Primitive Integral Solutions to x^2 + y^3 = z^{10}

TL;DR

This paper classifies all primitive integer solutions to the equation x^2 + y^3 = z^{10} using a combination of advanced number-theoretic techniques, including modular, local, and Chabauty methods over number fields.

Contribution

It introduces a novel combination of modular, number field, and Chabauty techniques to solve a specific Diophantine equation for primitive solutions.

Findings

Complete classification of primitive solutions to x^2 + y^3 = z^{10}

Development of hybrid methods combining modular and number field techniques

Application of Chabauty methods over number fields for Diophantine equations

Abstract

We classify primitive integer solutions to x^2 + y^3 = z^10. The technique is to combine modular methods at the prime 5, number field enumeration techniques in place of modular methods at the prime 2, Chabauty techniques for elliptic curves over number fields, and local methods.