A note on balancing non-Wieferich primes in arithmetic progressions

Abstract

A prime p is called balancing non-Wieferich prime if B p−( 8 p ) ≡ 0 (mod p 2 ), where B n be the n-th balancing number and 8 p denotes the Jacobi symbol. Under the assumption of the abc conjecture for the number field Q[ √ 2], S. S. Rout proved that there are at least O(log x/ log log x) said primes p ≡ 1 (mod r), where r > 2 be any fixed integer. In this paper, we improve the lower bound such that for any given integer r > 2 there are ≫ log x primes p ≤ x satisfies B p−( 8p ) ≡ 0 (mod p 2 ) and p ≡ 1 (mod r), under the abc conjecture for the number field. This improves the recent result of Y. Wang and Y. Ding by adding an additional condition that the primes p are in arithmetic progression.