All Even (Unitary) Perfect Polynomials Over $\mathbb{F_2}$ with Only Mersenne Primes as Odd Divisors


Luis H. Gallardo, Olivier Rahavandrainy




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 $h$.