A Note on Integer Solutions of the Diophantine Equation x2-dy2=1

In the equation dis any positive integer which is not a perfect square. For convenience we shall consider only those solutions of (1) for which x and yare both positive. All the others can be obtained from these. In fact, it is well known that if (x0, y0) is the minimum positive integer solution of (1), then all integer solutions (x, y) are given by and, in particular, all positive integer solutions are given by