All even (unitary) perfect polynomials over $\F_2$ with only Mersenne primes as odd divisors

Abstract

We address an arithmetic problem in the ring $\F_2[x]$ related to the fixed points of the sum of divisors function. We study some binary polynomials $A$ such that $\sigma (A)/A$ is still a binary polynomial. Technically, we prove that the only (unitary) perfect polynomials over $\F_2$ that are products of $x$, $x+1$ and of Mersenne primes are precisely the nine (resp. nine "classes") known ones. This follows from a new result about the factorization of $M^{2h+1}+1$, for a Mersenne prime $M$ and for a positive integer $h$.

16 pages. arXiv admin note: text overlap with arXiv:1908.00106