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

FINDING: The abc conjecture (Masser–Oesterlé–Szpiro) over number fields implies infinitely many non-Fibonacci-Wieferich primes; only two Wieferich primes (1093, 3511) are known below ~6.7×10¹⁵. | MATH: Wieferich condition: 2^(p−1) ≡ 1 (mod p²); Fibonacci-Wieferich: F_(p−(5/p)) ≡ 0 (mod p²), where (5/p) is the Legendre symbol. Heuristic count below x: N(x) ~ c·log log x (Wieferich), and for Fibonacci-Wieferich N_F(x) ~ c_F·log log x. abc: for ε>0, ∃K(ε): max(|a|,|b|,|c|) ≤ K(ε)·rad(abc)^(1+ε), for coprime a+b=c. | CONNECTION: The Fibonacci recurrence F_(n+1)=F_n+F_(n−1) has characteristic root φ=(1+√5)/2=1.6180339887…, with conjugate ψ=(1−√5)/2=−0.6180339887…; the ratio F_(n+1)/F_n→φ. The Legendre symbol (5/p) encodes whether √5∈F_p, i.e. whether φ exists in F_p — a direct arithmetic manifestation of the golden ratio's algebraic degree (ℚ(√5), discriminant 5, the pentagonal/icosahedral root system H₂/H₃). The log log x growth is the same double-logarithmic law governing Wieferich, Merse Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com