Leading Digits of Mersenne Numbers

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 $P(d)={\hspace{0.17em}\text{log}\hspace{0.17em}}_{10}(1+1/d)$, $d=1,2,\dots ,9$. In this article, 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 $\left\{M_n\right\}$ has worse distribution properties than “smooth” sequences with similar rates of growth, such as $\left\{2^{n\hspace{0.17em}\text{log}\hspace{0.17em}n}\right\}$. Surprisingly, the opposite seems to be true; indeed, we present data, based on the first billion terms of the sequence $\left\{M_n\right\}$, 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.

Keywords:

• Benford’s law
• Mersenne numbers
• distribution of primes

1991 MATHEMATICS SUBJECT CLASSIFICATION:

• 11K31 (11K06
• 11N05
• 11Y55)

Acknowledgments

We thank the referees for a careful reading of the paper and a number of comments that helped improve the paper.

Notes

1 In general, for sequences of the form $\left\{a^n\right\}$ the quality of the agreement between the actual and predicted leading digit counts is related to diophantine approximation properties of the number ${\hspace{0.17em}\text{log}\hspace{0.17em}}_{10}a$ (see Proposition 4.2 below and [Kuipers and Niederreiter 74, Chapter 2, Theorem 3.2]), but it is also affected by any linear relations between the numbers ${\hspace{0.17em}\text{log}\hspace{0.17em}}_{10}a$ and ${\hspace{0.17em}\text{log}\hspace{0.17em}}_{10}d,\hspace{0.17em}d=1,2,\dots ,9$. For the sequence $\left\{2^n\right\}$, the latter aspect comes into play and accounts in part for the small errors observed in Table 1; see [Cai, Faust et al. 18].

2 We emphasize that we do not require M_n to be prime, but we do require the exponent, p_n, to be prime. In other words, the sequence $\left\{M_n\right\}$ is the sequence of candidates for Mersenne primes.

3 That the digit 1 waiting times for the sequence $\left\{2^n\right\}$ are either 3 or 4 is easy to see by tracking leading digits after successive multiplications by 2; for the other three sequences, however, the set of possible values of the waiting times seems to have a much more complicated structure.

4 This result could also be derived from [Cai, Hildebrand et al. in press, Theorem 2.8], a general result on the local Benford distribution properties of a large class of arithmetic sequences. We present here an elementary argument that does not depend on the theory of uniform distribution modulo 1 and which suffices to obtain the assertions of the theorem.

5 Note that the conditions (2.9) and (2.10) of the theorem are independent of the base of the logarithm chosen in the definition of the Δ operator. For our proof it is convenient to work with base 10 logarithms instead of natural logarithms, so we have defined u_n in terms of the base 10 logarithm.