Example 1 Find all solutions to the indefinite equations
$$x^{2}-73 y^{2}=-1$$

and
$$x^{2}-73 y^{2}=1$$
From Example 3 of §5, we know that $\sqrt{73}=\langle 8, \overline{1,1,5,5,1,1,16}\rangle$. The period is 7. Therefore, by Theorem 1 and Theorem 3, the smallest positive solution to the indeterminate equation (19) is $x=h_{6}, y=k_{6}$. It is not difficult to find that
$$h_{6} / k_{6}=\langle 8,1,1,5,5,1,1\rangle=1068 / 125$$

Thus, by Corollary 4, all solutions to (19) are
$$x+y \sqrt{73}= \pm(1068 \pm 125 \sqrt{73})^{j}, \quad j=1,3,5,7, \cdots$$

All solutions to the indeterminate equation (20) are
$$x+y \sqrt{73}= \pm(1068 \pm 125 \sqrt{73})^{j}, \quad j=0,2,4,8, \cdots .$$