Two results on $x^r + y^r = dz^p$

TL;DR

This paper proves that for infinitely many integers d, the Fermat-type equation x^r + y^r = dz^p has no non-trivial primitive solutions with certain divisibility conditions, using modular methods and symplectic arguments.

Contribution

It establishes new non-existence results for solutions of Fermat-type equations with specific divisibility constraints, for infinitely many d and a set of exponents p of positive density.

Findings

No non-trivial primitive solutions for infinitely many d under certain divisibility conditions.

Uses modular method with symplectic argument to prove these results.

Results apply to a set of exponents p with positive density.

Abstract

This note proves two theorems regarding Fermat-type equation $x^r + y^r = dz^p$ where $r \geq 5$ 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.