On the extendibility of the set \(\{1,2,5\}\) (Q2707847)

Let \(t\) be a nonzero integer and \(S\) a set of positive integers. \(S\) is called a \(P_t\)-set if, for any two distinct elements \(x\) and \(y\) of \(S\), the integer \(xy+t\) is a perfect square. A \(P_t\)-set is called extendible if there exists a positive integer \(a\not\in S\) such that \(S\cup \{a\}\) is still a \(P_t\)-set. In this paper it is shown by elementary methods that the \(P_{-1}\)-set \(\{1,2,5\}\) is nonextendible.