Leading Digits of Mersenne Numbers

TL;DR

This paper investigates the leading digit distribution of Mersenne numbers, revealing they follow Benford's Law more regularly than expected despite prime irregularities, supported by extensive computational data and heuristic explanations.

Contribution

It demonstrates that Mersenne numbers' leading digits exhibit more regularity than similar smooth sequences, contrary to initial expectations, supported by large-scale computational evidence.

Findings

Leading digits of Mersenne numbers follow Benford's Law.

Mersenne numbers show more regular digit distribution than smooth sequences.

Empirical data based on the first billion terms supports these observations.

Abstract

It has long been known that sequences such as the powers of $2$ and the factorials satisfy Benford's Law; that is, leading digits in these sequences occur with frequencies given by $P(d)=\log_{10}(1+1/d)$, $d=1,2,\dots,9$. In this paper, we consider the leading digits of the Mersenne numbers $M_n=2^{p_n}-1$, where $p_n$ is the $n$-th prime. In light of known irregularities in the distribution of primes, one might expect that the leading digit sequence of $\{M_n\}$ has \emph{worse} distribution properties than "smooth" sequences with similar rates of growth, such as $\{2^{n\log n}\}$. Surprisingly, the opposite seems to be the true; indeed, we present data, based on the first billion terms of the sequence $\{M_n\}$, showing that leading digits of Mersenne numbers behave in many respects \emph{more regularly} than those in the above smooth sequences. We state several conjectures to this effect, and we provide an heuristic explanation for the observed phenomena based on classic models for the distribution of primes.