Search for Primitive Trinomials (mod 2)

See my old trinomial page for an introduction to trinomials over GF(2) and some history.

For a summary and the history of the search, see The Great Trinomial Hunt.

A summary of the primitive trinomials for known Mersenne exponents is here.


A certificate is simply a list of smallest irreducible factors for trinomials of given degree r. In our certificates the factors are encoded in hexadecimal.

Certificates are available for degrees 1279, 2281, 3217, 4423, 9689, 19937, 23209, 44497, 110503, 132049, 756839, 859433, 3021377, 6972593, 24036583, 25964951, 30402457, 32582657, 42643801, 43112609, 57885161, 74207281.

More details are available here.

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:
magma check.magma;

Alternatively, download the check-ntl.c file, compile it with NTL, and run check-ntl 32582657 < i32582657.log-ext (for example). The check-ntl program is much faster for large exponents than the check.magma program.

Status of new Mersenne exponents

Status of new Mersenne exponents (those larger than 6972593):


Further Information

A paper is available here.

See my old trinomial page for some history, and also Paul Zimmermann's page.

Richard Brent (in collaboration with Paul Zimmermann,
with CPU cycles contributed by Dan Bernstein, Bill Hart and Tanja Lange)

Last revised 4 April 2016.

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