Extensions of some parametric families of \(D(16)\)-triples (Q925329)

A set \(D\) of \(m\) positive integers is called a \(D(n)\)-\(m\)-tuple if for each pair \((a,b)\in D^2\) we have \(ab+n\) is a perfect square. The author considers families of \(D(16)\)-triples and shows that they can be extended only in a unique way to \(D(16)\)-quadruples. In particular he proves that for \(k\geq 5\) the triple \(\{k-4,k+4,4k\}\) can be extended only by \(d=k^3-4k\) and for \(k>5\) the triple \(\{k-4,4k,9k-12\}\) can be extended only by \(d=9k^3-48k^2+76k-32\). Moreover the triple \(\{1,20,33\}\) has only the extensions \(\{1,20,33,105\}\) and \(\{1,20,33,273\}\).