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Let  C :  y  2 = 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 = 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  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.

Partial descent on hyperelliptic curves and the generalized Fermat equation x 3 + y 4 + z 5 =0