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Original scientific paper

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

Polynomial \(D(4)\)-quadruples over Gaussian integers

Marija Bliznac Trebješanin orcid id orcid.org/0000-0003-0640-5407 ; Faculty of Science, University of Split, Ruđera Boškovića 33, 21 000 Split, Croatia
Sanda Bujačić Babić orcid id orcid.org/0000-0001-8842-3830 ; Faculty of Mathematics, University of Rijeka, Radmile Matejčić 2, 51 000 Rijeka, Croatia


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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 \(\mathbb{Z}[i][X]\).

Keywords

Diophantine \(m\)-tuples, polynomials, regular quadruples

Hrčak ID:

318142

URI

https://hrcak.srce.hr/318142

Publication date:

3.12.2024.

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