Arithmetic Bias in 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 variations within this asymptotic framework. We introduce the normalized divisor parameter $S(p)=\log τ(p-1)/\log\log p$, which measures the divisor complexity of p-1, including prime multiplicities. Using the currently known Mersenne prime exponents (excluding small cases), we compare $S(p)$ against nearby prime controls of comparable size. Across several complementary distribution-free methods, including percentile analysis, conditional likelihood estimation, and permutation tests, Mersenne prime exponents consistently exhibit elevated values of $S(p)$. To interpret this effect, we develop a heuristic framework based on the cyclotomic decomposition $2^{p-1}-1=\prod_{d|(p-1)}Φ_d(2)$, in which divisors of $p-1$ generate effective modular constraint layers. This motivates a heuristic refinement of the Wagstaff model of the form $\Pr(M_p\ \text{prime}) \approx C(\log p)^{S(p)}/p$. The proposed refinement preserves the classical Wagstaff scale in the typical regime $S(p)\approx 1$, while suggesting that the finite-scale distribution of Mersenne prime exponents exhibits a weak arithmetic bias linked to the divisor structure of $p-1$.