A D(n)-m-tuple, where n is a non-zero integer, is a set of m distinct elements in a commutative ring R
such that the product of any two distinct elements plus n is a perfect square in R. In this paper, we prove
that there does not exist a D(−1)-quadruple {a,b,c,d}\documentclass[12pt]{minimal}
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\begin{document}$$\{a,b,c,d\}$$\end{document} in the ring Z[-k]\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb{Z}[\sqrt{-k}]$$\end{document}, k≥2\documentclass[12pt]{minimal}
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\begin{document}$$k\ge 2$$\end{document} with positive
integers
a<b<16a2-a-2+2k(8a2+3a+1)\documentclass[12pt]{minimal}
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\begin{document}$$a<b< 16a^2-a-2+2\sqrt{k(8a^2+3a+1)}$$\end{document} and integers c and d satisfying d<0<c\documentclass[12pt]{minimal}
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\begin{document}$$d<0<c$$\end{document}. By combining that
result with
[14, Theorem 1.1] we were able to obtain a general result on the non-existence of a D(−1)-quadruple
{a,b,c,d}\documentclass[12pt]{minimal}
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\begin{document}$$\{a,b,c,d\}$$\end{document} in Z[-k]\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb{Z}[\sqrt{-k}]$$\end{document} with integers a, b, c, d satisfying a<b≤8a-3\documentclass[12pt]{minimal}
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\begin{document}$$a<b\le 8a-3$$\end{document}. Furthermore, for a non-negative integer
i and a positive integer j, we apply the obtained results in proving of the non-existence of D(−1)-quadruples containing powers of primes pi,qj\documentclass[12pt]{minimal}
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\begin{document}$$p^i, q^j$$\end{document} with an arbitrary different primes p and q.
