On the generalized Fermat equation $x^{13}+y^{13}=z^n$

Abstract

Let $n\in {\mathbb{Z}}_{\ge 2}$. We study the generalized Fermat equation \[x^{13}+y^{13}=z^n, \quad x,y,z \in \mathbb{Z}, \quad \gcd(x,y,z)=1.\] Using a combination of techniques, including the modular method, classical descent, unit sieves, and Chabauty and Mordell--Weil sieve methods over number fields, we show that for $n=5$ all its solutions $(a,b,c)$ are trivial, i.e. satisfy $abc=0$. Under the assumption of GRH, we also show that for $n=7$ there are only trivial solutions. Furthermore, we provide partial results towards solving the equation for general $n\in {\mathbb{Z}}_{\ge 2}$, in particular that any solution $(a,b,c)$ with $13\mid c$ is trivial.