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 ${a}^{2}+{b}^{6}={c}^{n}$ has no nontrivial positive integer solutions with $\left(a,b\right)=1$ 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