mathschallenge.net

Problem

Consider the Pell equation, $x^2 - 2y^2 = 1$, where $x$ and $y$ are positive integers.

The smallest solution is (3,2), with subsequent solutions being (17,12), (99,70), (577,408), ...

What is most interesting is that the sequence of solutions produce convergents for $\sqrt{2} = 1.414213...$

$\begin{align}\dfrac{3}{2} &= 1.5\\\dfrac{17}{12} &= 1.416666...\\\dfrac{99}{70} &= 1.414285...\\\dfrac{577}{408} &= 1.414215....\end{align}$

Assuming that at least one solution exists, prove that the ordered solutions of the equation $x^2 - dy^2 = 1$ produce convergents for $\sqrt{d}$.