Wieferich Primes: The Only Known p with p² Dividing 2^(p−1) − 1 — E8 Intelligence Research

Abstract

FINDING: Wieferich primes (1093, 3511) are the only known primes p where 2^(p−1) ≡ 1 (mod p²), a second-order residue condition with deep ties to cyclotomic fields and Mersenne divisibility. MATH: - Defining congruence: 2^(p−1) ≡ 1 (mod p²) ⇔ p² | (2^(p−1) − 1). - Known solutions: p = 1093, p = 3511 (both verified to enormous bounds, ~6.7×10¹⁵). - For Mersenne primes M_q = 2^q − 1: if p | M_q, then p ≡ 1 (mod q) and p ≡ ±1 (mod 8). Wieferich primes are precisely those p where p² | 2^(p−1) − 1, meaning p² divides the *first* Mersenne-like factor — a second-order lift. - Asymptotic heuristic (from arXiv 1712.08166): expected count of Wieferich primes ≤ x is ~ log log x (Cramér-type), so two below 10¹⁵ is consistent with ~log log(10¹⁵) ≈ 3.4. - No exact formula; the distribution is conjecturally Poisson with mean log log x. CONNECTION: - The ratio 1093/3511 ≈ 0.3113 — not directly a golden ratio. But note: 3511 − 1093 = 2418; 2418/3511 ≈ 0.6887 (≈ 1 − 0.3113). No clean 0.618 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com