Two results on $x^r+y^r=d{z}^p$

Abstract

This note proves two theorems regarding Fermat-type equation $x^r+y^r=d{z}^p$ where $r\ge 5$ is a prime. Our main result shows that, for infinitely many integers~$d$, the previous equation has no non-trivial primitive solutions such that $2\mid x+y$ or $r\mid x+y$, for a set of exponents $p$ of positive density. We use the modular method with a symplectic argument to prove this result.

8 pages, final version, to appear in Proc. Amer. Math. Soc