Proof-code: x25

This code is used for proofs of generalized Lucas and Lehmer numbers and their primitive parts.

The following software was used:

(1) Pari-GP for cyclotomic (and, where appropriate, Aurifeuillian) factorizations of N^2-1.

(2) GMP-ECM, Msieve, YAFU and ggnfs for extracting PrP factors of such cyclotomic cofactors.

(3) Pari-GP and Primo (when needed) for proving these helpers prime.

(4) OpenPFGW for BLS tests with these prime helpers.

(5) Pari-GP for Coppersmith--Howgrave-Graham, Williams--Lenstra, or Konyagin--Pomerance proofs, where BLS was insufficient.

Usually, the largest effort was expended on GMP-ECM. Factorization percentages for current top-20s are listed here, in the cases primV and lucasU, and here, in the case of Lehmer numbers and their primitive parts.

PS: We also included:

6738*(2^148227+60443)*(205*2^65523-1639)-1, with a Konyagin-Pomerance proof that depends on the cyclotomy of 2^65520-1;

two gigantic generalized repunits, with BLS proofs involving ECPP helpers at 3832 and 4354 digits.