\(D(-1)\) tuples in imaginary quadratic fields
This paper deals with a generalization of classical Diophantine \(m\)-tuples in the ring of integers of a number field. Let \(K\) be a number field, and let \(O_K\) be its ring of integers. A set \(\{a_1, \ldots, a_m\} \subset O_K \setminus \{0\}\) is said to be a Diophantine \(m\)-tuple in \(O_K\) with the property \(D(n)\), where \(n \in O_K\), if there exists \(x_{ij} \in O_K\) such that \(a_ia_j + n = x^2_{ij}\), for all \(1 \leq i < j \leq m\). In this paper, it is proved that in case where \(K\) is an imaginary quadratic number field, for any integer \(m > 36\) there does not exist any Diophantine \(m\)-tuple in \(O_K\) with the property \(D(-1)\).
scientific article; zbMATH DE number 7377607