Vol. 6, No. 4, 2012

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Multi-Frey $\mathbb{Q}$-curves and the Diophantine equation $a^2+b^6=c^n$

Michael A. Bennett and Imin Chen

Vol. 6 (2012), No. 4, 707–730
Abstract

We show that the equation a2 + b6 = cn has no nontrivial positive integer solutions with (a,b) = 1 via a combination of techniques based upon the modularity of Galois representations attached to certain -curves, corresponding surjectivity results of Ellenberg for these representations, and extensions of multi-Frey curve arguments of Siksek.

Keywords
Fermat equations, Galois representations, $\mathbb{Q}$-curves, multi-Frey techniques
Mathematical Subject Classification 2010
Primary: 11D41
Secondary: 11D61, 11G05, 14G05
Supplementary material

Zip file containing programs, data, and output files for the computations in the article

Milestones
Received: 25 October 2010
Revised: 23 February 2011
Accepted: 1 April 2011
Published: 25 July 2012
Authors
Michael A. Bennett
Department of Mathematics
University of British Columbia
1984 Mathematics Road
Vancouver, BC  V6T 1Z2
Canada
Imin Chen
Department of Mathematics
Simon Fraser University
8888 University Drive
Burnaby, BC  V5A 1S6
Canada