Let $\theta$ be a real number with continued fraction expansion $\left[a_{0}, a_{1}, a_{2}, \ldots\right]$. Define the convergents $p_{n} / q_{n}$ (by means of recurrence relations) and show that for $\beta>0$ we have

$$\left[a_{0}, a_{1}, \ldots, a_{n-1}, \beta\right]=\frac{\beta p_{n-1}+p_{n-2}}{\beta q_{n-1}+q_{n-2}}$$

Show that

$$\left|\theta-\frac{p_{n}}{q_{n}}\right|<\frac{1}{q_{n} q_{n+1}}$$

and deduce that $p_{n} / q_{n} \rightarrow \theta$ as $n \rightarrow \infty$.

By computing a suitable continued fraction expansion, find solutions in positive integers $x$ and $y$ to each of the equations $x^{2}-53 y^{2}=4$ and $x^{2}-53 y^{2}=-7$.