Multi-Frey ℚ-curves and the Diophantine equation a2 + b6 = cn

Bennett, Michael A.; Chen, Imin

doi:10.2140/ant.2012.6.707

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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

DOI: 10.2140/ant.2012.6.707

Abstract

We show that the equation $a^{2} + b^{6} = c^{n}$ 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 $ℚ$ -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

Vol. 6, No. 4, 2012