On prime factors of Mersenne numbers

Abstract

Let $(M_n)_{n\geq0}$ be the Mersenne sequence defined by $M_n=2^n-1$. Let $\omega(n)$ be the number of distinct prime divisors of $n.$ In this short note, we present a description of the Mersenne numbers satisfying $\omega(M_n)\leq3$. Moreover, we prove that the inequality, given $\epsilon>0$, $\omega(M_n)> 2^{(1-\epsilon)\log\log n} -3 $ holds for almost all positive integers $n$. Besides, we present the integer solutions $(m,n,a)$ of the equation $M_m+M_n=2p^a$ with $m,n\geq2$, $p$ an odd prime number and $a$ a positive integer.