Check that $r = 649 + 180\sqrt{13}$ produces a solution of $x^2 − 13y^2 = 1$.
Determine another positive solution of this equation $x^2 − 13y^2 = 1$.
From $r=649+180\sqrt{13}$ we have $x=649$ and $y=180$.
$$ x^2 - 13y^2 = 649^2 - 13(180)^2 = 421201 - 421200 = 1$$
By Theorem (8.21) all other positive solutions are produced by $r^n$, where $n$ is a natural number. We can try $n=2$.
$$ r^2 = (649+180\sqrt{13})^2 = 842401 + 233640\sqrt{13} $$
That is $(x,y)=(842401, 233640)$.
Let's verify this solution
$$ x^2 - 13y^2 = 842401^2 - 13(233640)^2 = 709639444801 - 709639444800 = 1$$