### Cai, Z, Faust, M, Hildebrand, AJ, Li, J and Zhang, Y (2017)

#### Leading Digits of Mersenne Numbers

Preprint in arXiv:1712.04425 [math.NT]; last accessed October 23, 2018.

**ISSN/ISBN:** Not available at this time.
**DOI:** Not available at this time.

**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)=log10(1+1/d), d=1,2,…,9. In this paper, we consider the leading digits of the Mersenne numbers Mn=2pn−1, where pn is the n-th prime. In light of known irregularities in the distribution of primes, one might expect that the leading digit sequence of {Mn} has \emph{worse} distribution properties than "smooth" sequences with similar rates of growth, such as {2nlogn}. Surprisingly, the opposite seems to be the true; indeed, we present data, based on the first billion terms of the sequence {Mn}, 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.

**Bibtex:**

```
@ARTICLE{,
author = {{Cai}, Zhaodong and {Faust}, Matthew and {Hildebrand}, A.~J. and {Li}, Junxian and
{Zhang}, Yuang},
title = "{Leading Digits of Mersenne Numbers}",
journal = {ArXiv e-prints},
archivePrefix = {arXiv},
eprint = {1712.04425},
primaryClass = {math.NT},
year = {2017},
month = {dec},
url = {https://arxiv.org/abs/1712.04425},
}
```

**Reference Type:** Preprint

**Subject Area(s):** Number Theory