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arXiv · 1112.4521

Fermat-type equations of signature (13,13,p) via Hilbert cuspforms

Luis Dieulefait·Nuno Freitas
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Abstract

In this paper we prove that equations of the form $x^{13} + y^{13} = Cz^{p}$ have no non-trivial primitive solutions (a,b,c) such that $13 \nmid c$ if $p > 4992539$ for an infinite family of values for $C$. Our method consists in relating a solution (a,b,c) to the previous equation to a solution (a,b,c_1) of another Diophantine equation with coefficients in $\Q(\sqrt{13})$. We then construct Frey-curves associated with (a,b,c_1) and we prove modularity of them in order to apply the modular approach via Hilbert cusp forms over $\Q(\sqrt{13})$. We also prove a modularity result for elliptic curves over totally real cyclic number fields of interest by itself.

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Luis Dieulefait, Nuno Freitas. 2012-01-29. Fermat-type equations of signature (13,13,p) via Hilbert cuspforms. https://arxiv.org/abs/1112.4521

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