Lucas non-Wieferich primes in arithmetic progressions (Q2422597)

First, by the help of Lucas sequences, the author introduces the notion of so-called Lucas-Wieferich primes. It is not known whether there are infinitely many Wieferich primes -- and the same is true for non-Wieferich primes. However, conditionally it is proved that there are infinitely many non-Wieferich ones. In the present paper it is proved that under a suitable version of the $abc$ conjecture for number fields, for any $k\geq 2$ there are infinitely many Lucas non-Wieferich primes $p$ with $p\equiv 1\pmod{k}$. In fact, a lower bound for the number of such primes is also given.