Divisor Structure of p-1 in Mersenne Prime Exponents

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 $S(p)=\log τ(p-1)/\log\log p$, 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 elevated 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 compatible with the classical asymptotic behavior. An analytic mechanism has not yet been established, and a theoretical explanation of the observed phenomenon remains open.