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

2011/12/19 by Dieulefait, Luis, Freitas, Nuno · 2 citations
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1112.4521

Abstract

In this paper we prove that equations of the form x¹³ + y¹³ = 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,c₁) of another Diophantine equation with coefficients in \Q(√(13)). We then construct Frey-curves associated with (a,b,c₁) and we prove modularity of them in order to apply the modular approach via Hilbert cusp forms over \Q(√(13)). We also prove a modularity result for elliptic curves over totally real cyclic number fields of interest by itself.

Cited by

• A survey on the generalized Fermat equation of various signatures over totally real fields
• On the generalized Fermat equation x¹³ + y¹³ = zⁿ

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