Arithmetic Bias in the Distribution of Mersenne Prime Exponents and the Divisor Structure of p-1

Abstract

According to the classical Wagstaff heuristic, the probability that a Mersenne number $M_p=2^p-1$ is prime depends primarily on the size of the exponent $p$. We investigate whether the divisor structure of $p-1$ produces detectable secondary variation within this aggregate probability scale. We introduce the normalized divisor parameter $S(p)=\frac{\log\tau(p-1)}{\log\log p}$, which provides a scale-adjusted measure of the divisor complexity of $p-1$. Using the currently known Mersenne prime exponents, excluding the smallest cases, we compare $S(p)$ against nearby prime controls of comparable size. Across several complementary statistical analyses, Mersenne prime exponents exhibit elevated values of $S(p)$. To interpret this empirical bias, we develop a reduced coarse-grained structural model based on the cyclotomic decomposition $2^{p-1}-1=\prod_{d\mid(p-1)}\Phi_d(2)$. The divisor structure of \(p-1\) generates cyclotomic layers associated with modular constraints on candidate factorizations. Their cumulative filtering effect motivates a structural refinement of the classical Wagstaff heuristic of the form $\mathbb P(M_p\ {\rm prime}\mid S)\approx C(p,S)\frac{\log p}{p}$, where $C(p,S)$ denotes the finite-scale structural factor. The resulting model predicts a redistribution toward higher values of $S(p)$, consistent with the observed imbalance across the explored exponent ranges, while preserving the aggregate Wagstaff probability scale after marginalization over $S$. The proposed framework is heuristic and finite-scale, and is intended as a possible structural interpretation of the observed arithmetic bias rather than as a derivation from first principles or a modification of the classical Wagstaff asymptotic scale.