On prime factors of Mersenne numbers

Abstract

Let $(M_n{)}_{n\ge 0}$ 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 $\omega (M_n)\le 3$. 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 $m,n\ge 2$, $p$ an odd prime number and $a$ a positive integer.

to appear in Palestine Journal of Mathematics