  Consider the power pseudorandom-number generator in a finite field ${\mathbb
F}_q$. That is, for some integer $e\ge2$, one considers the sequence
$u,u^e,u^{e^2},\dots$ in ${\mathbb F}_q$ for a given seed $u\in {\mathbb
F}_q^\times$. This sequence is eventually periodic. One can consider the number
of cycles that exist as the seed $u$ varies over ${\mathbb F}_q^\times$. This
is the same as the number of cycles in the functional graph of the map
$x\mapsto x^e$ in ${\mathbb F}_q^\times$. We prove some estimates for the
maximal and average number of cycles in the case of prime finite fields.
