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2012 Multi-Frey $\mathbb{Q}$-curves and the Diophantine equation $a^2+b^6=c^n$
Michael A. Bennett, Imin Chen
Algebra Number Theory 6(4): 707-730 (2012). DOI: 10.2140/ant.2012.6.707

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.

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Michael A. Bennett. Imin Chen. "Multi-Frey $\mathbb{Q}$-curves and the Diophantine equation $a^2+b^6=c^n$." Algebra Number Theory 6 (4) 707 - 730, 2012. https://doi.org/10.2140/ant.2012.6.707

Information

Received: 25 October 2010; Revised: 23 February 2011; Accepted: 1 April 2011; Published: 2012
First available in Project Euclid: 20 December 2017

zbMATH: 1264.11022
MathSciNet: MR2966716
Digital Object Identifier: 10.2140/ant.2012.6.707

Subjects:
Primary: 11D41
Secondary: 11D61 , 11G05 , 14G05

Keywords: $\mathbb{Q}$-curves , Fermat equations , Galois representations , multi-Frey techniques

Rights: Copyright © 2012 Mathematical Sciences Publishers

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Vol.6 • No. 4 • 2012
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