The extension of the $$D(-k)$$-pair $$\{k,k+1\}$$ to a quadruple

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 method), but also some less typical elementary arguments dealing recurrences, as well relatively new method determination integral points hyperelliptic curves.