Divisor Structure of p−1 Modulates Mersenne Prime Exponent Bias — E8 Intelligence Research

FINDING: Mersenne prime exponents p may exhibit secondary arithmetic bias tied to the divisor structure of p−1, modulating the classical Wagstaff probability scale. | MATH: Euclid–Euler theorem: N = 2^(p−1)(2^p − 1) is perfect iff 2^p − 1 is prime (p prime). Wagstaff heuristic: P(M_p prime) ≈ (e^γ / ln 2) · (1/p) · ln(p·ln 2), with e^γ ≈ 1.7811. Candidate refinement: probability depends on τ(p−1), σ(p−1), or factorization of p−1. | CONNECTION: No direct link to 0.618/1.618 ratios in the cited evidence. Structural resonance only: Mersenne exponents p are themselves prime, so p−1 is always even and divisible by 2 — a forced binary structure. Triangular-number link: every even perfect number N = 2^(p−1)(2^p−1) is also the p-th triangular number T_p = p(p+1)/2 only when p=2 (N=6); general perfect numbers are hexagonal numbers H_n = n(2n−1) with n = 2^(p−1). This is the only clean geometric-harmony connection supported. | DEPTH: 6 --- **Detailed extraction:** **1. Key mathematical insigh Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com