2011/12/19 by Luis Dieulefait, Luís Dieulefait, Nuno Freitas +2 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1112.4521
We improve the lower bound for the prime p in the exponent of the diophantine equation in the main theorem, thanks to new computations of coefficients of Hilbert newforms performed by John Voight
openalex publication_date 2011/12/19 · arxiv created 2012/01/29 · arxiv updated 2012/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we prove that equations of the form x13 + y13 = Czp 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.