# Partial descent on hyperelliptic curves and the generalized Fermat equation x3+y4+z5=0

@article{Siksek2011PartialDO, title={Partial descent on hyperelliptic curves and the generalized Fermat equation x3+y4+z5=0}, author={Samir Siksek and Michael Stoll}, journal={Bulletin of the London Mathematical Society}, year={2011}, volume={44} }

Let C: y2=f(x) be a hyperelliptic curve defined over ℚ. Let K be a number field and suppose f factors over K as a product of irreducible polynomials f=f1 f2 … fr. We shall define a ‘Selmer set’ corresponding to this factorization with the property that if it is empty, then C(ℚ)=∅. 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.

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