On prime factors of Mersenne numbers

2016/06/28 by Ady Cambraia, Michael Knapp, Cambraia, Ady +7
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories #Advanced Mathematical Theories and Applications #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1606.08690

openalex publication_date 2016/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (M_n)_n≥0 be the Mersenne sequence defined by M_n=2ⁿ-1. Let ω(n) be the number of distinct prime divisors of n. In this short note, we present a description of the Mersenne numbers satisfying ω(M_n)≤3. Moreover, we prove that the inequality, given ε>0, ω(M_n)> 2^{(1-ε)loglog 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≥2, p an odd prime number and a a positive integer.

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