Let $d$ be a positive integer which is not a square. Assume that the continued fraction expansion of $\sqrt{d}$ takes the form $\left[a_{0}, \overline{a_{1}, a_{2}, \ldots, a_{m}}\right]$.

(a) Define the convergents $p_{n} / q_{n}$, and show that $p_{n}$ and $q_{n}$ are coprime.

(b) The complete quotients $\theta_{n}$ may be written in the form $\left(\sqrt{d}+r_{n}\right) / s_{n}$, where $r_{n}$ and $s_{n}$ are rational numbers. Use the relation

$$\sqrt{d}=\frac{\theta_{n} p_{n-1}+p_{n-2}}{\theta_{n} q_{n-1}+q_{n-2}}$$

to find formulae for $r_{n}$ and $s_{n}$ in terms of the $p$ 's and $q$ 's. Deduce that $r_{n}$ and $s_{n}$ are integers.

(c) Prove that Pell's equation $x^{2}-d y^{2}=1$ has infinitely many solutions in integers $x$ and $y$.

(d) Find integers $x$ and $y$ satisfying $x^{2}-67 y^{2}=-2$.