The extendibility of $D(4)$-pairs

Abstract

The set of $m$ positive integers with the property that the product of any two of them increased by $4$ is a perfect square is called a $D(4)$-$m$-tuple. In this paper we consider the extensibility of general $D(4)$-pair $\{a,b\}$ and prove some results that support the conjecture that there does not exist a $D(4)$-quintuple.