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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
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