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The Diophantine equation $$X^3=u+27v$$ over real quadratic fields. (English) Zbl 1227.11052
Let $$k={\mathbb Q}(\sqrt{6})$$ or $$k={\mathbb Q}(\sqrt{3p})$$, where $$p$$ is a prime number, $$p\neq 3$$, $$p\equiv 3(\bmod 4)$$, $${\mathcal O}_k$$ the integers of $$k$$, $${\mathcal O}_k^\times$$ the units. The author proves that if the equation $$X^3=u+27v$$ has solutions $$X\in {\mathcal O}_k$$, $$u,v\in{\mathcal O}_k^\times$$, then $$k={\mathbb Q}(\sqrt{6})$$ or $$k={\mathbb Q}(\sqrt{33})$$, and the only solutions are $(X,u,v)=(w_1(4\pm \sqrt{6}),w_1^3,w_1^3(5\pm 2\sqrt{6}))$ for any $$w_1\in{\mathcal O}_{{\mathbb Q}(\sqrt{6})}^\times$$, or $(X,u,v)=(w_2(5\pm \sqrt{33}),-w_2^3,w_2^3(23\pm 4\sqrt{33}))$ for any $$w_2\in{\mathcal O}_{{\mathbb Q}(\sqrt{33})}^\times$$.
##### MSC:
 11D25 Cubic and quartic Diophantine equations 11R11 Quadratic extensions
##### Keywords:
real quadratic field; cubic Diophantine equation
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##### References:
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