  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.
