ABC Implies There are Infinitely Many non-Fibonacci-Wieferich Primes - An Application of ABC Conjecture over Number Fields

TL;DR

This paper introduces a generalized concept of Fibonacci-Wieferich primes based on a set of algebraic numbers and demonstrates, assuming the abc conjecture over number fields, that infinitely many such primes are non-Wieferich, with additional conjectures and heuristics.

Contribution

It extends the notion of Fibonacci-Wieferich primes to an X-base setting and proves infinitude under the abc conjecture, also proposing new conjectures and heuristic perspectives.

Findings

Infinitely many non-X-base Fibonacci-Wieferich primes assumed under abc conjecture.

Introduction of a new conjecture on the rank of the free part of an abelian group generated by X.

Heuristic analysis from arithmetic and geometric viewpoints.

Abstract

In this paper, we define $X$-base Fibonacci-Wieferich prime which is a generalized Wieferich prime where $X$ is a finite set of algebraic numbers. We are going to show that there are infinitely many non-$X$-base Fibonacci-Wieferich primes assuming the $abc$-conjecture of Masser-Oesterl\'{e}-Szpiro for number fields. We also provide a new conjecture concerning the rank of free part of abelian group generated by all elements in $X$, and we will use the arithmetic point of view and geometric point of view to give heuristic.