On the number of prime factors of Mersenne numbers

Abílio Lemos, Ady Cambraia Junior

Published 2016-06-28Version 1

Let $n$ a positive integer, $M_n=2^n-1$ the $n$th Mersenne number and $\omega(n)$ the number of distinct prime divisors of $n$. We present a description of the Mersenne numbers satisfying $\omega(M_n)\leq3$. Moreover, we prove that the inequality, for $\epsilon>0$, $\omega(M_n)> 2^{(1-\epsilon)\log\log n} -3 $ holds almost all positive integer $n$.