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Diophantine quadruples in \(\mathbb {Z}[i][X]\)

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Abstract

In this paper, we prove that every Diophantine quadruple in \(\mathbb {Z}[i][X]\) is regular. More precisely, we prove that if \(\{a, b, c, d\}\) is a set of four non-zero polynomials from \(\mathbb {Z}[i][X]\), not all constant, such that the product of any two of its distinct elements increased by 1 is a square of a polynomial from \(\mathbb {Z}[i][X]\), then

$$\begin{aligned} (a+b-c-d)^2=4(ab+1)(cd+1). \end{aligned}$$

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Notes

  1. In an (improper) D(1)-quadruple, we cannot have two equal non-constant polynomials because then it would be for example \((b-w)(b+w)=-1\) which is not possible since both factors on the left hand side of this equation cannot be constant.

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Acknowledgements

The authors were supported by the Croatian Science Foundation under the Project No. IP-2018-01-1313.

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Authors and Affiliations

  1. Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kačića-Miošića 26, 10000, Zagreb, Croatia

    Alan Filipin

  2. Department of Mathematics, University of Rijeka, Radmile Matejčić 2, 51000, Rijeka, Croatia

    Ana Jurasić

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  1. Alan Filipin
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  2. Ana Jurasić
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Correspondence to Ana Jurasić.

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Filipin, A., Jurasić, A. Diophantine quadruples in \(\mathbb {Z}[i][X]\). Period Math Hung 82, 198–212 (2021). https://doi.org/10.1007/s10998-020-00353-y

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