  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.
