Non-trivial integer solutions of xr+yr=Dzp

2025/11/20 by Yasemin Kara, Diana Mocanu, Ekin Özman
Physics and Astronomy · Mathematics · #Advanced Mathematical Theories and Applications #Analytic Number Theory Research #Advanced Mathematical Identities

paper · doi:10.1016/j.indag.2025.11.001

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 ≥ 5 is a fixed prime, whenever p is large enough. For r ≡ 3 ( mod 4 ) , we get the same result only assuming the Weak Frey–Mazur Conjecture.

Citations

• Elliptic curves over real quadratic fields are modular
• Modular Elliptic Curves and Fermat's Last Theorem

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