We address an arithmetic problem in the ring F 2 [x]. 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.