Non-Wieferich primes in number fields and ABC conjecture

TL;DR

This paper demonstrates, assuming the abc conjecture for number fields, that there are infinitely many non-Wieferich primes related to specific units in algebraic number fields of class number one.

Contribution

It establishes the infinitude of non-Wieferich primes in number fields under the abc conjecture, extending classical results to a broader algebraic setting.

Findings

Infinitely many non-Wieferich primes exist under the abc conjecture.

Results apply to algebraic number fields with class number one.

Connects non-Wieferich primes to units in number fields.

Abstract

Let $K/\mathbb{Q}$ be an algebraic number field of class number one and $\mathcal{O}_K$ be its ring of integers. We show that there are infinitely many non-Wieferich primes with respect to certain units in $\mathcal{O}_K$ under the assumption of the \textit{abc} conjecture for number fields.