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 $$(a+b-c-d{)}^2=(ab+4)(cd+4)$$ 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

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

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