Non-trivial Integer Solutions of $x^r+y^r=D{z}^p$

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 $x^r+y^r=D{z}^p$ do not have any non-trivial primitive integer solutions, where $r\ge 5$ is a fixed prime, whenever $p$ is large enough. For $r\equiv 3(\mathrm{mod}4)$, we get the same result with only assuming the Weak Frey--Mazur Conjecture.

12 pages, 1 table. Shortened one argument and corrected typos

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