Divisor Structure of p-1 in Mersenne Prime Exponents

TL;DR

This paper explores the divisor structure of p-1 in Mersenne prime exponents, finding that these exponents tend to have larger divisor-structure measures compared to nearby primes, suggesting a potential link to prime occurrence.

Contribution

It introduces a scale-invariant divisor-structure parameter for p-1 and demonstrates its correlation with Mersenne prime exponents using statistical analysis.

Findings

Mersenne prime exponents have larger divisor-structure values than nearby primes.

The effect is moderate but consistent across various non-parametric tests.

Results align with classical asymptotic behavior, but lack an analytic explanation.

Abstract

According to the Wagstaff heuristic, the probability that a Mersenne number $M_p = 2^p-1$ is prime mainly depends on the size of the exponent $p$. We investigate whether the secondary arithmetic structure in $p-1$ is linked to noticeable variation among Mersenne prime exponents, motivated by the cyclotomic decomposition of $2^{p-1}-1$. We introduce the normalized divisor-structure parameter $S(p)=\log \tau(p-1)/\log\log p$, which is a scale-invariant way to measure the divisor structure of $p-1$. which is a scale-invariant way to measure the divisor structure of $p-1$. Using the currently known Mersenne prime exponents (excluding small cases), we compare $S(p)$ with nearby prime controls through percentile analysis, stratified conditional likelihood estimation and stratified permutation testing. Across all methods, Mersenne prime exponents tend to exhibit larger values of $S(p)$ within local exponent windows compared to nearby primes of comparable size. The observed effect is moderate and remains stable across different non-parametric tests. It is also consistent with the classical asymptotic behavior. At present, no analytic explanation for the observed phenomenon is known.