Glasnik Matematicki, Vol. 46, No. 2 (2011), 333338.
A REMARK ON THE DIOPHANTINE EQUATION f(x)=g(y)
Ivica Gusić
Faculty of Chemical Engineering and Technology, University of Zagreb, Marulićev trg 19, 10000 Zagreb, Croatia
email: igusic@fkit.hr
Abstract. 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 K such that the equation dy^{2}=x^{3}+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 2Selmer 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 K^{*}.
2000 Mathematics Subject Classification.
11G05, 14G05.
Key words and phrases. Elliptic curve, quadratic twist, 2Selmer rank, number field.
Full text (PDF) (free access)
DOI: 10.3336/gm.46.2.05
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