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

Abstract

Abstract We prove the lower bound for the number of Lucas non-Wieferich primes in arithmetic progressions. More precisely, for any given integer k ≥ 2 k\ge 2 , there are ≫ log x \gg \hspace{0.25em}\log x Lucas non-Wieferich primes p ≤ x p\le x such that p ≡ ± 1 ( mod k ) p\equiv \pm 1\hspace{0.25em}\left({\rm{mod}}\hspace{0.33em}k) , assuming the a b c abc conjecture for number fields.