(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 primality tests (https://en.wikipedia.org/wiki/Probabilistic_algorithm) 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, http://factordb.com/prooffailed.php, https://sites.google.com/view/strong-pseudoprime, https://sites.google.com/view/bases-strong-pseudoprime, https://oeis.org/A001262, https://oeis.org/A020229, https://oeis.org/A020231, https://oeis.org/A020233, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773, https://oeis.org/A014233, https://oeis.org/A006945, https://oeis.org/A089825, https://oeis.org/A181782, https://oeis.org/A071294, https://oeis.org/A141768, https://oeis.org/A195328, https://oeis.org/A329759, https://oeis.org/A298756) 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, http://factordb.com/prooffailed.php, https://sites.google.com/view/strong-pseudoprime, https://sites.google.com/view/bases-strong-pseudoprime, https://oeis.org/A001262, https://oeis.org/A020229, https://oeis.org/A020231, https://oeis.org/A020233, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773, https://oeis.org/A014233, https://oeis.org/A006945, https://oeis.org/A089825, https://oeis.org/A181782, https://oeis.org/A071294, https://oeis.org/A141768, https://oeis.org/A195328, https://oeis.org/A329759, https://oeis.org/A298756) 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, 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 > 10^3000000, 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), 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://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)) of x⁴+4×y⁴ or x⁶+27×y⁶, 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, http://www.mersenne.org/report_ECM/, https://www.mersenne.ca/userfactors/ecm/1, https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, https://kurtbeschorner.de/ecm-efforts.htm, http://www.rechenkraft.net/yoyo//y_factors_ecm.php, http://www.rechenkraft.net/yoyo/y_status_ecm.php, http://www.wraithx.net/math/ecmprobs/ecmprobs.html) (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×n!±c and a×p#±c (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 b^3×n/2, thus, for the Cunningham number bⁿ±1, if the primitive part (i.e. Φ_n(b) for bⁿ−1 or Φ_2×n(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. Φ_n(b) for bⁿ−1 or Φ_2×n(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 L_n(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://www.rieselprime.de/ziki/Distributed_computing) 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) (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, 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://sites.google.com/view/fermat-pseudoprime, https://sites.google.com/view/bases-fermat-pseudoprime, https://oeis.org/A001567, https://oeis.org/A005935, https://oeis.org/A005936, https://oeis.org/A005938, https://oeis.org/A052155, https://oeis.org/A083737, https://oeis.org/A083739, https://oeis.org/A083876, https://oeis.org/A271221, https://oeis.org/A348258, https://oeis.org/A181780, https://oeis.org/A063994, https://oeis.org/A194946, https://oeis.org/A195327, https://oeis.org/A002997, https://oeis.org/A191311, https://oeis.org/A090086, https://oeis.org/A007535) 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, http://factordb.com/prooffailed.php, https://sites.google.com/view/strong-pseudoprime, https://sites.google.com/view/bases-strong-pseudoprime, https://oeis.org/A001262, https://oeis.org/A020229, https://oeis.org/A020231, https://oeis.org/A020233, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773, https://oeis.org/A014233, https://oeis.org/A006945, https://oeis.org/A089825, https://oeis.org/A181782, https://oeis.org/A071294, https://oeis.org/A141768, https://oeis.org/A195328, https://oeis.org/A329759, https://oeis.org/A298756) to all prime bases 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://sites.google.com/view/fermat-pseudoprime, https://sites.google.com/view/bases-fermat-pseudoprime, https://oeis.org/A001567, https://oeis.org/A005935, https://oeis.org/A005936, https://oeis.org/A005938, https://oeis.org/A052155, https://oeis.org/A083737, https://oeis.org/A083739, https://oeis.org/A083876, https://oeis.org/A271221, https://oeis.org/A348258, https://oeis.org/A181780, https://oeis.org/A063994, https://oeis.org/A194946, https://oeis.org/A195327, https://oeis.org/A002997, https://oeis.org/A191311, https://oeis.org/A090086, https://oeis.org/A007535) 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)

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, http://factordb.com/prooffailed.php, https://sites.google.com/view/strong-pseudoprime, https://sites.google.com/view/bases-strong-pseudoprime, https://oeis.org/A001262, https://oeis.org/A020229, https://oeis.org/A020231, https://oeis.org/A020233, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773, https://oeis.org/A014233, https://oeis.org/A006945, https://oeis.org/A089825, https://oeis.org/A181782, https://oeis.org/A071294, https://oeis.org/A141768, https://oeis.org/A195328, https://oeis.org/A329759, https://oeis.org/A298756) 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, 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 b^c−1 == 1 mod c for half of the bases b coprime to c, the Carmichael numbers are the composite numbers c such that b^c−1 == 1 mod c for all bases b coprime to c)) (and the numbers in https://oeis.org/A191311 but not in https://oeis.org/A129521 are the numbers in https://oeis.org/A191592, the smallest such number (except the trivial number 4, which is the only one such even number, the only one such non-squarefree number, and the only one such semiprime, note that all Carmichael numbers are odd, squarefree, and have at least 3 prime factors) is 11305 = 5 × 7 × 17 × 19). e.g. 15₁₅ (base 16) is Fermat pseudoprime to base b (assuming b is coprime to the number) if and only if b is cubic residue (https://en.wikipedia.org/wiki/Cubic_residue, https://mathworld.wolfram.com/CubicResidue.html) mod the prime 2³¹−1, thus 1/3 of the bases coprime to the number; 15₃₀ (base 16) is Fermat pseudoprime to base b (assuming b is coprime to the number) if and only if b is cubic residue (https://en.wikipedia.org/wiki/Cubic_residue, https://mathworld.wolfram.com/CubicResidue.html) mod the prime 2⁶¹−1, thus 1/3 of the bases coprime to the number; 15₆₃ (base 16) is Fermat pseudoprime to base b (assuming b is coprime to the number) if and only if b is cubic residue (https://en.wikipedia.org/wiki/Cubic_residue, https://mathworld.wolfram.com/CubicResidue.html) mod the prime 2¹²⁷−1, thus 1/3 of the bases coprime to the number; 85₆ (base 16) is Fermat pseudoprime to base b (assuming b is coprime to the number) if and only if b is cubic residue (https://en.wikipedia.org/wiki/Cubic_residue, https://mathworld.wolfram.com/CubicResidue.html) mod the prime 5×2¹²−1, thus 1/3 of the bases coprime to the number; 85₃₆ (base 16) is Fermat pseudoprime to base b (assuming b is coprime to the number) if and only if b is cubic residue (https://en.wikipedia.org/wiki/Cubic_residue, https://mathworld.wolfram.com/CubicResidue.html) mod the prime 5×2⁷²−1, thus 1/3 of the bases coprime to the number; 1₁₃ (base 9) is Fermat pseudoprime to base b (assuming b is coprime to the number) if and only if b is quadratic residue (https://en.wikipedia.org/wiki/Quadratic_residue, https://t5k.org/glossary/xpage/QuadraticResidue.html, https://www.rieselprime.de/ziki/Quadratic_residue, https://mathworld.wolfram.com/QuadraticResidue.html) mod the prime (3¹³−1)/2, thus 1/2 of the bases coprime to the number.

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:

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).

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

All these numbers are strong probable primes (https://t5k.org/prove/prove2_3.html, https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test, https://en.wikipedia.org/wiki/Strong_pseudoprime, https://t5k.org/glossary/xpage/MillersTest.html, https://t5k.org/glossary/xpage/StrongPRP.html, https://www.rieselprime.de/ziki/Miller-Rabin_pseudoprimality_test, https://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html, https://mathworld.wolfram.com/StrongPseudoprime.html, http://www.numericana.com/answer/pseudo.htm#rabin, http://www.numericana.com/answer/pseudo.htm#strong, http://www.javascripter.net/math/primes/millerrabinprimalitytest.htm, http://ntheory.org/data/spsps.txt, https://faculty.lynchburg.edu/~nicely/misc/mpzspsp.html, http://factordb.com/prooffailed.php, https://sites.google.com/view/strong-pseudoprime, https://sites.google.com/view/bases-strong-pseudoprime, https://oeis.org/A001262, https://oeis.org/A020229, https://oeis.org/A020231, https://oeis.org/A020233, https://oeis.org/A072276, https://oeis.org/A056915, https://oeis.org/A074773, https://oeis.org/A014233, https://oeis.org/A006945, https://oeis.org/A089825, https://oeis.org/A181782, https://oeis.org/A071294, https://oeis.org/A141768, https://oeis.org/A195328, https://oeis.org/A329759, https://oeis.org/A298756) to all prime bases ≤ 64 (i.e. bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61) (see https://oeis.org/A014233 and https://oeis.org/A141768 and https://oeis.org/A001262 and https://oeis.org/A074773 and http://ntheory.org/data/psps.txt), and strong Lucas probable primes (https://en.wikipedia.org/wiki/Strong_Lucas_pseudoprime, https://mathworld.wolfram.com/StrongLucasPseudoprime.html) with parameters (P, Q) defined by Selfridge's Method A (see https://oeis.org/A217255 and http://ntheory.org/data/slpsps-baillie.txt), and trial factored (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) to 10¹⁶ (thus, all these numbers are Baillie–PSW probable primes (https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test, https://mathworld.wolfram.com/Baillie-PSWPrimalityTest.html, http://pseudoprime.com/dopo.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_318.pdf))), and since all of them are > 10²⁵⁰⁰⁰, thus the property that they are in fact composite is less than 10⁻²⁰⁰⁰ (but still not zero!), see https://t5k.org/notes/prp_prob.html and https://www.ams.org/journals/mcom/1989-53-188/S0025-5718-1989-0982368-4/S0025-5718-1989-0982368-4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_22.pdf) (bases 11, 13, 16, 22, 30 have no unsolved families but have unproven probable primes, thus, we can 99.999999...999999% (with >2000 9's) say that the minimal sets for bases 11, 13, 16, 22, 30 are these five sets, but we still cannot 100% say this)

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

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

For the files in this page:

• File "unprovenPRP b": The unproven probable primes which are minimal primes in base b (assuming their primality), expressed as base b strings with x(y^n)z format (which in the README files it is denoted as "xy_nz") (y^n means n copies of y, e.g. 34(5^6)78 is 3455555578, this n is written in decimal)
• File "unprovenPRPalgebraic b": The algebraic ((a×bⁿ+c)/gcd(a+c,b−1)) form of the unproven probable primes which are minimal primes in base b (assuming their primality)
• File "primoinput b n": The input file for Primo for the nth minimal prime in base b (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://t5k.org/bios/page.php?id=46, https://www.rieselprime.de/ziki/Primo, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/primo-433-lx64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/Primo309) (local copy from factordb (http://factordb.com/, https://www.rieselprime.de/ziki/Factoring_Database)), they should be renamed to ".in" files before inputting to Primo, e.g. file "primoinput11_1068" is the input file for Primo for the 1068th minimal prime in base 11, i.e. the input file for Primo for the probable prime 57₆₂₆₆₈ in base 11, which equals the probable prime (57×11⁶²⁶⁶⁸−7)/10 (for more input files for Primo in factordb, see http://factordb.com/primobatch.php and http://factordb.com/primobatch.php?digits=300&files=10&parts=1&start=Generate+zip and http://factordb.com/primobatch.php?digits=300&files=32000&parts=32&start=Generate+zip). (Note: There is no input file for Primo for the probable prime 95₁₉₇₄₂₀ in base 13 (which equals the probable prime (113×13¹⁹⁷⁴²⁰−5)/12) and A3₅₉₂₁₉₇A in base 13 (which equals the probable prime (41×13⁵⁹²¹⁹⁸+27)/4) since these two probable primes are too large (>10¹⁹⁹⁹⁹⁹) to be PRP-tested in factordb, and factordb does not have input file for Primo for numbers with status (http://factordb.com/status.html, http://factordb.com/distribution.php) "U" (i.e. in http://factordb.com/listtype.php?t=2), factordb only has input file for Primo for numbers with status "PRP" (i.e. in http://factordb.com/listtype.php?t=1))