Twelve new primitive binary trinomials

TL;DR

This paper reports twelve new primitive binary trinomials of record degrees over GF(2), and discusses the existence of primitive trinomials for certain Mersenne exponents, completing the search for known Mersenne primes.

Contribution

The paper introduces twelve new primitive trinomials of record degrees and provides the first Mersenne exponent without a corresponding primitive trinomial, completing the search for known Mersenne primes.

Findings

Twelve new primitive trinomials found at record degrees

First Mersenne exponent without a primitive trinomial identified

Completes the search for known Mersenne prime exponents

Abstract

We exhibit twelve new primitive trinomials over GF(2) of record degrees $42643801$, $43112609$, and $74207281$. In addition we report the first Mersenne exponent not ruled out by Swan's theorem - namely $57885161$ - for which no primitive trinomial exists. This completes the search for the currently known Mersenne prime exponents.