Suppose that \(x_1+y_1\sqrt{d}\) and \(x_2+y_2\sqrt{d}\) give solutions to Pell’s equation \(x^2-dy^2=1\) and \(x_1,x_2,y_1,y_2\geq 0\). Show that the following are equivalent:
\(x_1+y_1\sqrt{d} < x_2+y_2\sqrt{d}\),
\(x_1<x_2\) and \(y_1<y_2\),
\(x_1<x_2\) or \(y_1<y_2\).
If Pell’s equation \(x^2-dy^2 = 1\) has a nontrivial solution \((x_1,y_1)\), show that it has infinitely many distinct solutions.
Show that there are infinitely many triples of consecutive integers, each of which is a sum of the square of two integers.
Suppose that Pell’s equation \(x^2-dy^2=1\) has a solution \((x_1,y_1)\) where \(x_1,y_1\) are positive and \(y_1\) is minimal among all solutions with positive \(x,y\). Show that if \(x+y\sqrt{d}\) gives a solution to \(x^2-dy^2=1\), then \(x+y\sqrt{d}=\pm(x_1+y_1\sqrt{d})^k\) for some integer \(k\).