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 ≤ 10 unsolved families)
The large minimal primes in base *b* are of the form (*a*×*b**n*+*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, 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*×*b**n*±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://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), 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) 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 *b**n*±1, see https://en.wikipedia.org/wiki/Cunningham_number, https://mathworld.wolfram.com/CunninghamNumber.html, 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, http://myfactors.mooo.com/) 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) 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, 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://factordb.com/nmoverview.php?method=2) (the *N*+1 case), if either *N*−1 or *N*+1 (or both) can be ≥ 1/3 factored but neither can be ≥ 1/2 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: (thus these numbers also do not need primality certificates)
The Cunningham numbers *b**n*±1 has algebraic factorization to product of *Φ**d*(*b*) with positive integers *d* dividing *n* (the *b**n*−1 case) or positive integers *d* dividing 2×*n* but not dividing *n* (the *b**n*+1 case), where *Φ* is the cyclotomic polynomial (https://en.wikipedia.org/wiki/Cyclotomic_polynomial, https://mathworld.wolfram.com/CyclotomicPolynomial.html) (see https://stdkmd.net/nrr/repunit/repunitnote.htm and 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))
(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) in base *b* with length *n*, i.e. *R**n*(*b*) = (*b**n*−1)/(*b*−1), "*S**n*(*b*)" means *b**n*+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×*R*308(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×*R*564(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×*R*576(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×*R*344(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×*R*354(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×*S*2319(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×*R*1504(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×*S*3543(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×*R*1886(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 *n*th 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.