Lucas non-Wieferich primes in arithmetic progressions and the $abc$ conjecture

Abstract

We prove the lower bound for the number of Lucas non-Wieferich primes in arithmetic progressions. More precisely, for any given integer $k\ge 2$ there are $\gg \log x$ Lucas non-Wieferich primes $p\le x$ such that $p\equiv \pm 1(\mathrm{mod}k)$, assuming the $abc$ conjecture for number fields. Further, we discuss some applications of Lucas sequences in Cryptography.

12 pages, added some applications in cryptography