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

TL;DR

This paper classifies all even (unitary) perfect polynomials over  with only Mersenne primes as odd divisors, revealing exactly nine such polynomials and establishing a new factorization result for Mersenne primes.

Contribution

It provides a complete classification of certain perfect polynomials over  and introduces a new factorization theorem for Mersenne primes.

Findings

Exactly nine such perfect polynomials exist.

New factorization result for Mersenne primes: M^{2h+1} + 1.

Classification based on divisors x, x+1, and Mersenne primes.

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 $h$.