On digits of Mersenne numbers

2020/01/10 by Bryce Kerr, Kerr, Bryce, László Mérai +3
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2001.03380

openalex publication_date 2020/01/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Motivated by recently developed interest to the distribution of q-ary digits of Mersenne numbers M_p = 2^p-1, where p is prime, we estimate rational exponential sums with M_p, p ≤ X, modulo a large power of a fixed odd prime q. In turn this immediately implies the normality of strings of q-ary digits amongst about (log X)^3/2+o(1) rightmost digits of M_p, p ≤ X. Previous results imply this only for about (log X)^1+o(1) rightmost digits.

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