The negative Pell equation and Pythagorean triples (Q1587886)

Let \(d\) be a positive integer and \((A,B,C)\) be a primitive Pythagorean triple. In this paper, the authors prove that the Pell equation \(x^2-dy^2=-1\) is solvable in integers \(x\) and \(y\) if and only if there exist positive integers \(a,b\) and an \((A,B,C)\) such that \(d=a^2+b^2\) and \(|aA-bB|=1\). In this case, \(x=|aB+bA|\) and \(y=C\). Similarly, the authors also prove that the number \(\epsilon=(x+y\sqrt d)/2\), where \(x,y\) are relatively prime integers and \(x\) is odd, is the fundamental unit of \(\mathbb{Q}(\sqrt d)\) with the negative norm if and only if there exist positive integers \(a,b\) and an \((A,B,C)\) such that \(d=a^2+b^2\) and \(|aA-bB|=2\), and \((a,b)=(A,B)=1\). The method is elementary.