In this paper, we prove that every Diophantine quadruple in $$\mathbb {Z}[i][X]$$ is regular. More precisely, we prove that if $$\{a, b, c, d\}$$ is a set of four non-zero polynomials from $$\mathbb {Z}[i][X]$$ , not all constant, such that the product of any two of its distinct elements increased by 1 is a square of a polynomial from $$\mathbb {Z}[i][X]$$ , then $$\begin{aligned} (a+b-c-d)^2=4(ab+1)(cd+1). \end{aligned}$$

中文翻译：

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$$\mathbb {Z}[i][X]$$ 中的丢番图四元组

在本文中，我们证明 $$\mathbb {Z}[i][X]$$ 中的每个丢番图四元组都是正则的。更准确地说，我们证明如果 $$\{a, b, c, d\}$$ 是来自 $$\mathbb {Z}[i][X]$$ 的四个非零多项式的集合，不是全部常数，使得任何两个不同元素的乘积增加 1 是从 $$\mathbb {Z}[i][X]$$ 的多项式的平方，然后 $$\begin{aligned} (a+ bcd)^2=4(ab+1)(cd+1)。\end{对齐}$$