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) (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) implementation) primality certificates (https://en.wikipedia.org/wiki/Primality_certificate, https://primes.utm.edu/glossary/xpage/Certificate.html, https://mathworld.wolfram.com/PrimalityCertificate.html) for the minimal primes > 10300 in bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 28, 30 (local copy from factordb (http://factordb.com/)).
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.
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) or N+1 test (https://primes.utm.edu/prove/prove3_2.html), 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
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://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 ≥ 1/4 factored (https://en.wikipedia.org/wiki/Integer_factorization, https://www.rieselprime.de/ziki/Factorization, https://mathworld.wolfram.com/PrimeFactorization.html), see the article http://www.ams.org/journals/mcom/1975-29-130/S0025-5718-1975-0384673-1/S0025-5718-1975-0384673-1.pdf for the case that either N−1 or N+1 (or both) can be ≥ 1/3 factored, 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), 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 bn±1 has algebraic factorization to product of Φd(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) (see https://stdkmd.net/nrr/repunit/repunitnote.htm and https://www.mersenneforum.org/attachment.php?attachmentid=7727&d=1330555980 and https://homes.cerias.purdue.edu/~ssw/cun/mine.pdf)
(below, "Rn(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. Rn(b) = (bn−1)/(b−1), "Sn(b)" means bn+1, the special cases of Rn(10) and Sn(10) are in https://stdkmd.net/nrr/repunit/ and https://stdkmd.net/nrr/repunit/10001.htm, respectively, in fact, Rn(b) and Sn(b) are 111...111 and 1000...0001 in base b, respectively)
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=
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=