Non-trivial Integer Solutions of $x^r+y^r=Dz^p$

TL;DR

This paper proves that for fixed prime r ≥ 5, infinitely many equations of the form x^r + y^r = D z^p lack non-trivial primitive integer solutions when p is large, using advanced modular methods and conjectures.

Contribution

It introduces a novel approach combining modular methods over totally real fields with conjectures to establish non-existence results for these exponential equations.

Findings

Infinitely many equations have no non-trivial primitive solutions for large p.

Results depend on conjectures, with some cases requiring only the Weak Frey--Mazur Conjecture.

The method applies to equations with fixed prime r ≥ 5 and specific residue conditions.

Abstract

In this paper, we use the modular method over totally real fields together with some standard conjectures (the Weak Frey--Mazur Conjecture and the Eichler--Shimura Conjecture) to prove that infinitely many equations of the type $x^r+y^r=Dz^p$ do not have any non-trivial primitive integer solutions, where $r \geq 5$ is a fixed prime, whenever $p$ is large enough. For $r \equiv 3 \pmod 4$, we get the same result with only assuming the Weak Frey--Mazur Conjecture.