Find the integer, \(n\), that satisfies \(n^2<33127<(n+1)^2\). Find also a small integer \(m\) such that \((n+m)^2-33127\) is a perfect square. Hence express \(33127\) in the form \(pq\), where \(p\) and \(q\) are integers greater than \(1\).
By considering the possible factorisations of \(33127\), show that there are exactly two positive values of \(m\) for which \((n+m)^2-33127\) is a perfect square, and find the other value.
