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

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Siksek, Samir ; Stoll, Michael:
Partial descent on hyperelliptic curves and the generalized Fermat equation x³+y⁴+z⁵=0.
In: The Bulletin of the London Mathematical Society. Bd. 44 (2012) Heft 1 . - S. 151-166.
ISSN 0024-6093
DOI: https://doi.org/10.1112/blms/bdr086

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Abstract

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.