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On the extensions of the Diophantine triples in Gaussian integers

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A Diophantine m-tuple is a set of m distinct integers such that the product of any two distinct elements plus one is a perfect square. In this paper we study the extensibility of a Diophantine triple \(\{k-1, k+1, 16k^3-4k\}\) in Gaussian integers \({\mathbb {Z}}{[i]}\) to a Diophantine quadruple. Similar one-parameter family, \(\{k-1, k+1, 4k\}\), was studied in Franušić (Glasnik matematički 43(2):265–291, 2008), where it was shown that the extension to a Diophantine quadruple is unique (with an element \(16k^3-4k\)). The family of the triples of the same form \(\{k-1, k+1, 16k^3-4k\}\) was studied in rational integers in Bugeaud et al. (Glasg Math J 49:333–344, 2007). It appeared as a special case while solving the extensibility problem of Diophantine pair \(\{k-1, k+1\}\), in which it was not possible to use the same method as in the other cases. As authors (Bugeaud, Dujella and Mignotte) point out, the difficulty appears because the gap between \(k+1\) and \(16k^3-4k\) is not sufficiently large. We find the same difficulty here while trying to use Diophantine approximations. Then we partially solve this problem by using linear forms in logarithms.

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N. A. and A. F. were supported by the Croatian Science Foundation under the project no. IP-2018-01-1313. Z. F. acknowledges 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. The authors thank the referee for constructive suggestions on how to improve this text.

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Correspondence to Nikola Adžaga.

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Communicated by Adrian Constantin.

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Adžaga, N., Filipin, A. & Franušić, Z. On the extensions of the Diophantine triples in Gaussian integers. Monatsh Math 197, 535–563 (2022).

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