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arXiv · 1606.08690

On prime factors of Mersenne numbers

Ady Cambraia Jr·Michael P. Knapp·Abílio Lemos·B. K. Moriya·Paulo H. A. Rodrigues
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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.

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Ady Cambraia Jr, Michael P. Knapp, Abílio Lemos, B. K. Moriya, Paulo H. A. Rodrigues. 2016-06-28. On prime factors of Mersenne numbers. https://arxiv.org/abs/1606.08690

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