Example 2 Find all positive integer solutions $(x, y, z)$ for $8^{x}+15^{y}=17^{x}$.
Solution: Taking $\bmod 8$ we get: $(-1)^{y} \equiv 1(\bmod 8)$, $y$ is even, taking $\bmod 7$ we get: $2 \equiv 3^{x}(\bmod 7)$, $z$ is even, let $y=$ $2 m, z=2 t$, then
$$
\begin{array}{l}
2^{3 x}=17^{2 t}-15^{2 m}=\left(17^{t}-15^{m}\right)\left(17^{t}+15^{m}\right), \\
\therefore 17^{t}-15^{m}=2^{s}, 17^{t}+15^{m}=2^{3 x-s},
\end{array}
$$

Solving we get $17^{t}=2^{5-1}+2^{3 x-5-1}$, since $17$ is odd,
$\therefore s=1$ (because $3 x-s=1$ is impossible by the latter equation),
$$
17^{t}-15^{m}=2 \text {, }
$$

When $m \geqslant 2$, $(-1)^{t}-(-3)^{m} \equiv(-1)^{t} \equiv 2$ $(\bmod 9)$, contradiction!
$\therefore m=1, t=1, x=y=z=2$, all positive integer solutions $(x, y, z)=(2,2,2)$.