Partial Descent on Hyperelliptic Curves and the Generalized Fermat Equation x^3+y^4+z^5=0

TL;DR

This paper introduces a new method using partial descent on hyperelliptic curves to solve the generalized Fermat equation with signature (3,4,5), overcoming limitations of previous techniques.

Contribution

The authors develop a novel partial descent approach on hyperelliptic curves and define a Selmer set that effectively determines the existence of rational points, solving previously intractable cases.

Findings

Successfully solved the generalized Fermat equation x^3 + y^4 + z^5 = 0

Demonstrated the effectiveness of the new method on complex Diophantine equations

Established a framework for analyzing rational points via Selmer sets

Abstract

Let C : y^2=f(x) be a hyperelliptic curve defined over the rationals. Let K be a number field and suppose f factors over K as a product of irreducible polynomials f=f_1 f_2...f_r. We shall define a "Selmer set" corresponding to this factorization with the property that if it is empty then the curve C has no rational points. We shall demonstrate the effectiveness of our new method by solving the generalized Fermat equation with signature (3,4,5), which is unassailable via the previously existing methods.