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Non-existence of certain Diophantine quadruples in rings of integers of pure cubic fields. (English) Zbl 1284.11057
A $$D(w)$$-$$m$$-tuple is an $$m$$-tuple $$(a_1,\dots,a_m)$$ of positive integers such that for all $$1\leq i<j\leq m$$ $$a_ia_j+w$$ is a perfect square. Considering $$D(w)$$-quadruples A. Dujella [Acta Arith. 65, No. 1, 15–27 (1993; Zbl 0849.11018)] proved that no $$D(4k+2)$$-quadruple exists. Of course the problem can be stated in more general rings $$R$$, e.g. $$R$$ is the maximal order of some number field. In the paper under review such a generalization is considered. In particular the case that $$R$$ is the maximal order of pure cubic fields $$\mathbb Q(\root 3\of d)$$ is studied and the author obtains a result analogous to Dujella’s result in the integer case.
##### MSC:
 11D09 Quadratic and bilinear Diophantine equations 11R16 Cubic and quartic extensions 11D79 Congruences in many variables
##### Keywords:
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