Glasnik Matematicki, Vol. 59, No. 1 (2024), 1-31. \( \)

POLYNOMIAL \(D(4)\)-QUADRUPLES OVER GAUSSIAN INTEGERS

Marija Bliznac Trebješanin and Sanda Bujačić Babić

Faculty of Science, University of Split, Ruđera Boškovića 33, 21 000 Split, Croatia
e-mail:marbli@pmfst.hr

Faculty of Mathematics, University of Rijeka, Radmile Matejčić 2, 51 000 Rijeka, Croatia
e-mail:sbujacic@math.uniri.hr

Abstract.   A set \(\{a, b, c, d\}\) of four non-zero distinct polynomials in \(\mathbb{Z}[i][X]\) is said to be a Diophantine \(D(4)\)-quadruple if the product of any two of its distinct elements increased by 4 is a square of some polynomial in \(\mathbb{Z}[i][X]\). In this paper we prove that every \(D(4)\)-quadruple in \(\mathbb{Z}[i][X]\) is regular, or equivalently that the equation \[ (a+b-c-d)^2=(ab+4)(cd+4)\hspace{20ex} \] holds for every \(D(4)\)-quadruple in \(\mathbb{Z}[i][X]\).

2020 Mathematics Subject Classification.   11D09, 11D45

Key words and phrases.   Diophantine \(m\)-tuples, polynomials, regular quadruples

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https://doi.org/10.3336/gm.59.1.01

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