See Richard Brent page for more information.

Certificates are available for degree 1279, 2281, 3217, 4423, 9689, 19937, 23209, 44497, 110503, 132049, 756839, 859433, 3021377, 6972593, 24036583, 25964951, 30402457, 32582657, 42643801, 43112609, 57885161, 74207281 (a copy of those log files can be obtained on Richard Brent's log page).

How to check a certificate?. First download the certificate, uncompress it (gunzip ixxx.log-ext.gz), download the check.magma program, replace the value of r and the file name in the first lines, then start Magma and type:
load "check.magma";

Note: due to a bug in Magma up to version V2.21-3, some trinomials are not correctly checked.

Alternatively, you can download the check-ntl.c file, compile it with NTL, and run check-ntl 32582657 < i32582657.log-ext (for example). Note: this works only up from 859433, since other certificates do not contain factors of small degree.

Status of new Mersenne exponents: