On the generalized Fermat equation $x^{13} + y^{13} = z^n$

TL;DR

This work addresses the generalized Fermat equation $x^{13}+y^{13}=z^{n}$ by blending modular methods, multi-Frey techniques, and descent with unit sieves to handle two challenging exponent cases. The authors prove that for $n=5$ (unconditional) and (conditionally on GRH) $n=7$, all primitive solutions are trivial, and they also establish that whenever $13igm|c$, any solution is trivial for all $nigge 2$. A key strategy reduces the problem to rational points on hyperelliptic curves over the cubic field $K$, then prunes possible unit contributions via a novel unit sieve, and finally applies Chabauty–Mordell–Weil computations to enumerate $K$-rational points in the relevant cases. The results illustrate how a carefully orchestrated combination of modern Diophantine tools can push beyond previous limits in solving generalized Fermat equations, and they lay groundwork for tackling similar equations with large exponents using a hybrid modular-descent framework.

Abstract

Let $n \in \mathbb{Z}_{\geq 2}$. We study the generalized Fermat equation \[x^{13}+y^{13}=z^n, \quad x,y,z \in \mathbb{Z}, \quad \gcd(x,y,z)=1.\] 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 $n \in \mathbb{Z}_{\geq 2}$, in particular that any solution $(a,b,c)$ with $13\mid c$ is trivial.

Theorems & Definitions (9)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Theorem 2.3
• proof
• Remark 2.4