Powers in Lucas Sequences via Galois Representations

2013/07/18 by Jesse Silliman, Isabel Vogt, Silliman, Jesse +1
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1307.5078

openalex publication_date 2013/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Perfect powers in second order linear recurrences
• Modular Elliptic Curves and Fermat's Last Theorem

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• Caractère d'isogénie et critères d'irréductibilité
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