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

Abstract

The classical Wagstaff heuristic models the primality of $M_p=2^p-1$ primarily through the size of the prime exponent $p$, at the aggregate scale $C\log p/p$. We test whether the divisor structure of $p-1$ produces detectable secondary variation, using $S(p)=\log \tau (p-1)/\log \log p$. Known Mersenne-prime exponents $p\ge 13$ show systematically elevated values of $S$ relative to nearby prime exponents. In the population-level Wagstaff-balanced design over all $5,761,450$ prime exponents $13\le p<10^8$, $32$ of the $47$ known cases lie in the higher-$S$ regions; the exact conditional randomization gives the two-sided reference value $p_{\mathrm{two}}=0.0115$. We develop a structural model from the cyclotomic and residue constraints generated by the divisors of $p-1$. Within a declared factorwise representation, its local response is determined by the exact modular geometry, while subsequent effective aggregation remains model-dependent and transmission to primality is heuristic. The modular census is constructed without using $S(p)$ or $\tau (p-1)$ as summary predictors and without using Mersenne-prime labels. Applied to the same population, it gives a higher-$S$ allocation of $28.61$ on the conditioned 47-case scale, compared with $23.5$ under the balanced Wagstaff reference. A low-dimensional reduction with reference anchoring gives $32.28$, close to the $32$ observed. This agreement provides a quantitative consistency check for the structural model. For the prospectively fixed interval $10^8\le p<10^9$, the same global reduction assigns a conditioned higher-$S$ share of $73.88\%$.

arXiv commentRevised version (39 pages, 15 tables). The structural model and its limitations have been clarified and the presentation streamlined. An error in the reproducibility archive added in v6 has been corrected without affecting the reported results