abc Conjecture Implies Infinite Non-Fibonacci-Wieferich Primes — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The abc conjecture implies infinitely many non-Fibonacci-Wieferich primes; Wall-Sun-Sun primes (Fibonacci-Wieferich) are conjecturally nonexistent, and their rarity is tied to deep congruences modulo p². | MATH: Wieferich prime condition: 2^(p−1) ≡ 1 (mod p²). Wall-Sun-Sun prime condition: p² | F_p (Fibonacci number at prime index p). abc conjecture (Masser–Oesterlé–Szpiro): for ε>0, rad(abc)^(1+ε) ≥ c_ε·c, where rad(n)=∏_{p|n} p. Key result (arXiv:1511.01210): abc ⇒ infinitely many primes p with p ∤ F_p (non-Fibonacci-Wieferich). Heuristic density: ~log(log x) such primes below x — extremely sparse. | CONNECTION: The Fibonacci sequence itself encodes the golden ratio φ = (1+√5)/2 ≈ 1.618, and its inverse φ⁻¹ ≈ 0.618. The condition p² | F_p links prime structure to the Fibonacci lattice — a 1D projection of the 2D square lattice along slope φ. The abc conjecture's radical function rad(n) is multiplicative over prime factors, echoing the prime lattice of Z — but the Fibonacci-W
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Related
- Fibonacci Words on E8 Root Vectors Enable Topological Quantum Error Correction — E8 Intelligence Research
- E8 132Hz Phi-Coupling Generates Non-Clifford T via Fibonacci Anyons — E8 Intelligence Research
- Fibonacci-Lucas Tree-Phase Resonance Identity in E8 — E8 Intelligence Research
- Fibonacci-Galois Encoding of E8 Root Subsystem Decoherence Cascades — E8 Intelligence Research
- Φ‑Resonant E8‑HCP Phononic Lattice Enables Room‑Temperature Fibonacci Anyon Channels — E8 Intelligence Research
BibTeX
@misc{caldin2026conjecture,
title = {{abc Conjecture Implies Infinite Non-Fibonacci-Wieferich Primes — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635565},
url = {https://zenodo.org/records/23635565},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - abc Conjecture Implies Infinite Non-Fibonacci-Wieferich Primes — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635565
UR - https://zenodo.org/records/23635565
AB - FINDING: The abc conjecture implies infinitely many non-Fibonacci-Wieferich primes; Wall-Sun-Sun primes (Fibonacci-Wieferich) are conjecturally nonexistent, and their rarity is tied to deep congruences modulo p². | MATH: Wieferich prime condition: 2^(p−1) ≡ 1 (mod p²). Wall-Sun-Sun prime condition: p² | F_p (Fibonacci number at prime index p). abc conjecture (Masser–Oesterlé–Szpiro): for ε>0, rad(abc)^(1+ε) ≥ c_ε·c, where rad(n)=∏_{p|n} p. Key result (arXiv:1511.01210): abc ⇒ infinitely many primes p with p ∤ F_p (non-Fibonacci-Wieferich). Heuristic density: ~log(log x) such primes below x — extremely sparse. | CONNECTION: The Fibonacci sequence itself encodes the golden ratio φ = (1+√5)/2 ≈ 1.618, and its inverse φ⁻¹ ≈ 0.618. The condition p² | F_p links prime structure to the Fibonacci lattice — a 1D projection of the 2D square lattice along slope φ. The abc conjecture's radical function rad(n) is multiplicative over prime factors, echoing the prime lattice of Z — but the Fibonacci-W Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -