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The Diophantine equation \(X^3=u+v\) over real quadratic fields. (English) Zbl 1228.11039
Let \(k\) be a real quadratic field, \({\mathcal O}_k\) the ring of integers, \({\mathcal O}_k^*\) the group of units. The author solves the Diophantine equation \(X^3=u+v\), \(X\in{\mathcal O}_k\), \(u,v\in{\mathcal O}_k^*\). He presents sufficient conditions for the non-existence of elliptic curves with everywhere good reduction over \(k\).

MSC:
11D25 Cubic and quartic Diophantine equations
11G05 Elliptic curves over global fields
Software:
ecdata
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