Abstract
For a nonzero integer n, a set of m distinct nonzero integers \(\{a_1,a_2,\ldots ,a_m\}\) such that \(a_ia_j+n\) is a perfect square for all \(1\le i<j\le m\), is called a D(n)-m-tuple. In this paper, by using properties of so-called regular Diophantine m-tuples and certain family of elliptic curves, we show that there are infinitely many essentially different sets consisting of perfect squares which are simultaneously \(D(n_1)\)-quadruples and \(D(n_2)\)-quadruples with distinct nonzero squares \(n_1\) and \(n_2\).
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Acknowledgements
The authors want to thank to Matija Kazalicki and the referees for a careful reading of our paper and for many valuable suggestions which improved the quality of the paper. The authors were supported by the Croatian Science Foundation under the Project No. IP-2018-01-1313. The authors acknowledge support from the QuantiXLie Center of Excellence, a project co-financed by the Croatian Government and European Union through the European Regional Development Fund - the Competitiveness and Cohesion Operational Programme (Grant KK.01.1.1.01.0004).
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Dujella, A., Petričević, V. Doubly regular Diophantine quadruples. RACSAM 114, 189 (2020). https://doi.org/10.1007/s13398-020-00921-4
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DOI: https://doi.org/10.1007/s13398-020-00921-4