Let \(n,m\) be integers with \(n\neq 0\) and \(m\geq 2\). Then a \(D(n)\)-\(m\)-tuple (say over a ring \(R\)) is a set of \(m\) non-zero elements in \(R\) such that the product of any two distinct elements plus \(n\) is a perfect square in \(R\). The authors show that there does not exist a \(D(-1)\)-quadruple \(\{a, b, c, d\}\) in \(\mathbb{Z}[\sqrt{-k}]\) with integer \(k\geq 2\), such that \(a,b\) are positive integers with \(a<b\leq 8a-3\) and \(c,d\) are negative integers. As a corollary, it is shown that such a \(D(-1)\)-pair \(\{a, b\}\) cannot be extended to a \(D(-1)\)-quintuple \(\{a, b, c, d, e\}\) in \(\mathbb{Z}[\sqrt{-k}]\) with integers \(c,d,e\). Further, the latter result is applied to the \(D(-1)\)-pair \(\{p^i, q^j\}\) with \(p,q\) being different primes and \(i,j\) positive integers, and it is shown that this pair cannot be extended to a \(D(-1)\)-quintuple in \(\mathbb{Z}[\sqrt{-k}]\) at all.
In the proofs several things are combined, among others various gap principles and systems of Pellian equations.