Fermat-type equations of signature $$(13,13,p)$$ via Hilbert cuspforms
In this paper we prove that for \(p > 13649\) equations of the form \(x^{13} + y^{13} = Cz^{p}\) have no non-trivial primitive solutions \((a,b,c)\) such that \(13 \not \mid c\) for an infinite family of values for \(C\). Our method consists on relating a solution \((a,b,c)\) to the previous equation to a solution \((a,b,c_1)\) of another Diophantine equation with coefficients in \(\mathbb Q (\sqrt{13})\). Then we attach to \((a,b,c_1)\) a Frey curve \(E_{(a,b)}\) defined over \(\mathbb Q (\sqrt{13})\) that is not a \(\mathbb Q \)-curve. We prove a modularity result of independent interest for certain elliptic curves over totally real abelian number fields satisfying some local conditions at \(3\). This theorem, in particular, implies modularity of \(E_{(a,b)}\). This enables us to use level lowering results and apply the modular approach via Hilbert cuspforms over \(\mathbb Q (\sqrt{13})\) to prove the non-existence of \((a,b,c_1)\) and, consequently, of \((a,b,c)\).