  In this paper, we prove the lower bound for the number of balancing
non-Wieferich primes in arithmetic progressions. More precisely, for any given
integer $r\geq2$ there are $\gg\log x$ balancing non-Wieferich primes $p\leq x$
such that $p\equiv\pm1 \pmod{r}$, under the assumption of the $abc$ conjecture
for the number field $\mathbb{Q}(\sqrt{2})$. Further, we discuss some
applications of balancing sequences in cryptography.
