The extendibility of Diophantine pairs with property $D(-1)$

Abstract

A set $\{a_1, a_2, \cdots, a_m\}$ of $m$ distinct positive integers is called a $D(-1)$-$m$-tuple if the product of any distinct two elements in the set decreased by one is a perfect square. In this paper, we find a solution of Pellian equations which is constructed by $D(-1)$-triples and using this result, we prove the extendibility of $D(-1)$-pair with some conditions.