engleski

A remark on the Diophantine equation f(x)=g(y)

Let K be an algebraic number field, and let h(x)=x^3+ax be a polynomial over K. We prove that there exists infinitely many b in K such that the equation dy^2=x^3+ax+b has no solutions over K for infinitely many d in K*/K* 2. The proof is based on recent results of B. Mazur and K. Rubin on the 2-Selmer rank in families of quadratic twists of elliptic curves over number fields. On the other side, it is known that if the parity conjecture is valid, then there exist a number field K and a cubic polynomial f irreducible over K, such that the equation dy^2=f(x) has infinitely many solutions for each d in K*.

Elliptic curve; quadratic twist; 2-Selmer rank; number field

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