pell's equation converge

Question

Explain Why pell's equation $x_n+ny_n=1$, $(x_n/y_n)^2$ is coverge to n as n increase

For example, n=11 the answer $(x_n/y_n)^2$ is very close to 11 when n increase.

Answer 1

Let $\mathcal{P}_d(\mathbb{Z})^+$ be the set of the positive integer solutions to the Pell equation $x^2-dy^2=1$, we have that, if $(a,b)\in\mathcal{P}_d(\mathbb{Z})^+$, then, $a/b$ is a convergent of $\sqrt d$, when you square $a/b$ you are getting ridiculously good rational approximations of $d$, remember convergents give you the "best" rational approximations