abc Conjecture Implies Infinitely Many Non-X-Base Fibonacci-Wieferich Primes — E8 Intelligence Research

FINDING: The abc conjecture (Masser–Oesterlé–Szpiro) implies infinitely many non-X-base Fibonacci-Wieferich primes over number fields, generalizing classical Wieferich primes via a Fibonacci-base analogue. | MATH: Wieferich condition: a^(p−1) ≡ 1 (mod p²); Fibonacci-Wieferich: F_{p−(5/p)} ≡ 0 (mod p²), where (5/p) is the Legendre symbol (5/p) = ±1. abc conjecture: for ε>0, ∃K(ε) such that for coprime a+b=c, max(|a|,|b|,|c|) ≤ K(ε)·rad(abc)^{1+ε}. rad(n) = ∏_{p|n} p. | CONNECTION: Fibonacci ratios F_{n+1}/F_n → φ = 1.6180339887...; discriminant 5 in (5/p) ties to √5 = φ + 1/φ = 2φ − 1. The Fibonacci-Wieferich modulus p² involves the quadratic field ℚ(√5), whose ring of integers has the golden-ratio unit φ. No direct base-60 or crystallographic symmetry appears in the cited result. | DEPTH: 7 Notes on scope (evidence-limited): - The arXiv paper (1511.05645) proves the implication for non-X-base Fibonacci-Wieferich primes; it does not prove infinitude of Fibonacci-Wieferich primes themse Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com