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 $$(\log x/ \log \log x)(\log \log \log x)^{M}$$ .