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unproven-probable-primes/README.md

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 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 > 10<sup>299</sup> (base *b* = 26 has 82 known minimal (probable) primes > 10<sup>299</sup> and 4 unsolved families, base *b* = 36 has 75 known minimal (probable) primes > 10<sup>299</sup> and 4 unsolved families, base *b* = 17 has 99 known minimal (probable) primes > 10<sup>299</sup> and 18 unsolved families, base *b* = 21 has 80 known minimal (probable) primes > 10<sup>299</sup> and 12 unsolved families, base *b* = 19 has 201 known minimal (probable) primes > 10<sup>299</sup> 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 > 10<sup>100000</sup>, and longer than the age of the universe (https://en.wikipedia.org/wiki/Age_of_the_universe) for numbers > 10<sup>500000</sup>, and longer than one quettasecond (https://en.wikipedia.org/wiki/Quetta-) for numbers > 10<sup>3000000</sup>, even if we can do 10<sup>9</sup> 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*)<sup>12</sup>) (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 10<sup>9</sup> 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 5.422859049×10<sup>39</sup> seconds (https://en.wikipedia.org/wiki/Second), or 1.719577324×10<sup>32</sup> 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*)<sup>*log*(*log*(*log*(*n*)))</sup>) 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 > 10<sup>25000</sup>, 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 *Φ*<sub>4807</sub>(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 1<sub>4807</sub>01, the large prime factor of 15<sub>2196</sub>1 is related to the prime 93<sub>2195</sub>07, the large prime factor of 16<sub>4295</sub>7 is related to the prime 3<sub>12890</sub>7, the large prime factor of 201<sub>2692</sub> is related to the prime 201<sub>2693</sub>, the large prime factor of *Φ*<sub>5014</sub>(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 23<sub>10027</sub>09, the large prime factor of *Φ*<sub>7884</sub>(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 37<sub>15767</sub>3, the large prime factor of 6805<sub>3738</sub>7 is related to the prime 272<sub>3740</sub>7, the large prime factor of *Φ*<sub>1283</sub>(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 68<sub>5131</sub>3, the large prime factor of *Φ*<sub>2907</sub>(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 79<sub>11627</sub>21, the large prime factor of *Φ*<sub>11470</sub>(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 80<sub>5734</sub>81, the large prime factor of 83<sub>542</sub>16<sub>542</sub>7 (a number in a non-simple family 8{3}1{6}7) is related to the prime 1<sub>3256</sub>03), 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 10<sup>16</sup>) 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-*n*th-powers factorization with *n* > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html), sum/difference-of-two-*n*th-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 *x*<sup>4</sup>+4×*y*<sup>4</sup> or *x*<sup>6</sup>+27×*y*<sup>6</sup>, 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)) *b*<sup>*n*</sup>±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, 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)) (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 10<sup>20</sup> and 10<sup>90</sup>) 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 10<sup>100</sup> and 10<sup>400</sup>), special number field sieve can only be used on a number of special form, e.g. numbers of the form *a*×*b*<sup>*n*</sup>±*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*<sup>3×*n*/2</sup>, thus, for the Cunningham number *b*<sup>*n*</sup>±1, if the primitive part (i.e. *Φ*<sub>*n*</sub>(*b*) for *b*<sup>*n*</sup>−1 or *Φ*<sub>2×*n*</sub>(*b*) for *b*<sup>*n*</sup>+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*<sub>*n*</sub>(*b*) for *b*<sup>*n*</sup>−1 or *Φ*<sub>2×*n*</sub>(*b*) for *b*<sup>*n*</sup>+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*<sub>*n*</sub>(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.
+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 *Φ*<sub>4807</sub>(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 1<sub>4807</sub>01, the large prime factor of 15<sub>2196</sub>1 is related to the prime 93<sub>2195</sub>07, the large prime factor of 16<sub>4295</sub>7 is related to the prime 3<sub>12890</sub>7, the large prime factor of 201<sub>2692</sub> is related to the prime 201<sub>2693</sub>, the large prime factor of *Φ*<sub>5014</sub>(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 23<sub>10027</sub>09, the large prime factor of *Φ*<sub>7884</sub>(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 37<sub>15767</sub>3, the large prime factor of 6805<sub>3738</sub>7 is related to the prime 272<sub>3740</sub>7, the large prime factor of *Φ*<sub>1283</sub>(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 68<sub>5131</sub>3, the large prime factor of *Φ*<sub>2907</sub>(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 79<sub>11627</sub>21, the large prime factor of *Φ*<sub>11470</sub>(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 80<sub>5734</sub>81, the large prime factor of 83<sub>542</sub>16<sub>542</sub>7 (a number in a non-simple family 8{3}1{6}7) is related to the prime 1<sub>3256</sub>03), 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 10<sup>16</sup>) 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-*n*th-powers factorization with *n* > 1 (https://en.wikipedia.org/wiki/Binomial_number, https://mathworld.wolfram.com/BinomialNumber.html), sum/difference-of-two-*n*th-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 *x*<sup>4</sup>+4×*y*<sup>4</sup> or *x*<sup>6</sup>+27×*y*<sup>6</sup>, 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)) *b*<sup>*n*</sup>±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, 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)) (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 10<sup>20</sup> and 10<sup>90</sup>) 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 10<sup>100</sup> and 10<sup>400</sup>), special number field sieve can only be used on a number of special form, e.g. numbers of the form *a*×*b*<sup>*n*</sup>±*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*<sup>3×*n*/2</sup>, thus, for the Cunningham number *b*<sup>*n*</sup>±1, if the primitive part (i.e. *Φ*<sub>*n*</sub>(*b*) for *b*<sup>*n*</sup>−1 or *Φ*<sub>2×*n*</sub>(*b*) for *b*<sup>*n*</sup>+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*<sub>*n*</sub>(*b*) for *b*<sup>*n*</sup>−1 or *Φ*<sub>2×*n*</sub>(*b*) for *b*<sup>*n*</sup>+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*<sub>*n*</sub>(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 > 10<sup>25000</sup> 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 10<sup>16</sup> 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 15<sub>63</sub> (its value is (4×16<sup>63</sup>−1)/3) and 85<sub>36</sub> (its value is (25×16<sup>36</sup>−1)/3), and our list for base 9 minimal primes would wrongly include the composite 1<sub>13</sub> (its value is (9<sup>13</sup>−1)/8) (and hence would wrongly exclude the prime 561<sub>36</sub>, since this prime has 1<sub>13</sub> 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).