Glasnik Matematicki, Vol. 46, No. 2 (2011), 333-338.

A REMARK ON THE DIOPHANTINE EQUATION f(x)=g(y)

Ivica Gusić

Abstract.   Let K be an algebraic number field, and let h(x)=x³+ax be a polynomial over K. We prove that there exists infinitely many b K such that the equation dy²=x³+ax+b has no solutions over K for infinitely many d 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 dy²=f(x) has infinitely many solutions for each d K^*.

2000 Mathematics Subject Classification.   11G05, 14G05.

Key words and phrases.   Elliptic curve, quadratic twist, 2-Selmer rank, number field.

DOI: 10.3336/gm.46.2.05

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