Powers in Lucas sequences via Galois representations

Silliman, Jesse; Vogt, Isabel

doi:10.1090/S0002-9939-2014-12316-1

Powers in Lucas sequences via Galois representations
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by Jesse Silliman and Isabel Vogt

Proc. Amer. Math. Soc. 143 (2015), 1027-1041

DOI: https://doi.org/10.1090/S0002-9939-2014-12316-1

Published electronically: November 5, 2014

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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.

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