Wall's Conjecture and the ABC Conjecutre

TL;DR

This paper demonstrates that the abc conjecture implies infinitely many non-Fibonacci-Wieferich primes and offers a new heuristic estimate for their distribution.

Contribution

It establishes a link between the abc conjecture and the existence of infinitely many non-Fibonacci-Wieferich primes, providing new insights into their distribution.

Findings

Infinite non-Fibonacci-Wieferich primes implied by abc conjecture

New heuristic for counting such primes

Connection between abc conjecture and prime properties

Abstract

We show that the $abc$ conjecture of Masser-Oesterl\'{e}-Szpiro for number fields implies that there are infinitely many non-Fibonacci-Wieferich primes. We also provide a new heuristic for the number of such primes beneath a certain value.