Balancing non-Wieferich primes in arithmetic progression and $abc$ conjecture

Abstract

In this note, we shall define the balancing Wieferich prime which is an analogue of the famous Wieferich primes. We prove that, under the $abc$ conjecture for the number field $\mathbf{Q}(\sqrt{2})$, there are infinitely many balancing non-Wieferich primes. In particular, under the assumption of the $abc$ conjecture for the number field $\mathbf{Q}(\sqrt{2})$ there are at least $O(\log x/{\log \log x})$ such primes $p \equiv 1(\mathrm{mod}\ k)$ for any fixed integer $k> 2$.