Let $$n\ne 0$$ be an integer. A set of m distinct positive integers $$\{a_1,a_2,\ldots ,a_m\}$$ is called a D(n)-m-tuple if $$a_ia_j + n$$ perfect square for all $$1\le i < j \le m$$ . k In this paper, we prove that $$\{k,k+1,c,d\}$$ $$D(-k)$$ -quadruple with $$c>1$$ , then $$d=1$$ The proof relies not only on standard methods in field (Baker’s linear forms logarithms and the hypergeometric met...