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

Understand

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.