13. (1) Try to find a set of positive integer solutions for the equation $x^{3}+y^{4}=z^{5}$;
(2) Is the set of positive integer solutions for this equation finite or infinite.
13. Since $2^{24}+2^{24}=2^{25}$, it follows that $\left(2^{8}, 2^{6}, 2^{5}\right)=(256,64,32)$ is a set of positive integer solutions. If $\left(x_{0}, y_{0}, z_{0}\right)$ is a set of positive integer solutions, then for $k \in \mathrm{N},\left(x_{0} k^{20}, y_{0} k^{5}, z_{0} k^{12}\right)$ is also a set of positive integer solutions to the original equation, so the set of positive integer solutions to the original equation is infinite.