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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
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