These are the *Primo* (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) (an elliptic curve primality proving (https://t5k.org/prove/prove4_2.html, https://en.wikipedia.org/wiki/Elliptic_curve_primality, https://t5k.org/glossary/xpage/ECPP.html, https://mathworld.wolfram.com/EllipticCurvePrimalityProving.html, https://t5k.org/top20/page.php?id=27, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/~morain/Prgms/ecpp.english.html, https://www.multiprecision.org/cm/ecpp.html) implementation) primality certificates (https://en.wikipedia.org/wiki/Primality_certificate, https://t5k.org/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/Labo/Francois.Morain/Primes/myprimes.html, https://stdkmd.net/nrr/cert/, https://alfredreichlg.de/10w7/certifiedprimes.html, http://xenon.stanford.edu/~tjw/pp/index.html, http://factordb.com/certoverview.php) for the minimal primes > 10299 and < 1025000 (primes < 10299 are automatically proven primes in *factordb*, and primes < 10299 can be verified in a few seconds, proof of their primality is not included here, in order to save space, larger primes can take from hours to months to prove, unless their *N*−1 (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1) or/and *N*+1 (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2) can be ≥ 1/4 factored) in bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 (the solved or near-solved bases, i.e. the bases *b* with ≤ 6 unsolved families)
The large minimal primes in base *b* are of the form (*a*×*b**n*+*c*)/*gcd*(*a*+*c*,*b*−1) for some *a*, *b*, *c*, *n* such that *a* ≥ 1, *b* ≥ 2 (*b* is the base), *c* ≠ 0, *gcd*(*a*,*c*) = 1, *gcd*(*b*,*c*) = 1, the large numbers (i.e. the numbers with large *n*, say *n* > 1000) can be easily proven primes using *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://factordb.com/nmoverview.php?method=1) or *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://factordb.com/nmoverview.php?method=2) if and only if *c* = ±1 and *gcd*(*a*+*c*,*b*−1) = 1, except this special case of *c* = ±1 and *gcd*(*a*+*c*,*b*−1) = 1, such numbers need primality certificates to be proven primes (otherwise, they will only be 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://stdkmd.net/nrr/records.htm#probableprimenumbers, https://stdkmd.net/nrr/repunit/prpfactors.htm, https://www.alfredreichlg.de/10w7/prp.html, http://factordb.com/listtype.php?t=1)), and elliptic curve primality proving are used for these numbers.
The case *c* = 1 and *gcd*(*a*+*c*,*b*−1) = 1 (corresponding to generalized Proth prime (https://en.wikipedia.org/wiki/Proth_prime, https://t5k.org/glossary/xpage/ProthPrime.html, https://www.rieselprime.de/ziki/Proth_prime, https://mathworld.wolfram.com/ProthNumber.html, https://www.numbersaplenty.com/set/Proth_number/, https://web.archive.org/web/20230706041914/https://pzktupel.de/Primetables/TableProthTOP10KK.php, https://pzktupel.de/Primetables/ProthK.txt, https://pzktupel.de/Primetables/TableProthTOP10KS.php, https://pzktupel.de/Primetables/ProthS.txt, https://pzktupel.de/Primetables/TableProthGen.php, https://pzktupel.de/Primetables/TableProthGen.txt, https://sites.google.com/view/proth-primes, https://t5k.org/primes/search_proth.php, https://t5k.org/top20/page.php?id=66, https://www.primegrid.com/forum_thread.php?id=2665, https://www.primegrid.com/stats_pps_llr.php, https://www.primegrid.com/stats_ppse_llr.php, https://www.primegrid.com/stats_mega_llr.php) base *b*: *a*×*b**n*+1, they are related to generalized Sierpinski conjecture base *b* (http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Sierp-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.utm.edu/staff/caldwell/preprints/2to100.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_3.pdf), http://www.bitman.name/math/article/1259 (in Italian))) can be easily proven prime using Pocklington *N*−1 method (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1), and the case *c* = −1 and *gcd*(*a*+*c*,*b*−1) = 1 (corresponding to generalized Riesel prime (https://www.rieselprime.de/ziki/Riesel_prime, https://web.archive.org/web/20230628151418/https://pzktupel.de/Primetables/TableRieselTOP10KK.php, https://pzktupel.de/Primetables/RieselK.txt, https://pzktupel.de/Primetables/TableRieselTOP10KS.php, https://pzktupel.de/Primetables/RieselS.txt, https://pzktupel.de/Primetables/TableRieselGen.php, https://pzktupel.de/Primetables/TableRieselGen.txt, https://sites.google.com/view/proth-primes, https://t5k.org/primes/search_proth.php) base *b*: *a*×*b**n*−1, they are related to generalized Riesel conjecture base *b* (http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm, http://www.noprimeleftbehind.net/crus/Riesel-conjecture-reserves.htm, http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt)) can be easily proven prime using Morrison *N*+1 method (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2), these primes can be proven prime using Yves Gallot's *Proth.exe* (https://t5k.org/programs/gallot/, https://t5k.org/bios/page.php?id=411, https://www.rieselprime.de/ziki/Proth.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/proth), these primes can also be proven prime using Jean Penné's *LLR* (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64), also 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 see https://stdkmd.net/nrr/prime/primesize.txt and https://stdkmd.net/nrr/prime/primesize.zip (see which numbers are proven primes and which numbers are only probable primes), also see https://stdkmd.net/nrr/records.htm (compare the sections "Prime numbers" and "Probable prime numbers").
Primes which either *N*−1 or *N*+1 is trivially fully factored (i.e. primes of the form *k*×*b**n*±1, with small *k*) do not need primality certificates, since they can be easily proven primes using *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://factordb.com/nmoverview.php?method=1) or *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://factordb.com/nmoverview.php?method=2), these primes are: (i.e. their *N*−1 or *N*+1 are smooth numbers (https://en.wikipedia.org/wiki/Smooth_number, https://mathworld.wolfram.com/SmoothNumber.html, https://oeis.org/A003586, https://oeis.org/A051037, https://oeis.org/A002473, https://oeis.org/A051038, https://oeis.org/A080197, https://oeis.org/A080681, https://oeis.org/A080682, https://oeis.org/A080683))
* the 3176th minimal prime in base 13, 810104151, which equals 17746×13416+1, *N*−1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000003590431555, for the factorization of *N*−1 in *factordb* see http://factordb.com/index.php?id=1100000003590431556&open=ecm
* the 3177th minimal prime in base 13, 81104351, which equals 1366×13436+1, *N*−1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000002373259109, for the factorization of *N*−1 in *factordb* see http://factordb.com/index.php?id=1100000002373259124&open=ecm
* the 3188th minimal prime in base 13, 93015511, which equals 120×131552+1, *N*−1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000000765961452, for the factorization of *N*−1 in *factordb* see http://factordb.com/index.php?id=1100000000765961453&open=ecm
* the 3191st minimal prime in base 13, 39062661, which equals 48×136267+1, *N*−1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000000765961441, for the factorization of *N*−1 in *factordb* see http://factordb.com/index.php?id=1100000000765961451&open=ecm
* the 649th minimal prime in base 14, 34D708, which equals 47×14708−1, *N*+1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000001540144903, for the factorization of *N*+1 in *factordb* see http://factordb.com/index.php?id=1100000001540144907&open=ecm
* the 650th minimal prime in base 14, 4D19698, which equals 5×1419698−1, *N*+1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000000884560233, for the factorization of *N*+1 in *factordb* see http://factordb.com/index.php?id=1100000000884560625&open=ecm
* the 2335th minimal prime in base 16, 88F545, which equals 137×16545−1, *N*+1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000000413679658, for the factorization of *N*+1 in *factordb* see http://factordb.com/index.php?id=1100000000413877337&open=ecm
* the 3310th minimal prime in base 20, JCJ629, which equals 393×20629−1, *N*+1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000001559454258, for the factorization of *N*+1 in *factordb* see http://factordb.com/index.php?id=1100000001559454271&open=ecm
* the 3408th minimal prime in base 24, 88N5951, which equals 201×245951−1, *N*+1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000003593275880, for the factorization of *N*+1 in *factordb* see http://factordb.com/index.php?id=1100000003593373246&open=ecm
* the 25509th minimal prime in base 28, EB04051, which equals 403×28406+1, *N*−1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000001534442374, for the factorization of *N*−1 in *factordb* see http://factordb.com/index.php?id=1100000001534442380&open=ecm
* the 2616th minimal prime in base 30, C010221, which equals 12×301023+1, *N*−1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000000785448736, for the factorization of *N*−1 in *factordb* see http://factordb.com/index.php?id=1100000000785448737&open=ecm
* the 2619th minimal prime in base 30, OT34205, which equals 25×3034205−1, *N*+1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000000800812865, for the factorization of *N*+1 in *factordb* see http://factordb.com/index.php?id=1100000000819405041&open=ecm
* the 35237th minimal prime in base 36, P8Z390, which equals 909×36390−1, *N*+1 is trivially fully factored, for its helper file in *factordb* see http://factordb.com/helper.php?id=1100000000764100228, for the factorization of *N*+1 in *factordb* see http://factordb.com/index.php?id=1100000000764100231&open=ecm
(these primes can be proven prime using Yves Gallot's *Proth.exe* (https://t5k.org/programs/gallot/, https://t5k.org/bios/page.php?id=411, https://www.rieselprime.de/ziki/Proth.exe, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/proth), these primes can also be proven prime using Jean Penné's *LLR* (http://jpenne.free.fr/index2.html, https://t5k.org/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64), see the *README* file for *LLR* (https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/llr403win64/Readme.txt, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/llr403linux64/Readme.txt, http://jpenne.free.fr/index2.html))
Also, there are no primality certificates for these primes in *factordb* because although they are > 10299, but their *N*−1 or *N*+1 is fully factored (but not trivially fully factored, however, only need 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://oeis.org/A189172) to 108) and the largest prime factor is < 10299 (primes < 10299 are automatically proven primes in *factordb*): (i.e. their *N*−1 or *N*+1 are product of a 108-smooth number (https://en.wikipedia.org/wiki/Smooth_number, https://mathworld.wolfram.com/SmoothNumber.html, https://oeis.org/A003586, https://oeis.org/A051037, https://oeis.org/A002473, https://oeis.org/A051038, https://oeis.org/A080197, https://oeis.org/A080681, https://oeis.org/A080682, https://oeis.org/A080683) and a prime < 10299)
* the 2328th minimal prime in base 16, 8802467, with 300 decimal digits, *N*−1 is 23 × 3 × 7 × 13 × 25703261 × (289-digit prime)
* the 25174th minimal prime in base 26, OL0214M9, with 309 decimal digits, *N*−1 is 22 × 52 × 7 × 223 × 42849349 × (296-digit prime)
* the 25485th minimal prime in base 28, JN206, with 300 decimal digits, *N*−1 is 2 × 1061 × 1171 × 74311 × (289-digit prime)
The helper file for the 2328th minimal prime in base 16 (8802467) in *factordb*: http://factordb.com/helper.php?id=1100000002468140199
The helper file for the 25174th minimal prime in base 26 (OL0214M9) in *factordb*: http://factordb.com/helper.php?id=1100000000840631576
The helper file for the 25485th minimal prime in base 28 (JN206) in *factordb*: http://factordb.com/helper.php?id=1100000002611724435
Factorization of *N*−1 for the 2328th minimal prime in base 16 (8802467) in *factordb*: http://factordb.com/index.php?id=1100000002468140641&open=ecm
Factorization of *N*−1 for the 25174th minimal prime in base 26 (OL0214M9) in *factordb*: http://factordb.com/index.php?id=1100000000840631577&open=ecm
Factorization of *N*−1 for the 25485th minimal prime in base 28 (JN206) in *factordb*: http://factordb.com/index.php?id=1100000002611724440&open=ecm
Also the case where *N*−1 or *N*+1 is product of a Cunningham number (of the form *b**n*±1, see 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://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)) and a small number (either a small integer or a fraction whose numerator and denominator are both small), and this Cunningham number is ≥ 1/3 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) or this Cunningham number is ≥ 1/4 factored and the number is not very large (say not > 10100000). If either *N*−1 or *N*+1 (or both) can be ≥ 1/2 factored, then we can use the Pocklington *N*−1 primality test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, 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) (the *N*−1 case) or the Morrison *N*+1 primality test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2) (the *N*+1 case); if either *N*−1 or *N*+1 (or both) can be ≥ 1/3 factored, then we can use the Brillhart-Lehmer-Selfridge primality test (https://www.ams.org/journals/mcom/1975-29-130/S0025-5718-1975-0384673-1/S0025-5718-1975-0384673-1.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_23.pdf), https://en.wikipedia.org/wiki/Pocklington_primality_test#Extensions_and_variants); if either *N*−1 or *N*+1 (or both) can be ≥ 1/4 factored but neither can be ≥ 1/3 factored, then we need to use *CHG* (https://mersenneforum.org/attachment.php?attachmentid=21133&d=1571237465, https://t5k.org/bios/page.php?id=797, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/CHG) to prove its primality (see https://mersenneforum.org/showpost.php?p=528149&postcount=3 and https://mersenneforum.org/showpost.php?p=603181&postcount=438), however, unlike Brillhart-Lehmer-Selfridge primality test for the numbers *N* such that *N*−1 or *N*+1 (or both) can be ≥ 1/3 factored can run for arbitrarily large numbers *N* (thus, there are no unproven probable primes *N* such that *N*−1 or *N*+1 (or both) can be ≥ 1/3 factored), *CHG* for the numbers *N* such that either *N*−1 or *N*+1 (or both) can be ≥ 1/4 factored but neither can be ≥ 1/3 factored cannot run for very large *N* (say > 10100000), for the examples of the numbers which are proven prime by *CHG*, see https://t5k.org/primes/page.php?id=126454, https://t5k.org/primes/page.php?id=131964, https://t5k.org/primes/page.php?id=123456, https://t5k.org/primes/page.php?id=130933, https://stdkmd.net/nrr/cert/1/ (search for "CHG"), https://stdkmd.net/nrr/cert/2/ (search for "CHG"), https://stdkmd.net/nrr/cert/3/ (search for "CHG"), https://stdkmd.net/nrr/cert/4/ (search for "CHG"), https://stdkmd.net/nrr/cert/5/ (search for "CHG"), https://stdkmd.net/nrr/cert/6/ (search for "CHG"), https://stdkmd.net/nrr/cert/7/ (search for "CHG"), https://stdkmd.net/nrr/cert/8/ (search for "CHG"), https://stdkmd.net/nrr/cert/9/ (search for "CHG"), http://xenon.stanford.edu/~tjw/pp/index.html (search for "CHG"), however, *factordb* (http://factordb.com/, https://www.rieselprime.de/ziki/Factoring_Database) lacks the ability to verify *CHG* proofs, see https://mersenneforum.org/showpost.php?p=608362&postcount=165; if neither *N*−1 nor *N*+1 can be ≥ 1/4 factored but *N**n*−1 can be ≥ 1/3 factored for a small *n*, then we can use the cyclotomy primality test (https://t5k.org/glossary/xpage/Cyclotomy.html, https://t5k.org/prove/prove3_3.html, http://factordb.com/nmoverview.php?method=3) (however, this situation does not exist for these numbers, since only one of *N*−1 and *N*+1 is product of a Cunningham number and a small number, the only exception is the numbers in the family {2}1 in base *b*, in such case both *N*−1 and *N*+1 are products of a Cunningham number and a small number, thus for the numbers in the family {2}1 in base *b*, maybe factorization of *N*2−1 can be used)): (thus these numbers also do not need primality certificates)
(for the examples of generalized repunit primes (*all* generalized repunit primes base *b* have that *N*−1 is product of a Cunningham number (base *b*, the −1 side) and a small number (namely *b*/(*b*−1))), see https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html and https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html and https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html and http://xenon.stanford.edu/~tjw/pp/index.html)
(for the references of factorization of *b**n*±1, see: https://homes.cerias.purdue.edu/~ssw/cun/index.html (2 ≤ *b* ≤ 12), https://homes.cerias.purdue.edu/~ssw/cun/pmain423.txt (2 ≤ *b* ≤ 12), https://doi.org/10.1090/conm/022 (2 ≤ *b* ≤ 12), 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) (2 ≤ *b* ≤ 12), https://web.archive.org/web/20190315214330/http://cage.ugent.be/~jdemeyer/cunningham/ (2 ≤ *b* ≤ 12), https://maths-people.anu.edu.au/~brent/factors.html (13 ≤ *b* ≤ 99), https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html (2 ≤ *b* ≤ 999), https://stdkmd.net/nrr/repunit/ (*b* = 10), https://stdkmd.net/nrr/repunit/10001.htm (*b* = 10), https://stdkmd.net/nrr/repunit/phin10.htm (*b* = 10, only primitive factors), https://stdkmd.net/nrr/repunit/Phin10.txt (*b* = 10, only primitive factors), https://stdkmd.net/nrr/repunit/Phin10.txt.lz (*b* = 10, only primitive factors), https://stdkmd.net/nrr/repunit/Phin10.txt.gz (*b* = 10, only primitive factors), https://kurtbeschorner.de/ (*b* = 10), https://kurtbeschorner.de/fact-2500.htm (*b* = 10), https://repunit-koide.jimdofree.com/ (*b* = 10), https://repunit-koide.jimdofree.com/app/download/10034950550/Repunit100-20230630.pdf?t=1688135997 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_242.pdf) (*b* = 10), https://gmplib.org/~tege/repunit.html (*b* = 10), https://gmplib.org/~tege/fac10m.txt (*b* = 10), https://gmplib.org/~tege/fac10p.txt (*b* = 10), https://web.archive.org/web/20120426061657/http://oddperfect.org/ (prime *b*), https://web.archive.org/web/20081006071311/http://www-staff.maths.uts.edu.au/~rons/fact/fact.htm (2 ≤ *b* ≤ 9973, prime *b*), http://myfactors.mooo.com/ (any *b*), http://myfactorcollection.mooo.com:8090/dbio.html (any *b*), http://www.asahi-net.or.jp/~KC2H-MSM/cn/old/index.htm (any *b*, only primitive factors), http://www.asahi-net.or.jp/~KC2H-MSM/cn/index.htm (any *b*, only primitive factors), https://web.archive.org/web/20050922233702/http://user.ecc.u-tokyo.ac.jp/~g440622/cn/index.html (any *b*, only primitive factors), also for the factors of *b**n*±1 with 2 ≤ *b* ≤ 100 and 1 ≤ *n* ≤ 100 see http://myfactorcollection.mooo.com:8090/cgi-bin/showFullRep?FBase=2&TBase=100&FExp=1&TExp=100&c0=&EN=&LM= (all factors) and http://myfactorcollection.mooo.com:8090/cgi-bin/showPrimFactors?FBase=2&TBase=100&FExp=1&TExp=100&c0=&LM= (only primitive factors))
The Cunningham numbers *b**n*±1 has algebraic factorization to product of *Φ**d*(*b*) with positive integers *d* dividing *n* (the *b**n*−1 case) (see https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization) or positive integers *d* dividing 2×*n* but not dividing *n* (the *b**n*+1 case) (see https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization), 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) (see https://stdkmd.net/nrr/repunit/repunitnote.htm and 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) and 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) and http://homes.cerias.purdue.edu/~ssw/cun1.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_71.pdf))
(below, "*R**n*(*b*)" means the repunit (https://en.wikipedia.org/wiki/Repunit, https://t5k.org/glossary/xpage/Repunit.html, https://t5k.org/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://pzktupel.de/Primetables/TableRepunit.php, https://pzktupel.de/Primetables/TableRepunitGen.php, https://pzktupel.de/Primetables/TableRepunitGen.txt, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html, https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/, https://web.archive.org/web/20120227163614/http://phi.redgolpe.com/5.asp, https://web.archive.org/web/20120227163508/http://phi.redgolpe.com/4.asp, https://web.archive.org/web/20120227163610/http://phi.redgolpe.com/3.asp, https://web.archive.org/web/20120227163512/http://phi.redgolpe.com/2.asp, https://web.archive.org/web/20120227163521/http://phi.redgolpe.com/1.asp, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.fermatquotient.com/PrimSerien/GenRepuP.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/article/380/231/, http://www.bitman.name/math/table/379, http://www.bitman.name/math/table/488, https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_5.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf), https://t5k.org/top20/page.php?id=57, https://t5k.org/top20/page.php?id=16, https://oeis.org/A002275, https://oeis.org/A004022, https://oeis.org/A053696, https://oeis.org/A085104, https://oeis.org/A179625) in base *b* with length *n*, i.e. *R**n*(*b*) = (*b**n*−1)/(*b*−1) (see https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization), "*S**n*(*b*)" means *b**n*+1 (see https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization), the special cases of *R**n*(10) and *S**n*(10) are in https://stdkmd.net/nrr/repunit/ and https://stdkmd.net/nrr/repunit/10001.htm, respectively, in fact, *R**n*(*b*) and *S**n*(*b*) are 111...111 and 1000...0001 in base *b*, respectively)
* the 3168th minimal prime in base 13, 93081, *N*−1 is 117×*R*308(13), thus factor *N*−1 is equivalent to factor 13308−1, and for the algebraic factors of 13308−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&Exp=308&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 13308−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN=&LM=
* the 3179th minimal prime in base 13, B563C, *N*−1 is 11×*R*564(13), thus factor *N*−1 is equivalent to factor 13564−1, and for the algebraic factors of 13564−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&Exp=564&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 13564−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN=&LM=
* the 3180th minimal prime in base 13, 1B576, *N*−1 is 23×*R*576(13), thus factor *N*−1 is equivalent to factor 13576−1, and for the algebraic factors of 13576−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&Exp=576&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 13576−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN=&LM=
* the 25199th minimal prime in base 26, 9K343AP, *N*+1 is 6370×*R*344(26), thus factor *N*+1 is equivalent to factor 26344−1, and for the algebraic factors of 26344−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=26&Exp=344&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 26344−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&Exp=344&c0=-&EN=&LM=
* the 25200th minimal prime in base 26, 83541, *N*−1 is 208×*R*354(26), thus factor *N*−1 is equivalent to factor 26354−1, and for the algebraic factors of 26354−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=26&Exp=354&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 26354−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&Exp=354&c0=-&EN=&LM=
The helper file for the 3168th minimal prime in base 13 (93081) in *factordb*: http://factordb.com/helper.php?id=1100000000840126705
The helper file for the 3179th minimal prime in base 13 (B563C) in *factordb*: http://factordb.com/helper.php?id=1100000000000217927
The helper file for the 3180th minimal prime in base 13 (1B576) in *factordb*: http://factordb.com/helper.php?id=1100000002321021456
The helper file for the 25199th minimal prime in base 26 (9K343AP) in *factordb*: http://factordb.com/helper.php?id=1100000000840632228
The helper file for the 25200th minimal prime in base 26 (83541) in *factordb*: http://factordb.com/helper.php?id=1100000000840632517
Factorization status (and *ECM* efforts for the prime factors between 1024 and 10100) of *N*−1 for the 3168th minimal prime in base 13 (93081) in *factordb*: http://factordb.com/index.php?id=1100000000840126706&open=ecm
Factorization status (and *ECM* efforts for the prime factors between 1024 and 10100) of *N*−1 for the 3179th minimal prime in base 13 (B563C) in *factordb*: http://factordb.com/index.php?id=1100000000271764311&open=ecm
Factorization status (and *ECM* efforts for the prime factors between 1024 and 10100) of *N*−1 for the 3180th minimal prime in base 13 (1B576) in *factordb*: http://factordb.com/index.php?id=1100000002321021531&open=ecm
Factorization status (and *ECM* efforts for the prime factors between 1024 and 10100) of *N*+1 for the 25199th minimal prime in base 26 (9K343AP) in *factordb*: http://factordb.com/index.php?id=1100000000840632232&open=ecm
Factorization status (and *ECM* efforts for the prime factors between 1024 and 10100) of *N*−1 for the 25200th minimal prime in base 26 (83541) in *factordb*: http://factordb.com/index.php?id=1100000000840632623&open=ecm
(in the tables below, *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))
(for the prime factors > 1024 (other than the ultimate one) in the tables below, "*ECM*" means 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://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, https://stdkmd.net/nrr/records.htm#largefactorecm, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=ecm&maxrows=100, http://maths-people.anu.edu.au/~brent/factors.html, http://maths-people.anu.edu.au/~brent/ftp/champs.txt, https://homes.cerias.purdue.edu/~ssw/cun/press/tech.html, https://homes.cerias.purdue.edu/~ssw/cun/press/nontech.html, https://kurtbeschorner.de/ecm-efforts.htm, http://www.rechenkraft.net/yoyo//y_factors_ecm.php, http://www.rechenkraft.net/yoyo/y_status_ecm.php, https://www.alpertron.com.ar/ECM.HTM), "*P*−1" means 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, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=p-1&maxrows=100, http://www.numericana.com/answer/factoring.htm#p-1, http://factordb.com/listecm.php?c=2), "*P*+1" means 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, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=p%2b1&maxrows=100, http://www.numericana.com/answer/factoring.htm#p+1, http://factordb.com/listecm.php?c=3), "*SNFS*" means the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://www.rieselprime.de/ziki/SNFS_polynomial_selection, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=snfs&maxrows=100, https://stdkmd.net/nrr/wanted.htm#smallpolynomial), "*GNFS*" means 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, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=gnfs&maxrows=100, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs))
For the number 13308−1, it is the product of *Φ**d*(13) with positive integers *d* dividing 308 (i.e. *d* = 1, 2, 4, 7, 11, 14, 22, 28, 44, 77, 154, 308), and the factorization of *Φ**d*(13) for these positive integers *d* are:
|from|prime factorization|
|---|---|
|*Φ*1(13)|22 × 3|
|*Φ*2(13)|2 × 7|
|*Φ*4(13)|2 × 5 × 17|
|*Φ*7(13)|5229043|
|*Φ*11(13)|23 × 419 × 859 × 18041|
|*Φ*14(13)|7 × 29 × 22079|
|*Φ*22(13)|128011456717|
|*Φ*28(13)|23161037562937|
|*Φ*44(13)|5281 × 3577574298489429481|
|*Φ*77(13)|624958606550654822293 × (47-digit prime)|
|*Φ*154(13)|78947177 × (59-digit prime)|
|*Φ*308(13)|7393 × 1702933 × 150324329 × 718377597171850001 × 4209006442599882158485591696242263069 × (61-digit prime)|
For the number 13564−1, it is the product of *Φ**d*(13) with positive integers *d* dividing 564 (i.e. *d* = 1, 2, 3, 4, 6, 12, 47, 94, 141, 188, 282, 564), and the factorization of *Φ**d*(13) for these positive integers *d* are:
|from|prime factorization|
|---|---|
|*Φ*1(13)|22 × 3|
|*Φ*2(13)|2 × 7|
|*Φ*3(13)|3 × 61|
|*Φ*4(13)|2 × 5 × 17|
|*Φ*6(13)|157|
|*Φ*12(13)|28393|
|*Φ*47(13)|183959 × 19216136497 × 534280344481909234853671069326391741|
|*Φ*94(13)|498851139881 × 3245178229485124818467952891417691434077|
|*Φ*141(13)|283 × 1693 × 1924651 × 455036140638637 × (76-digit prime)|
|*Φ*188(13)|36097 × 75389 × 99886248944632632917 × (74-digit prime)|
|*Φ*282(13)|590202369266263393 × (85-digit prime)|
|*Φ*564(13)|233628485003849577181 × 94531330515097101267386264339794253977 (*ECM*, *B1* = 3000000, *Sigma* = 2146847123) × 27969827431131578608318126024627616357147784803797 (*GNFS*) × (98-digit prime)|
For the number 13576−1, it is the product of *Φ**d*(13) with positive integers *d* dividing 576 (i.e. *d* = 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, 288, 576), and the factorization of *Φ**d*(13) for these positive integers *d* are:
|from|prime factorization|
|---|---|
|*Φ*1(13)|22 × 3|
|*Φ*2(13)|2 × 7|
|*Φ*3(13)|3 × 61|
|*Φ*4(13)|2 × 5 × 17|
|*Φ*6(13)|157|
|*Φ*8(13)|2 × 14281|
|*Φ*9(13)|3 × 1609669|
|*Φ*12(13)|28393|
|*Φ*16(13)|2 × 407865361|
|*Φ*18(13)|19 × 271 × 937|
|*Φ*24(13)|815702161|
|*Φ*32(13)|2 × 2657 × 441281 × 283763713|
|*Φ*36(13)|37 × 428041 × 1471069|
|*Φ*48(13)|1009 × 659481276875569|
|*Φ*64(13)|2 × 193 × 1601 × 10433 × 68675120456139881482562689|
|*Φ*72(13)|73 × 4177 × 181297 × 9818892432332713|
|*Φ*96(13)|97 × 88993 × 127028743393 × 403791981344275297|
|*Φ*144(13)|3889 × 680401 × 29975087953 × 6654909974864689 × 558181416418089697|
|*Φ*192(13)|1153 × 11352931040252580224415980746369 × 14977427998321433931503086910333672833|
|*Φ*288(13)|2017 × 47521 × 54721 × 1590049 × 8299042833797200969471889569 × (61-digit prime)|
|*Φ*576(13)|577 × 6337 × 5247817273269739636080024961 × 5497355933986265726220616321 × 1032606621363411464640473542092061600217962755283816476128113983937 (*GNFS*) × (86-digit prime)|
For the number 26344−1, it is the product of *Φ**d*(26) with positive integers *d* dividing 344 (i.e. *d* = 1, 2, 4, 8, 43, 86, 172, 344), and the factorization of *Φ**d*(26) for these positive integers *d* are:
|from|prime factorization|
|---|---|
|*Φ*1(26)|52|
|*Φ*2(26)|33|
|*Φ*4(26)|677|
|*Φ*8(26)|17 × 26881|
|*Φ*43(26)|(60-digit prime)|
|*Φ*86(26)|681293 × (54-digit prime)|
|*Φ*172(26)|173 × 66221 × 97942133 × 338286119038330712762413 × 290239124722842089063959709049053 × (48-digit prime)|
|*Φ*344(26)|259295161 × 14470172263033 × (217-digit prime)|
For the number 26354−1, it is the product of *Φ**d*(26) with positive integers *d* dividing 354 (i.e. *d* = 1, 2, 3, 6, 59, 118, 177, 354), and the factorization of *Φ**d*(26) for these positive integers *d* are:
|from|prime factorization|
|---|---|
|*Φ*1(26)|52|
|*Φ*2(26)|33|
|*Φ*3(26)|19 × 37|
|*Φ*6(26)|3 × 7 × 31|
|*Φ*59(26)|3541 × 334945708538658924935948356996883525107 × 10265667109489266992108219345733472151257|
|*Φ*118(26)|254250862891621 × (68-digit prime)|
|*Φ*177(26)|47791 × 1311074895191091284466533625050044762267011115706300424823729 × (100-digit prime)|
|*Φ*354(26)|709 × 16441898216641 × (149-digit prime)|
Although these numbers also have *N*−1 or *N*+1 is product of a Cunningham number and a small number, but since the corresponding Cunningham numbers are < 25% factored, and the partial factorizations of them are insufficient for any of the proving methods that could make use of them, like the numbers (131193−1)/12 (see https://web.archive.org/web/20020809125049/http://www.users.globalnet.co.uk/~aads/C0131193.html and its *factordb* entry http://factordb.com/index.php?id=1000000000043597217&open=prime and its primality certificate http://factordb.com/cert.php?id=1000000000043597217 and its helper file http://factordb.com/helper.php?id=1000000000043597217 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000271071123&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=1192&c0=-&EN=&LM=) and (55839−1)/54 (see https://web.archive.org/web/20020821230129/http://www.users.globalnet.co.uk/~aads/C0550839.html and its *factordb* entry http://factordb.com/index.php?id=1100000000672342180&open=prime and its primality certificate http://factordb.com/cert.php?id=1100000000672342180 and its helper file http://factordb.com/helper.php?id=1100000000672342180 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000674669599&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=55&Exp=838&c0=-&EN=&LM=) and (701013−1)/69 (see https://web.archive.org/web/20020825072348/http://www.users.globalnet.co.uk/~aads/C0701013.html and its *factordb* entry http://factordb.com/index.php?id=1100000000599116446&open=prime and its primality certificate http://factordb.com/cert.php?id=1100000000599116446 and its helper file http://factordb.com/helper.php?id=1100000000599116446 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000599116447&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=70&Exp=1012&c0=-&EN=&LM=) and (79659−1)/78 (see https://web.archive.org/web/20020825073634/http://www.users.globalnet.co.uk/~aads/C0790659.html and its *factordb* entry http://factordb.com/index.php?id=1100000000235993821&open=prime and its primality certificate http://factordb.com/cert.php?id=1100000000235993821 and its helper file http://factordb.com/helper.php?id=1100000000235993821 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000271854142&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=79&Exp=658&c0=-&EN=&LM=) and (7116384+1)/2 (see section "Faktorisieren der Zahl (71^16384+1)/2-1" of http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt and its *factordb* entry http://factordb.com/index.php?id=1100000000213085670&open=prime and its primality certificate http://factordb.com/cert.php?id=1100000000213085670 and its helper file http://factordb.com/helper.php?id=1100000000213085670 and factorization status of its *N*−1 http://factordb.com/index.php?id=1100000000710475165&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=71&Exp=16384&c0=-&EN=&LM=), thus we treat these numbers as integers with no special form (i.e. ordinary primes (https://t5k.org/glossary/xpage/OrdinaryPrime.html)) and prove its primality with *Primo* (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), and these numbers still need primality certificates:
* the 151st minimal prime in base 9, 30115811, *N*−1 is 9×*S*2319(3), thus factor *N*−1 is equivalent to factor 32319+1, *N*−1 is only 12.693% factored (see http://factordb.com/index.php?id=1100000002376318423&open=prime), and for the algebraic factors of 32319+1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=3&Exp=2319&LBIDPMList=B&LBIDLODList=D, and for the prime factorization of 32319+1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=3&Exp=2319&c0=%2B&EN=&LM=
* the 3187th minimal prime in base 13, 715041, *N*−1 is 91×*R*1504(13), thus factor *N*−1 is equivalent to factor 131504−1, *N*−1 is only 28.604% factored (see http://factordb.com/index.php?id=1100000002320890755&open=prime) (since 28.604% is between 1/4 and 1/3, *CHG* proof is possible, however, since *factordb* (http://factordb.com/, https://www.rieselprime.de/ziki/Factoring_Database) lacks the ability to verify *CHG* proofs, thus there is still primality certificate in *factordb*), and for the algebraic factors of 131504−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&Exp=1504&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 131504−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=1504&c0=-&EN=&LM=
* the 2342nd minimal prime in base 16, 90354291, *N*−1 is 144×*S*3543(16), thus factor *N*−1 is equivalent to factor 163543+1, *N*−1 is only 1.255% factored (see http://factordb.com/index.php?id=1100000000633424191&open=prime), and for the algebraic factors of 163543+1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=16&Exp=3543&LBIDPMList=B&LBIDLODList=D, and for the prime factorization of 163543+1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=16&Exp=3543&c0=%2B&EN=&LM=
* the 25240th minimal prime in base 26, 518854P, *N*+1 is 130×*R*1886(26), thus factor *N*+1 is equivalent to factor 261886−1, *N*+1 is only 7.262% factored (see http://factordb.com/index.php?id=1100000003850155314&open=prime), and for the algebraic factors of 261886−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=26&Exp=1886&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 261886−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&Exp=1886&c0=-&EN=&LM=
* the 35277th minimal prime in base 36, OZ3932AZ, *N*+1 is 31500×*R*3933(36), thus factor *N*+1 is equivalent to factor 363933−1, *N*+1 is only 16.004% factored (see http://factordb.com/index.php?id=1100000000840634476&open=prime), and for the algebraic factors of 363933−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=36&Exp=3933&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 363933−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=36&Exp=3933&c0=-&EN=&LM=
The helper file for the 151st minimal prime in base 9 (30115811) in *factordb*: http://factordb.com/helper.php?id=1100000002376318423
The helper file for the 3187th minimal prime in base 13 (715041) in *factordb*: http://factordb.com/helper.php?id=1100000002320890755
The helper file for the 2342nd minimal prime in base 16 (90354291) in *factordb*: http://factordb.com/helper.php?id=1100000000633424191
The helper file for the 25240th minimal prime in base 26 (518854P) in *factordb*: http://factordb.com/helper.php?id=1100000003850155314
The helper file for the 35277th minimal prime in base 36 (OZ3932AZ) in *factordb*: http://factordb.com/helper.php?id=1100000000840634476
Factorization status (and *ECM* efforts for the prime factors between 1024 and 10100) of *N*−1 for the 151st minimal prime in base 9 (30115811) in *factordb*: http://factordb.com/index.php?id=1100000002376318436&open=ecm
Factorization status (and *ECM* efforts for the prime factors between 1024 and 10100) of *N*−1 for the 3187th minimal prime in base 13 (715041) in *factordb*: http://factordb.com/index.php?id=1100000002320890782&open=ecm
Factorization status (and *ECM* efforts for the prime factors between 1024 and 10100) of *N*−1 for the 2342nd minimal prime in base 16 (90354291) in *factordb*: http://factordb.com/index.php?id=1100000000633424203&open=ecm
Factorization status (and *ECM* efforts for the prime factors between 1024 and 10100) of *N*+1 for the 25240th minimal prime in base 26 (518854P) in *factordb*: http://factordb.com/index.php?id=1100000003850159350&open=ecm
Factorization status (and *ECM* efforts for the prime factors between 1024 and 10100) of *N*+1 for the 35277th minimal prime in base 36 (OZ3932AZ) in *factordb*: http://factordb.com/index.php?id=1100000000840634478&open=ecm
(in the tables below, *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))
(for the prime factors > 1024 (other than the ultimate one) in the tables below, "*ECM*" means 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://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, https://stdkmd.net/nrr/records.htm#largefactorecm, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=ecm&maxrows=100, http://maths-people.anu.edu.au/~brent/factors.html, http://maths-people.anu.edu.au/~brent/ftp/champs.txt, https://homes.cerias.purdue.edu/~ssw/cun/press/tech.html, https://homes.cerias.purdue.edu/~ssw/cun/press/nontech.html, https://kurtbeschorner.de/ecm-efforts.htm, http://www.rechenkraft.net/yoyo//y_factors_ecm.php, http://www.rechenkraft.net/yoyo/y_status_ecm.php, https://www.alpertron.com.ar/ECM.HTM), "*P*−1" means 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, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=p-1&maxrows=100, http://www.numericana.com/answer/factoring.htm#p-1, http://factordb.com/listecm.php?c=2), "*P*+1" means 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, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=p%2b1&maxrows=100, http://www.numericana.com/answer/factoring.htm#p+1, http://factordb.com/listecm.php?c=3), "*SNFS*" means the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://www.rieselprime.de/ziki/SNFS_polynomial_selection, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=snfs&maxrows=100, https://stdkmd.net/nrr/wanted.htm#smallpolynomial), "*GNFS*" means 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, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=gnfs&maxrows=100, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs))
For the number 32319+1, it is the product of *Φ**d*(3) with positive integers *d* dividing 4638 but not dividing 2319 (i.e. *d* = 2, 6, 1546, 4638), and the factorization of *Φ**d*(3) for these positive integers *d* are: (since 6 and 4638 are == 6 mod 12, thus for these two positive integers *d*, *Φ**d*(3) has 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, http://pagesperso-orange.fr/colin.barker/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.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)), and *Φ**dL*(3) and *Φ**dM*(3) are their Aurifeuillean *L* and *M* factors, respectively)
|from|currently known prime factorization|
|---|---|
|*Φ*2(3)|22|
|*Φ*6*L*(3)|*1* (empty product (https://en.wikipedia.org/wiki/Empty_product))|
|*Φ*6*M*(3)|7|
|*Φ*1546(3)|1182691 × 454333843 × 7175619780295897339 × 219067434459114063477547 × 650663511671253931884619 × (288-digit composite with no known proper factor, *SNFS* difficulty is 370, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=3&Exp=773&c0=%2B&LM=&SA=)|
|*Φ*4638*L*(3)|18553 × 2957658597967379799686737984695290731543 × (325-digit composite with no known proper factor, *SNFS* difficulty is 370, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=3&Exp=2319&c0=%2B&LM=L&SA=)|
|*Φ*4638*M*(3)|4639 × 6716055901 × (356-digit composite with no known proper factor, *SNFS* difficulty is 370, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=3&Exp=2319&c0=%2B&LM=M&SA=)|
For the number 131504−1, it is the product of *Φ**d*(13) with positive integers *d* dividing 1504 (i.e. *d* = 1, 2, 4, 8, 16, 32, 47, 94, 188, 376, 752, 1504), and the factorization of *Φ**d*(13) for these positive integers *d* are:
|from|currently known prime factorization|
|---|---|
|*Φ*1(13)|22 × 3|
|*Φ*2(13)|2 × 7|
|*Φ*4(13)|2 × 5 × 17|
|*Φ*8(13)|2 × 14281|
|*Φ*16(13)|2 × 407865361|
|*Φ*32(13)|2 × 2657 × 441281 × 283763713|
|*Φ*47(13)|183959 × 19216136497 × 534280344481909234853671069326391741|
|*Φ*94(13)|498851139881 × 3245178229485124818467952891417691434077|
|*Φ*188(13)|36097 × 75389 × 99886248944632632917 × (74-digit prime)|
|*Φ*376(13)|41737 × 553784729353 × 188172028979257 × 398225319299696783138113 × 7663511503164270157006126605793 × 8935170451146532986983277856738508374630999814576686938913 × (62-digit prime)|
|*Φ*752(13)|13537 × 1232912541076129 × (391-digit composite with no known proper factor, *SNFS* difficulty is 422, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=13&Exp=376&c0=%2B&LM=&SA=)|
|*Φ*1504(13)|4513 × 9426289921 × (807-digit composite with no known proper factor, *SNFS* difficulty is 838, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=13&Exp=752&c0=%2B&LM=&SA=)|
For the number 163543+1 = 214172+1, it is the product of *Φ**d*(2) with positive integers *d* dividing 28344 but not dividing 14172 (i.e. *d* = 8, 24, 9448, 28344), and the factorization of *Φ**d*(2) for these positive integers *d* are:
|from|currently known prime factorization|
|---|---|
|*Φ*8(2)|17|
|*Φ*24(2)|241|
|*Φ*9448(2)|107083633 × 7076306353 × 2428629073416562046689 × (1382-digit composite with no known proper factor, *SNFS* difficulty is 1423, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=2&Exp=4724&c0=%2B&LM=&SA=)|
|*Φ*28344(2)|265073089 × (2834-digit composite with no known proper factor, *SNFS* difficulty is too large to handle for the script, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=2&Exp=14172&c0=%2B&LM=&SA=)|
For the number 261886−1, it is the product of *Φ**d*(26) with positive integers *d* dividing 1886 (i.e. *d* = 1, 2, 23, 41, 46, 82, 943, 1886), and the factorization of *Φ**d*(26) for these positive integers *d* are:
|from|currently known prime factorization|
|---|---|
|*Φ*1(26)|52|
|*Φ*2(26)|33|
|*Φ*23(26)|13709 × 1086199 × 1528507873 × 615551139461|
|*Φ*41(26)|83 × 2633923 × (49-digit prime)|
|*Φ*46(26)|47 × 1157729 × 378673381 × 629584013567417|
|*Φ*82(26)|9677 × 1532581 × (47-digit prime)|
|*Φ*943(26)|384118835398327 × (1231-digit composite with no known proper factor, *SNFS* difficulty is 1335, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=26&Exp=943&c0=-&LM=&SA=)|
|*Φ*1886(26)|(1246-digit composite with no known proper factor, *SNFS* difficulty is 1335, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=26&Exp=943&c0=%2B&LM=&SA=)|
For the number 363933−1 = 67866−1, it is the product of *Φ**d*(6) with positive integers *d* dividing 7866 (i.e. *d* = 1, 2, 3, 6, 9, 18, 19, 23, 38, 46, 57, 69, 114, 138, 171, 207, 342, 414, 437, 874, 1311, 2622, 3933, 7866), and the factorization of *Φ**d*(6) for these positive integers *d* are:
|from|currently known prime factorization|
|---|---|
|*Φ*1(6)|5|
|*Φ*2(6)|7|
|*Φ*3(6)|43|
|*Φ*6(6)|31|
|*Φ*9(6)|19 × 2467|
|*Φ*18(6)|46441|
|*Φ*19(6)|191 × 638073026189|
|*Φ*23(6)|47 × 139 × 3221 × 7505944891|
|*Φ*38(6)|1787 × 48713705333|
|*Φ*46(6)|113958101 × 990000731|
|*Φ*57(6)|47881 × 820459 × 219815829325921729|
|*Φ*69(6)|11731 × 1236385853432057889667843739281|
|*Φ*114(6)|457 × 137713 × 190324492938225748951|
|*Φ*138(6)|24648570768391 × 816214079084081564521|
|*Φ*171(6)|19 × 25896916098621777025320461067950269867 × (46-digit prime)|
|*Φ*207(6)|399097 × (98-digit prime)|
|*Φ*342(6)|62174327387790051073 × (65-digit prime)|
|*Φ*414(6)|4811469913 × 61040960263 × 25280883279243199352415750302719 × (51-digit prime)|
|*Φ*437(6)|989723472495640900314985156529340457 × (273-digit composite with no known proper factor, *SNFS* difficulty is 341, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=6&Exp=437&c0=-&LM=&SA=)|
|*Φ*874(6)|(309-digit prime, for its primality certificate see http://factordb.com/cert.php?id=1100000000019287760)|
|*Φ*1311(6)|100745107 × 1719861571 × 2376829061449 × (587-digit composite with no known proper factor, *SNFS* difficulty is 682, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=6&Exp=1311&c0=-&LM=&SA=)|
|*Φ*2622(6)|41953 × 266030354191322260711 × (592-digit composite with no known proper factor, *SNFS* difficulty is 682, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=6&Exp=1311&c0=%2B&LM=&SA=)|
|*Φ*3933(6)|7867 × (1845-digit composite with no known proper factor, *SNFS* difficulty is too large to handle for the script, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=6&Exp=3933&c0=-&LM=&SA=)|
|*Φ*7866(6)|(1849-digit composite with no known proper factor, *SNFS* difficulty is too large to handle for the script, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=6&Exp=3933&c0=%2B&LM=&SA=)|
For the files in this page:
* File "certificate *b* *n*": The primality certificate for the *n*th minimal prime in base *b* (local copy from *factordb* (http://factordb.com/, https://www.rieselprime.de/ziki/Factoring_Database)), after downloading these files, these files should be renamed to ".out" files, e.g. file "certificate9_149" is the primality certificate for the 149th minimal prime in base 9, i.e. the primality certificate for the prime 763292 in base 9, which equals the prime (31×9330−19)/4.