On the generalized Fermat equation x¹³ + y¹³ = zⁿ

2025/10/14 by Alex J. Best, Best, Alex J., Sander R. Dahmen +3 · 1 citation
Mathematics · #11G05 #11G10 #11G30 #Algebraic Geometry and Number Theory #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT) #Primary 11D41 #Secondary 11F80

paper · pdf · doi:10.48550/arxiv.2510.12092

openalex publication_date 2025/10/14 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28

Abstract

Let n ∈ ℤ_{≥ 2}. We study the generalized Fermat equation x¹³+y¹³=zⁿ, x,y,z ∈ ℤ, gcd(x,y,z)=1. Using a combination of techniques, including the modular method, classical descent, unit sieves, and Chabauty and Mordell--Weil sieve methods over number fields, we show that for n=5 all its solutions (a,b,c) are trivial, i.e. satisfy abc=0. Under the assumption of GRH, we also show that for n=7 there are only trivial solutions. Furthermore, we provide partial results towards solving the equation for general n ∈ ℤ_{≥ 2}, in particular that any solution (a,b,c) with 13| c is trivial.

Citations

• A modular approach to Fermat equations of signature (p,p,5) using Frey hyperelliptic curves
• Some extensions of the modular method and Fermat equations of signature (13,13,n)
• Perfect powers expressible as sums of two fifth or seventh powers
• Recipes to Fermat-type equations of the form x^r + y^r = Cz^p
• Fermat-type equations of signature (13,13,p) via Hilbert cuspforms
• Explicit Chabauty over Number Fields

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• Revisiting the Fermat-type equation x¹³ + y¹³ = 3z⁷

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