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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.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:

• the 3176th minimal prime in base 13, 81010₄₁₅1, which equals 17746×13⁴¹⁶+1, N−1 is trivially fully factored
• the 3177th minimal prime in base 13, 8110₄₃₅1, which equals 1366×13⁴³⁶+1, N−1 is trivially fully factored
• the 3188th minimal prime in base 13, 930₁₅₅₁1, which equals 120×13¹⁵⁵²+1, N−1 is trivially fully factored
• the 3191st minimal prime in base 13, 390₆₂₆₆1, which equals 48×13⁶²⁶⁷+1, N−1 is trivially fully factored
• the 649th minimal prime in base 14, 34D₇₀₈, which equals 47×14⁷⁰⁸−1, N+1 is trivially fully factored
• the 650th minimal prime in base 14, 4D₁₉₆₉₈, which equals 5×14¹⁹⁶⁹⁸−1, N+1 is trivially fully factored
• the 2335th minimal prime in base 16, 88F₅₄₅, which equals 137×16⁵⁴⁵−1, N+1 is trivially fully factored
• the 3310th minimal prime in base 20, JCJ₆₂₉, which equals 393×20⁶²⁹−1, N+1 is trivially fully factored
• the 3408th minimal prime in base 24, 88N₅₉₅₁, which equals 201×24⁵⁹⁵¹−1, N+1 is trivially fully factored
• the 25509th minimal prime in base 28, EB0₄₀₅1, which equals 403×28⁴⁰⁶+1, N−1 is trivially fully factored
• the 2616th minimal prime in base 30, C0₁₀₂₂1, which equals 12×30¹⁰²³+1, N−1 is trivially fully factored
• the 2619th minimal prime in base 30, OT₃₄₂₀₅, which equals 25×30³⁴²⁰⁵−1, N+1 is trivially fully factored
• the 35237th minimal prime in base 36, P8Z₃₉₀, which equals 909×36³⁹⁰−1, N+1 is trivially fully factored

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):

• 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)

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/The_Cunningham_project, https://primes.utm.edu/glossary/xpage/CunninghamProject.html, https://www.rieselprime.de/ziki/Cunningham_project, https://homes.cerias.purdue.edu/~ssw/cun/index.html, https://maths-people.anu.edu.au/~brent/factors.html, https://web.archive.org/web/20190315214330/http://cage.ugent.be/~jdemeyer/cunningham/, http://myfactors.mooo.com/, https://doi.org/10.1090/conm/022, https://www.mersenneforum.org/attachment.php?attachmentid=7727&d=1330555980 (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_7.pdf), https://homes.cerias.purdue.edu/~ssw/cun/mine.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_6.pdf), http://homes.cerias.purdue.edu/~ssw/cun1.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_71.pdf)) and a small number, and this Cunningham number is ≥ 1/3 factored (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html, https://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html, http://www.numericana.com/answer/factoring.htm) or this Cunningham number is ≥ 1/4 factored and the number is not very large (say not > 10¹⁰⁰⁰⁰⁰). If either N−1 or N+1 (or both) can be ≥ 1/2 factored, then we can use the Pocklington N−1 primality test (https://primes.utm.edu/prove/prove3_1.html, http://bln.curtisbright.com/2013/10/09/the-n-1-and-n1-primality-tests/, https://en.wikipedia.org/wiki/Pocklington_primality_test, https://www.rieselprime.de/ziki/Pocklington%27s_theorem, https://mathworld.wolfram.com/PocklingtonsTheorem.html, https://stdkmd.net/nrr/pock/, http://factordb.com/nmoverview.php?method=1) (the N−1 case) or the Morrison N+1 primality test (https://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) (the N+1 case); if either N−1 or N+1 (or both) can be ≥ 1/3 factored, then we can use the Brillhart-Lehmer-Selfridge primality test (https://www.ams.org/journals/mcom/1975-29-130/S0025-5718-1975-0384673-1/S0025-5718-1975-0384673-1.pdf (cached copy at https://github.com/xayahrainie4793/pdf-files-cached-copy/blob/main/pdf_23.pdf), https://en.wikipedia.org/wiki/Pocklington_primality_test#Extensions_and_variants); if either N−1 or N+1 (or both) can be ≥ 1/4 factored but neither can be ≥ 1/3 factored, then we need to use CHG (https://mersenneforum.org/attachment.php?attachmentid=21133&d=1571237465, https://primes.utm.edu/bios/page.php?id=797, https://github.com/xayahrainie4793/prime-programs-cached-copy/tree/main/CHG) to prove its primality (see https://mersenneforum.org/showpost.php?p=528149&postcount=3 and https://mersenneforum.org/showpost.php?p=603181&postcount=438), however, unlike Brillhart-Lehmer-Selfridge primality test for the numbers N such that N−1 or N+1 (or both) can be ≥ 1/3 factored can run for arbitrarily large numbers N (thus, there are no unproven probable primes N such that N−1 or N+1 (or both) can be ≥ 1/3 factored), CHG for the numbers N such that either N−1 or N+1 (or both) can be ≥ 1/4 factored but neither can be ≥ 1/3 factored cannot run for very large N (say > 10¹⁰⁰⁰⁰⁰), for the examples of the numbers which are proven prime by CHG, see https://primes.utm.edu/primes/page.php?id=126454, https://primes.utm.edu/primes/page.php?id=131964, https://primes.utm.edu/primes/page.php?id=123456, https://primes.utm.edu/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"), however, factordb (http://factordb.com/) lacks the ability to verify CHG proofs, see https://mersenneforum.org/showpost.php?p=608362&postcount=165; if neither N−1 nor N+1 can be ≥ 1/4 factored but Nⁿ−1 can be ≥ 1/3 factored for a small n, then we can use the cyclotomy primality test (https://primes.utm.edu/glossary/xpage/Cyclotomy.html, https://primes.utm.edu/prove/prove3_3.html, http://factordb.com/nmoverview.php?method=3) (however, this situation does not exist for these numbers, since only one of N−1 and N+1 is product of a Cunningham number and a small number, the only exception is the numbers in the family {2}1 in base b, in such case both N−1 and N+1 are products of a Cunningham number and a small number, thus for the numbers in the family {2}1 in base b, maybe factorization of N²−1 can be used)): (thus these numbers also do not need primality certificates)

The Cunningham numbers bⁿ±1 has algebraic factorization to product of Φ_d(b) with positive integers d dividing n (the bⁿ−1 case) or positive integers d dividing 2×n but not dividing n (the bⁿ+1 case), 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) (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://www.numbersaplenty.com/set/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, http://www.fermatquotient.com/PrimSerien/GenRepuP.txt, http://www.primenumbers.net/Henri/us/MersFermus.htm, 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://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), "S_n(b)" means bⁿ+1, 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 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 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 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 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 factorization of 26³⁵⁴−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&Exp=354&c0=-&EN=

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, and for the 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 (since 28.604% is between 1/4 and 1/3, CHG proof is possible, however, since factordb (http://factordb.com/) lacks the ability to verify CHG proofs, thus there is still primality certificate in factordb), and for the 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, and for the 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, and for the factorization of 26¹⁸⁸⁶−1, see http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=26&Exp=1886&c0=-&EN=

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/)), 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.