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.
- 1050000+65859 is the smallest prime with 50001 digits; at the time of its announcement to the Number Theory List on 2022-05-05 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.
- 1040000+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.
has developed an algorithm to
q-coefficients of the modular j-function,
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 τ-function looks at q-coefficients 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 off-by-1 error: The table entry p, q actually stands for τ (pq-1).
- τ (474176) with 38404 digits: This was the first open case and is now settled: PARI/GP, Primo; Factordb; PrimePages.
- τ (1994518) 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 R86453 = (1086453 - 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.