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×bⁿ+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://pzktupel.de/Primetables/TableProthTOP10KK.php, https://pzktupel.de/Primetables/TableProthTOP10KS.php, https://pzktupel.de/Primetables/TableProthGen.php, 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×bⁿ+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://pzktupel.de/Primetables/TableRieselTOP10KK.php, https://pzktupel.de/Primetables/TableRieselTOP10KS.php, https://pzktupel.de/Primetables/TableRieselGen.php, https://sites.google.com/view/proth-primes, https://t5k.org/primes/search_proth.php) base b: a×bⁿ−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×bⁿ±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 bⁿ±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 Nⁿ−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 N²−1 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 bⁿ±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://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/Repunit100-20221222.pdf?t=1671715731 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_26.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), 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 bⁿ±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 bⁿ±1 has algebraic factorization to product of Φ_d(b) with positive integers d dividing n (the bⁿ−1 case) (see https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization) or positive integers d dividing 2×n but not dividing n (the bⁿ+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://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) in base b with length n, i.e. R_n(b) = (bⁿ−1)/(b−1) (see https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization), "S_n(b)" means bⁿ+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×R₃₀₈(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×R₅₆₄(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×R₅₇₆(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×R₃₄₄(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×R₃₅₄(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:

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:

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:

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:

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:

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×S₂₃₁₉(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×R₁₅₀₄(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×S₃₅₄₃(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×R₁₈₈₆(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×R₃₉₃₃(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)

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:

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:

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:

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:

For the files in this page:

• File "certificate b n": The primality certificate for the nth 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.