On prime factors of Mersenne numbers

TL;DR

This paper investigates the prime factorization properties of Mersenne numbers, characterizes those with up to three distinct prime factors, and explores solutions to a related exponential equation involving Mersenne numbers.

Contribution

It provides a classification of Mersenne numbers with at most three prime factors and analyzes the typical growth of their prime divisors, also solving a specific exponential Diophantine equation.

Findings

Mersenne numbers with up to three prime factors are characterized.

For almost all n, the number of prime divisors of M_n exceeds a bound related to log log n.

Solutions to M_m + M_n = 2p^a are explicitly described.

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.