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Glasnik matematički, Vol. 59 No. 1, 2024.

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:

30.12.2024.

Visits: 398 *

Publication date: 20 June 2024

Volume: Vol 59

Issue: Svezak 1

Pages: 1-31

DOI: 10.3336/gm.59.1.01

Article Information (continued)

Categories:

Subject: Izvorni znanstveni članak

Keywords:

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

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

Marija Bliznac Trebješanin[1]

Email: marbli@pmfst.hr

Sanda Bujačić Babić[2]

Email: sbujacic@math.uniri.hr

 

[1] Faculty of Science, University of Split, Ruđera Boškovića 33, 21 000 Split, Croatia

[2] Faculty of Mathematics, University of Rijeka, Radmile Matejčić 2, 51 000 Rijeka, Croatia

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

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