Non-Wieferich primes under the abc conjecture

Abstract

Assuming the abc conjecture, Silverman proved that, for any given positive integer a⩾2, there are ≫log⁡x primes p⩽x such that ap−1≢1(modp2). In this paper, we show that, for any given integers a⩾2 and k⩾2, there still are ≫log⁡x primes p⩽x satisfying ap−1≢1(modp2) and p≡1(modk), under the assumption of the abc conjecture. This improves a recent result of Chen and Ding.