(Edit: Now you can click http://5.199.134.130/certificates.tar.xz and http://5.199.134.130/certificates.tar.xz.SHA256SUM and http://5.199.134.130/certificates.tar.xz.par2 and http://5.199.134.130/certificates.tar.xz.vol00+10.par2 to download all primality certificates in factordb (which includes the primality certificates of the minimal primes in bases 2 ≤ b ≤ 36 in factordb), also the .xz files in http://5.199.134.130/certificates/ ("n.xz" for the primality certificates of the definitely primes which are proven primes by primality certificates (i.e. not proven primes by N−1 or N+1 or combine of N−1 and N+1) with n decimal digits for n ≥ 300) (pixz is recommended for unpacking. "pixz -d < primes.tar.xz | tar -xv"))

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

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, http://irvinemclean.com/maths/pfaq7.htm, https://t5k.org/top20/page.php?id=27, https://t5k.org/primes/search.php?Comment=ECPP&OnList=all&Number=1000000&Style=HTML, http://www.ellipsa.eu/public/primo/records.html, http://www.lix.polytechnique.fr/~morain/Prgms/ecpp.english.html, https://www.multiprecision.org/cm/ecpp.html, https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1199989-X/S0025-5718-1993-1199989-X.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_256.pdf), https://arxiv.org/pdf/2404.05506.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_428.pdf)) 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, http://5.199.134.130/certificates.tar.xz, http://5.199.134.130/certificates.tar.xz.SHA256SUM, http://5.199.134.130/certificates.tar.xz.par2, http://5.199.134.130/certificates.tar.xz.vol00+10.par2, http://5.199.134.130/certificates/) 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 (for primes ≤ the 50000000th prime (i.e. 982451653), we check the online list of the first 50000000 primes in https://t5k.org/lists/small/millions/ (i.e. we simply use table lookup), and for the primes > the 50000000th prime (i.e. 982451653) and < 10¹⁶, we simply use the sieve of Eratosthenes (https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes, https://t5k.org/glossary/xpage/SieveOfEratosthenes.html, https://www.rieselprime.de/ziki/Sieve_of_Eratosthenes, https://mathworld.wolfram.com/SieveofEratosthenes.html, https://oeis.org/A083221, https://oeis.org/A083140, https://oeis.org/A145583, https://oeis.org/A145540, https://oeis.org/A145538, https://oeis.org/A145539, https://oeis.org/A227155, https://oeis.org/A227797, https://oeis.org/A227798, https://oeis.org/A227799, https://oeis.org/A145584, https://oeis.org/A145585, https://oeis.org/A145586, https://oeis.org/A145587, https://oeis.org/A145588, https://oeis.org/A145589, https://oeis.org/A145590, https://oeis.org/A145591, https://oeis.org/A145592, https://oeis.org/A145532, https://oeis.org/A145533, https://oeis.org/A145534, https://oeis.org/A145535, https://oeis.org/A145536, https://oeis.org/A145537) (in fact, we use trial division (https://en.wikipedia.org/wiki/Trial_division, https://t5k.org/glossary/xpage/TrialDivision.html, https://www.rieselprime.de/ziki/Trial_factoring, https://mathworld.wolfram.com/TrialDivision.html, http://www.numericana.com/answer/factoring.htm#trial, https://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) with all 11-rough numbers (https://en.wikipedia.org/wiki/Rough_number, https://mathworld.wolfram.com/RoughNumber.html, https://oeis.org/A007310, https://oeis.org/A007775, https://oeis.org/A008364, https://oeis.org/A008365, https://oeis.org/A008366, https://oeis.org/A166061, https://oeis.org/A166063) > 1 and ≤ sqrt(p) (the square root (https://en.wikipedia.org/wiki/Square_root, https://www.rieselprime.de/ziki/Square_root, https://mathworld.wolfram.com/SquareRoot.html) of the prime), i.e. we use the wheel factorization (https://en.wikipedia.org/wiki/Wheel_factorization, https://t5k.org/glossary/xpage/WheelFactorization.html) with modulo 210 = 2×3×5×7 (the primorial (https://en.wikipedia.org/wiki/Primorial, https://t5k.org/glossary/xpage/Primorial.html, https://mathworld.wolfram.com/Primorial.html, https://www.numbersaplenty.com/set/primorial/, https://oeis.org/A002110) of the prime 7), to save time), see https://t5k.org/prove/prove2_1.html, and for the primes > 10¹⁶ and < 10²⁹⁹, we use the Adleman–Pomerance–Rumely primality test (https://en.wikipedia.org/wiki/Adleman%E2%80%93Pomerance%E2%80%93Rumely_primality_test, https://www.rieselprime.de/ziki/Adleman%E2%80%93Pomerance%E2%80%93Rumely_primality_test, https://mathworld.wolfram.com/Adleman-Pomerance-RumelyPrimalityTest.html, https://t5k.org/prove/prove4_1.html, https://t5k.org/primes/search.php?Comment=APR-CL%20assisted&OnList=all&Number=1000000&Style=HTML), this primality test can verify the primes with such size in less than one second, see https://t5k.org/prove/prove2_1.html, and no need to use elliptic curve primality proving for the primes with such size), 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://irvinemclean.com/maths/pfaq4.htm, 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://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) can be ≥ 1/4 factored (i.e. the product of the known prime factors of N−1 or/and N+1 is ≥ the fourth root of it)) in bases b = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 26, 28, 30, 36 (the "easy" bases (bases b with ≤ 150 minimal primes > 10²⁹⁹ (base b = 26 has 82 known minimal (probable) primes > 10²⁹⁹ and 4 unsolved families, base b = 36 has 75 known minimal (probable) primes > 10²⁹⁹ and 4 unsolved families, base b = 17 has 99 known minimal (probable) primes > 10²⁹⁹ and 18 unsolved families, base b = 21 has 80 known minimal (probable) primes > 10²⁹⁹ and 12 unsolved families, base b = 19 has 201 known minimal (probable) primes > 10²⁹⁹ and 23 unsolved families))).

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 (i.e. they are the near-Cunningham numbers (http://factordb.com/tables.php?open=4, https://oeis.org/wiki/OEIS_sequences_needing_factors#Near_powers.2C_factorials.2C_and_primorials (sections "near-powers with b = 2" and "near-powers with b = 3" and "near-powers with b = 5" and "near-powers with b = 6" and "near-powers with b = 7" and "near-powers with b = 10" and "near-powers with b > 10")), 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://irvinemclean.com/maths/pfaq4.htm, 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://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2) if and only if c = ±1 and gcd(a+c,b−1) = 1 (if this large minimal prime in base b is xy_nz (where x and z are strings (may be empty) of digits in base b, y is a digit in base b) in base b, then c = 1 and gcd(a+c,b−1) = 1 if and only if the digit y is 0 and the string z is 1, and c = −1 and gcd(a+c,b−1) = 1 if and only if the digit y is b−1 and the string z is 𝜆 (the empty string (https://en.wikipedia.org/wiki/Empty_string)), if we reduce these families by removing all trailing digits y from x, and removing all leading digits y from z, to make the families be easier, e.g. family 12333{3}33345 in base b is reduced to family 12{3}45 in base b, since they are in fact the same family), except this special case (https://en.wikipedia.org/wiki/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://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm#probableprimenumbers, https://stdkmd.net/nrr/repunit/prpfactors.htm, https://www.mersenne.ca/prp.php?show=1, https://www.alfredreichlg.de/10w7/prp.html, http://factordb.com/listtype.php?t=1)), and elliptic curve primality proving are used for these numbers.

There are also other versions of the N−1 and N+1 tests, using primitive roots (https://en.wikipedia.org/wiki/Primitive_root_modulo_n, https://mathworld.wolfram.com/PrimitiveRoot.html, http://www.bluetulip.org/2014/programs/primitive.html, http://www.numbertheory.org/php/lprimroot.html, http://www.numbertheory.org/php/lprimrootneg.html, http://www.numbertheory.org/php/lprimroot_generator.html, http://www.numbertheory.org/php/lprimrootneg_generator.html, https://oeis.org/A046147, https://oeis.org/A060749, https://oeis.org/A046144, https://oeis.org/A008330, https://oeis.org/A046145, https://oeis.org/A001918, https://oeis.org/A046146, https://oeis.org/A071894, https://oeis.org/A002199, https://oeis.org/A033948, https://oeis.org/A033949), see https://www.mathpages.com/home/kmath473/kmath473.htm for the N−1 test and see https://bln.curtisbright.com/2013/11/23/a-variant-n1-primality-test/ for the N+1 test.

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, http://www.prothsearch.com/frequencies.html, http://www.prothsearch.com/history.html, https://www.rieselprime.de/Data/PStatistics.htm, https://www.rieselprime.de/Data/PRanges50.htm, https://www.rieselprime.de/Data/PRanges300.htm, https://www.rieselprime.de/Data/PRanges1200.htm, http://irvinemclean.com/maths/pfaq6.htm, https://www.numbersaplenty.com/set/Proth_number/, https://web.archive.org/web/20230706041914/https://pzktupel.de/Primetables/TableProthTOP10KK.php, https://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, https://www.primegrid.com/primes/primes.php?project=PPS&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=PPSE&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=MEG&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, http://boincvm.proxyma.ru:30080/test4vm/public/pps_dc_status.php, https://www.rieselprime.de/ziki/PrimeGrid_Proth_Prime_Search, https://www.rieselprime.de/ziki/PrimeGrid_Proth_Prime_Search_Extended, https://www.rieselprime.de/ziki/PrimeGrid_Proth_Mega_Prime_Search) 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, https://web.archive.org/web/20230928115832/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17598&d=1516910519, https://web.archive.org/web/20221230035429/https://sites.google.com/site/robertgerbicz/sierpinski.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4557&d=1263456866, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, 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/, http://irvinemclean.com/maths/pfaq4.htm, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1), 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://www.rieselprime.de/Data/Statistics.htm, http://irvinemclean.com/maths/pfaq6.htm, https://web.archive.org/web/20230628151418/https://pzktupel.de/Primetables/TableRieselTOP10KK.php, https://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, http://www.noprimeleftbehind.net/stats/index.php?content=prime_list, https://t5k.org/primes/search_proth.php, http://www.noprimeleftbehind.net:9000/all.html, http://www.noprimeleftbehind.net:2000/all.html, http://www.noprimeleftbehind.net:1468/all.html, http://www.noprimeleftbehind.net:1400/all.html, https://www.rieselprime.de/ziki/NPLB_Drive_17, https://www.rieselprime.de/ziki/NPLB_Drive_18, https://www.rieselprime.de/ziki/NPLB_Drive_19, https://www.rieselprime.de/ziki/NPLB_Drive_High-n) 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, https://web.archive.org/web/20230928115850/http://www.noprimeleftbehind.net/crus/vstats_new/all_ck_riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=17597&d=1516910519, https://web.archive.org/web/20221230035558/https://sites.google.com/site/robertgerbicz/riesel.txt, https://www.mersenneforum.org/attachment.php?attachmentid=4558&d=1263456995, https://www.mersenneforum.org/showthread.php?t=10910, https://www.mersenneforum.org/showthread.php?t=25177, http://www.bitman.name/math/article/2005 (in Italian))) 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://irvinemclean.com/maths/pfaq4.htm, 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), you should know the difference of probable primes and definitely primes (see https://www.mersenneforum.org/showpost.php?p=651069&postcount=3 and https://www.mersenneforum.org/showpost.php?p=572047&postcount=239), you can compare the top definitely primes page (https://t5k.org/primes/lists/all.txt) and the top probable primes page (http://www.primenumbers.net/prptop/prptop.php), also you can compare the definitely primes with ≥ 100000 decimal digits in factordb (http://factordb.com/listtype.php?t=4&mindig=100000&perpage=5000&start=0) and the probable primes with ≥ 100000 decimal digits in factordb (http://factordb.com/listtype.php?t=1&mindig=100000&perpage=5000&start=0), http://factordb.com/nmoverview.php?method=1&digits=100000&perpage=500&skip=0 is the primes with ≥ 100000 decimal digits in factordb which are proven primes by the N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1), http://factordb.com/nmoverview.php?method=2&digits=100000&perpage=500&skip=0 is the primes with ≥ 100000 decimal digits in factordb which are proven primes by the N+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), also see https://web.archive.org/web/20240305200806/https://stdkmd.net/nrr/prime/primesize.txt and https://web.archive.org/web/20240305201054/https://stdkmd.net/nrr/prime/primesize.zip (see which numbers have "-proven" or "+proven" in the "note" column), also see https://stdkmd.net/nrr/prime/prime_all.htm and https://stdkmd.net/nrr/prime/prime_all.txt (see which numbers have "pr" in the "status" column), also see https://web.archive.org/web/20240202224722/https://stdkmd.net/nrr/records.htm (compare the sections "Prime numbers" and "Probable prime numbers").

(for more examples of numbers which are proven primes using the N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) or the N+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), see https://www.mersenneforum.org/showthread.php?t=16209)

Primes which either N−1 or N+1 is trivially (https://en.wikipedia.org/wiki/Triviality_(mathematics), https://mathworld.wolfram.com/Trivial.html) 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 the N−1 test (https://t5k.org/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=1) or the N+1 test (https://t5k.org/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://irvinemclean.com/maths/pfaq4.htm, http://factordb.com/nmoverview.php?method=2), 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)) (i.e. the greatest prime factor (http://mathworld.wolfram.com/GreatestPrimeFactor.html, https://oeis.org/A006530) of N−1 or N+1 is small)

• 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 10317th minimal prime in base 17, 5A70₂₇₄1, which equals 1622×17²⁷⁵+1, N−1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000003782940709, for the factorization of N−1 in factordb see http://factordb.com/index.php?id=1100000003782941930&open=ecm
• the 10359th minimal prime in base 17, 9D0₁₀₆₇1, which equals 166×17¹⁰⁶⁸+1, N−1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000000765961369, for the factorization of N−1 in factordb see http://factordb.com/index.php?id=1100000000765961370&open=ecm
• the 10370th minimal prime in base 17, A0₁₃₅₅1, which equals 10×17¹³⁵⁶+1, N−1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000000034167087, for the factorization of N−1 in factordb see http://factordb.com/index.php?id=1100000000271866825&open=ecm
• the 10386th minimal prime in base 17, 530₄₈₆₇1, which equals 88×17⁴⁸⁶⁸+1, N−1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000000762660735, for the factorization of N−1 in factordb see http://factordb.com/index.php?id=1100000000762660737&open=ecm
• the 10408th minimal prime in base 17, 570₅₁₃₁₀1, which equals 92×17⁵¹³¹¹+1, N−1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000000765961389, for the factorization of N−1 in factordb see http://factordb.com/index.php?id=1100000000785469616&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 13373rd minimal prime in base 21, 5D0₁₉₈₄₈1, which equals 118×21¹⁹⁸⁴⁹+1, N−1 is trivially fully factored, for its helper file in factordb see http://factordb.com/helper.php?id=1100000000777265872, for the factorization of N−1 in factordb see http://factordb.com/index.php?id=1100000000785469310&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 (https://en.wikipedia.org/wiki/Triviality_(mathematics), https://mathworld.wolfram.com/Trivial.html) 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://www.mersenne.ca/tf1G/, https://www.mersenne.ca/tfmissed.php, https://oeis.org/A189172) to 10¹²) 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 10¹²-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) (i.e. the greatest prime factor (http://mathworld.wolfram.com/GreatestPrimeFactor.html, https://oeis.org/A006530) of N−1 or N+1 is < 10²⁹⁹, and the second-greatest prime factor (https://oeis.org/A087039, https://stdkmd.net/nrr/records.htm#BIGFACTOR, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "Largest penultimate prime factor (ultimate factor shown also):")) of this number (N−1 or N+1) is < 10¹²)

• the 2328th minimal prime in base 16, 880₂₄₆7, with 300 decimal digits, N−1 is 2³ × 3 × 7 × 13 × 25703261 × (289-digit prime)
• the 10311st minimal prime in base 17, 85A₂₄₁55, with 302 decimal digits, N+1 is 2 × 1291 × 942385161439 × (286-digit prime)
• the 10312nd minimal prime in base 17, 90₂₄₂701, with 303 decimal digits, N−1 is 2⁶ × 17² × 1773259 × 4348181 × 603217519 × (277-digit prime)
• the 10315th minimal prime in base 17, E7₂₅₅A, with 317 decimal digits, N−1 is 2⁴ × 283 × 619471 × 62754967151 × (296-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 10311st minimal prime in base 17 (85A₂₄₁55) in factordb: http://factordb.com/helper.php?id=1100000003782940703

The helper file for the 10312nd minimal prime in base 17 (90₂₄₂701) in factordb: http://factordb.com/helper.php?id=1100000003782940704

The helper file for the 10315th minimal prime in base 17 (E7₂₅₅A) in factordb: http://factordb.com/helper.php?id=1100000003782940707

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 10311st minimal prime in base 17 (85A₂₄₁55) in factordb: http://factordb.com/index.php?id=1100000003782944423&open=ecm

Factorization of N−1 for the 10312nd minimal prime in base 17 (90₂₄₂701) in factordb: http://factordb.com/index.php?id=1100000003782941925&open=ecm

Factorization of N−1 for the 10315th minimal prime in base 17 (E7₂₅₅A) in factordb: http://factordb.com/index.php?id=1100000003782941928&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://oeis.org/wiki/OEIS_sequences_needing_factors#Cunningham_numbers (sections "b = 2" and "b = 3" and "b = 10" and "other integer b"), https://homes.cerias.purdue.edu/~ssw/cun/index.html, https://maths-people.anu.edu.au/~brent/factors.html, https://web.archive.org/web/20190315214330/http://cage.ugent.be/~jdemeyer/cunningham/, http://myfactors.mooo.com/, https://doi.org/10.1090/conm/022, https://www.mersenneforum.org/attachment.php?attachmentid=7727&d=1330555980 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_7.pdf), https://homes.cerias.purdue.edu/~ssw/cun/mine.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_6.pdf), http://homes.cerias.purdue.edu/~ssw/cun1.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_71.pdf)) 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) (i.e. the product of the known prime factors of this Cunningham number is ≥ the cube root (https://en.wikipedia.org/wiki/Cube_root, https://mathworld.wolfram.com/CubeRoot.html) of it) or this Cunningham number is ≥ 1/4 factored (i.e. the product of the known prime factors of this Cunningham number is ≥ the fourth root of it) and the number is not very large (say not > 10¹⁰⁰⁰⁰⁰). If either N−1 or N+1 (or both) can be ≥ 1/2 factored (i.e. the product of the known prime factors of either N−1 or N+1 (or both) is ≥ the square root (https://en.wikipedia.org/wiki/Square_root, https://www.rieselprime.de/ziki/Square_root, https://mathworld.wolfram.com/SquareRoot.html) of it), 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/, http://irvinemclean.com/maths/pfaq4.htm, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1) (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://irvinemclean.com/maths/pfaq4.htm, 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 (i.e. the product of the known prime factors of either N−1 or N+1 (or both) is ≥ the cube root (https://en.wikipedia.org/wiki/Cube_root, https://mathworld.wolfram.com/CubeRoot.html) of it), 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 (i.e. the product of the known prime factors of either N−1 or N+1 (or both) is ≥ the fourth root of it) but neither can be ≥ 1/3 factored (i.e. the products of the known prime factors of both N−1 and N+1 are < the cube roots (https://en.wikipedia.org/wiki/Cube_root, https://mathworld.wolfram.com/CubeRoot.html) of them), then we need to use CHG (https://www.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://www.mersenneforum.org/showpost.php?p=528149&postcount=3 and https://www.mersenneforum.org/showpost.php?p=603181&postcount=438 and https://www.mersenneforum.org/showpost.php?p=277617&postcount=7), however, unlike Brillhart-Lehmer-Selfridge primality test for the numbers N such that either N−1 or N+1 (or both) can be ≥ 1/3 factored (i.e. the product of the known prime factors of either N−1 or N+1 (or both) is ≥ the cube root (https://en.wikipedia.org/wiki/Cube_root, https://mathworld.wolfram.com/CubeRoot.html) of it) can run for arbitrarily large numbers N (thus, there are no unproven probable primes N such that either N−1 or N+1 (or both) can be ≥ 1/3 factored (i.e. the product of the known prime factors of either N−1 or N+1 (or both) is ≥ the cube root (https://en.wikipedia.org/wiki/Cube_root, https://mathworld.wolfram.com/CubeRoot.html) of it)), CHG for the numbers N such that either N−1 or N+1 (or both) can be ≥ 1/4 factored (i.e. the product of the known prime factors of either N−1 or N+1 (or both) is ≥ the fourth root of it) but neither can be ≥ 1/3 factored (i.e. the products of the known prime factors of both N−1 and N+1 are < the cube roots (https://en.wikipedia.org/wiki/Cube_root, https://mathworld.wolfram.com/CubeRoot.html) of them) 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://www.mersenneforum.org/showpost.php?p=608362&postcount=165; if neither N−1 nor N+1 can be ≥ 1/4 factored (i.e. the products of the known prime factors of both N−1 and N+1 are < the fourth roots of them) but Nⁿ−1 can be ≥ 1/3 factored (i.e. the product of the known prime factors of Nⁿ−1 is ≥ the cube root (https://en.wikipedia.org/wiki/Cube_root, https://mathworld.wolfram.com/CubeRoot.html) of it) 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, https://t5k.org/primes/search.php?Comment=Cyclotomy&OnList=all&Number=1000000&Style=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 more examples see https://web.archive.org/web/20240305200806/https://stdkmd.net/nrr/prime/primesize.txt and https://web.archive.org/web/20240305201054/https://stdkmd.net/nrr/prime/primesize.zip (see which numbers have "proven@" in the "note" column), also see https://stdkmd.net/nrr/cert/1/#CERT_11101_4809 and https://stdkmd.net/nrr/cert/1/#CERT_15551_2197 and https://stdkmd.net/nrr/cert/1/#CERT_16667_4296 and https://stdkmd.net/nrr/cert/2/#CERT_20111_2692 and https://stdkmd.net/nrr/cert/2/#CERT_23309_10029 and https://stdkmd.net/nrr/cert/3/#CERT_37773_15768 and https://stdkmd.net/nrr/cert/6/#CERT_6805W7_3739 and https://stdkmd.net/nrr/cert/6/#CERT_68883_5132 and https://stdkmd.net/nrr/cert/7/#CERT_79921_11629 and https://stdkmd.net/nrr/cert/8/#CERT_80081_5736 and https://stdkmd.net/nrr/cert/8/#CERT_83W16W7_543 and https://stdkmd.net/nrr/cert/9/#CERT_93307_2197 for the related numbers (although not all of them are related to Cunningham numbers), e.g. "11101_4809" (decimal (base b = 10) form: 1₄₈₀₇01, algebraic form: (10⁴⁸⁰⁹−91)/9) is related to "Phi_4807_10" (the number Φ₄₈₀₇(10), where Φ is the cyclotomic polynomial), "15551_2197" (decimal (base b = 10) form: 15₂₁₉₆1, algebraic form: (14×10²¹⁹⁷−41)/9, the prime is a cofactor of it (divided it by 11×23×167)) is related to "93307_2197" (decimal (base b = 10) form: 93₂₁₉₅07, algebraic form: (28×10²¹⁹⁷−79)/3), "16667_4296" (decimal (base b = 10) form: 16₄₂₉₅7, algebraic form: (5×10⁴²⁹⁶+1)/3, the prime is a cofactor of it (divided it by 347×821×140235709×806209146522749)) is related to "33337_12891" (decimal (base b = 10) form: 3₁₂₈₉₀7, algebraic form: (10¹²⁸⁹¹+11)/3), "20111_2692" (decimal (base b = 10) form: 201₂₆₉₂, algebraic form: (181×10²⁶⁹²−1)/9, the prime is a cofactor of it (divided it by 3×43)) is related to "20111_2693" (decimal (base b = 10) form: 201₂₆₉₃, algebraic form: (181×10²⁶⁹³−1)/9), "23309_10029" (decimal (base b = 10) form: 23₁₀₀₂₇09, algebraic form: (7×10¹⁰⁰²⁹−73)/3) is related to "Phi_5014_10" (the number Φ₅₀₁₄(10), where Φ is the cyclotomic polynomial), "37773_15768" (decimal (base b = 10) form: 37₁₅₇₆₇3, algebraic form: (34×10¹⁵⁷⁶⁸−43)/9) is related to "Phi_7884_10" (the number Φ₇₈₈₄(10), where Φ is the cyclotomic polynomial), "6805w7_3739" (decimal (base b = 10) form: 6805₃₇₃₈7, algebraic form: (6125×10³⁷³⁹+13)/9, the prime is a cofactor of it (divided it by 3²)) is related to "27227_3741" (decimal (base b = 10) form: 272₃₇₄₀7, algebraic form: (245×10³⁷⁴¹+43)/9), "68883_5132" (decimal (base b = 10) form: 68₅₁₃₁3, algebraic form: (62×10⁵¹³²−53)/9) is related to "Phi_1283_10" (the number Φ₁₂₈₃(10), where Φ is the cyclotomic polynomial), "79921_11629" (decimal (base b = 10) form: 79₁₁₆₂₇21, algebraic form: 8×10¹¹⁶²⁹−79) is related to "Phi_2907_10" (the number Φ₂₉₀₇(10), where Φ is the cyclotomic polynomial), "80081_5736" (decimal (base b = 10) form: 80₅₇₃₄81, algebraic form: 8×10⁵⁷³⁶+81) is related to "Phi_11470_10" (the number Φ₁₁₄₇₀(10), where Φ is the cyclotomic polynomial), "83w16w7_543" (decimal (base b = 10) form: 83₅₄₂16₅₄₂7, algebraic form: (25×10¹⁰⁸⁶−5×10⁵⁴³+1)/3, the prime is a cofactor of it (divided it by 7×109×563041×869047141×147372142447)) is related to "11103_3258" (decimal (base b = 10) form: 1₃₂₅₆03, algebraic form: (10³²⁵⁸−73)/9), etc. the N−1 of "11101_4809" is 100 × R₄₈₀₇(10) (which is equivalent to the Cunningham number 10⁴⁸⁰⁷−1) and Φ₄₈₀₇(10) is an algebraic factor of the Cunningham number 10⁴⁸⁰⁷−1, the N−1 of "93307_2197" is 6 × "15551_2197", the N−1 of "33337_12891" has sum-of-two-cubes factorization and an algebraic factor is 2 × "16667_4296", the N−1 of "20111_2693" is 10 × "20111_2692", the N+1 of "23309_10029" is 210 × R₁₀₀₂₈(10) (which is equivalent to the Cunningham number 10¹⁰⁰²⁸−1) and Φ₅₀₁₄(10) is an algebraic factor of the Cunningham number 10¹⁰⁰²⁸−1, the N+1 of "37773_15768" is 34 × R₁₅₇₆₈(10) (which is equivalent to the Cunningham number 10¹⁵⁷⁶⁸−1) and Φ₇₈₈₄(10) is an algebraic factor of the Cunningham number 10¹⁵⁷⁶⁸−1, the N+1 of "27227_3741" is 4 × "6805w7_3739", the N−1 of "68883_5132" is 62 × R₅₁₃₂(10) (which is equivalent to the Cunningham number 10⁵¹³²−1) and Φ₁₂₈₃(10) is an algebraic factor of the Cunningham number 10⁵¹³²−1, the N−1 of "79921_11629" is 720 × R₁₁₆₂₈(10) (which is equivalent to the Cunningham number 10¹¹⁶²⁸−1) and Φ₂₉₀₇(10) is an algebraic factor of the Cunningham number 10¹¹⁶²⁸−1, the N−1 of "80081_5736" is 80 × S₅₇₃₅(10) (which is equivalent to the Cunningham number 10⁵⁷³⁵+1) and Φ₁₁₄₇₀(10) is an algebraic factor of the Cunningham number 10⁵⁷³⁵+1, the N+1 of "11103_3258" has difference-of-two-6th-powers factorization and an algebraic factor is 4 × "83w16w7_543", etc.)

(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/pmain1123.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), http://myfactorcollection.mooo.com:8090/cgi-bin/showCustomRep?CustomList=B&EN=&LM= (2 ≤ b ≤ 12), http://myfactorcollection.mooo.com:8090/cgi-bin/showREGComps?REGCompList=F&REGSortList=A&LabelList=E&REGHeader=&REGExp= (2 ≤ b ≤ 12), https://maths-people.anu.edu.au/~brent/factors.html (13 ≤ b ≤ 99), https://arxiv.org/pdf/1004.3169.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_206.pdf) (13 ≤ b ≤ 99), https://maths-people.anu.edu.au/~brent/pd/rpb134t.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_208.pdf) (13 ≤ b ≤ 99), http://myfactorcollection.mooo.com:8090/cgi-bin/showANCustomRep?LevelList=A&BaseRangeList=A&EN=&LM= (13 ≤ b ≤ 99), http://myfactorcollection.mooo.com:8090/cgi-bin/showANCustomRep?LevelList=B&BaseRangeList=A&EN=&LM= (13 ≤ b ≤ 99), https://web.archive.org/web/20220513215832/http://myfactorcollection.mooo.com:8090/cgi-bin/showCustomRep?CustomList=A&EN=&LM= (13 ≤ b ≤ 99), http://myfactorcollection.mooo.com:8090/cgi-bin/showANComps?LevelList=A&BaseRangeList=A&REGSortList=A&LabelList=E&REGHeader=&REGExp= (13 ≤ b ≤ 99), http://myfactorcollection.mooo.com:8090/cgi-bin/showANComps?LevelList=B&BaseRangeList=A&REGSortList=A&LabelList=E&REGHeader=&REGExp= (13 ≤ b ≤ 99), http://maths-people.anu.edu.au/~brent/ftp/rpb200t.txt.gz (13 ≤ b ≤ 99), http://maths-people.anu.edu.au/~brent/ftp/factors/comps.gz (13 ≤ b ≤ 99), https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html (2 ≤ b ≤ 999), https://mers.sourceforge.io/factoredM.txt (b = 2), https://oeis.org/A250197/a250197_2.txt (b = 2), https://web.archive.org/web/20130530210800/http://www.euronet.nl/users/bota/medium-p.htm (b = 2), https://www.mersenne.org/report_exponent/ (b = 2, −1 side, prime n), https://www.mersenne.org/report_factors/ (b = 2, −1 side, prime n), https://www.mersenne.org/report_exponent/?exp_lo=2&exp_hi=1000&full=1&ancient=1&expired=1&ecmhist=1&swversion=1 (b = 2, −1 side, prime n), https://www.mersenne.org/report_exponent/?exp_lo=1001&exp_hi=2000&full=1&ancient=1&expired=1&ecmhist=1&swversion=1 (b = 2, −1 side, prime n), https://www.mersenne.org/report_factors/?dispdate=1&exp_hi=999999937 (b = 2, −1 side, prime n), https://www.mersenne.ca/prp.php?show=2 (b = 2, −1 side, prime n), https://www.mersenne.ca/exponent/browse/1/9999 (b = 2, −1 side, prime n), https://web.archive.org/web/20211128174912/http://mprime.s3-website.us-west-1.amazonaws.com/mersenne/MERSENNE_FF_with_factors.txt (b = 2, −1 side, prime n), https://web.archive.org/web/20210726214248/http://mprime.s3-website.us-west-1.amazonaws.com/wagstaff/WAGSTAFF_FF_with_factors.txt (b = 2, +1 side, prime n), https://www-users.york.ac.uk/~ss44/cyc/m/mersenne.htm (b = 2, −1 side, prime n, n ≤ 263), https://planetmath.org/tableoffactorsofsmallmersennenumbers (b = 2, −1 side, prime n, n ≤ 199), https://web.archive.org/web/20190211112446/http://home.earthlink.net/~elevensmooth/ (b = 2, n divides 1663200), 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://stdkmd.net/nrr/repunit/Phin10ex.txt (b = 10, only primitive factors), https://stdkmd.net/nrr/repunit/Phin10ex.txt.lz (b = 10, only primitive factors), https://stdkmd.net/nrr/repunit/Phin10ex.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://web.archive.org/web/20160906031334/http://www.h4.dion.ne.jp/~rep/ (b = 10), https://repunit-koide.jimdofree.com/app/download/10034950550/Repunit100-20240104.pdf?t=1705060986 (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), http://chesswanks.com/pxp/repfactors.html (b = 10), https://web.archive.org/web/20120426061657/http://oddperfect.org/ (prime b, −1 side, prime n), http://myfactorcollection.mooo.com:8090/oddperfect/Jan27_2023/opfactors.gz (prime b, −1 side, prime n, bⁿ < 10⁸⁵⁰), https://web.archive.org/web/20081006071311/http://www-staff.maths.uts.edu.au/~rons/fact/fact.htm (2 ≤ b ≤ 9973, prime b), http://myfactorcollection.mooo.com:8090/cgi-bin/showCROPComps?OPCompList=A&OPSortList=A&LabelList=E&OPHeader=&OPExp= (2 ≤ b ≤ 9973, prime b, −1 side, prime n), http://myfactorcollection.mooo.com:8090/cgi-bin/showCROPComps?OPCompList=B&OPSortList=A&LabelList=E&OPHeader=&OPExp= (2 ≤ b ≤ 9973, prime b, −1 side, prime n), http://myfactors.mooo.com/ (2 ≤ b ≤ 1100000), http://myfactorcollection.mooo.com:8090/dbio.html (2 ≤ b ≤ 1100000), http://myfactorcollection.mooo.com:8090/interactive.html (2 ≤ b ≤ 1100000) (the lattices saparated to two lattices means the number has Aurifeuillean factorization, and for such lattices, the left lattice is for the Aurifeuillean L part, and the right lattice is for the Aurifeuillean M part), http://myfactorcollection.mooo.com:8090/brentdata/Mar31_2024/factors.gz (2 ≤ b ≤ 1100000), http://maths-people.anu.edu.au/~brent/ftp/factors/factors.gz (2 ≤ b ≤ 9999, only prime factors > 10⁹), http://www.asahi-net.or.jp/~KC2H-MSM/cn/old/index.htm (2 ≤ b ≤ 1000, only primitive factors), http://www.asahi-net.or.jp/~KC2H-MSM/cn/index.htm (2 ≤ b ≤ 1000, only primitive factors), https://web.archive.org/web/20050922233702/http://user.ecc.u-tokyo.ac.jp/~g440622/cn/index.html (2 ≤ b ≤ 1000, only primitive factors), https://web.archive.org/web/20070629012309/http://subsite.icu.ac.jp/people/mitsuo/enbunsu/table.html (2 ≤ b ≤ 1000, only primitive factors), also for the factors of bⁿ±1 with 2 ≤ b ≤ 400 and 1 ≤ n ≤ 400 and for the first holes of bⁿ±1 with 2 ≤ b ≤ 400 see the links in the list below)