minimal-elements-of-the-prime-numbers/primality-certificates at main · xayahrainie4793/minimal-elements-of-the-prime-numbers

These are the Primo (http://www.ellipsa.eu/public/primo/primo.html, http://www.rieselprime.de/dl/Primo309.zip, https://primes.utm.edu/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://primes.utm.edu/prove/prove4_2.html, https://en.wikipedia.org/wiki/Elliptic_curve_primality, https://primes.utm.edu/glossary/xpage/ECPP.html, https://mathworld.wolfram.com/EllipticCurvePrimalityProving.html, https://primes.utm.edu/top20/page.php?id=27) implementation) primality certificates (https://en.wikipedia.org/wiki/Primality_certificate, https://primes.utm.edu/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/, http://factordb.com/certoverview.php) for the minimal primes > 10²⁹⁹ and < 10²⁵⁰⁰⁰ (primes < 10²⁹⁹ are automatically proven primes in factordb) in bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 (the solved or near-solved bases, i.e. the bases b with ≤ 6 unsolved families)

The large minimal primes in base b are of the form (a×bⁿ+c)/gcd(a+c,b−1) for some a, b, c, n such that a ≥ 1, b ≥ 2 (b is the base), c ≠ 0, gcd(a,c) = 1, gcd(b,c) = 1, the large numbers (i.e. the numbers with large n, say n > 1000) can be easily proven primes using N−1 test (https://primes.utm.edu/prove/prove3_1.html, http://factordb.com/nmoverview.php?method=1) or N+1 test (https://primes.utm.edu/prove/prove3_2.html, http://factordb.com/nmoverview.php?method=2) if and only if c = ±1 and gcd(a+c,b−1) = 1, except this special case of c = ±1 and gcd(a+c,b−1) = 1, such numbers need primality certificates to be proven primes (otherwise, they will only be probable primes (https://en.wikipedia.org/wiki/Probable_prime, https://primes.utm.edu/glossary/xpage/PRP.html, https://www.rieselprime.de/ziki/Probable_prime, https://mathworld.wolfram.com/ProbablePrime.html, https://stdkmd.net/nrr/records.htm#probableprimenumbers, https://stdkmd.net/nrr/repunit/prpfactors.htm, http://factordb.com/listtype.php?t=1)), and elliptic curve primality proving are used for these numbers.

Primes which either N−1 or N+1 is trivially fully factored (i.e. primes of the form k×bⁿ±1, with small k) do not need primality certificates, since they can be easily proven primes using N−1 test (https://primes.utm.edu/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=1) or N+1 test (https://primes.utm.edu/prove/prove3_2.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, http://factordb.com/nmoverview.php?method=2), these primes are: (i.e. their N−1 or N+1 are smooth numbers (https://en.wikipedia.org/wiki/Smooth_number, https://mathworld.wolfram.com/SmoothNumber.html))

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
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
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
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
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
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
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
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
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
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
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
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
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

(these primes can be proven prime using Yves Gallot's Proth.exe (https://primes.utm.edu/programs/gallot/, https://primes.utm.edu/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://primes.utm.edu/bios/page.php?id=431, https://www.rieselprime.de/ziki/LLR, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403win64, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/llr403linux64), see the README file for LLR (https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/llr403win64/Readme.txt, https://github.com/xayahrainie4793/prime-programs-cached-copy/blob/main/llr403linux64/Readme.txt, http://jpenne.free.fr/index2.html))

Also, there are no primality certificates for these primes in factordb because although they are > 10²⁹⁹, but their N−1 or N+1 is fully factored (but not trivially fully factored, however, only need trial division (https://en.wikipedia.org/wiki/Trial_division, https://primes.utm.edu/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) to 10⁸) and the largest prime factor is < 10²⁹⁹ (primes < 10²⁹⁹ are automatically proven primes in factordb): (i.e. their N−1 or N+1 are product of a 10⁸-smooth number (https://en.wikipedia.org/wiki/Smooth_number, https://mathworld.wolfram.com/SmoothNumber.html) and a prime < 10²⁹⁹)

the 2328th minimal prime in base 16, 880₂₄₆7, with 300 decimal digits, N−1 is 2³ × 3 × 7 × 13 × 25703261 × (289-digit prime)
the 25174th minimal prime in base 26, OL0₂₁₄M9, with 309 decimal digits, N−1 is 2² × 5² × 7 × 223 × 42849349 × (296-digit prime)
the 25485th minimal prime in base 28, JN₂₀₆, with 300 decimal digits, N−1 is 2 × 1061 × 1171 × 74311 × (289-digit prime)

The helper file for the 2328th minimal prime in base 16 (880₂₄₆7) in factordb: http://factordb.com/helper.php?id=1100000002468140199

The helper file for the 25174th minimal prime in base 26 (OL0₂₁₄M9) in factordb: http://factordb.com/helper.php?id=1100000000840631576

The helper file for the 25485th minimal prime in base 28 (JN₂₀₆) in factordb: http://factordb.com/helper.php?id=1100000002611724435

Factorization of N−1 for the 2328th minimal prime in base 16 (880₂₄₆7) in factordb: http://factordb.com/index.php?id=1100000002468140641

Factorization of N−1 for the 25174th minimal prime in base 26 (OL0₂₁₄M9) in factordb: http://factordb.com/index.php?id=1100000000840631577

Factorization of N−1 for the 25485th minimal prime in base 28 (JN₂₀₆) in factordb: http://factordb.com/index.php?id=1100000002611724440

(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)

The Cunningham numbers bⁿ±1 has algebraic factorization to product of Φ_d(b) with positive integers d dividing n (the bⁿ−1 case) (see https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization) or positive integers d dividing 2×n but not dividing n (the bⁿ+1 case) (see https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization), where Φ is the cyclotomic polynomial (https://en.wikipedia.org/wiki/Cyclotomic_polynomial, https://mathworld.wolfram.com/CyclotomicPolynomial.html, http://www.numericana.com/answer/polynomial.htm#cyclotomic, https://stdkmd.net/nrr/repunit/repunitnote.htm#cyclotomic) (see https://stdkmd.net/nrr/repunit/repunitnote.htm and https://doi.org/10.1090/conm/022, https://www.mersenneforum.org/attachment.php?attachmentid=7727&d=1330555980 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_7.pdf) and https://homes.cerias.purdue.edu/~ssw/cun/mine.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_6.pdf) and http://homes.cerias.purdue.edu/~ssw/cun1.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_71.pdf))

(below, "R_n(b)" means the repunit (https://en.wikipedia.org/wiki/Repunit, https://primes.utm.edu/glossary/xpage/Repunit.html, https://primes.utm.edu/glossary/xpage/GeneralizedRepunitPrime.html, https://www.rieselprime.de/ziki/Repunit, https://mathworld.wolfram.com/Repunit.html, https://mathworld.wolfram.com/RepunitPrime.html, https://pzktupel.de/Primetables/TableRepunit.php, https://pzktupel.de/Primetables/TableRepunitGen.php, https://www.numbersaplenty.com/set/repunit/, https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit, https://web.archive.org/web/20021001222643/http://www.users.globalnet.co.uk/~aads/index.html, https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html, https://web.archive.org/web/20021114005730/http://www.users.globalnet.co.uk/~aads/titans.html, https://web.archive.org/web/20021015210104/http://www.users.globalnet.co.uk/~aads/faclist.html, http://www.elektrosoft.it/matematica/repunit/repunit.htm, http://www.fermatquotient.com/PrimSerien/GenRepu.txt (in German), http://www.fermatquotient.com/PrimSerien/GenRepuP.txt (in German), http://www.primenumbers.net/Henri/us/MersFermus.htm, http://www.bitman.name/math/article/380/231/, http://www.bitman.name/math/table/379, http://www.bitman.name/math/table/488, https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185243-9/S0025-5718-1993-1185243-9.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_4.pdf), https://cs.uwaterloo.ca/journals/JIS/VOL3/DUBNER/dubner.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_5.pdf), https://www.ams.org/journals/mcom/1979-33-148/S0025-5718-1979-0537980-7/S0025-5718-1979-0537980-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_119.pdf), https://primes.utm.edu/top20/page.php?id=57, https://primes.utm.edu/top20/page.php?id=16, https://oeis.org/A002275) in base b with length n, i.e. R_n(b) = (bⁿ−1)/(b−1) (see https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization), "S_n(b)" means bⁿ+1 (see https://stdkmd.net/nrr/repunit/repunitnote.htm#repunit_factorization), the special cases of R_n(10) and S_n(10) are in https://stdkmd.net/nrr/repunit/ and https://stdkmd.net/nrr/repunit/10001.htm, respectively, in fact, R_n(b) and S_n(b) are 111...111 and 1000...0001 in base b, respectively)

the 3168th minimal prime in base 13, 9₃₀₈1, N−1 is 117×R₃₀₈(13), thus factor N−1 is equivalent to factor 13³⁰⁸−1, and for the algebraic factors of 13³⁰⁸−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&Exp=308&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 13³⁰⁸−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN=
the 3179th minimal prime in base 13, B₅₆₃C, N−1 is 11×R₅₆₄(13), thus factor N−1 is equivalent to factor 13⁵⁶⁴−1, and for the algebraic factors of 13⁵⁶⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&Exp=564&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 13⁵⁶⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN=
the 3180th minimal prime in base 13, 1B₅₇₆, N−1 is 23×R₅₇₆(13), thus factor N−1 is equivalent to factor 13⁵⁷⁶−1, and for the algebraic factors of 13⁵⁷⁶−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&Exp=576&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 13⁵⁷⁶−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN=
the 25199th minimal prime in base 26, 9K₃₄₃AP, N+1 is 6370×R₃₄₄(26), thus factor N+1 is equivalent to factor 26³⁴⁴−1, and for the algebraic factors of 26³⁴⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=26&Exp=344&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 26³⁴⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&Exp=344&c0=-&EN=
the 25200th minimal prime in base 26, 8₃₅₄1, N−1 is 208×R₃₅₄(26), thus factor N−1 is equivalent to factor 26³⁵⁴−1, and for the algebraic factors of 26³⁵⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=26&Exp=354&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 26³⁵⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&Exp=354&c0=-&EN=

The helper file for the 3168th minimal prime in base 13 (9₃₀₈1) in factordb: http://factordb.com/helper.php?id=1100000000840126705

The helper file for the 3179th minimal prime in base 13 (B₅₆₃C) in factordb: http://factordb.com/helper.php?id=1100000000000217927

The helper file for the 3180th minimal prime in base 13 (1B₅₇₆) in factordb: http://factordb.com/helper.php?id=1100000002321021456

The helper file for the 25199th minimal prime in base 26 (9K₃₄₃AP) in factordb: http://factordb.com/helper.php?id=1100000000840632228

The helper file for the 25200th minimal prime in base 26 (8₃₅₄1) in factordb: http://factordb.com/helper.php?id=1100000000840632517

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N−1 for the 3168th minimal prime in base 13 (9₃₀₈1) in factordb: http://factordb.com/index.php?id=1100000000840126706&open=ecm

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N−1 for the 3179th minimal prime in base 13 (B₅₆₃C) in factordb: http://factordb.com/index.php?id=1100000000271764311&open=ecm

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N−1 for the 3180th minimal prime in base 13 (1B₅₇₆) in factordb: http://factordb.com/index.php?id=1100000002321021531&open=ecm

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N+1 for the 25199th minimal prime in base 26 (9K₃₄₃AP) in factordb: http://factordb.com/index.php?id=1100000000840632232&open=ecm

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N−1 for the 25200th minimal prime in base 26 (8₃₅₄1) in factordb: http://factordb.com/index.php?id=1100000000840632623&open=ecm

(in the tables below, Φ is the cyclotomic polynomial (https://en.wikipedia.org/wiki/Cyclotomic_polynomial, https://mathworld.wolfram.com/CyclotomicPolynomial.html, http://www.numericana.com/answer/polynomial.htm#cyclotomic, https://stdkmd.net/nrr/repunit/repunitnote.htm#cyclotomic))

(for the prime factors > 10²⁴ (other than the ultimate one) in the tables below, "ECM" means the elliptic-curve factorization method (https://en.wikipedia.org/wiki/Lenstra_elliptic-curve_factorization, https://www.rieselprime.de/ziki/Elliptic_curve_method, https://mathworld.wolfram.com/EllipticCurveFactorizationMethod.html, http://www.numericana.com/answer/factoring.htm#ecm, http://factordb.com/listecm.php?c=1, http://www.loria.fr/~zimmerma/ecmnet/, http://www.loria.fr/~zimmerma/records/ecmnet.html, http://www.loria.fr/~zimmerma/records/factor.html, http://www.loria.fr/~zimmerma/records/top50.html, https://stdkmd.net/nrr/records.htm#largefactorecm, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=ecm&maxrows=100, http://maths-people.anu.edu.au/~brent/factors.html, http://maths-people.anu.edu.au/~brent/ftp/champs.txt, https://homes.cerias.purdue.edu/~ssw/cun/press/tech.html, https://homes.cerias.purdue.edu/~ssw/cun/press/nontech.html, https://kurtbeschorner.de/ecm-efforts.htm, http://www.rechenkraft.net/yoyo//y_factors_ecm.php, http://www.rechenkraft.net/yoyo/y_status_ecm.php, https://www.alpertron.com.ar/ECM.HTM), "P−1" means the Pollard P−1 method (https://en.wikipedia.org/wiki/Pollard%27s_p_%E2%88%92_1_algorithm, https://www.rieselprime.de/ziki/P-1_factorization_method, https://mathworld.wolfram.com/Pollardp-1FactorizationMethod.html, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=p-1&maxrows=100, http://www.numericana.com/answer/factoring.htm#p-1, http://factordb.com/listecm.php?c=2), "P+1" means the Williams P+1 method (https://en.wikipedia.org/wiki/Williams%27s_p_%2B_1_algorithm, https://www.rieselprime.de/ziki/P%2B1_factorization_method, https://mathworld.wolfram.com/WilliamspPlus1FactorizationMethod.html, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=p%2b1&maxrows=100, http://www.numericana.com/answer/factoring.htm#p+1, http://factordb.com/listecm.php?c=3), "SNFS" means the special number field sieve (https://en.wikipedia.org/wiki/Special_number_field_sieve, https://www.rieselprime.de/ziki/Special_number_field_sieve, https://www.rieselprime.de/ziki/SNFS_polynomial_selection, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGSNFS, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=snfs&maxrows=100, https://stdkmd.net/nrr/wanted.htm#smallpolynomial), "GNFS" means the general number field sieve (https://en.wikipedia.org/wiki/General_number_field_sieve, https://www.rieselprime.de/ziki/General_number_field_sieve, https://mathworld.wolfram.com/NumberFieldSieve.html, https://stdkmd.net/nrr/records.htm#BIGGNFS, http://mklasson.com/factors/viewlog.php?hide_algebraic=true&sort=2&order=desc&method=gnfs&maxrows=100, https://stdkmd.net/nrr/wanted.htm#suitableforgnfs))

For the number 13³⁰⁸−1, it is the product of Φ_d(13) with positive integers d dividing 308 (i.e. d = 1, 2, 4, 7, 11, 14, 22, 28, 44, 77, 154, 308), and the factorization of Φ_d(13) for these positive integers d are:

from	prime factorization
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 Φ_d(13) with positive integers d dividing 564 (i.e. d = 1, 2, 3, 4, 6, 12, 47, 94, 141, 188, 282, 564), and the factorization of Φ_d(13) for these positive integers d are:

from	prime factorization
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, B1 = 3000000, Sigma = 2146847123) × 27969827431131578608318126024627616357147784803797 (GNFS) × (98-digit prime)

For the number 13⁵⁷⁶−1, it is the product of Φ_d(13) with positive integers d dividing 576 (i.e. d = 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, 288, 576), and the factorization of Φ_d(13) for these positive integers d are:

from	prime factorization
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 26³⁴⁴−1, it is the product of Φ_d(26) with positive integers d dividing 344 (i.e. d = 1, 2, 4, 8, 43, 86, 172, 344), and the factorization of Φ_d(26) for these positive integers d are:

from	prime factorization
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 Φ_d(26) with positive integers d dividing 354 (i.e. d = 1, 2, 3, 6, 59, 118, 177, 354), and the factorization of Φ_d(26) for these positive integers d are:

from	prime factorization
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, these numbers still need primality certificates:

the 151st minimal prime in base 9, 30₁₁₅₈11, N−1 is 9×S₂₃₁₉(3), thus factor N−1 is equivalent to factor 3²³¹⁹+1, N−1 is only 12.693% factored (see http://factordb.com/index.php?id=1100000002376318423&open=prime), and for the algebraic factors of 3²³¹⁹+1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=3&Exp=2319&LBIDPMList=B&LBIDLODList=D, and for the prime factorization of 3²³¹⁹+1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=3&Exp=2319&c0=%2B&EN=
the 3187th minimal prime in base 13, 7₁₅₀₄1, N−1 is 91×R₁₅₀₄(13), thus factor N−1 is equivalent to factor 13¹⁵⁰⁴−1, N−1 is only 28.604% factored (see http://factordb.com/index.php?id=1100000002320890755&open=prime) (since 28.604% is between 1/4 and 1/3, CHG proof is possible, however, since factordb (http://factordb.com/, https://www.rieselprime.de/ziki/Factoring_Database) lacks the ability to verify CHG proofs, thus there is still primality certificate in factordb), and for the algebraic factors of 13¹⁵⁰⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=13&Exp=1504&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 13¹⁵⁰⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=1504&c0=-&EN=
the 2342nd minimal prime in base 16, 90₃₅₄₂91, N−1 is 144×S₃₅₄₃(16), thus factor N−1 is equivalent to factor 16³⁵⁴³+1, N−1 is only 1.255% factored (see http://factordb.com/index.php?id=1100000000633424191&open=prime), and for the algebraic factors of 16³⁵⁴³+1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=16&Exp=3543&LBIDPMList=B&LBIDLODList=D, and for the prime factorization of 16³⁵⁴³+1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=16&Exp=3543&c0=%2B&EN=
the 25240th minimal prime in base 26, 5₁₈₈₅4P, N+1 is 130×R₁₈₈₆(26), thus factor N+1 is equivalent to factor 26¹⁸⁸⁶−1, N+1 is only 7.262% factored (see http://factordb.com/index.php?id=1100000003850155314&open=prime), and for the algebraic factors of 26¹⁸⁸⁶−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=26&Exp=1886&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 26¹⁸⁸⁶−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&Exp=1886&c0=-&EN=
the 35277th minimal prime in base 36, OZ₃₉₃₂AZ, N+1 is 31500×R₃₉₃₃(36), thus factor N+1 is equivalent to factor 36³⁹³³−1, N+1 is only 16.004% factored (see http://factordb.com/index.php?id=1100000000840634476&open=prime), and for the algebraic factors of 36³⁹³³−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showExplorer?Base=36&Exp=3933&LBIDPMList=A&LBIDLODList=D, and for the prime factorization of 36³⁹³³−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=36&Exp=3933&c0=-&EN=

The helper file for the 151st minimal prime in base 9 (30₁₁₅₈11) in factordb: http://factordb.com/helper.php?id=1100000002376318423

The helper file for the 3187th minimal prime in base 13 (7₁₅₀₄1) in factordb: http://factordb.com/helper.php?id=1100000002320890755

The helper file for the 2342nd minimal prime in base 16 (90₃₅₄₂91) in factordb: http://factordb.com/helper.php?id=1100000000633424191

The helper file for the 25240th minimal prime in base 26 (5₁₈₈₅4P) in factordb: http://factordb.com/helper.php?id=1100000003850155314

The helper file for the 35277th minimal prime in base 36 (OZ₃₉₃₂AZ) in factordb: http://factordb.com/helper.php?id=1100000000840634476

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N−1 for the 151st minimal prime in base 9 (30₁₁₅₈11) in factordb: http://factordb.com/index.php?id=1100000002376318436&open=ecm

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N−1 for the 3187th minimal prime in base 13 (7₁₅₀₄1) in factordb: http://factordb.com/index.php?id=1100000002320890782&open=ecm

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N−1 for the 2342nd minimal prime in base 16 (90₃₅₄₂91) in factordb: http://factordb.com/index.php?id=1100000000633424203&open=ecm

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N+1 for the 25240th minimal prime in base 26 (5₁₈₈₅4P) in factordb: http://factordb.com/index.php?id=1100000003850159350&open=ecm

Factorization status (and ECM efforts for the prime factors between 10²⁴ and 10¹⁰⁰) of N+1 for the 35277th minimal prime in base 36 (OZ₃₉₃₂AZ) in factordb: http://factordb.com/index.php?id=1100000000840634478&open=ecm

For the number 3²³¹⁹+1, it is the product of Φ_d(3) with positive integers d dividing 4638 but not dividing 2319 (i.e. d = 2, 6, 1546, 4638), and the factorization of Φ_d(3) for these positive integers d are: (since 6 and 4638 are == 6 mod 12, thus for these two positive integers d, Φ_d(3) has Aurifeuillian factorization (https://en.wikipedia.org/wiki/Aurifeuillean_factorization, https://www.rieselprime.de/ziki/Aurifeuillian_factor, https://mathworld.wolfram.com/AurifeuilleanFactorization.html, http://www.numericana.com/answer/numbers.htm#aurifeuille, http://pagesperso-orange.fr/colin.barker/lpa/cycl_fac.htm, http://list.seqfan.eu/oldermail/seqfan/2017-March/017363.html, http://myfactorcollection.mooo.com:8090/source/cyclo.cpp, http://myfactorcollection.mooo.com:8090/LCD_2_199, http://myfactorcollection.mooo.com:8090/LCD_2_998, https://stdkmd.net/nrr/repunit/repunitnote.htm#aurifeuillean, https://www.unshlump.com/hcn/aurif.html, https://www.ams.org/journals/mcom/2006-75-253/S0025-5718-05-01766-7/S0025-5718-05-01766-7.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_138.pdf)), and Φ_dL(3) and Φ_dM(3) are their Aurifeuillian L and M factors, respectively)

from	currently known prime factorization
Φ₂(3)	2²
Φ_6L(3)	1 (empty product (https://en.wikipedia.org/wiki/Empty_product))
Φ_6M(3)	7
Φ₁₅₄₆(3)	1182691 × 454333843 × 7175619780295897339 × 219067434459114063477547 × 650663511671253931884619 × (288-digit composite with no known proper factor, SNFS difficulty is 370, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=3&Exp=773&c0=%2B&LM=&SA=)
Φ_4638L(3)	18553 × 2957658597967379799686737984695290731543 × (325-digit composite with no known proper factor, SNFS difficulty is 370, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=3&Exp=2319&c0=%2B&LM=L&SA=)
Φ_4638M(3)	4639 × 6716055901 × (356-digit composite with no known proper factor, SNFS difficulty is 370, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=3&Exp=2319&c0=%2B&LM=M&SA=)

For the number 13¹⁵⁰⁴−1, it is the product of Φ_d(13) with positive integers d dividing 1504 (i.e. d = 1, 2, 4, 8, 16, 32, 47, 94, 188, 376, 752, 1504), and the factorization of Φ_d(13) for these positive integers d are:

from	currently known prime factorization
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 422, 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 838, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=13&Exp=752&c0=%2B&LM=&SA=)

For the number 16³⁵⁴³+1 = 2¹⁴¹⁷²+1, it is the product of Φ_d(2) with positive integers d dividing 28344 but not dividing 14172 (i.e. d = 8, 24, 9448, 28344), and the factorization of Φ_d(2) for these positive integers d are:

from	currently known prime factorization
Φ₈(2)	17
Φ₂₄(2)	241
Φ₉₄₄₈(2)	107083633 × 7076306353 × 2428629073416562046689 × (1382-digit composite with no known proper factor, SNFS difficulty is 1423, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=2&Exp=4724&c0=%2B&LM=&SA=)
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 26¹⁸⁸⁶−1, it is the product of Φ_d(26) with positive integers d dividing 1886 (i.e. d = 1, 2, 23, 41, 46, 82, 943, 1886), and the factorization of Φ_d(26) for these positive integers d are:

from	currently known prime factorization
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 1335, 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 1335, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=26&Exp=943&c0=%2B&LM=&SA=)

For the number 36³⁹³³−1 = 6⁷⁸⁶⁶−1, it is the product of Φ_d(6) with positive integers d dividing 7866 (i.e. d = 1, 2, 3, 6, 9, 18, 19, 23, 38, 46, 57, 69, 114, 138, 171, 207, 342, 414, 437, 874, 1311, 2622, 3933, 7866), and the factorization of Φ_d(6) for these positive integers d are:

from	currently known prime factorization
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 341, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=6&Exp=437&c0=-&LM=&SA=)
Φ₈₇₄(6)	(309-digit prime, for its primality certificate see http://factordb.com/cert.php?id=1100000000019287760)
Φ₁₃₁₁(6)	100745107 × 1719861571 × 2376829061449 × (587-digit composite with no known proper factor, SNFS difficulty is 682, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=6&Exp=1311&c0=-&LM=&SA=)
Φ₂₆₂₂(6)	41953 × 266030354191322260711 × (592-digit composite with no known proper factor, SNFS difficulty is 682, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSNFS?Number=&Base=6&Exp=1311&c0=%2B&LM=&SA=)
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.
File "helper b n" The helper file for the nth minimal prime in base b (local copy from factordb (http://factordb.com/, https://www.rieselprime.de/ziki/Factoring_Database)), only include the primes listed above (i.e. primes which either N−1 or N+1 is trivially fully factored, primes which either N−1 or N+1 is fully factored (but not trivially fully factored, however, only need trial division to 10⁸) and the largest prime factor is < 10²⁹⁹, primes which either N−1 or N+1 is product of a Cunningham number and a small number (no matter this Cunningham number is > 1/3 factored, between 1/4 and 1/3 factored, or < 1/4 factored)).

Latest commit

Git stats

Files

README.md