary/xpage/JacobiSymbol.html, https://mathworld.wolfram.com/JacobiSymbol.html, http://www.numericana.com/answer/reciprocity.htm#legendre, http://math.fau.edu/richman/jacobi.htm, https://oeis.org/A110242, https://oeis.org/A110247, https://oeis.org/A157412) (D|N) = −1 (and thus these numbers are only probable primes and not definitely primes (https://en.wikipedia.org/wiki/Provable_prime, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#primenumbers, http://factordb.com/listtype.php?t=4), i.e. they might be pseudoprimes (https://en.wikipedia.org/wiki/Pseudoprime, https://t5k.org/glossary/xpage/Pseudoprime.html, https://www.rieselprime.de/ziki/Pseudoprime, https://mathworld.wolfram.com/Pseudoprime.html, http://ntheory.org/pseudoprimes.html, http://www.numericana.com/answer/pseudo.htm, http://www.pseudoprime.com/pseudo.html, https://www.mathpages.com/home/kmath003/kmath003.htm, https://www.mersenneforum.org/showthread.php?t=28839, https://www.mersenneforum.org/showthread.php?t=10476)), because of their sizes and neither N−1 (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) nor N+1 (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) can be ≥ 1/4 factored (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) (i.e. the products of the known prime factors of both N−1 and N+1 are < the fourth roots of them) (i.e. they are ordinary primes (https://t5k.org/glossary/xpage/OrdinaryPrime.html)), all known primality tests (https://en.wikipedia.org/wiki/Primality_test, https://www.rieselprime.de/ziki/Primality_test, https://mathworld.wolfram.com/PrimalityTest.html) run far too slowly (the CPU time (https://en.wikipedia.org/wiki/CPU_time) is longer than the life expectancy of human (https://en.wikipedia.org/wiki/Life_expectancy) for numbers > 10100000, and longer than the age of the universe (https://en.wikipedia.org/wiki/Age_of_the_universe) for numbers > 10500000, and longer than one quettasecond (https://en.wikipedia.org/wiki/Quetta-) for numbers > 103000000, even if we can do 109 bitwise operations (https://en.wikipedia.org/wiki/Bitwise_operation) per second (https://en.wikipedia.org/wiki/Second), see https://www.mersenneforum.org/showpost.php?p=627117&postcount=1) to run on these numbers, the only known primality test with polynomial time (https://en.wikipedia.org/wiki/Polynomial_time, https://mathworld.wolfram.com/PolynomialTime.html) of the number of digits is Agrawal–Kayal–Saxena primality test (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)), but it has time complexity (https://en.wikipedia.org/wiki/Time_complexity) O(log(n)12) (where 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), 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)) and if we can do 109 bitwise operations (https://en.wikipedia.org/wiki/Bitwise_operation) per second (https://en.wikipedia.org/wiki/Second), use this test to prove the primality of a 5000-digit (in decimal) prime needs 5.422859049×1039 seconds (https://en.wikipedia.org/wiki/Second), or 1.719577324×1032 years (https://en.wikipedia.org/wiki/Year), much longer than the age of the universe (https://en.wikipedia.org/wiki/Age_of_the_universe), thus to do this test is still impractically, also a near-polynomial time (although not completely polynomial time) primality test, Adleman–Pomerance–Rumely primality test (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), which has time complexity O(log(n)log(log(log(n)))) and also too large, for the difference of the large definitely primes (https://en.wikipedia.org/wiki/Provable_prime, https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#primenumbers, http://factordb.com/listtype.php?t=4) and the large probable primes (https://en.wikipedia.org/wiki/Probable_prime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Probable_prime, https://mathworld.wolfram.com/ProbablePrime.html, http://www.primenumbers.net/prptop/prptop.php, https://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), you should know the difference of probable primes and definitely primes (see https://www.mersenneforum.org/showpost.php?p=651069&postcount=3 and https://www.mersenneforum.org/showpost.php?p=572047&postcount=239), you can compare the top definitely primes page (https://t5k.org/primes/lists/all.txt) and the top probable primes page (http://www.primenumbers.net/prptop/prptop.php), also you can compare the definitely primes with ≥ 100000 decimal digits in factordb (http://factordb.com/listtype.php?t=4&mindig=100000&perpage=5000&start=0) and the probable primes with ≥ 100000 decimal digits in factordb (http://factordb.com/listtype.php?t=1&mindig=100000&perpage=5000&start=0), http://factordb.com/nmoverview.php?method=1&digits=100000&perpage=500&skip=0 is the primes with ≥ 100000 decimal digits in factordb which are proven primes by the N−1 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), http://factordb.com/nmoverview.php?method=2&digits=100000&perpage=500&skip=0 is the primes with ≥ 100000 decimal digits in factordb which are proven primes by the N+1 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), also see https://web.archive.org/web/20240305200806/https://stdkmd.net/nrr/prime/primesize.txt and https://web.archive.org/web/20240305201054/https://stdkmd.net/nrr/prime/primesize.zip (see which numbers have "-", "+", "-proven", "+proven" in the "note" column), also see https://stdkmd.net/nrr/prime/prime_all.htm and https://stdkmd.net/nrr/prime/prime_all.txt (see which numbers have "pr" in the "status" column), also see https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm (compare the sections "Prime numbers" and "Probable prime numbers")), all of these probable primes are > 1025000, if they are in fact primes, then they are minimal primes to the corresponding bases.

If we want to use the classical tests (https://t5k.org/prove/prove3.html) to prove the primality of a large probable prime N, then we must factor N−1 or N+1 to the factored part ≥ 1/4, see https://www.mersenneforum.org/showpost.php?p=529633&postcount=410 and https://www.mersenneforum.org/showpost.php?p=534290&postcount=412 and https://www.mersenneforum.org/showpost.php?p=538954&postcount=414 and https://www.mersenneforum.org/showpost.php?p=564758&postcount=428 and https://stdkmd.net/nrr/repunit/changes200401.htm (the related numbers for the known repunit probable primes) and https://stdkmd.net/nrr/cert/1/ and https://stdkmd.net/nrr/cert/2/ and https://stdkmd.net/nrr/cert/3/ and https://stdkmd.net/nrr/cert/4/ and https://stdkmd.net/nrr/cert/5/ and https://stdkmd.net/nrr/cert/6/ and https://stdkmd.net/nrr/cert/7/ and https://stdkmd.net/nrr/cert/8/ and https://stdkmd.net/nrr/cert/9/ (e.g. the large prime factor of Φ4807(10) (where Φ is the cyclotomic polynomial (https://en.wikipedia.org/wiki/Cyclotomic_polynomial, https://mathworld.wolfram.com/CyclotomicPolynomial.html, http://www.numericana.com/answer/polynomial.htm#cyclotomic, https://stdkmd.net/nrr/repunit/repunitnote.htm#cyclotomic, https://oeis.org/A013595, https://oeis.org/A013596, https://oeis.org/A253240) is related to the prime 1480701, the large prime factor of 1521961 is related to the prime 93219507, the large prime factor of 1642957 is related to the prime 3128907, the large prime factor of 2012692 is related to the prime 2012693, the large prime factor of Φ5014(10) (where Φ is the cyclotomic polynomial (https://en.wikipedia.org/wiki/Cyclotomic_polynomial, https://mathworld.wolfram.com/CyclotomicPolynomial.html, http://www.numericana.com/answer/polynomial.htm#cyclotomic, https://stdkmd.net/nrr/repunit/repunitnote.htm#cyclotomic, https://oeis.org/A013595, https://oeis.org/A013596, https://oeis.org/A253240) is related to the prime 231002709, the large prime factor of Φ7884(10) (where Φ is the cyclotomic polynomial (https://en.wikipedia.org/wiki/Cyclotomic_polynomial, https://mathworld.wolfram.com/CyclotomicPolynomial.html, http://www.numericana.com/answer/polynomial.htm#cyclotomic, https://stdkmd.net/nrr/repunit/repunitnote.htm#cyclotomic, https://oeis.org/A013595, https://oeis.org/A013596, https://oeis.org/A253240) is related to the prime 37157673, the large prime factor of 680537387 is related to the prime 27237407, the large prime factor of Φ1283(10) (where Φ is the cyclotomic polynomial (https://en.wikipedia.org/wiki/Cyclotomic_polynomial, https://mathworld.wolfram.com/CyclotomicPolynomial.html, http://www.numericana.com/answer/polynomial.htm#cyclotomic, https://stdkmd.net/nrr/repunit/repunitnote.htm#cyclotomic, https://oeis.org/A013595, https://oeis.org/A013596, https://oeis.org/A253240) is related to the prime 6851313, the large prime factor of Φ2907(10) (where Φ is the cyclotomic polynomial (https://en.wikipedia.org/wiki/Cyclotomic_polynomial, https://mathworld.wolfram.com/CyclotomicPolynomial.html, http://www.numericana.com/answer/polynomial.htm#cyclotomic, https://stdkmd.net/nrr/repunit/repunitnote.htm#cyclotomic, https://oeis.org/A013595, https://oeis.org/A013596, https://oeis.org/A253240) is related to the prime 791162721, the large prime factor of Φ11470(10) (where Φ is the cyclotomic polynomial (https://en.wikipedia.org/wiki/Cyclotomic_polynomial, https://mathworld.wolfram.com/CyclotomicPolynomial.html, http://www.numericana.com/answer/polynomial.htm#cyclotomic, https://stdkmd.net/nrr/repunit/repunitnote.htm#cyclotomic, https://oeis.org/A013595, https://oeis.org/A013596, https://oeis.org/A253240) is related to the prime 80573481, the large prime factor of 83542165427 (a number in a non-simple family 8{3}1{6}7) is related to the prime 1325603), and except 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) with the primes up to certain limit (say 1016) and the algebra factors (https://en.wikipedia.org/w/index.php?title=Factorization&oldid=1143370673#Factoring_other_polynomials, https://mathworld.wolfram.com/PolynomialFactorization.html, http://www.numericana.com/answer/factoring.htm#special, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization, https://stdkmd.net/nrr/1/10004.htm#about_algebraic, https://stdkmd.net/nrr/1/10008.htm#about_algebraic, https://stdkmd.net/nrr/1/13333.htm#about_algebraic, https://stdkmd.net/nrr/3/39991.htm#about_algebraic, https://stdkmd.net/nrr/4/40001.htm#about_algebraic, https://stdkmd.net/nrr/4/49992.htm#about_algebraic, https://stdkmd.net/nrr/5/53333.htm#about_algebraic, https://stdkmd.net/nrr/5/53335.htm#about_algebraic, https://stdkmd.net/nrr/5/54444.htm#about_algebraic, https://stdkmd.net/nrr/5/55552.htm#about_algebraic, https://stdkmd.net/nrr/7/71111.htm#about_algebraic, https://stdkmd.net/nrr/7/79999.htm#about_algebraic, https://stdkmd.net/nrr/8/83333.htm#about_algebraic, https://stdkmd.net/nrr/8/83336.htm#about_algebraic, https://stdkmd.net/nrr/8/88878.htm#about_algebraic, https://stdkmd.net/nrr/8/88889.htm#about_algebraic, https://stdkmd.net/nrr/8/89996.htm#about_algebraic, https://stdkmd.net/nrr/8/89999.htm#about_algebraic, https://stdkmd.net/nrr/9/99919.htm#about_algebraic, https://stdkmd.net/nrr/9/99991.htm#about_algebraic, https://stdkmd.net/nrr/9/99992.htm#about_algebraic, http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=algebraic&maxrows=10000, https://sites.google.com/view/algebraic-factors-of-xn-kyn, https://sites.google.com/view/factorsofk2n-1foroddk20000, https://brnikat.com/nums/cullen_woodall/algebraic.txt, https://www.mersenneforum.org/showpost.php?p=96560&postcount=99, https://www.mersenneforum.org/showpost.php?p=96651&postcount=101, https://www.mersenneforum.org/showthread.php?t=21916, https://www.mersenneforum.org/showpost.php?p=196598&postcount=492, https://www.mersenneforum.org/showpost.php?p=203083&postcount=149, https://www.mersenneforum.org/showpost.php?p=206065&postcount=192, https://www.mersenneforum.org/showpost.php?p=208044&postcount=260, https://www.mersenneforum.org/showpost.php?p=210533&postcount=336, https://www.mersenneforum.org/showpost.php?p=452132&postcount=66, https://www.mersenneforum.org/showpost.php?p=451337&postcount=32, https://www.mersenneforum.org/showpost.php?p=208852&postcount=227, https://www.mersenneforum.org/showpost.php?p=232904&postcount=604, https://www.mersenneforum.org/showpost.php?p=383690&postcount=1, https://www.mersenneforum.org/showpost.php?p=207886&postcount=253, https://www.mersenneforum.org/showpost.php?p=452819&postcount=1445, https://www.numberempire.com/factoringcalculator.php, https://www.alpertron.com.ar/POLFACT.HTM, https://www.emathhelp.net/calculators/algebra-2/factoring-calculator/) (e.g. difference-of-two-squares factorization (https://en.wikipedia.org/wiki/Difference_of_two_squares), sum/difference-of-two-cubes factorization (https://en.wikipedia.org/wiki/Sum_of_two_cubes), Sophie Germain's identity (https://en.wikipedia.org/wiki/Sophie_Germain%27s_identity, https://www.theoremoftheday.org/Binomial/GermainId/TotDGermainIdentity.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_478.pdf)), difference-of-two-nth-powers factorization with n > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html), sum/difference-of-two-nth-powers factorization with odd n > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html), Aurifeuillean factorization (https://en.wikipedia.org/wiki/Aurifeuillean_factorization, https://www.rieselprime.de/ziki/Aurifeuillian_factor, https://mathworld.wolfram.com/AurifeuilleanFactorization.html, http://www.numericana.com/answer/numbers.htm#aurifeuille, https://web.archive.org/web/20231002141924/http://colin.barker.pagesperso-orange.fr/lpa/cycl_fac.htm, http://list.seqfan.eu/oldermail/seqfan/2017-March/017363.html, http://myfactorcollection.mooo.com:8090/source/cyclo.cpp, http://myfactorcollection.mooo.com:8090/LCD_2_199, http://myfactorcollection.mooo.com:8090/LCD_2_998, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/source/cyclo.cpp, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/LucasCD/LCD_2_199, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/LucasCD/LCD_2_998, https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean, https://www.unshlump.com/hcn/aurif.html, https://www.mersenneforum.org/showthread.php?t=10439, https://www.mersenneforum.org/showpost.php?p=515828&postcount=8, https://maths-people.anu.edu.au/~brent/pd/rpb135.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_97.pdf), https://www.ams.org/journals/mcom/2006-75-253/S0025-5718-05-01766-7/S0025-5718-05-01766-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_138.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb127.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_164.pdf), https://www.jams.jp/scm/contents/Vol-2-3/2-3-16.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_167.pdf), https://web.archive.org/web/20130702000532/http://xyyxf.at.tut.by/aurifeuillean.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_443.pdf)), and the algebra factors of the Cunningham number (https://en.wikipedia.org/wiki/Cunningham_number, https://mathworld.wolfram.com/CunninghamNumber.html, https://www.numbersaplenty.com/set/Cunningham_number/, https://en.wikipedia.org/wiki/Cunningham_Project, https://t5k.org/glossary/xpage/CunninghamProject.html, https://www.rieselprime.de/ziki/Cunningham_project, https://oeis.org/wiki/OEIS_sequences_needing_factors#Cunningham_numbers (sections with integer b), https://homes.cerias.purdue.edu/~ssw/cun/index.html, https://maths-people.anu.edu.au/~brent/factors.html, https://web.archive.org/web/20190315214330/http://cage.ugent.be/~jdemeyer/cunningham/, http://myfactors.mooo.com/, https://doi.org/10.1090/conm/022, https://www.mersenneforum.org/attachment.php?attachmentid=7727&d=1330555980 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_7.pdf), https://homes.cerias.purdue.edu/~ssw/cun/mine.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_6.pdf), http://homes.cerias.purdue.edu/~ssw/cun1.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_71.pdf)) bn±1 (see https://stdkmd.net/nrr/repunit/repunitnote.htm and https://brnikat.com/nums/cullen_woodall/algebraic.txt)), we can use the elliptic-curve factorization method (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, https://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), https://arxiv.org/pdf/1004.3366.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_492.pdf)) (elliptic-curve factorization method need to calculate the group order, for the tools to calculate the group order, see http://myfactorcollection.mooo.com:8090/calculators.html (section "Group Order") and http://factordb.com/groupcalc.php and https://www.mersenneforum.org/showthread.php?t=14184 (a Magma script, you can use online Magma calculator (http://magma.maths.usyd.edu.au/calc/) to run)) or the Pollard P−1 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) or the Williams P+1 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) or the Pollard rho method (https://en.wikipedia.org/wiki/Pollard%27s_rho_algorithm, https://www.rieselprime.de/ziki/Rho_factorization_method, https://mathworld.wolfram.com/PollardRhoFactorizationMethod.html) or the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, 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/) or the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, 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/) or the quadratic sieve (https://en.wikipedia.org/wiki/Quadratic_sieve, https://www.rieselprime.de/ziki/Self-initializing_quadratic_sieve, https://www.rieselprime.de/ziki/Multiple_polynomial_quadratic_sieve, https://mathworld.wolfram.com/QuadraticSieve.html, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "Quadratic sieve method:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=qs&maxrows=10000), to factor these numbers (see http://www.numericana.com/answer/factoring.htm), elliptic-curve factorization method and Pollard P−1 method and Williams P+1 method are methods which find a non-large (say between 1020 and 1090) prime factor, an elliptic-curve factorization program is GMP-ECM (https://web.archive.org/web/20210803045418/https://gforge.inria.fr/projects/ecm, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/ecm704dev-svn2990-win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/ecm704dev-svn2990-linux64, https://www.rieselprime.de/ziki/GMP-ECM), special number field sieve and general number field sieve are methods which factor a large number (say between 10100 and 10400), special number field sieve can only be used on a number of special form, e.g. numbers of the form a×bn±c with small a, b, c and large n, and cannot be used for general numbers such as a×nc and a×pc (where ! is the factorial (https://en.wikipedia.org/wiki/Factorial, https://t5k.org/glossary/xpage/Factorial.html, https://www.rieselprime.de/ziki/Factorial_number, https://mathworld.wolfram.com/Factorial.html, https://www.numbersaplenty.com/set/factorial/, https://oeis.org/A000142), # is the primorial (https://en.wikipedia.org/wiki/Primorial, https://t5k.org/glossary/xpage/Primorial.html, https://mathworld.wolfram.com/Primorial.html, https://www.numbersaplenty.com/set/primorial/, https://oeis.org/A002110)) with small a, c and (large n or large prime p), the difficulty (https://www.rieselprime.de/ziki/SNFS_polynomial_selection) of such a number is equivalent to general number field sieve for a general number around bn/2, thus, for the Cunningham number bn±1, if the primitive part (i.e. Φn(b) for bn−1 or Φn(b) for bn+1, where Φ is the cyclotomic polynomial (https://en.wikipedia.org/wiki/Cyclotomic_polynomial, https://mathworld.wolfram.com/CyclotomicPolynomial.html, http://www.numericana.com/answer/polynomial.htm#cyclotomic, https://stdkmd.net/nrr/repunit/repunitnote.htm#cyclotomic, https://oeis.org/A013595, https://oeis.org/A013596, https://oeis.org/A253240)) is > 1/3 factored (i.e. the product of the known prime factors of the primitive part is > the cube root (https://en.wikipedia.org/wiki/Cube_root, https://mathworld.wolfram.com/CubeRoot.html) of it) (by the elliptic-curve factorization method or the Pollard P−1 method or the Williams P+1 method), then general number field sieve is usually used for the unfactored part, if the primitive part (i.e. Φn(b) for bn−1 or Φn(b) for bn+1, where Φ is the cyclotomic polynomial (https://en.wikipedia.org/wiki/Cyclotomic_polynomial, https://mathworld.wolfram.com/CyclotomicPolynomial.html, http://www.numericana.com/answer/polynomial.htm#cyclotomic, https://stdkmd.net/nrr/repunit/repunitnote.htm#cyclotomic, https://oeis.org/A013595, https://oeis.org/A013596, https://oeis.org/A253240)) is < 1/3 factored (i.e. the product of the known prime factors of the primitive part is < the cube root (https://en.wikipedia.org/wiki/Cube_root, https://mathworld.wolfram.com/CubeRoot.html) of it) (by the elliptic-curve factorization method or the Pollard P−1 method or the Williams P+1 method), then special number field sieve is usually used for the unfactored part, for more information see https://escatter11.fullerton.edu/nfs/numbers.php (the status of numbers in NFS@HOME (http://escatter11.fullerton.edu/nfs/, https://en.wikipedia.org/wiki/NFS@Home)), for the calculator for special number field sieve and general number field sieve, see http://myfactorcollection.mooo.com:8090/calculators.html (section ".poly Maker"), a general number field sieve program is GGNFS (http://sourceforge.net/projects/ggnfs, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/GGNFS), also a (special or general) number field program is CADO-NFS (https://web.archive.org/web/20210506173015/http://cado-nfs.gforge.inria.fr/index.html, https://www.rieselprime.de/ziki/CADO-NFS, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/cado-nfs-2.3.0), however, all these factorization algorithms (https://en.wikipedia.org/wiki/Algorithm, https://www.rieselprime.de/ziki/Algorithm) take long time, i.e. they cannot be done in polynomial time (https://en.wikipedia.org/wiki/Polynomial_time, https://mathworld.wolfram.com/PolynomialTime.html), the best known running time is Ln(1/2,1+o(1)) (where o is the little o notation (https://en.wikipedia.org/wiki/Little_o_notation, https://t5k.org/glossary/xpage/LittleOh.html, https://mathworld.wolfram.com/Little-ONotation.html)), see https://www.ams.org/journals/jams/1992-05-03/S0894-0347-1992-1137100-0/S0894-0347-1992-1137100-0.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_230.pdf), also there are integer factorization records (https://en.wikipedia.org/wiki/Integer_factorization_records), also there are many OEIS sequences which need factors (see https://oeis.org/wiki/OEIS_sequences_needing_factors), also there is a World Integer Factorization Center page (see https://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/index.htm), also there is a NFS@home (http://escatter11.fullerton.edu/nfs/, https://en.wikipedia.org/wiki/NFS@Home, https://www.rieselprime.de/ziki/NFS@Home) distributed computing (https://en.wikipedia.org/wiki/Distributed_computing, https://en.wikipedia.org/wiki/List_of_distributed_computing_projects, https://www.rieselprime.de/ziki/Distributed_computing, https://www.rieselprime.de/ziki/Distributed_computing_project, http://www.distributedcomputing.info/ap-math.html) project which factors many integers with certain types (see https://escatter11.fullerton.edu/nfs/numbers.php and https://escatter11.fullerton.edu/nfs/crunching.php and https://escatter11.fullerton.edu/nfs/crunching_es.php and https://escatter11.fullerton.edu/nfs/crunching_e.php and https://escatter11.fullerton.edu/nfs/crunching_fs.php), also almost all numbers are not fully factored (i.e. almost all numbers are "C" or "CF" or "U" (instead of "FF" or "P" or "PRP") in factordb) when trial factor to a finite limit and ECM factor with a finite number of B1 and B2 (see http://factordb.com/distribution.php and https://www.mersenneforum.org/showthread.php?t=21301), also there are records for n consecutive numbers which are all fully factored (just like the records for n primes in arithmetic progression (https://en.wikipedia.org/wiki/Primes_in_arithmetic_progression, https://t5k.org/glossary/xpage/ArithmeticSequence.html, https://mathworld.wolfram.com/PrimeArithmeticProgression.html, https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem, https://mathworld.wolfram.com/Green-TaoTheorem.html, https://t5k.org/top20/page.php?id=14, https://t5k.org/primes/search.php?Comment=Arithmetic%20progression&OnList=all&Number=1000000&Style=HTML, https://www.primegrid.com/forum_thread.php?id=7022, https://www.primegrid.com/stats_ap26.php, https://www.pzktupel.de/JensKruseAndersen/aprecords.php, http://www.primerecords.dk/aprecords.htm, https://oeis.org/A133277, https://oeis.org/A113827, https://oeis.org/A005115, https://oeis.org/A093364, https://oeis.org/A133276, https://oeis.org/A033189, https://oeis.org/A113872, https://oeis.org/A033188, https://oeis.org/A231406, https://oeis.org/A113834, https://oeis.org/A088430) and the records for Cunningham chains with length n (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/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, http://www.primenumbers.net/Henri/us/NouvTh1us.htm, https://web.archive.org/web/20050406075848/http://ksc9.th.com/warut/cunningham.html, https://oeis.org/A005602, https://oeis.org/A005603, https://oeis.org/A057331, https://oeis.org/A057330) and the records for prime n-tuples (https://en.wikipedia.org/wiki/Prime_k-tuple, https://t5k.org/glossary/xpage/PrimeKTuplet.html, https://mathworld.wolfram.com/PrimeConstellation.html, https://t5k.org/top20/page.php?id=61, https://t5k.org/top20/page.php?id=55, https://t5k.org/top20/page.php?id=56, https://t5k.org/primes/search.php?Comment=plet&OnList=all&Number=1000000&Style=HTML, https://pzktupel.de/ktuplets.php, https://pzktupel.de/oldpage.htm, https://pzktupel.de/Prime%20k-tuplets_1997.htm, https://pzktupel.de/largest.php, https://pzktupel.de/ktpatt_hl.php, https://pzktupel.de/smarchive.php, https://pzktupel.de/SMArchiv/smadditions.php, https://pzktupel.de/smallest.php, https://web.archive.org/web/20211019145924/http://anthony.d.forbes.googlepages.com/ktuplets.htm, https://web.archive.org/web/20070702033150/http://www.ltkz.demon.co.uk/ktuplets.htm, http://www.opertech.com/primes/k-tuples.html, https://www.opertech.com/primes/k050.html, https://www.opertech.com/primes/k100.html, https://www.opertech.com/primes/k150.html, https://www.opertech.com/primes/k200.html, https://www.opertech.com/primes/modexample.html, https://www.opertech.com/primes/w3159.html, https://www.opertech.com/primes/residues.html, https://www.opertech.com/primes/residueclasses.html, https://oeis.org/A008407, https://oeis.org/A020497, https://oeis.org/A083409, https://oeis.org/A186634, https://oeis.org/A065688, https://oeis.org/A261324, https://oeis.org/A186702, https://oeis.org/A007529, https://oeis.org/A007530, https://oeis.org/A086140, https://oeis.org/A022008, https://oeis.org/A257124, https://oeis.org/A065706, https://oeis.org/A257125, https://oeis.org/A257127, https://oeis.org/A257129, https://oeis.org/A257131, https://oeis.org/A257135, https://oeis.org/A257166, https://oeis.org/A257169, https://oeis.org/A257308, https://oeis.org/A257373), they are "simultaneous primes", see https://www.pzktupel.de/JensKruseAndersen/simultprime.php), see http://www.primerecords.dk/consecutive_factorizations.htm and http://www.math.uni.wroc.pl/~jwr/cons-fac/ (unlike primality proving, when the numbers are sufficiently large, no efficient, non-quantum (https://en.wikipedia.org/wiki/Quantum_computer) integer factorization algorithm is known), i.e. integer factorization may be P-complete (https://en.wikipedia.org/wiki/P-complete) and NP-complete (https://en.wikipedia.org/wiki/NP-complete, https://mathworld.wolfram.com/NP-CompleteProblem.html) and NP-hard (https://en.wikipedia.org/wiki/NP-hard, https://mathworld.wolfram.com/NP-HardProblem.html) (thus, factor a large integer is much harder than determining whether the same integer is prime (determining whether an integer is prime and factor an integer are two completely different problems, we can quickly use Fermat primality test (https://t5k.org/prove/prove2_2.html, https://en.wikipedia.org/wiki/Fermat_primality_test, https://en.wikipedia.org/wiki/Fermat_pseudoprime, https://t5k.org/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Fermat_pseudoprimality_test, https://mathworld.wolfram.com/FermatPseudoprime.html, https://www.numbersaplenty.com/set/Poulet_number/, http://www.numericana.com/answer/pseudo.htm#pseudoprime, http://ntheory.org/data/psps.txt, https://www.cecm.sfu.ca/Pseudoprimes/psps-below-2-to-64.txt.bz2, https://www.cecm.sfu.ca/Pseudoprimes/factored-psps-below-2-to-64.txt.bz2, https://www.cecm.sfu.ca/Pseudoprimes/annotated-psps-below-2-to-64.txt.bz2, 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/A020136, https://oeis.org/A005936, https://oeis.org/A005937, https://oeis.org/A005938, https://oeis.org/A020137, https://oeis.org/A020138, https://oeis.org/A005939, https://oeis.org/A020139, https://oeis.org/A020140, https://oeis.org/A020141, https://oeis.org/A020142, https://oeis.org/A020143, https://oeis.org/A020144, https://oeis.org/A020145, https://oeis.org/A020146, https://oeis.org/A020147, https://oeis.org/A020148, https://oeis.org/A020149, https://oeis.org/A020150, https://oeis.org/A020151, https://oeis.org/A020152, https://oeis.org/A020153, https://oeis.org/A020154, https://oeis.org/A020155, https://oeis.org/A020156, https://oeis.org/A020157, https://oeis.org/A020158, https://oeis.org/A020159, https://oeis.org/A020160, https://oeis.org/A020161, https://oeis.org/A020162, https://oeis.org/A020163, https://oeis.org/A020164, https://oeis.org/A000864, 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/A211455, https://oeis.org/A211456, https://oeis.org/A211457, https://oeis.org/A211458, https://oeis.org/A063994, https://oeis.org/A105222, https://oeis.org/A194946, https://oeis.org/A195327, https://oeis.org/A002997, https://oeis.org/A191311, https://oeis.org/A090086, https://oeis.org/A007535, https://oeis.org/A090087, https://oeis.org/A090085, https://oeis.org/A090088, https://oeis.org/A090089, https://oeis.org/A253233, https://oeis.org/A271801) to prove that an integer is composite, although the most ancient 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) and sieve of Eratosthenes (https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes, https://t5k.org/glossary/xpage/SieveOfEratosthenes.html, https://www.rieselprime.de/ziki/Sieve_of_Eratosthenes, https://mathworld.wolfram.com/SieveofEratosthenes.html, https://oeis.org/A083221, https://oeis.org/A083140, https://oeis.org/A145583, https://oeis.org/A145540, https://oeis.org/A145538, https://oeis.org/A145539, https://oeis.org/A227155, https://oeis.org/A227797, https://oeis.org/A227798, https://oeis.org/A227799, https://oeis.org/A145584, https://oeis.org/A145585, https://oeis.org/A145586, https://oeis.org/A145587, https://oeis.org/A145588, https://oeis.org/A145589, https://oeis.org/A145590, https://oeis.org/A145591, https://oeis.org/A145592, https://oeis.org/A145532, https://oeis.org/A145533, https://oeis.org/A145534, https://oeis.org/A145535, https://oeis.org/A145536, https://oeis.org/A145537) can solving these two problems simultaneously), there are many numbers with 500 digits to 10000 digits which are known to be composite but do not have any known factors other than 1 and themselves). However, it has not been proven that no efficient algorithm exists (this is indeed an unsolved problem in computer science (https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_computer_science)). The presumed difficulty (https://en.wikipedia.org/wiki/Computational_hardness_assumption) of this problem is at the heart of widely used algorithms in cryptography (https://en.wikipedia.org/wiki/Cryptography, https://mathworld.wolfram.com/Cryptography.html) such as RSA (https://en.wikipedia.org/wiki/RSA_(cryptosystem), https://t5k.org/glossary/xpage/RSA.html, https://mathworld.wolfram.com/RSAEncryption.html, https://web.archive.org/web/20061209135708/http://www.rsasecurity.com/rsalabs/node.asp?id=2093), there are many large 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), called RSA numbers (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), which are very hard to factor and are part of the RSA Factoring Challenge (https://en.wikipedia.org/wiki/RSA_Factoring_Challenge), e.g. the RSA-640 number (http://factordb.com/index.php?id=1100000000193433853&open=ecm, https://en.wikipedia.org/wiki/RSA-640, http://mathworld.wolfram.com/news/2005-11-08/rsa-640/) and the RSA-230 number (http://factordb.com/index.php?id=1100000000104374171&open=ecm, https://en.wikipedia.org/wiki/RSA-230, https://web.archive.org/web/20210714184715/https://lists.gforge.inria.fr/pipermail/cado-nfs-discuss/2018-August/000926.html) and the RSA-768 number (http://factordb.com/index.php?id=1100000000193442616&open=ecm, https://en.wikipedia.org/wiki/RSA-768, http://eprint.iacr.org/2010/006.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_228.pdf)). Besides, integer factorization can be used for public-key cryptography (https://en.wikipedia.org/wiki/Public-key_cryptography, https://t5k.org/glossary/xpage/PublicKey.html, https://mathworld.wolfram.com/Public-KeyCryptography.html) is because it has no known polynomial time algorithm. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves (https://en.wikipedia.org/wiki/Elliptic_curve, https://mathworld.wolfram.com/EllipticCurve.html, http://www.numericana.com/answer/modularity.htm#elliptic), algebraic number theory (https://en.wikipedia.org/wiki/Algebraic_number_theory, https://mathworld.wolfram.com/AlgebraicNumberTheory.html), and quantum computing (https://en.wikipedia.org/wiki/Quantum_computing)), and hence to do this is impractically.

Our data assumes that a number > 1025000 which has passed the Miller–Rabin primality tests (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, https://miller-rabin.appspot.com/, http://www.pi-e.de/Miller-Rabin-Pseudoprimzahlen.htm (in German), 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/A020230, https://oeis.org/A020231, https://oeis.org/A020232, https://oeis.org/A020233, https://oeis.org/A020234, https://oeis.org/A020235, https://oeis.org/A020236, https://oeis.org/A020237, https://oeis.org/A020238, https://oeis.org/A020239, https://oeis.org/A020240, https://oeis.org/A020241, https://oeis.org/A020242, https://oeis.org/A020243, https://oeis.org/A020244, https://oeis.org/A020245, https://oeis.org/A020246, https://oeis.org/A020247, https://oeis.org/A020248, https://oeis.org/A020249, https://oeis.org/A020250, https://oeis.org/A020251, https://oeis.org/A020252, https://oeis.org/A020253, https://oeis.org/A020254, https://oeis.org/A020255, https://oeis.org/A020256, https://oeis.org/A020257, https://oeis.org/A020258, https://oeis.org/A020259, https://oeis.org/A020260, https://oeis.org/A020261, https://oeis.org/A020262, 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 p < 64 and has passed the Baillie–PSW primality test (https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test, https://mathworld.wolfram.com/Baillie-PSWPrimalityTest.html, http://pseudoprime.com/dopo.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_318.pdf)) and has 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 is in fact prime, since in some cases (e.g. b = 11) a candidate for minimal prime base b is too large to be proven prime rigorously, this candidate for minimal prime base 11 has 65263 decimal digits, while the top record ordinary prime (https://t5k.org/glossary/xpage/OrdinaryPrime.html) (i.e. neither N−1 (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) nor N+1 (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) can be ≥ 1/4 factored (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) (i.e. the products of the known prime factors of both N−1 and N+1 are < the fourth roots of them)) has 86453 decimal digits (the entry of this prime in top definitely primes is https://t5k.org/primes/page.php?id=136044), see https://t5k.org/top20/page.php?id=27 and https://t5k.org/primes/search.php?Comment=ECPP&OnList=all&Number=1000000&Style=HTML and http://factordb.com/certoverview.php?digits=300&perpage=1000&skip=0&descending=on), however, if we assume a number which has passed the Fermat primality tests (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://www.cecm.sfu.ca/Pseudoprimes/psps-below-2-to-64.txt.bz2, https://www.cecm.sfu.ca/Pseudoprimes/factored-psps-below-2-to-64.txt.bz2, https://www.cecm.sfu.ca/Pseudoprimes/annotated-psps-below-2-to-64.txt.bz2, 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/A020136, https://oeis.org/A005936, https://oeis.org/A005937, https://oeis.org/A005938, https://oeis.org/A020137, https://oeis.org/A020138, https://oeis.org/A005939, https://oeis.org/A020139, https://oeis.org/A020140, https://oeis.org/A020141, https://oeis.org/A020142, https://oeis.org/A020143, https://oeis.org/A020144, https://oeis.org/A020145, https://oeis.org/A020146, https://oeis.org/A020147, https://oeis.org/A020148, https://oeis.org/A020149, https://oeis.org/A020150, https://oeis.org/A020151, https://oeis.org/A020152, https://oeis.org/A020153, https://oeis.org/A020154, https://oeis.org/A020155, https://oeis.org/A020156, https://oeis.org/A020157, https://oeis.org/A020158, https://oeis.org/A020159, https://oeis.org/A020160, https://oeis.org/A020161, https://oeis.org/A020162, https://oeis.org/A020163, https://oeis.org/A020164, https://oeis.org/A000864, 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/A211455, https://oeis.org/A211456, https://oeis.org/A211457, https://oeis.org/A211458, https://oeis.org/A063994, https://oeis.org/A105222, https://oeis.org/A194946, https://oeis.org/A195327, https://oeis.org/A002997, https://oeis.org/A191311, https://oeis.org/A090086, https://oeis.org/A007535, https://oeis.org/A090087, https://oeis.org/A090085, https://oeis.org/A090088, https://oeis.org/A090089, https://oeis.org/A253233, https://oeis.org/A271801) to many bases is in fact prime, our list for base 16 minimal primes would wrongly include the composites 1563 (its value is (4×1663−1)/3) and 8536 (its value is (25×1636−1)/3), and our list for base 9 minimal primes would wrongly include the composite 113 (its value is (913−1)/8) (and hence would wrongly exclude the prime 56136, since this prime has 113 as subsequence (https://en.wikipedia.org/wiki/Subsequence, https://mathworld.wolfram.com/Subsequence.html)), although their corresponding families (1{5} in base 16, 8{5} in base 16, {1} in base 9, respectively) can be ruled out as only contain composite numbers (only count the numbers > b), and our data will be wrong for these bases, see https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test#The_danger_of_relying_only_on_Fermat_tests (only run Fermat tests are dangerous).

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 bthe smallest composite number which is strong pseudoprime to all these bases bprime factorization
2204723 × 89
3121112
434111 × 31
578111 × 71
62177 × 31
72552
8932
9917 × 13
10932
111337 × 19
12917 × 13
13855 × 17
14153 × 5
1516877 × 241
16153 × 5
17932
182552
19932
20213 × 7
2122113 × 17
22213 × 7
23169132
242552
252177 × 31
26932
27121112
28932
29153 × 5
304972
31153 × 5
322552
335455 × 109
34333 × 11
35932
36355 × 7
95189131 × 61
240199111 × 181
385189131 × 61
777154123 × 67
933138719 × 73
1320409717 × 241
471247117 × 673
5628562717 × 331
725272513 × 2417
785278513 × 2617
147879409972
173401026131 × 331
613801135937 × 307
787501374759 × 233
2549231829929 × 631
3776875329732
486605257613 × 31 × 277
1804842327611812
40950863832319 × 2017
1277234440501101 × 401
4216299597921181 × 541
92121172749141157 × 313
814494960528132239223 × 593
64390572806844161701101 × 1601
1769236083487960192001101 × 1901
1948244569546278212321241 × 881
349336087797801632182455 × 43649
62769592775616394272161127 × 2143
1264010713499945362918313 × 89 × 1093
9345883071009581737341531239 × 1429
2, 31373653829 × 1657
31, 7390801912131 × 4261
2, 29941719471033883 × 22051
2, 3, 5253260012251 × 11251
2, 9332593380103073083 × 12329
350, 39582815431705849617541 × 22621
2346211568, 30560936271766094417673 × 23017
660, 5692828722713264110657 × 21313
11000544, 3148110731634928112577 × 25153
91869414, 634612859812923446675818115277 × 30553
15, 7500684175255325209241418627 × 60383
15, 551185532110317762473242114431 × 43291
15, 1339301939619470171616930115451 × 46351
725270293939359937, 356981966704819837588559416913309 × 66541
336781006125, 9639812373923155105053550112251 × 85751
2, 3, 5, 73215031751151 × 751 × 28351
2, 7, 61475912314148781 × 97561
2, 379215, 45708375475792980677137653 × 550609
2, 1005905886, 1340600841105936894253230149 × 460297
2, 45650740, 3722628058109134866497110119 × 991063
15, 176006322, 4221622697154639673381160541 × 963241
15, 953185122, 158682512356199220146059407234599 × 938393
15, 7363882082, 211573017068182242175507817401809 × 602713
15, 7363882082, 992620450144556273919523041370081 × 740161
4230279247111683200, 14694767155120705706, 16641139526367750375350269456337197279 × 1775503
2, 13, 23, 16628031122004669633611557 × 1834669
2, 3, 5, 7, 1121523028987476763 × 10627 × 29947
2, 3, 5, 7, 11, 1334747496603831303 × 16927 × 157543
2, 1215, 34862, 574237825216526845022213290341 × 6580681
2, 642735, 553174392, 3046413974318583172186472822159 × 11288633
2, 2570940, 211991001, 3749873356476366229612014880401 × 9760801
2, 141889084524735, 1199124725622454117, 11096072698276303650552456424894513716371 × 14865481
2, 3, 5, 7, 11, 13, 1734155007172832110670053 × 32010157
2, 3, 5, 7, 11, 13, 17, 1934155007172832110670053 × 32010157
2, 75088, 642735, 203659041, 3613982119307183769235784924786439 × 123932191
2, 2570940, 880937, 610386380, 4130785767377057958215454730702523 × 122810089
2, 4130806001517, 149795463772692060, 186635894390467037, 396730417934771580579992521755828519227 × 894923 × 968731
2, 123635709730000, 9233062284813009, 43835965440333360, 761179012939631437, 1263739024124850375585226005592931977382500329 × 1530001313
2, 3, 5, 7, 11, 13, 17, 19, 233825123056546413051149491 × 747451 × 34233211
2, 3, 5, 7, 11, 13, 17, 19, 23, 293825123056546413051149491 × 747451 × 34233211
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 313825123056546413051149491 × 747451 × 34233211
2, 325, 9375, 28178, 450775, 9780504, 1795265022> 264?
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37318665857834031151167461399165290221 × 798330580441
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 4133170440646798873859619811287836182261 × 2575672364521
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 436003094289670105800312596501 (conjectured)54786377365501 × 109572754731001 (conjectured)
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 4759276361075595573263446330101 (conjectured)172157429516701 × 344314859033401 (conjectured)
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53564132928021909221014087501701 (conjectured)531099297693901 × 1062198595387801 (conjectured)
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59564132928021909221014087501701 (conjectured)531099297693901 × 1062198595387801 (conjectured)
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 611543267864443420616877677640751301 (conjectured)27778299663977101 × 55556599327954201 (conjectured)
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 671543267864443420616877677640751301 (conjectured)27778299663977101 × 55556599327954201 (conjectured)
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71> 1036 (conjectured)?
all primes pn> e(n/2)1/2 (under the generalized Riemann hypothesis)

Thus, for this minimal prime problem in base b, especially for square (https://en.wikipedia.org/wiki/Square_number, https://www.rieselprime.de/ziki/Square_number, https://mathworld.wolfram.com/SquareNumber.html, https://www.numbersaplenty.com/set/square_number/, https://oeis.org/A000290) base b, we should not assume a number which has passed the Fermat primality tests to many bases is in fact prime, also, there are Carmichael numbers (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, https://oeis.org/A002997) (i.e. composites which are Fermat pseudoprimes to all bases b coprime to them) which are strong pseudoprimes to several bases simultaneously, see 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), this article gives a 397-digit such number (factordb entry: http://factordb.com/index.php?id=1100000000708885054) which is strong pseudoprime to all bases b ≤ 306 (see https://www.mersenneforum.org/showpost.php?p=613381&postcount=6), another example is a 23707-digit number (https://t5k.org/curios/page.php?number_id=4265, http://factordb.com/index.php?id=1100000002517553325) which is strong pseudoprime to all bases b ≤ 101100, also see http://factordb.com/prooffailed.php (the factordb test failed page) (numbers passed Miller–Rabin primality tests (10 prime bases at least), but turned out to be composite) and https://faculty.lynchburg.edu/~nicely/misc/mpzspsp.html (the GNU GMP test failed page) (this page mentions a 337-digit number (http://factordb.com/index.php?id=1100000000047694476) which is strong pseudoprime to all bases b ≤ 210) (now GMP runs the Baillie–PSW primality test (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)) combine with some Miller–Rabin primality tests (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, https://miller-rabin.appspot.com/, http://www.pi-e.de/Miller-Rabin-Pseudoprimzahlen.htm (in German), 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/A020230, https://oeis.org/A020231, https://oeis.org/A020232, https://oeis.org/A020233, https://oeis.org/A020234, https://oeis.org/A020235, https://oeis.org/A020236, https://oeis.org/A020237, https://oeis.org/A020238, https://oeis.org/A020239, https://oeis.org/A020240, https://oeis.org/A020241, https://oeis.org/A020242, https://oeis.org/A020243, https://oeis.org/A020244, https://oeis.org/A020245, https://oeis.org/A020246, https://oeis.org/A020247, https://oeis.org/A020248, https://oeis.org/A020249, https://oeis.org/A020250, https://oeis.org/A020251, https://oeis.org/A020252, https://oeis.org/A020253, https://oeis.org/A020254, https://oeis.org/A020255, https://oeis.org/A020256, https://oeis.org/A020257, https://oeis.org/A020258, https://oeis.org/A020259, https://oeis.org/A020260, https://oeis.org/A020261, https://oeis.org/A020262, 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) combine with some trial divisions (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), see https://gmplib.org/manual/Number-Theoretic-Functions), thus we need to combine with strong Lucas primality tests (https://en.wikipedia.org/wiki/Strong_Lucas_pseudoprime, https://mathworld.wolfram.com/StrongLucasPseudoprime.html, http://ntheory.org/data/slpsps-baillie.txt, http://www.hoegge.dk/lucasselfridgeprps.txt, https://oeis.org/A217255), to do the Baillie–PSW primality test, only run strong tests is also a little dangerous (especially when the number is a semiprime (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), with both factors primes, such numbers are Fermat pseudoprimes of 1/2 and 1/3 of the bases coprime to them, respectively, and also strong pseudoprimes to many bases), however, there is no known overlap between these lists of strong pseudoprimes and strong Lucas pseudoprimes, and there is even evidence that the numbers in these lists tend to be different kinds of numbers. For example, Fermat pseudoprimes tend to fall into the residue class 1 (mod n) for many small n, whereas Lucas pseudoprimes tend to fall into the residue class −1 (mod n) for many small n. As a result, a number that passes both a strong Fermat and a strong Lucas test is very likely to be prime, indeed, no known composites which pass the Baillie–PSW probable prime test, and no composites < 264 pass the Baillie–PSW probable prime test (see http://ntheory.org/pseudoprimes.html (the box "#BPSW") and https://faculty.lynchburg.edu/~nicely/misc/bpsw.html).

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 bc−1 == 1 mod c for half of the bases b coprime to c, the Carmichael numbers are the composite numbers c such that bc−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.

(for more examples, see https://www.mersenneforum.org/showpost.php?p=659223&postcount=1 and https://www.mersenneforum.org/showpost.php?p=659421&postcount=17 and https://oeis.org/A014233 and https://faculty.lynchburg.edu/~nicely/misc/mpzspsp.html and https://miller-rabin.appspot.com/ and http://www.pi-e.de/Miller-Rabin-Pseudoprimzahlen.htm (in German) and http://factordb.com/prooffailed.php)

numberfactordb entrybases 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=10000000000435690552, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 29, 32, 36, 41, 48, 54, 58, 59, 6420 (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=10000000000435690852, 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, 6432 (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=10000000000435691512, 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, 6439 (61.90%)
856 in base b = 16http://factordb.com/index.php?id=1398101335, 7, 8, 12, 13, 17, 18, 25, 27, 35, 38, 40, 44, 46, 49, 56, 57, 59, 60, 6420 (31.75%)
8536 in base b = 16http://factordb.com/index.php?id=11000000003488293873, 5, 8, 9, 13, 15, 17, 22, 24, 25, 27, 28, 29, 39, 40, 41, 45, 46, 47, 51, 53, 62, 6423 (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=3177332285412, 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, 6450 (79.37%)
589 in base b = 25http://factordb.com/index.php?id=203450520833332, 4, 8, 13, 16, 26, 27, 31, 32, 43, 45, 47, 52, 54, 62, 63, 6417 (26.98%)
I0901 in base b = 26 (see https://github.com/curtisbright/mepn-data/commit/7565d197d7b438b437871bf71614a6f8914397f7)http://factordb.com/index.php?id=82316532, 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, 6440 (63.49%)
28462346×37+1 (see https://www.mersenneforum.org/showpost.php?p=137730&postcount=1)http://factordb.com/index.php?id=622471507033, 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, 6436 (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 numbersrecord numberfactordb entry of the record numbernumber of decimal digits of the record numberreference 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://www.cecm.sfu.ca/Pseudoprimes/psps-below-2-to-64.txt.bz2, https://www.cecm.sfu.ca/Pseudoprimes/factored-psps-below-2-to-64.txt.bz2, https://www.cecm.sfu.ca/Pseudoprimes/annotated-psps-below-2-to-64.txt.bz2, 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/A020136, https://oeis.org/A005936, https://oeis.org/A005937, https://oeis.org/A005938, https://oeis.org/A020137, https://oeis.org/A020138, https://oeis.org/A005939, https://oeis.org/A020139, https://oeis.org/A020140, https://oeis.org/A020141, https://oeis.org/A020142, https://oeis.org/A020143, https://oeis.org/A020144, https://oeis.org/A020145, https://oeis.org/A020146, https://oeis.org/A020147, https://oeis.org/A020148, https://oeis.org/A020149, https://oeis.org/A020150, https://oeis.org/A020151, https://oeis.org/A020152, https://oeis.org/A020153, https://oeis.org/A020154, https://oeis.org/A020155, https://oeis.org/A020156, https://oeis.org/A020157, https://oeis.org/A020158, https://oeis.org/A020159, https://oeis.org/A020160, https://oeis.org/A020161, https://oeis.org/A020162, https://oeis.org/A020163, https://oeis.org/A020164, https://oeis.org/A000864, 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/A211455, https://oeis.org/A211456, https://oeis.org/A211457, https://oeis.org/A211458, https://oeis.org/A063994, https://oeis.org/A105222, https://oeis.org/A194946, https://oeis.org/A195327, https://oeis.org/A002997, https://oeis.org/A191311, https://oeis.org/A090086, https://oeis.org/A007535, https://oeis.org/A090087, https://oeis.org/A090085, https://oeis.org/A090088, https://oeis.org/A090089, https://oeis.org/A253233, https://oeis.org/A271801)O(log(n))any number, but only probabilistic (https://en.wikipedia.org/wiki/Probabilistic_algorithm) primality testing(108177207−1)/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=prime8177207http://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, https://miller-rabin.appspot.com/, http://www.pi-e.de/Miller-Rabin-Pseudoprimzahlen.htm (in German), 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/A020230, https://oeis.org/A020231, https://oeis.org/A020232, https://oeis.org/A020233, https://oeis.org/A020234, https://oeis.org/A020235, https://oeis.org/A020236, https://oeis.org/A020237, https://oeis.org/A020238, https://oeis.org/A020239, https://oeis.org/A020240, https://oeis.org/A020241, https://oeis.org/A020242, https://oeis.org/A020243, https://oeis.org/A020244, https://oeis.org/A020245, https://oeis.org/A020246, https://oeis.org/A020247, https://oeis.org/A020248, https://oeis.org/A020249, https://oeis.org/A020250, https://oeis.org/A020251, https://oeis.org/A020252, https://oeis.org/A020253, https://oeis.org/A020254, https://oeis.org/A020255, https://oeis.org/A020256, https://oeis.org/A020257, https://oeis.org/A020258, https://oeis.org/A020259, https://oeis.org/A020260, https://oeis.org/A020261, https://oeis.org/A020262, 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(108177207−1)/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=prime8177207http://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, http://www.hoegge.dk/lucasselfridgeprps.txt, https://oeis.org/A217120)O(log(n))any number, but only probabilistic (https://en.wikipedia.org/wiki/Probabilistic_algorithm) primality testing(108177207−1)/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=prime8177207http://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, http://www.hoegge.dk/lucasselfridgeprps.txt, https://oeis.org/A217255)O(log(n))any number, but only probabilistic (https://en.wikipedia.org/wiki/Probabilistic_algorithm) primality testing(108177207−1)/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=prime8177207http://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(108177207−1)/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=prime8177207http://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/, https://64ordle.au/fermat/small/, https://www.primegrid.com/forum_thread.php?id=8778, https://www.primegrid.com/stats_div_llr.php, https://www.primegrid.com/primes/primes.php?project=DIV&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.rieselprime.de/ziki/PrimeGrid_Fermat_Divisor_Search, http://www.fermatsearch.org/factors/faclist.php, http://www.fermatsearch.org/factors/composite.php))http://factordb.com/index.php?id=65537&open=prime5http://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://en.wikipedia.org/wiki/List_of_Mersenne_primes_and_perfect_numbers, 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://www.rieselprime.de/ziki/List_of_known_Mersenne_primes, https://mathworld.wolfram.com/MersenneNumber.html, https://mathworld.wolfram.com/MersennePrime.html, https://pzktupel.de/Primetables/TableMersenne.php)282589933−1 (the largest known Mersenne prime (https://en.wikipedia.org/wiki/Mersenne_prime, https://en.wikipedia.org/wiki/List_of_Mersenne_primes_and_perfect_numbers, 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://www.rieselprime.de/ziki/List_of_known_Mersenne_primes, 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=prime24862048https://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_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=, http://boincvm.proxyma.ru:30080/test4vm/public/pps_dc_status.php, http://boincvm.proxyma.ru:30080/test4vm/user_profile/llr2_status.html, 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://web.archive.org/web/20210415051133/http://prpnet.primegrid.com:12001/, https://web.archive.org/web/20220115151556/http://prpnet.primegrid.com:12006/, https://www.rieselprime.de/ziki/321_Search, https://www.rieselprime.de/ziki/12121_Search, https://www.rieselprime.de/ziki/27121_Search, https://www.rieselprime.de/ziki/PrimeGrid_321_Prime_Search, https://www.rieselprime.de/ziki/PrimeGrid_27121_Prime_Search, 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"10223×231172165+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_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=, http://boincvm.proxyma.ru:30080/test4vm/public/pps_dc_status.php, http://boincvm.proxyma.ru:30080/test4vm/user_profile/llr2_status.html, 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://web.archive.org/web/20210415051133/http://prpnet.primegrid.com:12001/, https://web.archive.org/web/20220115151556/http://prpnet.primegrid.com:12006/, https://www.rieselprime.de/ziki/321_Search, https://www.rieselprime.de/ziki/12121_Search, https://www.rieselprime.de/ziki/27121_Search, https://www.rieselprime.de/ziki/PrimeGrid_321_Prime_Search, https://www.rieselprime.de/ziki/PrimeGrid_27121_Prime_Search, 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=prime9383761https://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, 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=, 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://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://web.archive.org/web/20210415051133/http://prpnet.primegrid.com:12001/, https://web.archive.org/web/20220115151556/http://prpnet.primegrid.com:12006/, https://www.rieselprime.de/ziki/321_Search, https://www.rieselprime.de/ziki/12121_Search, https://www.rieselprime.de/ziki/27121_Search, https://www.rieselprime.de/ziki/PrimeGrid_321_Prime_Search, https://www.rieselprime.de/ziki/PrimeGrid_27121_Prime_Search, 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"3×220928756−1 (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, 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=, 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://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://web.archive.org/web/20210415051133/http://prpnet.primegrid.com:12001/, https://web.archive.org/web/20220115151556/http://prpnet.primegrid.com:12006/, https://www.rieselprime.de/ziki/321_Search, https://www.rieselprime.de/ziki/12121_Search, https://www.rieselprime.de/ziki/27121_Search, https://www.rieselprime.de/ziki/PrimeGrid_321_Prime_Search, https://www.rieselprime.de/ziki/PrimeGrid_27121_Prime_Search, 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=prime6300184https://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)19637361048576+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_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=, http://boincvm.proxyma.ru:30080/test4vm/public/pps_dc_status.php, http://boincvm.proxyma.ru:30080/test4vm/user_profile/llr2_status.html, 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://web.archive.org/web/20210415051133/http://prpnet.primegrid.com:12001/, https://web.archive.org/web/20220115151556/http://prpnet.primegrid.com:12006/, https://www.rieselprime.de/ziki/321_Search, https://www.rieselprime.de/ziki/12121_Search, https://www.rieselprime.de/ziki/27121_Search, https://www.rieselprime.de/ziki/PrimeGrid_321_Prime_Search, https://www.rieselprime.de/ziki/PrimeGrid_27121_Prime_Search, 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=prime6598776https://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)3622×57558139−1 (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, 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=, 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://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://web.archive.org/web/20210415051133/http://prpnet.primegrid.com:12001/, https://web.archive.org/web/20220115151556/http://prpnet.primegrid.com:12006/, https://www.rieselprime.de/ziki/321_Search, https://www.rieselprime.de/ziki/12121_Search, https://www.rieselprime.de/ziki/27121_Search, https://www.rieselprime.de/ziki/PrimeGrid_321_Prime_Search, https://www.rieselprime.de/ziki/PrimeGrid_27121_Prime_Search, 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=prime5282917https://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 > 0any 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=prime86453https://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 = Ln(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 n21193−1http://factordb.com/index.php?id=1000000000000001193&open=ecm360 (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 = Ln(1/3, (64/9)1/3)any numberthe 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=ecm250 (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 factor9999999999999937 (the largest prime < 1016)http://factordb.com/index.php?id=9999999999999937&open=ecm16(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 960119−1http://factordb.com/index.php?id=1100000000216738427&open=ecm66http://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) L2366http://factordb.com/index.php?id=1100000000216896584&open=ecm60http://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, https://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), https://arxiv.org/pdf/1004.3366.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_492.pdf))e(21/2+o(1))×log(n)1/2×log(log(n))1/2 = Lp(1/2, 21/2)any prime factorprime factor of 7337+1http://factordb.com/index.php?id=1100000000632146801&open=ecm83http://www.loria.fr/~zimmerma/records/top50.html

(Since the largest prime factor found by Lenstra elliptic-curve (ECM) integer factoring has 83 decimal digits, and the largest prime factor found by Pollard (P−1) integer factoring has 66 decimal digits, and the largest prime factor found by Williams (P+1) integer factoring has 60 decimal digits, thus there are no numbers which are not fully factored (i.e. have status (http://factordb.com/status.html, http://factordb.com/distribution.php) "CF" instead of "FF" in factordb) and have a known prime factor (other than the algebraic factors, e.g. the repunit number R842 has a known 401-digit prime factor but is not fully factored (i.e. have status (http://factordb.com/status.html, http://factordb.com/distribution.php) "CF" instead of "FF" in factordb) (with a 413-digit composite factor without known proper factor > 1) (see http://factordb.com/index.php?id=1000000000043587866&open=ecm), but the repunit number R842 has three algebraic factors, namely Φ2(10) and Φ421(10) and Φ842(10) (where Φ is the cyclotomic polynomial (https://en.wikipedia.org/wiki/Cyclotomic_polynomial, https://mathworld.wolfram.com/CyclotomicPolynomial.html, http://www.numericana.com/answer/polynomial.htm#cyclotomic, https://stdkmd.net/nrr/repunit/repunitnote.htm#cyclotomic, https://oeis.org/A013595, https://oeis.org/A013596, https://oeis.org/A253240)), and the 401-digit prime factor is a prime factor of Φ421(10), but the 413-digit composite factor without known proper factor > 1 is a prime factor of Φ842(10), thus this number is not a counterexample) with > 83 decimal digits)

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 factorsPrimes 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://www.cecm.sfu.ca/Pseudoprimes/psps-below-2-to-64.txt.bz2, https://www.cecm.sfu.ca/Pseudoprimes/factored-psps-below-2-to-64.txt.bz2, https://www.cecm.sfu.ca/Pseudoprimes/annotated-psps-below-2-to-64.txt.bz2, 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/A020136, https://oeis.org/A005936, https://oeis.org/A005937, https://oeis.org/A005938, https://oeis.org/A020137, https://oeis.org/A020138, https://oeis.org/A005939, https://oeis.org/A020139, https://oeis.org/A020140, https://oeis.org/A020141, https://oeis.org/A020142, https://oeis.org/A020143, https://oeis.org/A020144, https://oeis.org/A020145, https://oeis.org/A020146, https://oeis.org/A020147, https://oeis.org/A020148, https://oeis.org/A020149, https://oeis.org/A020150, https://oeis.org/A020151, https://oeis.org/A020152, https://oeis.org/A020153, https://oeis.org/A020154, https://oeis.org/A020155, https://oeis.org/A020156, https://oeis.org/A020157, https://oeis.org/A020158, https://oeis.org/A020159, https://oeis.org/A020160, https://oeis.org/A020161, https://oeis.org/A020162, https://oeis.org/A020163, https://oeis.org/A020164, https://oeis.org/A000864, 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/A211455, https://oeis.org/A211456, https://oeis.org/A211457, https://oeis.org/A211458, https://oeis.org/A063994, https://oeis.org/A105222, https://oeis.org/A194946, https://oeis.org/A195327, https://oeis.org/A002997, https://oeis.org/A191311, https://oeis.org/A090086, https://oeis.org/A007535, https://oeis.org/A090087, https://oeis.org/A090085, https://oeis.org/A090088, https://oeis.org/A090089, https://oeis.org/A253233, https://oeis.org/A271801)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, http://www.hoegge.dk/lucasselfridgeprps.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, https://miller-rabin.appspot.com/, http://www.pi-e.de/Miller-Rabin-Pseudoprimzahlen.htm (in German), 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/A020230, https://oeis.org/A020231, https://oeis.org/A020232, https://oeis.org/A020233, https://oeis.org/A020234, https://oeis.org/A020235, https://oeis.org/A020236, https://oeis.org/A020237, https://oeis.org/A020238, https://oeis.org/A020239, https://oeis.org/A020240, https://oeis.org/A020241, https://oeis.org/A020242, https://oeis.org/A020243, https://oeis.org/A020244, https://oeis.org/A020245, https://oeis.org/A020246, https://oeis.org/A020247, https://oeis.org/A020248, https://oeis.org/A020249, https://oeis.org/A020250, https://oeis.org/A020251, https://oeis.org/A020252, https://oeis.org/A020253, https://oeis.org/A020254, https://oeis.org/A020255, https://oeis.org/A020256, https://oeis.org/A020257, https://oeis.org/A020258, https://oeis.org/A020259, https://oeis.org/A020260, https://oeis.org/A020261, https://oeis.org/A020262, 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, http://www.hoegge.dk/lucasselfridgeprps.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/A005277Dedekind 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/A019434Lucas–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://en.wikipedia.org/wiki/List_of_Mersenne_primes_and_perfect_numbers, 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://www.rieselprime.de/ziki/List_of_known_Mersenne_primes, 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/A060377Lucas–Lehmer primality test numbers: https://oeis.org/A003010
Residues of Pépin primality test for Fermat numbers: https://oeis.org/A152153Residues 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/A129802Possible 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_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=, http://boincvm.proxyma.ru:30080/test4vm/public/pps_dc_status.php, http://boincvm.proxyma.ru:30080/test4vm/user_profile/llr2_status.html, 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://web.archive.org/web/20210415051133/http://prpnet.primegrid.com:12001/, https://web.archive.org/web/20220115151556/http://prpnet.primegrid.com:12006/, https://www.rieselprime.de/ziki/321_Search, https://www.rieselprime.de/ziki/12121_Search, https://www.rieselprime.de/ziki/27121_Search, https://www.rieselprime.de/ziki/PrimeGrid_321_Prime_Search, https://www.rieselprime.de/ziki/PrimeGrid_27121_Prime_Search, 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/A080076Lucas–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, 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=, 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://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://web.archive.org/web/20210415051133/http://prpnet.primegrid.com:12001/, https://web.archive.org/web/20220115151556/http://prpnet.primegrid.com:12006/, https://www.rieselprime.de/ziki/321_Search, https://www.rieselprime.de/ziki/12121_Search, https://www.rieselprime.de/ziki/27121_Search, https://www.rieselprime.de/ziki/PrimeGrid_321_Prime_Search, https://www.rieselprime.de/ziki/PrimeGrid_27121_Prime_Search, 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/CurrentlyKnown-I.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/CurrentlyKnown-II.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/20240117130424/https://www.mersenneforum.org/321search/The%20status%20of%20the%20search.html, 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/20210412221221/http://www.15k.org/Summary10e04.htm, https://web.archive.org/web/20210412224110/http://www.15k.org/Summary10e05.htm, https://web.archive.org/web/20210412230926/http://www.15k.org/Summary10e06.htm, https://web.archive.org/web/20210412212330/http://www.15k.org/Summary10e07.htm, https://web.archive.org/web/20210412215840/http://www.15k.org/Summary10e08.htm, https://web.archive.org/web/20210412222809/http://www.15k.org/Summary10e09.htm, https://web.archive.org/web/20210715045529/http://www.15k.org/Summary10e10.htm, https://web.archive.org/web/20210718040304/http://www.15k.org/15k/stats.htm, https://web.archive.org/web/20210715013702/http://www.15k.org/lowweight.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/20240117130424/https://www.mersenneforum.org/321search/The%20status%20of%20the%20search.html, 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), http://jpenne.free.fr/Sierpeven/, 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 numbersRiesel problem (http://www.prothsearch.com/rieselprob.html, https://web.archive.org/web/20081120153544/http://www.15k.org/RieselProblem.htm, https://web.archive.org/web/20081120154015/http://www.15k.org/EvenRieselProblem.htm, 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))
Smallest k such that k×2n+1 is prime (https://oeis.org/A035050, https://oeis.org/A035089 (corresponding primes), https://oeis.org/A057778 (only odd k allowed), https://oeis.org/A057775 (only odd k allowed, corresponding primes), https://oeis.org/A051886 (only prime k allowed), https://oeis.org/A051900 (only prime k allowed, corresponding primes), https://oeis.org/A134855 (only odd prime k allowed), https://oeis.org/A134854 (only odd prime k allowed, corresponding primes))Smallest k such that k×2n−1 is prime (https://oeis.org/A085427, https://oeis.org/A127582 (corresponding primes), https://oeis.org/A126717 (only odd k allowed), https://oeis.org/A201914 (only odd k allowed, corresponding primes), https://oeis.org/A126715 (only odd prime k allowed))
Smallest k such that k×10n+1 is prime (https://oeis.org/A121172, https://oeis.org/A070854 (corresponding primes))Smallest k such that k×10n−1 is prime (https://oeis.org/A120642, https://oeis.org/A065582 (only k not divisible by 10 allowed, corresponding primes))
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 78557Real 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.primegrid.com/download/5ob_all.html, 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 78557Dual Riesel problem (https://www.mersenneforum.org/showthread.php?t=6545), finding and proving the smallest odd k such that |2nk| 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 2nk 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 10nk 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 2nk 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))
Smallest k such that 2n+k is prime (https://oeis.org/A013597, https://oeis.org/A092131 (k = 0 allowed), https://oeis.org/A014210 (corresponding primes), https://oeis.org/A104080 (k = 0 allowed, corresponding primes))Smallest k such that 2nk is prime (https://oeis.org/A013603 (k = 0 allowed), https://oeis.org/A014234 (k = 0 allowed, corresponding primes))
Smallest k such that 10n+k is prime (https://oeis.org/A033873, https://oeis.org/A003617 (corresponding primes, n replaced by n−1))Smallest k such that 10nk is prime (https://oeis.org/A033874, https://oeis.org/A003618 (corresponding primes))
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 ≥ 1Generalized 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, http://www.primenumbers.net/Henri/us/NouvTh1us.htm, https://web.archive.org/web/20050406075848/http://ksc9.th.com/warut/cunningham.html, 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, http://www.primenumbers.net/Henri/us/NouvTh1us.htm, https://web.archive.org/web/20050406075848/http://ksc9.th.com/warut/cunningham.html, 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://oeis.org/A014574, https://oeis.org/A002822, 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://sweet.ua.pt/tos/twin_gaps.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://web.archive.org/web/20081120155333/http://www.15k.org/FirstKTwin.htm, https://www.primepuzzles.net/problems/prob_049.htm, https://www.rieselprime.de/ziki/Twin_Prime_Search, https://www.rieselprime.de/ziki/RPS_Drive_9ks, 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, also all known large definitely bi-twin primes (https://en.wikipedia.org/wiki/Bi-twin_chain, https://mathworld.wolfram.com/BitwinChain.html, http://www.primenumbers.net/Henri/fr-us/BiTwinRec.htm, http://www.primenumbers.net/Henri/us/NouvChPus.htm, https://oeis.org/A066388, https://oeis.org/A076504, https://oeis.org/A177680), which contain two twin prime pairs and a Cunningham chain of the first kind with length 2 and Cunningham chain of the second kind with length 2, also contain a Sophie Germain prime and a safe prime and a Sophie Germain prime of the second kind and a safe prime of the second kind, are two primes p (in a Cunningham chain of the first kind with length 2) such that p+1 has many factors and two primes q (in a Cunningham chain of the second kind with length 2) 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, https://miller-rabin.appspot.com/, http://www.pi-e.de/Miller-Rabin-Pseudoprimzahlen.htm (in German), 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/A020230, https://oeis.org/A020231, https://oeis.org/A020232, https://oeis.org/A020233, https://oeis.org/A020234, https://oeis.org/A020235, https://oeis.org/A020236, https://oeis.org/A020237, https://oeis.org/A020238, https://oeis.org/A020239, https://oeis.org/A020240, https://oeis.org/A020241, https://oeis.org/A020242, https://oeis.org/A020243, https://oeis.org/A020244, https://oeis.org/A020245, https://oeis.org/A020246, https://oeis.org/A020247, https://oeis.org/A020248, https://oeis.org/A020249, https://oeis.org/A020250, https://oeis.org/A020251, https://oeis.org/A020252, https://oeis.org/A020253, https://oeis.org/A020254, https://oeis.org/A020255, https://oeis.org/A020256, https://oeis.org/A020257, https://oeis.org/A020258, https://oeis.org/A020259, https://oeis.org/A020260, https://oeis.org/A020261, https://oeis.org/A020262, 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, http://www.hoegge.dk/lucasselfridgeprps.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)

Two references of many types of pseudoprimes: http://ntheory.org/pseudoprimes.html and http://gilchrist.ca/jeff/factoring/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://ceur-ws.org/Vol-1326/020-Forisek.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_481.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, https://miller-rabin.appspot.com/, http://www.pi-e.de/Miller-Rabin-Pseudoprimzahlen.htm (in German), http://factordb.com/prooffailed.php, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773

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