From 3a15792c24573405c63abf1e72f57549bebbb93a Mon Sep 17 00:00:00 2001 From: xayahrainie4793 <66528374+xayahrainie4793@users.noreply.github.com> Date: Sat, 15 Jun 2024 14:03:04 +0800 Subject: [PATCH] Update README.md --- primality-certificates/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/primality-certificates/README.md b/primality-certificates/README.md index 56dcb428..3c7014d9 100644 --- a/primality-certificates/README.md +++ b/primality-certificates/README.md @@ -75,7 +75,7 @@ Also the case where *N*−1 or *N*+1 is product of a Cunningham number (of the f (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 *Φ*4807(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 *Φ*5014(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 *Φ*7884(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 *Φ*1283(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 *Φ*2907(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 *Φ*11470(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 × *R*4807(10) (which is equivalent to the Cunningham number 104807−1) and *Φ*4807(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 × *R*10028(10) (which is equivalent to the Cunningham number 1010028−1) and *Φ*5014(10) is an algebraic factor of the Cunningham number 1010028−1, the *N*+1 of "37773_15768" is 34 × *R*15768(10) (which is equivalent to the Cunningham number 1015768−1) and *Φ*7884(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 × *R*5132(10) (which is equivalent to the Cunningham number 105132−1) and *Φ*1283(10) is an algebraic factor of the Cunningham number 105132−1, the *N*−1 of "79921_11629" is 720 × *R*11628(10) (which is equivalent to the Cunningham number 1011628−1) and *Φ*2907(10) is an algebraic factor of the Cunningham number 1011628−1, the *N*−1 of "80081_5736" is 80 × *S*5735(10) (which is equivalent to the Cunningham number 105735+1) and *Φ*11470(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 *b**n*±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 2*n*−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/Repunit100-20240610.pdf?t=1718031438 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_480.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*, *b**n* < 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/May31_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 *b**n*±1 with 2 ≤ *b* ≤ 400 and 1 ≤ *n* ≤ 400 and for the first holes of *b**n*±1 with 2 ≤ *b* ≤ 400 see the links in the list below) +(for the references of factorization of *b**n*±1, see: https://homes.cerias.purdue.edu/~ssw/cun/index.html (2 ≤ *b* ≤ 12), https://homes.cerias.purdue.edu/~ssw/cun/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 2*n*−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*, *b**n* < 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/May31_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 *b**n*±1 with 2 ≤ *b* ≤ 400 and 1 ≤ *n* ≤ 400 and for the first holes of *b**n*±1 with 2 ≤ *b* ≤ 400 see the links in the list below) |range of bases *b*|the factors of *b**n*±1 with 1 ≤ *n* ≤ 100|the factors of *b**n*±1 with 101 ≤ *n* ≤ 200|the factors of *b**n*±1 with 201 ≤ *n* ≤ 300|the factors of *b**n*±1 with 301 ≤ *n* ≤ 400|the first holes of *b**n*±1 and their known prime factors| |---|---|---|---|---|---|