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The Generalized Fermat Equation $x^2 + y^3 = z^{25}$

Nuno Freitas, Michael Stoll

TL;DR

This work tackles the generalized Fermat equation $x^2+y^3=z^{25}$ by leveraging Edwards' complete parameterization of primitive solutions to $x^2+y^3=z^5$, reducing the problem to solving five equations of the form $H(u,v)=w^5$ with $H$ degree $10$ over sextic fields. Through a careful analysis of factorization types, local solubility, Frey-curve arguments, and Chabauty methods on associated genus-2 and genus-4 curves, the authors rule out many potential cases and obtain conditional and partial unconditional results toward classifying primitive solutions. Their main conditional conclusion is that if the five quintic-type equations have only the expected solutions, the only primitive solutions to $x^2+y^3=z^{25}$ are the trivial ones and the Catalan solution $(\pm3,-2,1)$; several factorization types are fully resolved, while others remain contingent on solving higher-dimensional rational points problems. The results illustrate the deep interplay between parametric Diophantine parameterizations, modular/elliptic techniques, and explicit arithmetic geometry in advancing the finite-solution program for generalized Fermat equations.

Abstract

We consider the generalized Fermat equation (*) $x^2 + y^3 = z^{25}$. Using the known parameterization of the primitive integral solutions to $x^2 + y^3 = z^5$ (due to Edwards), we reduce the solution of (*) to the solution of five specific equations of the form $H(u,v) = w^5$, where $H$ is homogeneous of degree $10$ with coefficients in a sextic number field $K$, $u$ and $v$ are coprime (rational) integers, and $w \in K$.

The Generalized Fermat Equation $x^2 + y^3 = z^{25}$

TL;DR

This work tackles the generalized Fermat equation by leveraging Edwards' complete parameterization of primitive solutions to , reducing the problem to solving five equations of the form with degree over sextic fields. Through a careful analysis of factorization types, local solubility, Frey-curve arguments, and Chabauty methods on associated genus-2 and genus-4 curves, the authors rule out many potential cases and obtain conditional and partial unconditional results toward classifying primitive solutions. Their main conditional conclusion is that if the five quintic-type equations have only the expected solutions, the only primitive solutions to are the trivial ones and the Catalan solution ; several factorization types are fully resolved, while others remain contingent on solving higher-dimensional rational points problems. The results illustrate the deep interplay between parametric Diophantine parameterizations, modular/elliptic techniques, and explicit arithmetic geometry in advancing the finite-solution program for generalized Fermat equations.

Abstract

We consider the generalized Fermat equation (*) . Using the known parameterization of the primitive integral solutions to (due to Edwards), we reduce the solution of (*) to the solution of five specific equations of the form , where is homogeneous of degree with coefficients in a sextic number field , and are coprime (rational) integers, and .

Paper Structure

This paper contains 9 sections, 8 theorems, 68 equations, 8 tables.

Table of Contents

  1. Introduction
  2. The Edwards Parameterization
  3. Factorization Types
  4. Local Primitive Solutions
  5. The Frey curve and $X_E^\pm(5)$
  6. Factorization type $[1,1,10]$ over $\mathbb{Q}$
  7. Factorization type $[4,8]$ over $\mathbb{Q}$
  8. Factorization type $[6,6]$ over $\mathbb{Q}(\sqrt{5})$
  9. Forms that are irreducible over $\mathbb{Q}(\sqrt{5})$

Key Result

Theorem 1.1

If five specific equations of the form where $H$ is homogeneous of degree $10$ with coefficients in a sextic number field $K$ (depending on the equation), $u$ and $v$ are coprime (rational) integers and $w \in K$, have only the expected solutions (see Table table:summary below), then the only primitive integer solutions to eqn:main are t (with $A \in L$), where $L$ is a number field of degree $12

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 2.1: Edwards Ed*pages 235--236
  • Remark 2.2
  • Lemma 4.1
  • proof
  • Lemma 5.1
  • proof
  • Lemma 8.1
  • proof
  • Lemma 8.2
  • ...and 5 more