Powers in Lucas Sequences via Galois Representations

TL;DR

This paper develops a systematic method to identify perfect powers in Lucas sequences and establishes bounds on prime powers assuming the Frey-Mazur conjecture, extending previous work in the field.

Contribution

It generalizes prior results to a broad class of Lucas sequences and connects the problem to Galois representations and the Frey-Mazur conjecture for the first time.

Findings

A systematic approach to find perfect powers in Lucas sequences.

A new bound on admissible prime powers assuming the Frey-Mazur conjecture.

Extension of previous specific results to a general framework.

Abstract

Let $u_n$ be a nondegenerate Lucas sequence. We generalize the results of Bugeaud, Mignotte, and Siksek, 2006 to give a systematic approach towards the problem of determining all perfect powers in any particular Lucas sequence. We then prove a general bound on admissible prime powers in a Lucas sequence assuming the Frey-Mazur conjecture on isomorphic mod $p$ Galois representations of elliptic curves.