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:

- r=13466917 (M39): ruled out by Swan theorem since r=5 (mod 8), and x^r+x^2+1 is divisible by x^2+x+1.
- r=20996011 (M40): ruled out by Swan theorem since r=3 (mod 8), and x^r+x^2+1 is divisible by x^2+x+1.
- r=24036583 (M41): search started on April 25, 2007,
completed on August 27, 2007: there are exactly
**two**primitive trinomials (with their reciprocal) which have been checked by Allan Steel using Magma:*x*^{24036583}+*x*^{8785528}+ 1,*x*^{24036583}+*x*^{8412642}+ 1. - r=25964951 (M42): search started on July 19, 2007,
completed on November 3, 2007: there are exactly
**four**primitive trinomials (with their reciprocal) which have been checked by Allan Steel using Magma:*x*^{25964951}+*x*^{880890}+ 1,*x*^{25964951}+*x*^{4627670}+ 1,*x*^{25964951}+*x*^{4830131}+ 1,*x*^{25964951}+*x*^{6383880}+ 1. - r=30402457 (M43): search started on October 22, 2007, completed on
December 12, 2007: there is exactly
**one**primitive trinomial (with its reciprocal), which has been checked by Allan Steel using Magma:*x*^{30402457}+*x*^{2162059}+ 1. - r=32582657 (M44): search started on November 30, 2007, completed on
January 24, 2008: there are exactly
**three**primitive trinomials (with their reciprocals), which have been checked by Allan Steel using Magma:*x*^{32582657}+*x*^{5110722}+ 1,*x*^{32582657}+*x*^{5552421}+ 1,*x*^{32582657}+*x*^{7545455}+ 1. - r=37156667 (M45): ruled out by Swan's theorem since r=3 mod 8, and x^r+x^2+1 is divisible by x^5+x^2+1.
- r=43,112,609 (M46): search started on September 18, 2008, completed on
May 8th, 2009: there are exactly
**four**primitive trinomials (with their reciprocals), which have been checked by Allan Steel using Magma (except the third one). This search was done with the help of Tanja Lange, Dan Bernstein, and TU Eindhoven's Coding and Cryptography Computer Cluster:*x*^{43112609}+*x*^{3569337}+ 1,*x*^{43112609}+*x*^{4463337}+ 1,*x*^{43112609}+*x*^{17212521}+ 1,*x*^{43112609}+*x*^{21078848}+ 1. - r=42,643,801 (M47): search started on June 12, 2009, completed on
September 16, 2009: there are exactly
**five**primitive trinomials (with their reciprocals), which have been checked by Allan Steel using Magma.*x*^{42643801}+*x*^{55981}+ 1,*x*^{42643801}+*x*^{3706066}+ 1,*x*^{42643801}+*x*^{3896488}+ 1,*x*^{42643801}+*x*^{12899278}+ 1,*x*^{42643801}+*x*^{20150445}+ 1. - r=57,885,161 (M48): search started on February 6, 2013, completed on May 13, 2013: no primitive trinomial exists of this degree. This is the first Mersenne exponent for which this occurs.
- r=74,207,281
(M49): search started on January 25, 2016, completed on April 5, 2016.
There are
**three**primitive trinomials (checked by GrĂ©goire Lecerf using some independent code):*x*^{74207281}+*x*^{9156813}+ 1,*x*^{74207281}+*x*^{9999621}+ 1,*x*^{74207281}+*x*^{30684570}+ 1. - r=77232917 (M50): ruled out by Swan theorem since r=5 (mod 8), and x^r+x^2+1 is divisible by x^3+x+1.
- r=82589933 (M51): ruled out by Swan theorem since r=5 (mod 8), and x^r+x^2+1 is divisible by x^3+x+1.

- Download irred-ntl-1.9.
This is a patch for NTL,
(version 5.3.1 or 5.3.2), which speeds up the multiplication
over GF(2)[x], in particular by implementing Toom-Cook's 3-way
algorithm.
With NTL 5.4, download irred-ntl-1.9.1-5.4.
These files are distributed under the terms of the
GNU General Public
License.
Or download improved implementations of
Toom-Cook 3-way and
Toom-Cook 4-way (both modified from
code from
Marco Bodrato).
Note:
`irred-ntl`is now superseeded by the new code in the gf2x library.