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

In this paper we prove that equations of the form $x^{13}+y^{13}=C{z}^p$ have no non-trivial primitive solutions (a,b,c) such that $13\nmid c$ if $p > 4992539$foraninfinitefamilyofvaluesfor$C$. 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 $\text{Q}(\sqrt{13})$. 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 $\text{Q}(\sqrt{13})$. We also prove a modularity result for elliptic curves over totally real cyclic number fields of interest by itself.