Balancing non-Wieferich primes in arithmetic progressions

Abstract

A prime is called a balancing non-Wieferich prime if it satisfies $B_{p - \genfrac(){}{}{8}{p}} \not \equiv 0\pmod {p^{2}},$ where $\genfrac(){}{}{8}{p}$ and $B_n$ denote the Jacobi symbol and the n-th balancing number respectively. For any positive integers $k > 2$ and $n > 1$, there are $\gg \log x / \log \log x$ balancing non-Wieferich primes $p \le x$ such that $p \equiv 1 \pmod {k}$ under the assumption of the abc conjecture for the number field $\mathbb {Q}(\sqrt{2})$ (Proc. Japan Acad. Ser. A 92 (2016) 112–116). In this paper, for any fixed M, the lower bound $\log x / \log \log x$ is improved to $(\log x/ \log \log x)(\log \log \log x)^{M}$.