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arXiv · 2210.10575

Polynomial $D(4)$-quadruples over Gaussian Integers

Marija Bliznac Trebješanin·Sanda Bujačić Babić
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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 $$(a+b-c-d)^2=(ab+4)(cd+4)$$ holds for every $D(4)$-quadruple in $\mathbb{Z}[i][X]$.

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Marija Bliznac Trebješanin, Sanda Bujačić Babić. 2022-10-19. Polynomial $D(4)$-quadruples over Gaussian Integers. https://doi.org/10.3336/gm.59.1.01

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