Extension of a parametric family of Diophantine triples in Gaussian integers (Q520001)

A Diophantine \(m\)-tuple in a ring \(R\) is a set of the form \(\{a_1,\dots,a_m\}\subset R\) with \(a_ia_j+1\) being a square in \(R\) for all \(i\neq j\). The authors prove that if \(\{k, 4k + 4, 9k + 6, d\}\), where \(k\in\mathbb{Z}[i]\), \(k\neq 0,-1\), is a Diophantine quadruple in the ring \({\mathbb Z}[i]\), then \(d = 144k^3 + 240k^2 + 124k + 20\).