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

TL;DR

Under the assumption of the $abc$ conjecture for number fields, the paper establishes a lower bound on the count of Lucas non-Wieferich primes in specific arithmetic progressions, with implications for cryptography.

Contribution

It proves a logarithmic lower bound on Lucas non-Wieferich primes in arithmetic progressions assuming the $abc$ conjecture, linking number theory and cryptography.

Findings

At least proportional to log x Lucas non-Wieferich primes in certain progressions

Conditional proof based on the $abc$ conjecture for number fields

Applications of Lucas sequences in cryptography

Abstract

We prove the lower bound for the number of Lucas non-Wieferich primes in arithmetic progressions. More precisely, for any given integer $k\geq 2$ there are $\gg \log x$ Lucas non-Wieferich primes $p\leq x$ such that $p\equiv\pm1\pmod{k}$, assuming the $abc$ conjecture for number fields. Further, we discuss some applications of Lucas sequences in Cryptography.