Example 3 Solve the indeterminate equations
$$x^{2}-29 y^{2}=-1$$

and
$$x^{2}-29 y^{2}=1$$

for all solutions.
For $y=1,2, \cdots, 12$, it is known by calculation that $\pm 1+29 y^{2}$ are not perfect squares. When $y=13$, $-1+29 \cdot 13^{2}=70^{2}$. Therefore, $x=70, y=13$ is the smallest positive solution of the indeterminate equation (23). By
$$(70+13 \sqrt{29})^{2}=9801+1820 \sqrt{29}$$

it is known that the smallest positive solution of the indeterminate equation (24) is $x=9801, y=1820$. By Corollary 4, all solutions of the indeterminate equation (23) are
$$x+y \sqrt{29}= \pm(70 \pm 13 \sqrt{29})^{j}, \quad j=1,3,5,7, \cdots$$

All solutions of the indeterminate equation (24) are
$$x+y \sqrt{29}= \pm(70 \pm 13 \sqrt{29})^{j}, \quad j=0,2,4,6, \cdots$$

Of course, using $\sqrt{29}=\langle 5, \overline{2,1,1,2,10}\rangle$ can yield the same result (left to the reader).