The \(abc\) conjecture implies infinitely many non-Wieferich places for fixed bases in number field

Abstract

Silverman showed that, assuming the \(abc\) conjecture, there are \({\gg} \log x\) non-Wieferich primes base \( a\) less than \(x\) [14], for all non-zero \(a\). This inspired Graves and Murty [8], Chen and Ding [1,2], and then Ding [4] to find growth results, assuming the \(abc\) conjecture, for non-Wieferich primes \(p\) base \( a\), where \(p \equiv 1 \pmod{k}\) for integers \(k \geq 2\). In light of Murty, Srinivas, and Subramani's recent work on 'the Wieferich primes conjecture' and Euclidean algorithms in number fields [12], number theorists need results on non-Wieferich places in number fields. We prove analogues of the results of Graves and Murty and Ding, and show Ding's result holds for all bases \(a\) in all imaginary quadratic fields' rings of integers, with 31 explicitly listed exceptions.

Along the way, we generalize useful results on rational integers to algebraic integers.