2025/03/24 by Graves, Hester, Weiss, Benjamin
#11A41 #11B25 #11R04 #11R11 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2503.19144
Silverman showed that, assuming the abc conjecture, there are ≫ log x non-Wieferich primes base a less than x \citesilverman, for all non-zero a. This inspired Graves and Murty \citeGraves, Chen and Ding \citeChen1 \citeChen2, and then Ding \citeDing to find growth results, assuming the abc conjecture, for non-Wieferich primes p base a, where p ≡ 1 \pmodk for integers k ≥ 2. In light of Murty, Srinivas, and Subramani's recent work on `the Wieferich primes conjecture' and Euclidean algorithms in number fields \citemurty, number theorists need results on non-Wieferich places in number fields. We prove analogues of the results of Graves & 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.