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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
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References:
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