ECPP
Since release 0.4.0, the CM software contains a fastECPP implementation; more details are available in the announcement of a new ECPP record certification. The implementation has been the topic of a Mersenne forum thread. It is referenced on the PrimePages, where it is proposed as a choice when creating a new proof code.
This page collects certificates for primes proved by the author, in PARI/GP and in Primo format. I submit them to Factordb unless I forget to, where they should appear under my user ID. On the PrimePages, they appear under the proof code E3.
 10^{50000}+65859 is the smallest prime with 50001 digits; at the time of its announcement to the Number Theory List on 20220505 it was the largest prime proved with ECPP, so that it is likely that it will remain for some time on the list of twenty largest primes proved with ECPP.
 10^{40000}+14253 is the smallest prime with 40001 digits; like the previous one, it is of no particular interest, but when starting with 50001 digits, there was not much missing to this one, so I thought I might as well finish the proof.

Besides random large numbers, I have had a look at particular numbers
that have received attention in the literature.
Fredrik Johansson
has developed an algorithm to
compute single
qcoefficients of the modular jfunction,
thereby disproving the conjecture that none of them are prime.
He found the following coefficients to be probable prime:
 457871 with 3689 digits: PARI/GP, Primo; this one was known previously to be prime, see Factordb
 685031 with 4513 digits: PARI/GP, Primo; Factordb
 1029071 with 5532 digits: PARI/GP, Primo; Factordb
 1101431 with 5723 digits: PARI/GP, Primo; Factordb
 9407831 with 16734 digits: PARI/GP, Primo; Factordb
 11769911 with 18718 digits: PARI/GP, Primo; Factordb
 18437999 with 23429 digits: PARI/GP, Primo; Factordb
 Similarly to coefficients of the modular function j, the Ramanujan τfunction looks at qcoefficients of the modular form Δ. Nik(os) Lygeros and Olivier Rozier have worked on prime values of τ and maintain a list of prime and probable prime values. One needs to take care to not make an offby1 error: The table entry p, q actually stands for τ (p^{q1}).
 τ (47^{4176}) with 38404 digits: This was the first open case and is now settled: PARI/GP, Primo; Factordb; PrimePages.
 τ (199^{4518}) with 57125 digits: This was a new ECPP record at the time of computation: PARI/GP, Primo; PrimePages.
 To see how far the code could be pushed, I have tried to establish a new record with the repunit R_{86453} = (10^{86453}  1) / 9. The first phase has been carried out with a varying number of cores, ranging from 759 to 2639; in total, it has taken 383 CPU years and and less than 4 months in real time. The certificate length is 2979. The second phase has been run on a machine with 96 cores; it has taken about 25 CPU years and also about 4 months in real time. Verifying the certificate with PARI/GP on the same machine with 96 cores takes 190 CPU days, or almost two days of real time.