### Cai, Z, Faust, M, Hildebrand, AJ, Li, J and Zhang, Y (2021)

#### Leading digits of Mersenne numbers

Experimental Mathematics 30(3), pp. 405–421.

**ISSN/ISBN:** Not available at this time.
**DOI:** 10.1080/10586458.2018.1551162

**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 article, 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 worse distribution properties than “smooth” sequences with similar rates of growth, such as {2nlogn}. Surprisingly, the opposite seems to be 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 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 = {Zhaodong Cai and Matthew Faust and A. J. Hildebrand and Junxian Li and Yuan Zhang},
title = {Leading Digits of Mersenne Numbers},
journal = {Experimental Mathematics},
volume = {30},
number = {3},
pages = {405--421},
year = {2021},
publisher = {Taylor & Francis},
doi = {10.1080/10586458.2018.1551162},
URL = {https://doi.org/10.1080/10586458.2018.1551162},
}
```

**Reference Type:** Journal Article

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