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

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

2011-03-10

arxiv.org/abs/1103.1979v1

Mathematics

Number Theory

Scientific paper

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.

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