# On (unitary) perfect polynomials over $\mathbb{F}_2$ with only Mersenne primes as odd divisors

## TL;DR

This paper characterizes all (unitary) perfect polynomials over the finite field _2 with only Mersenne primes as odd divisors, revealing that only nine such polynomials exist, based on new factorization results.

## Contribution

It provides a complete classification of certain perfect polynomials over _2 involving Mersenne primes, introducing new factorization results for polynomials of the form M^{2h+1} +1.

## Key findings

- Only nine (unitary) perfect polynomials over _2 are formed from x, x+1, and Mersenne primes.
- New factorization results for M^{2h+1} +1 over _2.
- Implications for the structure of odd perfect polynomials over _2.

## Abstract

The only (unitary) perfect polynomials over $\mathbb{F}_2$ that are products of $x$, $x+1$ and 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 $h$. Other consequences of such a factorization are new results about odd perfect polynomials.