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

2024/04/10 by Yasemin Kara, Diana Mocanu, Kara, Yasemin +3 · 1 citation
Mathematics · #11D41 #11G05 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2404.07319

openalex publication_date 2024/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ≥ 5 is a fixed prime, whenever p is large enough. For r ≡ 3 \pmod 4, we get the same result with only assuming the Weak Frey--Mazur Conjecture.

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• A survey on the generalized Fermat equation of various signatures over totally real fields

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