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Powers in Lucas sequences via Galois representations


Authors: Jesse Silliman and Isabel Vogt
Journal: Proc. Amer. Math. Soc. 143 (2015), 1027-1041
MSC (2010): Primary 11B39; Secondary 11G05
DOI: https://doi.org/10.1090/S0002-9939-2014-12316-1
Published electronically: November 5, 2014
MathSciNet review: 3293720
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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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Additional Information

Jesse Silliman
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
Email: silliman@stanford.edu

Isabel Vogt
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: ivogt@mit.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12316-1
Received by editor(s): July 18, 2013
Published electronically: November 5, 2014
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.