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≥2k\ge 2, there are ≫logx\gg \hspace{0.25em}\log x Lucas non-Wieferich primes p≤xp\le x such that p≡±1(modk)p\equiv \pm 1\hspace{0.25em}\left({\rm{mod}}\hspace{0.33em}k), assuming the abcabc conjecture for number fields.

Keywords

• abc conjecture
• arithmetic progressions
• lucas sequences
• lucas non-wieferich primes
• wieferich primes
• 11b25
• 11b39
• 11a41
• 11n13