Glasnik matematički, Vol. 46 No. 2, 2011.
Original scientific paper
https://doi.org/10.3336/gm.46.2.05
A remark on the Diophantine equation f(x)=g(y)
Ivica Gusic
; Faculty of Chemical Engineering and Technology, University of Zagreb, Marulićev trg 19, 10000 Zagreb, Croatia
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*.
Keywords
Elliptic curve; quadratic twist; 2-Selmer rank; number field
Hrčak ID:
74263
URI
Publication date:
23.11.2011.
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