# On digits of Mersenne numbers

### Bryce Kerr

Max Planck Institute for Mathematics, Bonn, Germany### László Mérai

Austrian Academy of Sciences, Linz, Austria### Igor E. Shparlinski

University of New South Wales, Sydney, Australia

## 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 $M_p$, $p \le X$, 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 $M_p$, $p \le X$. Previous results imply this only for about $(\log X)^{1+o(1)}$ rightmost digits.

## Cite this article

Bryce Kerr, László Mérai, Igor E. Shparlinski, On digits of Mersenne numbers. Rev. Mat. Iberoam. 38 (2022), no. 6, pp. 1901–1925

DOI 10.4171/RMI/1316