Factorization of cyclotomic polynomial values at Mersenne primes

Abstract

We get some results about the factorization of ${\varphi }_p(M)\in {\mathbb{F}}_2[x]$, where $p$ is a prime number, ${\varphi }_p$ is the corresponding cyclotomic polynomial and $M$ is a Mersenne prime (polynomial). By the way, we better understand the factorization of the sum of the divisors of $M^{2h}$, for a positive integer $h$.