(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/cert/certificates.tpm.html, 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 xynz (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 (e.g. the prime 8957 in base b = 20 (although this prime is not minimal prime in base b = 20), its algebraic form is (161×2057−9)/19, neither N−1 nor N+1 is trivially fully factored, see https://www.mersenneforum.org/showthread.php?t=29763) (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://web.archive.org/web/20231030081449/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_321_llr.php, 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=321&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=27&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=121&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, 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, http://boincvm.proxyma.ru:30080/test4vm/user_profile/llr2_status.html, https://web.archive.org/web/20231223043356/https://www.mersenneforum.org/321search/index.html, https://web.archive.org/web/20110601231527/http://www.bodang.com/12121/, https://web.archive.org/web/20100518081012/http://www.bodang.com/12121/27k/, https://web.archive.org/web/20210415051133/http://prpnet.primegrid.com:12001/, https://web.archive.org/web/20220115151556/http://prpnet.primegrid.com:12006/, https://www.rieselprime.de/ziki/321_Search, https://www.rieselprime.de/ziki/12121_Search, https://www.rieselprime.de/ziki/27121_Search, https://www.rieselprime.de/ziki/PrimeGrid_321_Prime_Search, https://www.rieselprime.de/ziki/PrimeGrid_27121_Prime_Search, 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://web.archive.org/web/20231011144408/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://web.archive.org/web/20231030081316/https://pzktupel.de/Primetables/RieselK.txt, https://pzktupel.de/Primetables/TableRieselTOP10KS.php, https://pzktupel.de/Primetables/RieselS.txt, https://pzktupel.de/Primetables/TableRieselGen.php, https://pzktupel.de/Primetables/TableRieselGen.txt, https://sites.google.com/view/proth-primes, https://www.primegrid.com/stats_321_llr.php, https://www.primegrid.com/primes/primes.php?project=321&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=27&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, https://www.primegrid.com/primes/primes.php?project=121&factors=XGF&only=ALL&announcements=ALL&sortby=size&dc=yes&search=, http://www.noprimeleftbehind.net/stats/index.php?content=prime_list, https://t5k.org/primes/search_proth.php, http://www.noprimeleftbehind.net/prpnet/, http://www.noprimeleftbehind.net:9000/all.html, http://www.noprimeleftbehind.net:4000/all.html, http://www.noprimeleftbehind.net:2000/all.html, http://www.noprimeleftbehind.net:1468/all.html, http://www.noprimeleftbehind.net:1400/all.html, https://web.archive.org/web/20231223043356/https://www.mersenneforum.org/321search/index.html, https://web.archive.org/web/20110601231527/http://www.bodang.com/12121/, https://web.archive.org/web/20100518081012/http://www.bodang.com/12121/27k/, https://web.archive.org/web/20210415051133/http://prpnet.primegrid.com:12001/, https://web.archive.org/web/20220115151556/http://prpnet.primegrid.com:12006/, https://www.rieselprime.de/ziki/321_Search, https://www.rieselprime.de/ziki/12121_Search, https://www.rieselprime.de/ziki/27121_Search, https://www.rieselprime.de/ziki/PrimeGrid_321_Prime_Search, https://www.rieselprime.de/ziki/PrimeGrid_27121_Prime_Search, 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/pmain524.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®SortList=A&LabelList=E®Header=®Exp= (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®SortList=A&LabelList=E®Header=®Exp= (13 ≤ b ≤ 99), http://myfactorcollection.mooo.com:8090/cgi-bin/showANComps?LevelList=B&BaseRangeList=A®SortList=A&LabelList=E®Header=®Exp= (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, only primitive factors), https://oeis.org/A250197/a250197_2.txt (b = 2, only primitive factors), https://web.archive.org/web/20130531074320/http://www.euronet.nl/users/bota/medium-p-odd.txt (b = 2, +1 side, "1201 ≤ n ≤ 1999, odd n" or "2003 ≤ n ≤ 9973, prime n"), https://web.archive.org/web/20120314040812/http://www.euronet.nl/users/bota/medium-p-even4k.txt (b = 2, +1 side, 1200 ≤ n ≤ 2400, n is divisible by 4), https://web.archive.org/web/20120313035810/http://www.euronet.nl/users/bota/medium-m-odd.txt (b = 2, −1 side, 1201 ≤ n ≤ 2499, odd n), 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_PRP/ (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.org/report_PRP/?swv=1 (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://2721.hddkillers.com/billion/controls/?candnum=1&exponent=1¤tdepth=1&activedepth=1&digits=1&factors=1&searcher=1&candtype=2&rowlimit=0&highstrength=0&highsearcher= (b = 2, −1 side, n ≥ 3321928097, prime n), https://web.archive.org/web/20190222204546/http://home.earthlink.net/~elevensmooth/Billion.html (b = 2, −1 side, n ≥ 3321928097, 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://web.archive.org/web/20231126215849/http://bearnol.is-a-geek.com/Mersenneplustwo/Mersenneplustwo.html (b = 2, +1 side, prime n such that 2n−1 is also prime), 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/20180525070353/http://home.earthlink.net/~elevensmooth/ElevenFactors.html (b = 2, n divides 1663200), https://web.archive.org/web/20180209113238/http://home.earthlink.net/~elevensmooth/Progress.html (b = 2, n divides 1663200), https://stdkmd.net/nrr/repunit/ (b = 10), https://stdkmd.net/nrr/repunit/10001.htm (b = 10, +1 side), 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, only primitive factors), https://kurtbeschorner.de/fact-2500.htm (b = 10, only primitive factors), https://repunit-koide.jimdofree.com/ (b = 10), https://repunit-koide.jimdofree.com/app/download/10317119350/Repunit100-20240612.pdf?t=1718186099 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_485.pdf) (b = 10), https://gmplib.org/~tege/repunit.html (b = 10, only primitive factors), https://gmplib.org/~tege/fac10m.txt (b = 10, −1 side, only primitive factors), https://gmplib.org/~tege/fac10p.txt (b = 10, +1 side, only primitive factors), https://web.archive.org/web/20070708171301/http://www.ludwigsgymnasium.de/unterr/mathe/prim/zehnp.htm (b = 10, +1 side, only primitive factors), http://chesswanks.com/pxp/repfactors.html (b = 10), https://web.archive.org/web/20191127073837/http://oddperfect.org/composites.html (prime b, −1 side, prime n), https://web.archive.org/web/20170701232547/http://oddperfect.org/cleared.html (prime b, −1 side, prime n), https://www.lirmm.fr/~ochem/opn/checkfacts.txt (prime b, −1 side, prime n), https://www.lirmm.fr/~ochem/opn/checkfacts.txt.gz (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), https://homes.cerias.purdue.edu/~ssw/bell/r1 (3 ≤ b ≤ 179, prime b, n = b, −1 side), https://homes.cerias.purdue.edu/~ssw/bell/r2 (3 ≤ b ≤ 179, prime b, n = b, +1 side), https://homes.cerias.purdue.edu/~ssw/bell/bell4.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_86.pdf) (11 ≤ b ≤ 173, prime b, n = b, −1 side), https://homes.cerias.purdue.edu/~ssw/bell/bell.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_87.pdf) (3 ≤ b ≤ 179, prime b, n = b), http://www.prothsearch.com/fermat.html (b = 2, +1 side, power-of-2 n), http://www.prothsearch.com/GFN03.html (b = 3, +1 side, power-of-2 n), http://www.prothsearch.com/GFN05.html (b = 5, +1 side, power-of-2 n), http://www.prothsearch.com/GFN06.html (b = 6, +1 side, power-of-2 n), http://www.prothsearch.com/GFN07.html (b = 7, +1 side, power-of-2 n), http://www.prothsearch.com/GFN10.html (b = 10, +1 side, power-of-2 n), http://www.prothsearch.com/GFN11.html (b = 11, +1 side, power-of-2 n), http://www.prothsearch.com/GFN12.html (b = 12, +1 side, power-of-2 n), http://www.prothsearch.com/GFNfacs.html ("3 ≤ b ≤ 12, +1 side, power-of-2 n" and "b = 2, +1 side, 3-times-power-of-2 n"), http://www.prothsearch.com/GFNsmall.html ("3 ≤ b ≤ 12, +1 side, power-of-2 n" and "b = 2, +1 side, 3-times-power-of-2 n"), http://www.prothsearch.com/OriginalGFNs.html ("3 ≤ b ≤ 12, +1 side, power-of-2 n" and "b = 2, +1 side, 3-times-power-of-2 n"), https://web.archive.org/web/20070910080730/http://members.cox.net/jfoug/GFNFacts_Riesel.html ("3 ≤ b ≤ 12, +1 side, power-of-2 n" and "b = 2, +1 side, 3-times-power-of-2 n"), https://web.archive.org/web/20070914091821/http://members.cox.net/jfoug/GFNFacts_SearchLimits.html ("3 ≤ b ≤ 12, +1 side, power-of-2 n" and "b = 2, +1 side, 3-times-power-of-2 n"), https://web.archive.org/web/20070914092135/http://members.cox.net/jfoug/GFNFacts_ECMComposites.html ("3 ≤ b ≤ 12, +1 side, power-of-2 n" and "b = 2, +1 side, 3-times-power-of-2 n"), https://mers.sourceforge.io/MMPstats.txt (b = 2, −1 side, prime n such that n+1 is power-of-2), http://www.doublemersennes.org/factors.php (b = 2, −1 side, prime n such that n+1 is power-of-2), http://www.doublemersennes.org/history.php (b = 2, −1 side, prime n such that n+1 is power-of-2), http://www.doublemersennes.org/sieving/validi.php (b = 2, −1 side, prime n such that n+1 is power-of-2), http://www.fermatsearch.org/factors/faclist.php (b = 2, +1 side, power-of-2 n), http://www.fermatsearch.org/factors/composite.php (b = 2, +1 side, power-of-2 n), https://www.alpertron.com.ar/MODFERM.HTM (b = 2, power-of-3 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/Aug31_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)
The Cunningham numbers bn±1 has algebraic factorization to product of the cyclotomic numbers Φd(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))
The Aurifeuillean factorization (https://en.wikipedia.org/wiki/Aurifeuillean_factorization, https://www.rieselprime.de/ziki/Aurifeuillian_factor, https://mathworld.wolfram.com/AurifeuilleanFactorization.html, http://www.numericana.com/answer/numbers.htm#aurifeuille, https://web.archive.org/web/20231002141924/http://colin.barker.pagesperso-orange.fr/lpa/cycl_fac.htm, http://list.seqfan.eu/oldermail/seqfan/2017-March/017363.html, http://myfactorcollection.mooo.com:8090/source/cyclo.cpp, http://myfactorcollection.mooo.com:8090/LCD_2_199, http://myfactorcollection.mooo.com:8090/LCD_2_998, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/source/cyclo.cpp, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/LucasCD/LCD_2_199, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/LucasCD/LCD_2_998, https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean, https://www.unshlump.com/hcn/aurif.html, https://www.mersenneforum.org/showthread.php?t=10439, https://www.mersenneforum.org/showpost.php?p=515828&postcount=8, https://maths-people.anu.edu.au/~brent/pd/rpb135.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_97.pdf), https://www.ams.org/journals/mcom/2006-75-253/S0025-5718-05-01766-7/S0025-5718-05-01766-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_138.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb127.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_164.pdf), https://www.jams.jp/scm/contents/Vol-2-3/2-3-16.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_167.pdf), https://web.archive.org/web/20130702000532/http://xyyxf.at.tut.by/aurifeuillean.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_443.pdf)) for the the cyclotomic numbers Φd(b) for bases 2 ≤ b ≤ 36 are: (for more information, see https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean_list and http://myfactorcollection.mooo.com:8090/LCD_2_199 and http://myfactorcollection.mooo.com:8090/LCD_2_998 and https://web.archive.org/web/20231002141924/http://colin.barker.pagesperso-orange.fr/lpa/cycl_fac.htm and https://en.wikipedia.org/wiki/Aurifeuillean_factorization#Examples (see the table) and https://www.rieselprime.de/ziki/Aurifeuillian_factor (see the table))
(Note: although there are no OEIS sequences of the Aurifeuillean factors of bn±1 for bases b > 12, but there are OEIS sequences of the Aurifeuillean factors of pp±1 for prime bases p, for primes p == 1 mod 4, pp−1 has Aurifeuillean factors, and the OEIS sequence of the Aurifeuillean L factor is https://oeis.org/A352711, and the OEIS sequence of the Aurifeuillean M factor is https://oeis.org/A352732, for primes p == 3 mod 4, pp+1 has Aurifeuillean factors, and the OEIS sequence of the Aurifeuillean L factor is https://oeis.org/A352400, and the OEIS sequence of the Aurifeuillean M factor is https://oeis.org/A352401)
| b | the algebraic factor of bn±1 which has Aurifeuillean factorization | the Aurifeuillean L factor | the Aurifeuillean M factor | examples of the Aurifeuillean factorization for exponent 2×r+1 = 37 (see the "Aurifeuillian" section) | OEIS sequence of the Aurifeuillean L factor | OEIS sequence of the Aurifeuillean M factor |
|---|---|---|---|---|---|---|
| 2, 4, 8, 16, 32 | Φ4(22×r+1) | 22×r+1−2r+1+1 | 22×r+1+2r+1+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=2&Exp=74&LBIDPMList=B&LBIDLODList=D | https://oeis.org/A092440 | https://oeis.org/A085601 |
| 3, 9, 27 | Φ6(32×r+1) | 32×r+1−3r+1+1 | 32×r+1+3r+1+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=3&Exp=111&LBIDPMList=B&LBIDLODList=D | https://oeis.org/A220978 | https://oeis.org/A198410 |
| 5, 25 | Φ5(52×r+1) | 54×r+2−53×r+2+3×52×r+1−5r+1+1 | 54×r+2+53×r+2+3×52×r+1+5r+1+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=5&Exp=185&LBIDPMList=A&LBIDLODList=D | https://oeis.org/A220979 | https://oeis.org/A220980 |
| 6, 36 | Φ12(62×r+1) | 64×r+2−63×r+2+3×62×r+1−6r+1+1 | 64×r+2+63×r+2+3×62×r+1+6r+1+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=6&Exp=222&LBIDPMList=B&LBIDLODList=D | https://oeis.org/A220981 | https://oeis.org/A220982 |
| 7 | Φ14(72×r+1) | 76×r+3−75×r+3+3×74×r+2−73×r+2+3×72×r+1−7r+1+1 | 76×r+3+75×r+3+3×74×r+2+73×r+2+3×72×r+1+7r+1+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=7&Exp=259&LBIDPMList=B&LBIDLODList=D | https://oeis.org/A220983 | https://oeis.org/A220984 |
| 10 | Φ20(102×r+1) | 108×r+4−107×r+4+5×106×r+3−2×105×r+3+7×104×r+2−2×103×r+2+5×102×r+1−10r+1+1 | 108×r+4+107×r+4+5×106×r+3+2×105×r+3+7×104×r+2+2×103×r+2+5×102×r+1+10r+1+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=10&Exp=370&LBIDPMList=B&LBIDLODList=D | https://oeis.org/A220985 | https://oeis.org/A220986 |
| 11 | Φ22(112×r+1) | 1110×r+5−119×r+5+5×118×r+4−117×r+4−116×r+3+115×r+3−114×r+2−113×r+2+5×112×r+1−11r+1+1 | 1110×r+5+119×r+5+5×118×r+4+117×r+4−116×r+3−115×r+3−114×r+2+113×r+2+5×112×r+1+11r+1+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=11&Exp=407&LBIDPMList=B&LBIDLODList=D | https://oeis.org/A220987 | https://oeis.org/A220988 |
| 12 | Φ6(122×r+1) | 122×r+1−6×12r+1 | 122×r+1+6×12r+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=12&Exp=111&LBIDPMList=B&LBIDLODList=D | https://oeis.org/A220989 | https://oeis.org/A220990 |
| 13 | Φ13(132×r+1) | 1312×r+6−1311×r+6+7×1310×r+5−3×139×r+5+15×138×r+4−5×137×r+4+19×136×r+3−5×135×r+3+15×134×r+2−3×133×r+2+7×132×r+1−13r+1+1 | 1312×r+6+1311×r+6+7×1310×r+5+3×139×r+5+15×138×r+4+5×137×r+4+19×136×r+3+5×135×r+3+15×134×r+2+3×133×r+2+7×132×r+1+13r+1+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&Exp=481&LBIDPMList=A&LBIDLODList=D | – | – |
| 14 | Φ28(142×r+1) | 1412×r+6−1411×r+6+7×1410×r+5−2×149×r+5+3×148×r+4+147×r+4−7×146×r+3+145×r+3+3×144×r+2−2×143×r+2+7×142×r+1−14r+1+1 | 1412×r+6+1411×r+6+7×1410×r+5+2×149×r+5+3×148×r+4-147×r+4-7×146×r+3-145×r+3+3×144×r+2+2×143×r+2+7×142×r+1+14r+1+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=14&Exp=518&LBIDPMList=B&LBIDLODList=D | – | – |
| 15 | Φ30(152×r+1) | 158×r+4−157×r+4+8×156×r+3−3×155×r+3+13×154×r+2−3×153×r+2+8×152×r+1−15r+1+1 | 158×r+4+157×r+4+8×156×r+3+3×155×r+3+13×154×r+2+3×153×r+2+8×152×r+1+15r+1+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=15&Exp=555&LBIDPMList=B&LBIDLODList=D | – | – |
| 17 | Φ17(172×r+1) | 1716×r+8−1715×r+8+9×1714×r+7−3×1713×r+7+11×1712×r+6−1711×r+6−5×1710×r+5+3×179×r+5−15×178×r+4+3×177×r+4−5×176×r+3−175×r+3+11×174×r+2−3×173×r+2+9×172×r+1−17r+1+1 | 1716×r+8+1715×r+8+9×1714×r+7+3×1713×r+7+11×1712×r+6+1711×r+6−5×1710×r+5−3×179×r+5−15×178×r+4−3×177×r+4−5×176×r+3+175×r+3+11×174×r+2+3×173×r+2+9×172×r+1+17r+1+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=17&Exp=629&LBIDPMList=A&LBIDLODList=D | – | – |
| 18 | Φ4(182×r+1) | 182×r+1−6×18r+1 | 182×r+1+6×18r+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=18&Exp=74&LBIDPMList=B&LBIDLODList=D | – | – |
| 19 | Φ38(192×r+1) | 1918×r+9−1917×r+9+9×1916×r+8−3×1915×r+8+17×1914×r+7−5×1913×r+7+27×1912×r+6−7×1911×r+6+31×1910×r+5−7×199×r+5+31×198×r+4−7×197×r+4+27×196×r+3−5×195×r+3+17×194×r+2−3×193×r+2+9×192×r+1−19r+1+1 | 1918×r+9+1917×r+9+9×1916×r+8+3×1915×r+8+17×1914×r+7+5×1913×r+7+27×1912×r+6+7×1911×r+6+31×1910×r+5+7×199×r+5+31×198×r+4+7×197×r+4+27×196×r+3+5×195×r+3+17×194×r+2+3×193×r+2+9×192×r+1+19r+1+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=19&Exp=703&LBIDPMList=B&LBIDLODList=D | – | – |
| 20 | Φ5(202×r+1) | 204×r+2−10×203×r+1+3×202×r+1−10×20r+1 | 204×r+2+10×203×r+1+3×202×r+1+10×20r+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=20&Exp=185&LBIDPMList=A&LBIDLODList=D | – | – |
| 21 | Φ21(212×r+1) | 2112×r+6−2111×r+6+10×2110×r+5−3×219×r+5+13×218×r+4−2×217×r+4+7×216×r+3−2×215×r+3+13×214×r+2−3×213×r+2+10×212×r+1−21r+1+1 | 2112×r+6+2111×r+6+10×2110×r+5+3×219×r+5+13×218×r+4+2×217×r+4+7×216×r+3+2×215×r+3+13×214×r+2+3×213×r+2+10×212×r+1+21r+1+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=21&Exp=777&LBIDPMList=A&LBIDLODList=D | – | – |
| 22 | Φ44(222×r+1) | 2220×r+10−2219×r+10+11×2218×r+9−4×2217×r+9+27×2216×r+8−7×2215×r+8+33×2214×r+7−6×2213×r+7+21×2212×r+6−3×2211×r+6+11×2210×r+5−3×229×r+5+21×228×r+4−6×227×r+4+33×226×r+3−7×225×r+3+27×224×r+2−4×223×r+2+11×222×r+1−22r+1+1 | 2220×r+10+2219×r+10+11×2218×r+9+4×2217×r+9+27×2216×r+8+7×2215×r+8+33×2214×r+7+6×2213×r+7+21×2212×r+6+3×2211×r+6+11×2210×r+5+3×229×r+5+21×228×r+4+6×227×r+4+33×226×r+3+7×225×r+3+27×224×r+2+4×223×r+2+11×222×r+1+22r+1+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=22&Exp=814&LBIDPMList=B&LBIDLODList=D | – | – |
| 23 | Φ46(232×r+1) | 2322×r+11−2321×r+11+11×2320×r+10−3×2319×r+10+9×2318×r+9+2317×r+9−19×2316×r+8+5×2315×r+8−15×2314×r+7−2313×r+7+25×2312×r+6−7×2311×r+6+25×2310×r+5−239×r+5−15×238×r+4+5×237×r+4−19×236×r+3+235×r+3+9×234×r+2−3×233×r+2+11×232×r+1−23r+1+1 | 2322×r+11+2321×r+11+11×2320×r+10+3×2319×r+10+9×2318×r+9−2317×r+9−19×2316×r+8−5×2315×r+8−15×2314×r+7+2313×r+7+25×2312×r+6+7×2311×r+6+25×2310×r+5+239×r+5−15×238×r+4−5×237×r+4−19×236×r+3−235×r+3+9×234×r+2+3×233×r+2+11×232×r+1+23r+1+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=23&Exp=851&LBIDPMList=B&LBIDLODList=D | – | – |
| 24 | Φ12(242×r+1) | 244×r+2−12×243×r+1+3×242×r+1−12×24r+1 | 244×r+2+12×243×r+1+3×242×r+1+12×24r+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=24&Exp=222&LBIDPMList=B&LBIDLODList=D | – | – |
| 26 | Φ52(262×r+1) | 2624×r+12−2623×r+12+13×2622×r+11−4×2621×r+11+19×2620×r+10−2619×r+10−13×2618×r+9+4×2617×r+9−11×2616×r+8−2615×r+8+13×2614×r+7−2×2613×r+7+7×2612×r+6−2×2611×r+6+13×2610×r+5−269×r+5−11×268×r+4+4×267×r+4−13×266×r+3−265×r+3+19×264×r+2−4×263×r+2+13×262×r+1−26r+1+1 | 2624×r+12+2623×r+12+13×2622×r+11+4×2621×r+11+19×2620×r+10+2619×r+10−13×2618×r+9−4×2617×r+9−11×2616×r+8+2615×r+8+13×2614×r+7+2×2613×r+7+7×2612×r+6+2×2611×r+6+13×2610×r+5+269×r+5−11×268×r+4−4×267×r+4−13×266×r+3+265×r+3+19×264×r+2+4×263×r+2+13×262×r+1+26r+1+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=26&Exp=962&LBIDPMList=B&LBIDLODList=D | – | – |
| 28 | Φ14(282×r+1) | 286×r+3−14×285×r+2+3×284×r+2−14×283×r+1+3×282×r+1−14×28r+1 | 286×r+3+14×285×r+2+3×284×r+2+14×283×r+1+3×282×r+1+14×28r+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=28&Exp=259&LBIDPMList=B&LBIDLODList=D | – | – |
| 29 | Φ29(292×r+1) | 2928×r+14−2927×r+14+15×2926×r+13−5×2925×r+13+33×2924×r+12−5×2923×r+12+13×2922×r+11−2921×r+11+15×2920×r+10−7×2919×r+10+57×2918×r+9−11×2917×r+9+45×2916×r+8−5×2915×r+8+19×2914×r+7−5×2913×r+7+45×2912×r+6−11×2911×r+6+57×2910×r+5−7×299×r+5+15×298×r+4−297×r+4+13×296×r+3−5×295×r+3+33×294×r+2−5×293×r+2+15×292×r+1−29r+1+1 | 2928×r+14+2927×r+14+15×2926×r+13+5×2925×r+13+33×2924×r+12+5×2923×r+12+13×2922×r+11+2921×r+11+15×2920×r+10+7×2919×r+10+57×2918×r+9+11×2917×r+9+45×2916×r+8+5×2915×r+8+19×2914×r+7+5×2913×r+7+45×2912×r+6+11×2911×r+6+57×2910×r+5+7×299×r+5+15×298×r+4+297×r+4+13×296×r+3+5×295×r+3+33×294×r+2+5×293×r+2+15×292×r+1+29r+1+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=29&Exp=1073&LBIDPMList=A&LBIDLODList=D | – | – |
| 30 | Φ60(302×r+1) | 3016×r+8−3015×r+8+15×3014×r+7−5×3013×r+7+38×3012×r+6−8×3011×r+6+45×3010×r+5−8×309×r+5+43×308×r+4−8×307×r+4+45×306×r+3−8×305×r+3+38×304×r+2−5×303×r+2+15×302×r+1−30r+1+1 | 3016×r+8+3015×r+8+15×3014×r+7+5×3013×r+7+38×3012×r+6+8×3011×r+6+45×3010×r+5+8×309×r+5+43×308×r+4+8×307×r+4+45×306×r+3+8×305×r+3+38×304×r+2+5×303×r+2+15×302×r+1+30r+1+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=30&Exp=1110&LBIDPMList=B&LBIDLODList=D | – | – |
| 31 | Φ62(312×r+1) | 3130×r+15−3129×r+15+15×3128×r+14−5×3127×r+14+43×3126×r+13−11×3125×r+13+83×3124×r+12−19×3123×r+12+125×3122×r+11−25×3121×r+11+151×3120×r+10−29×3119×r+10+169×3118×r+9−31×3117×r+9+173×3116×r+8−31×3115×r+8+173×3114×r+7−31×3113×r+7+169×3112×r+6−29×3111×r+6+151×3110×r+5−25×319×r+5+125×318×r+4−19×317×r+4+83×316×r+3−11×315×r+3+43×314×r+2−5×313×r+2+15×312×r+1−31r+1+1 | 3130×r+15+3129×r+15+15×3128×r+14+5×3127×r+14+43×3126×r+13+11×3125×r+13+83×3124×r+12+19×3123×r+12+125×3122×r+11+25×3121×r+11+151×3120×r+10+29×3119×r+10+169×3118×r+9+31×3117×r+9+173×3116×r+8+31×3115×r+8+173×3114×r+7+31×3113×r+7+169×3112×r+6+29×3111×r+6+151×3110×r+5+25×319×r+5+125×318×r+4+19×317×r+4+83×316×r+3+11×315×r+3+43×314×r+2+5×313×r+2+15×312×r+1+31r+1+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=31&Exp=1147&LBIDPMList=B&LBIDLODList=D | – | – |
| 33 | Φ33(332×r+1) | 3320×r+10−3319×r+10+16×3318×r+9−5×3317×r+9+37×3316×r+8−6×3315×r+8+19×3314×r+7+3313×r+7−32×3312×r+6+9×3311×r+6−59×3310×r+5+9×339×r+5−32×338×r+4+337×r+4+19×336×r+3−6×335×r+3+37×334×r+2−5×333×r+2+16×332×r+1−33r+1+1 | 3320×r+10+3319×r+10+16×3318×r+9+5×3317×r+9+37×3316×r+8+6×3315×r+8+19×3314×r+7−3313×r+7−32×3312×r+6−9×3311×r+6−59×3310×r+5−9×339×r+5−32×338×r+4−337×r+4+19×336×r+3+6×335×r+3+37×334×r+2+5×333×r+2+16×332×r+1+33r+1+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=33&Exp=1221&LBIDPMList=A&LBIDLODList=D | – | – |
| 34 | Φ68(342×r+1) | 3432×r+16−3431×r+16+17×3430×r+15−6×3429×r+15+59×3428×r+14−15×3427×r+14+119×3426×r+13−26×3425×r+13+181×3424×r+12−35×3423×r+12+221×3422×r+11−40×3421×r+11+243×3420×r+10−43×3419×r+10+255×3418×r+9−44×3417×r+9+257×3416×r+8−44×3415×r+8+255×3414×r+7−43×3413×r+7+243×3412×r+6−40×3411×r+6+221×3410×r+5−35×349×r+5+181×348×r+4−26×347×r+4+119×346×r+3−15×345×r+3+59×344×r+2−6×343×r+2+17×342×r+1−34r+1+1 | 3432×r+16+3431×r+16+17×3430×r+15+6×3429×r+15+59×3428×r+14+15×3427×r+14+119×3426×r+13+26×3425×r+13+181×3424×r+12+35×3423×r+12+221×3422×r+11+40×3421×r+11+243×3420×r+10+43×3419×r+10+255×3418×r+9+44×3417×r+9+257×3416×r+8+44×3415×r+8+255×3414×r+7+43×3413×r+7+243×3412×r+6+40×3411×r+6+221×3410×r+5+35×349×r+5+181×348×r+4+26×347×r+4+119×346×r+3+15×345×r+3+59×344×r+2+6×343×r+2+17×342×r+1+34r+1+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=34&Exp=1258&LBIDPMList=B&LBIDLODList=D | – | – |
| 35 | Φ70(352×r+1) | 3524×r+12−3523×r+12+18×3522×r+11−6×3521×r+11+48×3520×r+10−7×3519×r+10+11×3518×r+9+5×3517×r+9−55×3516×r+8+8×3515×r+8−11×3514×r+7−5×3513×r+7+47×3512×r+6−5×3511×r+6−11×3510×r+5+8×359×r+5−55×358×r+4+5×357×r+4+11×356×r+3−7×355×r+3+48×354×r+2−6×353×r+2+18×352×r+1−35r+1+1 | 3524×r+12+3523×r+12+18×3522×r+11+6×3521×r+11+48×3520×r+10+7×3519×r+10+11×3518×r+9−5×3517×r+9−55×3516×r+8−8×3515×r+8−11×3514×r+7+5×3513×r+7+47×3512×r+6+5×3511×r+6−11×3510×r+5−8×359×r+5−55×358×r+4−5×357×r+4+11×356×r+3+7×355×r+3+48×354×r+2+6×353×r+2+18×352×r+1+35r+1+1 | http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=35&Exp=1295&LBIDPMList=B&LBIDLODList=D | – | – |
If core(b) (the squarefree part (https://oeis.org/A007913, https://en.wikipedia.org/wiki/Square-free_integer#Squarefree_core, http://mathworld.wolfram.com/SquarefreePart.html, https://stdkmd.net/nrr/repunit/repunitnote.htm#core) of b) is == 1 mod 4 (and core(b) > 1, i.e. b is not a square number (https://en.wikipedia.org/wiki/Square_number, https://www.rieselprime.de/ziki/Square_number, https://mathworld.wolfram.com/SquareNumber.html, https://www.numbersaplenty.com/set/square_number/, https://oeis.org/A000290)), then bn−1 has Aurifeuillean factorization if and only if n is an odd multiple of core(b) (i.e. n == core(b) mod 2×core(b)) (and bn+1 has no Aurifeuillean factorization for any n), and if n is an odd multiple of core(b) (i.e. n == core(b) mod 2×core(b)), we write ΦnL(b) = gcd(Φn(b), the Aurifeuillean L factor of Φcore(b)(bn/core(b))) and ΦnM(b) = gcd(Φn(b), the Aurifeuillean M factor of Φcore(b)(bn/core(b))); if core(b) (the squarefree part (https://oeis.org/A007913, https://en.wikipedia.org/wiki/Square-free_integer#Squarefree_core, http://mathworld.wolfram.com/SquarefreePart.html, https://stdkmd.net/nrr/repunit/repunitnote.htm#core) of b) is == 2, 3 mod 4, then bn+1 has Aurifeuillean factorization if and only if n is an odd multiple of core(b) (i.e. n == core(b) mod 2×core(b)) (and bn−1 has no Aurifeuillean factorization for any n), and if n is an odd multiple of core(b) (i.e. n == core(b) mod 2×core(b)), we write Φ(2×n)L(b) = gcd(Φ2×n(b), the Aurifeuillean L factor of Φ2×core(b)(bn/core(b))) and Φ(2×n)M(b) = gcd(Φ2×n(b), the Aurifeuillean M factor of Φ2×core(b)(bn/core(b))). Thus, ΦnL(b) is "L'n" (if core(b) == 1 mod 4) or "L'n/2" (if core(b) == 2, 3 mod 4) in the section "Algebraic Factors for an±1" in http://myfactorcollection.mooo.com:8090/calculators.html (although that page uses "a" instead of "b" for the base), and ΦnM(b) is "M'n" (if core(b) == 1 mod 4) or "M'n/2" (if core(b) == 2, 3 mod 4) in the section "Algebraic Factors for an±1" in http://myfactorcollection.mooo.com:8090/calculators.html (although that page uses "a" instead of "b" for the base), for the example of base b = 10 see https://stdkmd.net/nrr/repunit/phin10.htm (the numbers Φ20L(10), Φ20M(10), Φ60L(10), Φ60M(10), Φ100L(10), Φ100M(10), Φ140L(10), Φ140M(10), Φ180L(10), Φ180M(10), ...) and https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean_repunit.
The degree of the SNFS polynomial of the number Φn(b) (or ΦnL(b) or ΦnM(b)) is: (see https://www.mersenneforum.org/showpost.php?p=628883&postcount=5 and https://www.mersenneforum.org/showpost.php?p=657762&postcount=55 and https://www.rieselprime.de/ziki/SNFS_polynomial_selection)
- n divisible by 3 but not by 5 or 7 for Φn(b) or ΦnL(b) or ΦnM(b) (the cases ΦnL(b) or ΦnM(b) are only for the squarefree part of b is 3, 5, 7): poly of any even degree
- n divisible by 5 but not by 3 for Φn(b) or ΦnL(b) or ΦnM(b) (the cases ΦnL(b) or ΦnM(b) are only for the squarefree part of b is 3, 5): quartic (degree 4) or octic (degree 8), but octic (degree 8) is only useful for difficulties > 260
- n divisible by 7 but not by 3 for Φn(b) or ΦnL(b) or ΦnM(b) (the cases ΦnL(b) or ΦnM(b) are only for the squarefree part of b is 2, 3, 6, 7, 14): sextic (degree 6)
- n divisible by 11 but not by 3 or 5 or 7 for Φn(b) or ΦnL(b) or ΦnM(b) (the cases ΦnL(b) or ΦnM(b) are only for the squarefree part of b is 2, 3, 6): quintic (degree 5)
- n divisible by 13 but not by 3 or 5 or 7 for Φn(b) or ΦnL(b) or ΦnM(b) (the cases ΦnL(b) or ΦnM(b) are only for the squarefree part of b is 2, 3, 6): sextic (degree 6)
- n divisible by 17 but not by 3 or 5 or 7 for Φn(b) or ΦnL(b) or ΦnM(b) (the cases ΦnL(b) or ΦnM(b) are only for the squarefree part of b is 2, 3, 6): octic (degree 8), but only useful for difficulties > 260
- n divisible by 3 and 5 for Φn(b) or ΦnL(b) or ΦnM(b) (the cases ΦnL(b) or ΦnM(b) are only for the squarefree part of b is 2, 3, 5, 6, 10): quartic (degree 4) or octic (degree 8), but octic (degree 8) is only useful for difficulties > 260
- n divisible by 3 and 7 for Φn(b) or ΦnL(b) or ΦnM(b) (the cases ΦnL(b) or ΦnM(b) are only for the squarefree part of b is 2, 3, 6, 7, 14): sextic (degree 6)
- n divisible by 3 but not by 9 for ΦnL(b) or ΦnM(b) and the squarefree part of b is 2, 6, 10, 14: quartic (degree 4) or octic (degree 8), but octic (degree 8) is only useful for difficulties > 260
- n divisible by 9 for ΦnL(b) or ΦnM(b) and the squarefree part of b is 2, 6, 10, 14: sextic (degree 6)
- n divisible by 5 but not by 3 for ΦnL(b) or ΦnM(b) and the squarefree part of b is 2, 6, 10: octic (degree 8), but only useful for difficulties > 260
- other cases: quartic (degree 4) for difficulties ≤ 105, quintic (degree 5) for 105 < difficulties ≤ 210, sextic (degree 6) for difficulties > 210 (see https://stdkmd.net/nrr/c.cgi?q=37771_259#howto and https://stdkmd.net/nrr/c.cgi?q=23333_233#howto)
The SNFS difficulty of Φn(b) (or ΦnL(b) or ΦnM(b)) is (no matter how much "log10 of the composite cofactor (which has no known proper factor)" is) (n−1/2×n×(n is divisible by 2)−1/3×n×(n is divisible by 3)−1/5×n×(n is divisible by 5)−1/7×n×(n is divisible by 7)−1/11×n×(n is divisible by 11 but not by 3 or 5 or 7)−1/13×n×(n is divisible by 13 but not by 3 or 5 or 7)−1/17×n×(n is divisible by 17 but not by 3 or 5 or 7))×log10(b) (note: for n divisible by 5×7 = 35, we can only minus one of "1/5×n×(n is divisible by 5)" and "1/7×n×(n is divisible by 7)", we cannot minus both; also, for n divisible by 11×13 = 143 or 11×17 = 187 or 13×17 = 221, we can only minus one of "1/11×n×(n is divisible by 11 but not by 3 or 5 or 7)" and "1/13×n×(n is divisible by 13 but not by 3 or 5 or 7)" and "1/17×n×(n is divisible by 17 but not by 3 or 5 or 7)"; also, for ΦnL(b) and ΦnM(b), the SNFS difficulty is half of it, but we cannot minus the numbers except "1/2×n×(n is divisible by 2)", unless the squarefree part of b is 2, 3, 5, 6, 7, 10, 14, in this case we can also minus the number 1/3×n×(n is divisible by 3), also if the squarefree part of b is 2, 3, 6, we can also minus the numbers "1/5×n×(n is divisible by 5)" and "1/7×n×(n is divisible by 7)", also if the squarefree part of b is 5, 10, we can also minus the numbers "1/5×n×(n is divisible by 5)", also if the squarefree part of b is 7, 14, we can also minus the numbers "1/7×n×(n is divisible by 7)", see https://www.mersenneforum.org/showpost.php?p=621569&postcount=13 and https://www.mersenneforum.org/showpost.php?p=628871&postcount=4), and the GNFS difficulty of Φn(b) (or ΦnL(b) or ΦnM(b)) is just the log10 of the composite cofactor (which has no known proper factor) of Φn(b) (or ΦnL(b) or ΦnM(b), and the SNFS difficulty 3×r is equivalent to GNFS difficulty 2×r for 80 ≤ r ≤ 120 but not for all r, see https://www.mersenneforum.org/showpost.php?p=620429&postcount=50, and if the equivalent GNFS difficulty of the SNFS difficulty of Φn(b) (or ΦnL(b) or ΦnM(b)) is smaller than the GNFS difficulty of Φn(b) (or ΦnL(b) or ΦnM(b)), then SNFS is better for the number Φn(b) (or ΦnL(b) or ΦnM(b)), and if the equivalent GNFS difficulty of the SNFS difficulty of Φn(b) (or ΦnL(b) or ΦnM(b)) is larger than the GNFS difficulty of Φn(b) (or ΦnL(b) or ΦnM(b)), then GNFS is better for the number Φn(b)) (note: for quartic (degree 4) and quintic (degree 5) and octic (degree 8), the numbers are more difficult than their SNFS-size suggests, see https://www.degruyter.com/document/doi/10.1515/JMC.2007.007/pdf?licenseType=open-access (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_99.pdf))
(below, "Rn(b)" means the repunit (https://en.wikipedia.org/wiki/Repunit, https://en.wikipedia.org/wiki/List_of_repunit_primes, https://t5k.org/glossary/xpage/Repunit.html, https://t5k.org/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://www.rieselprime.de/ziki/Generalized_Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://pzktupel.de/Primetables/TableRepunit.php, https://pzktupel.de/Primetables/TableRepunitGen.php, https://pzktupel.de/Primetables/TableRepunitGen.txt, https://stdkmd.net/nrr/prime/prime_rp.htm, https://stdkmd.net/nrr/prime/prime_rp.txt, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, https://web.archive.org/web/20131019185910/http://www.primes.viner-steward.org/andy/titans.html, https://web.archive.org/web/20120227163453/http://phi.redgolpe.com/, https://web.archive.org/web/20120227163614/http://phi.redgolpe.com/5.asp, https://web.archive.org/web/20120227163508/http://phi.redgolpe.com/4.asp, https://web.archive.org/web/20120227163610/http://phi.redgolpe.com/3.asp, https://web.archive.org/web/20120227163512/http://phi.redgolpe.com/2.asp, https://web.archive.org/web/20120227163521/http://phi.redgolpe.com/1.asp, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.fermatquotient.com/PrimSerien/GenRepuP.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, https://jpbenney.blogspot.com/2022/04/another-sequence-of-note.html, http://perplexus.info/show.php?pid=8661&cid=51696, https://benvitalenum3ers.wordpress.com/2013/07/24/repunit-11111111111111-in-other-bases/, 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://t5k.org/primes/search.php?Comment=^Repunit&OnList=all&Number=1000000&Style=HTML, https://t5k.org/primes/search.php?Comment=Generalized%20repunit&OnList=all&Number=1000000&Style=HTML, https://oeis.org/A002275, https://oeis.org/A004022, https://oeis.org/A053696, https://oeis.org/A085104, https://oeis.org/A179625) in base b with length n, i.e. Rn(b) = (bn−1)/(b−1) (see https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization), "Sn(b)" means bn+1 (see https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization), the special cases of Rn(10) and Sn(10) are in https://stdkmd.net/nrr/repunit/ and https://stdkmd.net/nrr/repunit/10001.htm, respectively, in fact, Rn(b) and Sn(b) are 111...111 and 1000...0001 in base b, respectively, also, Rn(b) and Sn(b) are the Lucas sequences (https://en.wikipedia.org/wiki/Lucas_sequence, https://mathworld.wolfram.com/LucasSequence.html, https://t5k.org/top20/page.php?id=23, https://t5k.org/primes/search.php?Comment=Generalized%20Lucas%20number&OnList=all&Number=1000000&Style=HTML) Un(b+1,b) and Vn(b+1,b), respectively)
- the 3168th minimal prime in base 13, 93081, N−1 is 117×R308(13), thus factor N−1 is equivalent to factor the Cunningham number 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 the Cunningham number 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 the Cunningham number 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 10320th minimal prime in base 17, 92921, N−1 is 153×R292(17), thus factor N−1 is equivalent to factor the Cunningham number 17292−1, and for the algebraic factors of 17292−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=17&Exp=292&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 17292−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=17&Exp=292&c0=-&EN=&LM=
- the 13304th minimal prime in base 21, 72301, N−1 is 147×R230(21), thus factor N−1 is equivalent to factor the Cunningham number 21230−1, and for the algebraic factors of 21230−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=21&Exp=230&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 21230−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=21&Exp=230&c0=-&EN=&LM=
- the 13355th minimal prime in base 21, 310632, N+1 is 3×R1064(21), thus factor N−1 is equivalent to factor the Cunningham number 211064−1, and for the algebraic factors of 211064−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=21&Exp=1064&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 211064−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=21&Exp=1064&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 the Cunningham number 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 the Cunningham number 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 10320th minimal prime in base 17 (92921) in factordb: http://factordb.com/helper.php?id=1100000000840355814
The helper file for the 13304th minimal prime in base 21 (72301) in factordb: http://factordb.com/helper.php?id=1100000002325398836
The helper file for the 13355th minimal prime in base 21 (310632) in factordb: http://factordb.com/helper.php?id=1100000002325396014
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 10320th minimal prime in base 17 (92921) in factordb: http://factordb.com/index.php?id=1100000000840355817&open=ecm
Factorization status (and ECM efforts for the prime factors between 1024 and 10100) of N−1 for the 13304th minimal prime in base 21 (72301) in factordb: http://factordb.com/index.php?id=1100000002325398854&open=ecm
Factorization status (and ECM efforts for the prime factors between 1024 and 10100) of N+1 for the 13355th minimal prime in base 21 (310632) in factordb: http://factordb.com/index.php?id=1100000002325396028&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 prime factor (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 each algebraic factor) 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://factordb.com/listecm.php?c=4, http://www.loria.fr/~zimmerma/ecmnet/, http://www.loria.fr/~zimmerma/records/ecmnet.html, http://www.loria.fr/~zimmerma/records/factor.html, http://www.loria.fr/~zimmerma/records/top50.html, http://www.loria.fr/~zimmerma/records/ecm/params.html, https://oeis.org/wiki/OEIS_sequences_needing_factors#ECM_efforts, https://stdkmd.net/nrr/records.htm#largefactorecm, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "Elliptic curve method:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=ecm&maxrows=10000, http://maths-people.anu.edu.au/~brent/factors.html, http://maths-people.anu.edu.au/~brent/ftp/champs.txt, https://www.alpertron.com.ar/ECM.HTM, https://www.alpertron.com.ar/ECMREC.HTM, https://homes.cerias.purdue.edu/~ssw/cun/press/tech.html, https://homes.cerias.purdue.edu/~ssw/cun/press/nontech.html, https://www.mersenne.org/report_ECM/, https://www.mersenne.ca/userfactors/ecm/1, https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, https://kurtbeschorner.de/ecm-efforts.htm, http://www.rechenkraft.net/yoyo//y_factors_ecm.php, http://www.rechenkraft.net/yoyo/y_status_ecm.php, http://www.wraithx.net/math/ecmprobs/ecmprobs.html, http://www.loria.fr/~zimmerma/papers/ecm-entry.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_460.pdf), https://arxiv.org/pdf/1004.3366.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_492.pdf)), "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, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "Pollard p-1:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=p-1&maxrows=10000, http://www.numericana.com/answer/factoring.htm#p-1, http://factordb.com/listecm.php?c=2, http://www.loria.fr/~zimmerma/records/Pminus1.html, https://web.archive.org/web/20021015212913/http://www.users.globalnet.co.uk/~aads/Pminus1.html, https://web.archive.org/web/20231002022529/https://colin.barker.pagesperso-orange.fr/lpa/big_pm1.htm, https://www.mersenne.org/report_pminus1/, https://www.mersenne.ca/userfactors/pm1/1, https://www.mersenne.ca/smooth.php, https://www.mersenne.ca/p1missed.php, https://www.mersenne.ca/prob.php), "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, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "p+1:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=p%2b1&maxrows=10000, http://www.numericana.com/answer/factoring.htm#p+1, http://factordb.com/listecm.php?c=3, http://www.loria.fr/~zimmerma/records/Pplus1.html, https://www.mersenne.org/report_pplus1/, https://www.mersenne.ca/userfactors/pp1/1, https://www.mersenne.ca/pplus1.php), "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, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (sections "Special number field sieve by size of number factored:" and "Special number field sieve by SNFS difficulty:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=snfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#smallpolynomial, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/), "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, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "General number field sieve by size of number factored:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=gnfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/))
For the number 13308−1, it is the product of Φd(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 Φd(13) for these positive integers d are:
| from | prime factorization |
|---|---|
| Φ1(13) | 22 × 3 |
| Φ2(13) | 2 × 7 |
| Φ4(13) | 2 × 5 × 17 |
| Φ7(13) | 5229043 |
| Φ11(13) | 23 × 419 × 859 × 18041 |
| Φ14(13) | 7 × 29 × 22079 |
| Φ22(13) | 128011456717 |
| Φ28(13) | 23161037562937 |
| Φ44(13) | 5281 × 3577574298489429481 |
| Φ77(13) | 624958606550654822293 × (47-digit prime) |
| Φ154(13) | 78947177 × (59-digit prime) |
| Φ308(13) | 7393 × 1702933 × 150324329 × 718377597171850001 × 4209006442599882158485591696242263069 × (61-digit prime) |
For the number 13564−1, it is the product of Φd(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 Φd(13) for these positive integers d are:
| from | prime factorization |
|---|---|
| Φ1(13) | 22 × 3 |
| Φ2(13) | 2 × 7 |
| Φ3(13) | 3 × 61 |
| Φ4(13) | 2 × 5 × 17 |
| Φ6(13) | 157 |
| Φ12(13) | 28393 |
| Φ47(13) | 183959 × 19216136497 × 534280344481909234853671069326391741 |
| Φ94(13) | 498851139881 × 3245178229485124818467952891417691434077 |
| Φ141(13) | 283 × 1693 × 1924651 × 455036140638637 × (76-digit prime) |
| Φ188(13) | 36097 × 75389 × 99886248944632632917 × (74-digit prime) |
| Φ282(13) | 590202369266263393 × (85-digit prime) |
| Φ564(13) | 233628485003849577181 × 94531330515097101267386264339794253977 (ECM (Montgomery curve), B1 = 3000000, Sigma = 2146847123, the prime factorization of the group order is 23 × 33 × 5 × 11 × 23 × 4871 × 10099 × 17207 × 1389277 × 2661643 × 110532803) × 27969827431131578608318126024627616357147784803797 (GNFS) × (98-digit prime) |
For the number 13576−1, it is the product of Φd(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 Φd(13) for these positive integers d are:
| from | prime factorization |
|---|---|
| Φ1(13) | 22 × 3 |
| Φ2(13) | 2 × 7 |
| Φ3(13) | 3 × 61 |
| Φ4(13) | 2 × 5 × 17 |
| Φ6(13) | 157 |
| Φ8(13) | 2 × 14281 |
| Φ9(13) | 3 × 1609669 |
| Φ12(13) | 28393 |
| Φ16(13) | 2 × 407865361 |
| Φ18(13) | 19 × 271 × 937 |
| Φ24(13) | 815702161 |
| Φ32(13) | 2 × 2657 × 441281 × 283763713 |
| Φ36(13) | 37 × 428041 × 1471069 |
| Φ48(13) | 1009 × 659481276875569 |
| Φ64(13) | 2 × 193 × 1601 × 10433 × 68675120456139881482562689 |
| Φ72(13) | 73 × 4177 × 181297 × 9818892432332713 |
| Φ96(13) | 97 × 88993 × 127028743393 × 403791981344275297 |
| Φ144(13) | 3889 × 680401 × 29975087953 × 6654909974864689 × 558181416418089697 |
| Φ192(13) | 1153 × 11352931040252580224415980746369 × 14977427998321433931503086910333672833 |
| Φ288(13) | 2017 × 47521 × 54721 × 1590049 × 8299042833797200969471889569 × (61-digit prime) |
| Φ576(13) | 577 × 6337 × 5247817273269739636080024961 × 5497355933986265726220616321 × 1032606621363411464640473542092061600217962755283816476128113983937 (GNFS) × (86-digit prime) |
For the number 17292−1, it is the product of Φd(17) with positive integers d dividing 292 (i.e. d = 1, 2, 4, 73, 146, 292), and the factorization of Φd(17) for these positive integers d are:
| from | prime factorization |
|---|---|
| Φ1(17) | 24 |
| Φ2(17) | 2 × 32 |
| Φ4(17) | 2 × 5 × 29 |
| Φ73(17) | 293 × 1621745371 × 3038535503 × 319344640907 × 596137412912777 × (42-digit prime) |
| Φ146(17) | 284117 × 1517302254487813 × (68-digit prime) |
| Φ292(17) | 877 × 4673 × 734854949942641407221086529797643582883477575817985261656573824687508604116290261093 × (87-digit prime) |
For the number 21230−1, it is the product of Φd(21) with positive integers d dividing 230 (i.e. d = 1, 2, 5, 10, 23, 46, 115, 230), and the factorization of Φd(21) for these positive integers d are:
| from | prime factorization |
|---|---|
| Φ1(21) | 22 × 5 |
| Φ2(21) | 2 × 11 |
| Φ5(21) | 5 × 40841 |
| Φ10(21) | 185641 |
| Φ23(21) | 47 × 19597 × 139870566115103282847737 |
| Φ46(21) | 277 × 461 × 599 × 691 × 2215825387044753577 |
| Φ115(21) | 1381 × 282924347471791 × 3394964812534556016503466037951 × (69-digit prime) |
| Φ230(21) | 2531 × 11731 × 22952851 × 595377311 × 688660481 × 58286351831 × 69727564981 × (63-digit prime) |
For the number 211064−1, it is the product of Φd(21) with positive integers d dividing 1064 (i.e. d = 1, 2, 4, 7, 8, 14, 19, 28, 38, 56, 76, 133, 152, 266, 532, 1064), and the factorization of Φd(21) for these positive integers d are:
| from | currently known prime factorization |
|---|---|
| Φ1(21) | 22 × 5 |
| Φ2(21) | 2 × 11 |
| Φ4(21) | 2 × 13 × 17 |
| Φ7(21) | 43 × 631 × 3319 |
| Φ8(21) | 2 × 97241 |
| Φ14(21) | 81867661 |
| Φ19(21) | 12061389013 × 54921106624003 |
| Φ28(21) | 29 × 3697 × 68454248717 |
| Φ38(21) | 609673 × 987749814642143197 |
| Φ56(21) | 617 × 912521 × 115593326297 × 831380909129 |
| Φ76(21) | 229 × 457 × (43-digit prime) |
| Φ133(21) | 948175293266954869500463698756935713088089028515629708586399 × (83-digit prime) |
| Φ152(21) | 136649 × 6629177 × 8871582886760161 × 4370570172021545617284038736601 × 4510053597010461591911520110711387257 |
| Φ266(21) | 4523 × 263478423344974307 × 39188712102054729290763779 × 1027619231425962708522784338595411210117 × (58-digit prime) |
| Φ532(21) | 1080514246723801 × 4598307023923376056176577 (P−1, B1 = 100000, B2 = 39772318, the prime factorization of P−1 is 26 × 3 × 7 × 11 × 19 × 23 × 241 × 1229 × 1697 × 7369 × 192161) × 173111326443349916878938361 × 8616925958054113732048023533 × (192-digit prime) |
| Φ1064(21) | 140449 × 723460417 × (558-digit composite with no known proper factor, SNFS difficulty is 602.932, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=21&Exp=532&c0=%2B&LM=&SA=) |
For the number 26344−1, it is the product of Φd(26) with positive integers d dividing 344 (i.e. d = 1, 2, 4, 8, 43, 86, 172, 344), and the factorization of Φd(26) for these positive integers d are:
| from | prime factorization |
|---|---|
| Φ1(26) | 52 |
| Φ2(26) | 33 |
| Φ4(26) | 677 |
| Φ8(26) | 17 × 26881 |
| Φ43(26) | (60-digit prime) |
| Φ86(26) | 681293 × (54-digit prime) |
| Φ172(26) | 173 × 66221 × 97942133 × 338286119038330712762413 × 290239124722842089063959709049053 × (48-digit prime) |
| Φ344(26) | 259295161 × 14470172263033 × (217-digit prime) |
For the number 26354−1, it is the product of Φd(26) with positive integers d dividing 354 (i.e. d = 1, 2, 3, 6, 59, 118, 177, 354), and the factorization of Φd(26) for these positive integers d are:
| from | prime factorization |
|---|---|
| Φ1(26) | 52 |
| Φ2(26) | 33 |
| Φ3(26) | 19 × 37 |
| Φ6(26) | 3 × 7 × 31 |
| Φ59(26) | 3541 × 334945708538658924935948356996883525107 × 10265667109489266992108219345733472151257 |
| Φ118(26) | 254250862891621 × (68-digit prime) |
| Φ177(26) | 47791 × 1311074895191091284466533625050044762267011115706300424823729 × (100-digit prime) |
| Φ354(26) | 709 × 16441898216641 × (149-digit prime) |
Although these numbers also have N−1 or N+1 is product of a Cunningham number and a small number, but since the corresponding Cunningham numbers are < 25% factored, and the partial factorizations of them are insufficient for any of the proving methods that could make use of them (unless 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 < 10100 (i.e. this number (N−1 or N+1) is product of a 10100-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) and is thus accessible by massive ECM (or P−1 or P+1) computations, or the product of the prime factors > 10100 of this number (N−1 or N+1) is < 10250 (i.e. this number (N−1 or N+1) is product of a 10100-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 number < 10250) and is thus accessible by combine of massive ECM (or P−1 or P+1) computations and GNFS, there is no chance for fully factoring this number (N−1 or N+1) using current publicly available factorization methods if the SNFS difficulty of this number (N−1 or N+1) is > 360, since the currently record of the SNFS factorization is difficulty 360 (which is the number 21193−1, see https://homes.cerias.purdue.edu/~ssw/cun/champ.txt) and the currently record of the GNFS factorization number has 250 decimal digits (which is the number RSA-250, see https://en.wikipedia.org/wiki/Integer_factorization_records#Numbers_of_a_general_form) and the currently record of the ECM prime factor has 83 decimal digits (which is a prime factor of 7337+1, see http://www.loria.fr/~zimmerma/records/top50.html) and the currently record of the P−1 prime factor has 66 decimal digits (which is a prime factor of 960119−1, see http://www.loria.fr/~zimmerma/records/Pminus1.html) and the currently record of the P+1 prime factor has 60 decimal digits (which is a prime factor of the Lucas number L2366, see http://www.loria.fr/~zimmerma/records/Pplus1.html), thus there is no change to prove their primality 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)), 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 certificate chain http://factordb.com/certchain.php?fid=1000000000043597217&action=all&fr=0&to=100 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 certificate chain http://factordb.com/certchain.php?fid=1100000000672342180&action=all&fr=0&to=100 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 certificate chain http://factordb.com/certchain.php?fid=1100000000599116446&action=all&fr=0&to=100 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 certificate chain http://factordb.com/certchain.php?fid=1100000000235993821&action=all&fr=0&to=100 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 (1049081−1)/9 (see https://www.mersenneforum.org/showthread.php?t=13435 and its factordb entry http://factordb.com/index.php?id=1100000000013937242&open=prime and its primality certificate http://factordb.com/cert.php?id=1100000000013937242 and its certificate chain http://factordb.com/certchain.php?fid=1100000000013937242&action=all&fr=0&to=100 and its helper file http://factordb.com/helper.php?id=1100000000013937242 and factorization status of its N−1 http://factordb.com/index.php?id=1100000000020361525&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=10&Exp=49080&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 certificate chain http://factordb.com/certchain.php?fid=1100000000213085670&action=all&fr=0&to=100 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=) and (130965953−1)/13095 (see https://web.archive.org/web/20131020160719/http://www.primes.viner-steward.org/andy/E/33281741.html and its factordb entry http://factordb.com/index.php?id=1100000000439222497&open=prime and its helper file http://factordb.com/helper.php?id=1100000000439222497 and factorization status of its N−1 http://factordb.com/index.php?id=1100000000489615397&open=ecm and http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13096&Exp=5952&c0=-&EN=&LM= (this prime is still "PRP" in factordb and thus has no primality certificate and certificate chain since its N−1 is only 28.057% factored and thus need to use CHG proof, however, factordb lacks the ability to verify CHG proofs, see https://www.mersenneforum.org/showpost.php?p=608362&postcount=165)) and 31681130+3445781+1 (see http://csic.som.emory.edu/~lzhou/blogs/?p=717 (although this page misses the Aurifeuillean factorization of Φ6×n(3) = Φ(6×n)L(3) × Φ(6×n)M(3) for odd n) and its factordb entry http://factordb.com/index.php?id=1100000003878235677&open=prime and factorization status of its N−1 http://factordb.com/index.php?id=1100000006004342031&open=ecm (this prime is "U" in factordb and thus has no primality certificate and certificate chain and helper file since this prime is too large (>10199999) to be PRP-tested in factordb, also there is no factorization status of its N−1 in http://myfactors.mooo.com/ (i.e. there is no factorization status of its 31235349+1 in http://myfactors.mooo.com/) since 31235349+1 is beyond the table limit in http://myfactors.mooo.com/, the table limit for bn±1 in http://myfactors.mooo.com/ is (only consider non-perfect power bases b) is n ≤ 2000000 for base b = 2, n ≤ 600000 for base b = 3, n ≤ 400000 for bases 5 ≤ b ≤ 7, n ≤ 300000 for base b = 10, n ≤ 100000 for bases 11 ≤ b ≤ 99, n ≤ 10000 for bases 101 ≤ b ≤ 9999, n ≤ 500 for bases 10001 ≤ b ≤ 20000, n ≤ 300 for bases 20001 ≤ b ≤ 1100000, see http://myfactorcollection.mooo.com:8090/news.html and http://myfactorcollection.mooo.com:8090/interactive.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 "-" or "+" in the "note" column), 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 the Cunningham number 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 the Cunningham number 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 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 the Cunningham number 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 10391st minimal prime in base 17, 1F7092, N−1 is 31×R7092(17), thus factor N−1 is equivalent to factor the Cunningham number 177092−1, N−1 is only 7.085% factored (see http://factordb.com/index.php?id=1100000000840355927&open=prime), and for the algebraic factors of 177092−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=17&Exp=7092&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 177092−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=17&Exp=7092&c0=-&EN=&LM=
- the 25240th minimal prime in base 26, 518854P, N+1 is 130×R1886(26), thus factor N+1 is equivalent to factor the Cunningham number 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 the Cunningham number 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 10391st minimal prime in base 17 (1F7092) in factordb: http://factordb.com/helper.php?id=1100000000840355927
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 10391st minimal prime in base 17 (1F7092) in factordb: http://factordb.com/index.php?id=1100000000840355928&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 prime factor (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 each algebraic factor) 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://factordb.com/listecm.php?c=4, http://www.loria.fr/~zimmerma/ecmnet/, http://www.loria.fr/~zimmerma/records/ecmnet.html, http://www.loria.fr/~zimmerma/records/factor.html, http://www.loria.fr/~zimmerma/records/top50.html, http://www.loria.fr/~zimmerma/records/ecm/params.html, https://oeis.org/wiki/OEIS_sequences_needing_factors#ECM_efforts, https://stdkmd.net/nrr/records.htm#largefactorecm, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "Elliptic curve method:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=ecm&maxrows=10000, http://maths-people.anu.edu.au/~brent/factors.html, http://maths-people.anu.edu.au/~brent/ftp/champs.txt, https://www.alpertron.com.ar/ECM.HTM, https://www.alpertron.com.ar/ECMREC.HTM, https://homes.cerias.purdue.edu/~ssw/cun/press/tech.html, https://homes.cerias.purdue.edu/~ssw/cun/press/nontech.html, https://www.mersenne.org/report_ECM/, https://www.mersenne.ca/userfactors/ecm/1, https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, https://kurtbeschorner.de/ecm-efforts.htm, http://www.rechenkraft.net/yoyo//y_factors_ecm.php, http://www.rechenkraft.net/yoyo/y_status_ecm.php, http://www.wraithx.net/math/ecmprobs/ecmprobs.html, http://www.loria.fr/~zimmerma/papers/ecm-entry.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_460.pdf), https://arxiv.org/pdf/1004.3366.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_492.pdf)), "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, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "Pollard p-1:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=p-1&maxrows=10000, http://www.numericana.com/answer/factoring.htm#p-1, http://factordb.com/listecm.php?c=2, http://www.loria.fr/~zimmerma/records/Pminus1.html, https://web.archive.org/web/20021015212913/http://www.users.globalnet.co.uk/~aads/Pminus1.html, https://web.archive.org/web/20231002022529/https://colin.barker.pagesperso-orange.fr/lpa/big_pm1.htm, https://www.mersenne.org/report_pminus1/, https://www.mersenne.ca/userfactors/pm1/1, https://www.mersenne.ca/smooth.php, https://www.mersenne.ca/p1missed.php, https://www.mersenne.ca/prob.php), "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, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "p+1:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=p%2b1&maxrows=10000, http://www.numericana.com/answer/factoring.htm#p+1, http://factordb.com/listecm.php?c=3, http://www.loria.fr/~zimmerma/records/Pplus1.html, https://www.mersenne.org/report_pplus1/, https://www.mersenne.ca/userfactors/pp1/1, https://www.mersenne.ca/pplus1.php), "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, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (sections "Special number field sieve by size of number factored:" and "Special number field sieve by SNFS difficulty:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=snfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#smallpolynomial, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/), "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, https://homes.cerias.purdue.edu/~ssw/cun/champ.txt (section "General number field sieve by size of number factored:"), http://mklasson.com/factors/viewlog.php?sort=2&order=desc&method=gnfs&maxrows=10000, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs, https://www.mersenne.ca/userfactors/nfs/1, http://escatter11.fullerton.edu/nfs/))
For the number 32319+1, it is the product of Φd(3) with positive integers d dividing 4638 but not dividing 2319 (i.e. d = 2, 6, 1546, 4638), and the factorization of Φd(3) for these positive integers d are: (since 6 and 4638 are == 6 mod 12, thus for these two positive integers d, Φd(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, https://web.archive.org/web/20231002141924/http://colin.barker.pagesperso-orange.fr/lpa/cycl_fac.htm, http://list.seqfan.eu/oldermail/seqfan/2017-March/017363.html, http://myfactorcollection.mooo.com:8090/source/cyclo.cpp, http://myfactorcollection.mooo.com:8090/LCD_2_199, http://myfactorcollection.mooo.com:8090/LCD_2_998, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/source/cyclo.cpp, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/LucasCD/LCD_2_199, https://raw.githubusercontent.com/JonathanCrombie/Cowcave/main/website/LucasCD/LCD_2_998, https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean, https://www.unshlump.com/hcn/aurif.html, https://www.mersenneforum.org/showthread.php?t=10439, https://www.mersenneforum.org/showpost.php?p=515828&postcount=8, https://maths-people.anu.edu.au/~brent/pd/rpb135.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_97.pdf), https://www.ams.org/journals/mcom/2006-75-253/S0025-5718-05-01766-7/S0025-5718-05-01766-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_138.pdf), https://maths-people.anu.edu.au/~brent/pd/rpb127.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_164.pdf), https://www.jams.jp/scm/contents/Vol-2-3/2-3-16.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_167.pdf), https://web.archive.org/web/20130702000532/http://xyyxf.at.tut.by/aurifeuillean.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_443.pdf)), and ΦdL(3) and ΦdM(3) are their Aurifeuillean L and M factors, respectively)
| from | currently known prime factorization |
|---|---|
| Φ2(3) | 22 |
| Φ6L(3) | 1 (empty product (https://en.wikipedia.org/wiki/Empty_product)) |
| Φ6M(3) | 7 |
| Φ1546(3) | 1182691 × 454333843 × 7175619780295897339 × 219067434459114063477547 × 650663511671253931884619 × (288-digit composite with no known proper factor, SNFS difficulty is 369.292, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=3&Exp=773&c0=%2B&LM=&SA=, this composite has already checked with P−1 to B1 = 50000 and 3 times P+1 to B1 = 150000 and 10 times ECM to B1 = 250000 (these can be checked for composites < 10300), see http://factordb.com/sequences.php?se=1&aq=%283%5E773%2B1%29%2F4&action=all&fr=0&to=100, the "Check for factors" box shows "Already checked", this is the ECM effort t-level (https://oeis.org/wiki/OEIS_sequences_needing_factors#T-levels, https://stdkmd.net/nrr/wanted.htm (the "ECM" column of the three tables), https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, http://myfactorcollection.mooo.com:8090/lists.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the lists in the boxes), http://myfactorcollection.mooo.com:8090/dbio.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the DB inputs/outputs in the boxes), https://github.com/brubsby/t-level, https://www.mersenneforum.org/showthread.php?t=29615) t30 (see http://www.loria.fr/~zimmerma/records/ecm/params.html and https://www.rieselprime.de/ziki/Elliptic_curve_method#Choosing_the_best_parameters_for_ECM), i.e. the prime factors of this composite number are probably > 1030) |
| Φ4638L(3) | 18553 × 2957658597967379799686737984695290731543 × (325-digit composite with no known proper factor, SNFS difficulty is 369.292, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=3&Exp=2319&c0=%2B&LM=L&SA=) |
| Φ4638M(3) | 4639 × 6716055901 × (356-digit composite with no known proper factor, SNFS difficulty is 369.292, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=3&Exp=2319&c0=%2B&LM=M&SA=) |
For the number 131504−1, it is the product of Φd(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 Φd(13) for these positive integers d are:
| from | currently known prime factorization |
|---|---|
| Φ1(13) | 22 × 3 |
| Φ2(13) | 2 × 7 |
| Φ4(13) | 2 × 5 × 17 |
| Φ8(13) | 2 × 14281 |
| Φ16(13) | 2 × 407865361 |
| Φ32(13) | 2 × 2657 × 441281 × 283763713 |
| Φ47(13) | 183959 × 19216136497 × 534280344481909234853671069326391741 |
| Φ94(13) | 498851139881 × 3245178229485124818467952891417691434077 |
| Φ188(13) | 36097 × 75389 × 99886248944632632917 × (74-digit prime) |
| Φ376(13) | 41737 × 553784729353 × 188172028979257 × 398225319299696783138113 × 7663511503164270157006126605793 × 8935170451146532986983277856738508374630999814576686938913 × (62-digit prime) |
| Φ752(13) | 13537 × 1232912541076129 × (391-digit composite with no known proper factor, SNFS difficulty is 421.071, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=13&Exp=376&c0=%2B&LM=&SA=) |
| Φ1504(13) | 4513 × 9426289921 × (807-digit composite with no known proper factor, SNFS difficulty is 837.685, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=13&Exp=752&c0=%2B&LM=&SA=) |
For the number 163543+1 = 214172+1, it is the product of Φd(2) with positive integers d dividing 28344 but not dividing 14172 (i.e. d = 8, 24, 9448, 28344), and the factorization of Φd(2) for these positive integers d are:
| from | currently known prime factorization |
|---|---|
| Φ8(2) | 17 |
| Φ24(2) | 241 |
| Φ9448(2) | 107083633 × 7076306353 × 2428629073416562046689 × (1382-digit composite with no known proper factor, SNFS difficulty is 1422.066, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=2&Exp=4724&c0=%2B&LM=&SA=) |
| Φ28344(2) | 265073089 × (2834-digit composite with no known proper factor, SNFS difficulty is too large to handle for the script, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=2&Exp=14172&c0=%2B&LM=&SA=) |
For the number 177092−1, it is the product of Φd(17) with positive integers d dividing 7092 (i.e. d = 1, 2, 3, 4, 6, 9, 12, 18, 36, 197, 394, 591, 788, 1182, 1773, 2364, 3546, 7092), and the factorization of Φd(17) for these positive integers d are:
| from | currently known prime factorization |
|---|---|
| Φ1(17) | 24 |
| Φ2(17) | 2 × 32 |
| Φ3(17) | 307 |
| Φ4(17) | 2 × 5 × 29 |
| Φ6(17) | 3 × 7 × 13 |
| Φ9(17) | 19 × 1270657 |
| Φ12(17) | 83233 |
| Φ18(17) | 3 × 1423 × 5653 |
| Φ36(17) | 37 × 109 × 181 × 2089 × 382069 |
| Φ197(17) | 646477768184104922935115731396719622668746018369021 × (191-digit prime) |
| Φ394(17) | 1720812337 × 120652139803422836046398107883 × 11854861245452004511262968204651829313761 × 930821833870171289422620828584179333038475130149 × (115-digit prime) |
| Φ591(17) | 150824383 × (475-digit composite with no known proper factor, SNFS difficulty is 487.258, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=17&Exp=591&c0=-&LM=&SA=) |
| Φ788(17) | (483-digit composite with no known proper factor, SNFS difficulty is 487.258, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=17&Exp=394&c0=%2B&LM=&SA=) |
| Φ1182(17) | 3547 × 1924297 × (473-digit composite with no known proper factor, SNFS difficulty is 487.258, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=17&Exp=591&c0=%2B&LM=&SA=) |
| Φ1773(17) | 99289 × (1443-digit composite with no known proper factor, SNFS difficulty is too large to handle for the script, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=17&Exp=1773&c0=-&LM=&SA=) |
| Φ2364(17) | 3557821 × (959-digit composite with no known proper factor, SNFS difficulty is 969.594, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=17&Exp=1182&c0=%2B&LM=&SA=) |
| Φ3546(17) | 420878287 × 5406628753 × 7195614121 × 32800804957 × (1409-digit composite with no known proper factor, SNFS difficulty is too large to handle for the script, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=17&Exp=1773&c0=%2B&LM=&SA=) |
| Φ7092(17) | 21277 × 1560241 × 2654148561193 × (2872-digit composite with no known proper factor, SNFS difficulty is too large to handle for the script, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=17&Exp=3546&c0=%2B&LM=&SA=) |
For the number 261886−1, it is the product of Φd(26) with positive integers d dividing 1886 (i.e. d = 1, 2, 23, 41, 46, 82, 943, 1886), and the factorization of Φd(26) for these positive integers d are:
| from | currently known prime factorization |
|---|---|
| Φ1(26) | 52 |
| Φ2(26) | 33 |
| Φ23(26) | 13709 × 1086199 × 1528507873 × 615551139461 |
| Φ41(26) | 83 × 2633923 × (49-digit prime) |
| Φ46(26) | 47 × 1157729 × 378673381 × 629584013567417 |
| Φ82(26) | 9677 × 1532581 × (47-digit prime) |
| Φ943(26) | 384118835398327 × (1231-digit composite with no known proper factor, SNFS difficulty is 1334.320, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=26&Exp=943&c0=-&LM=&SA=) |
| Φ1886(26) | (1246-digit composite with no known proper factor, SNFS difficulty is 1334.320, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=26&Exp=943&c0=%2B&LM=&SA=) |
For the number 363933−1 = 67866−1, it is the product of Φd(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 Φd(6) for these positive integers d are:
| from | currently known prime factorization |
|---|---|
| Φ1(6) | 5 |
| Φ2(6) | 7 |
| Φ3(6) | 43 |
| Φ6(6) | 31 |
| Φ9(6) | 19 × 2467 |
| Φ18(6) | 46441 |
| Φ19(6) | 191 × 638073026189 |
| Φ23(6) | 47 × 139 × 3221 × 7505944891 |
| Φ38(6) | 1787 × 48713705333 |
| Φ46(6) | 113958101 × 990000731 |
| Φ57(6) | 47881 × 820459 × 219815829325921729 |
| Φ69(6) | 11731 × 1236385853432057889667843739281 |
| Φ114(6) | 457 × 137713 × 190324492938225748951 |
| Φ138(6) | 24648570768391 × 816214079084081564521 |
| Φ171(6) | 19 × 25896916098621777025320461067950269867 × (46-digit prime) |
| Φ207(6) | 399097 × (98-digit prime) |
| Φ342(6) | 62174327387790051073 × (65-digit prime) |
| Φ414(6) | 4811469913 × 61040960263 × 25280883279243199352415750302719 × (51-digit prime) |
| Φ437(6) | 989723472495640900314985156529340457 × (273-digit composite with no known proper factor, SNFS difficulty is 340.830, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=6&Exp=437&c0=-&LM=&SA=, this composite has already checked with P−1 to B1 = 50000 and 3 times P+1 to B1 = 150000 and 10 times ECM to B1 = 250000 (these can be checked for composites < 10300), see http://factordb.com/sequences.php?se=1&aq=%286%5E437-1%29*5%2F%286%5E23-1%29%2F%286%5E19-1%29&action=all&fr=0&to=100, the "Check for factors" box shows "Already checked", this is the ECM effort t-level (https://oeis.org/wiki/OEIS_sequences_needing_factors#T-levels, https://stdkmd.net/nrr/wanted.htm (the "ECM" column of the three tables), https://stdkmd.net/nrr/c.cgi?q=37771_259#ecm, https://stdkmd.net/nrr/c.cgi?q=23333_233#ecm, http://myfactorcollection.mooo.com:8090/lists.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the lists in the boxes), http://myfactorcollection.mooo.com:8090/dbio.html (the labels "Brent Format with t-level" and "Wagstaff Format with t-level" of the DB inputs/outputs in the boxes), https://github.com/brubsby/t-level, https://www.mersenneforum.org/showthread.php?t=29615) t30 (see http://www.loria.fr/~zimmerma/records/ecm/params.html and https://www.rieselprime.de/ziki/Elliptic_curve_method#Choosing_the_best_parameters_for_ECM), i.e. the prime factors of this composite number are probably > 1030) |
| Φ874(6) | (309-digit prime, for its ECPP primality certificate see http://factordb.com/cert.php?id=1100000000019287760, and for its certificate chain see http://factordb.com/certchain.php?fid=1100000000019287760&action=all&fr=0&to=100) |
| Φ1311(6) | 100745107 × 1719861571 × 2376829061449 × (587-digit composite with no known proper factor, SNFS difficulty is 681.660, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=6&Exp=1311&c0=-&LM=&SA=) |
| Φ2622(6) | 41953 × 266030354191322260711 × (592-digit composite with no known proper factor, SNFS difficulty is 681.660, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=6&Exp=1311&c0=%2B&LM=&SA=) |
| Φ3933(6) | 7867 × (1845-digit composite with no known proper factor, SNFS difficulty is too large to handle for the script, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=6&Exp=3933&c0=-&LM=&SA=) |
| Φ7866(6) | (1849-digit composite with no known proper factor, SNFS difficulty is too large to handle for the script, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=6&Exp=3933&c0=%2B&LM=&SA=) |
For the files in this page:
- File "certificate b n": The primality certificate for the 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.