  An exact r-coloring of a set $S$ is a surjective function $c:S \rightarrow
\{1, 2, \ldots,r\}$. A rainbow solution to an equation over $S$ is a solution
such that all components are a different color. We prove that every 3-coloring
of $\mathbb{N}$ with an upper density greater than $(4^s-1)/(3 \cdot 4^s)$
contains a rainbow solution to $x-y=z^k$. The rainbow number for an equation in
the set $S$ is the smallest integer $r$ such that every exact $r$-coloring has
a rainbow solution. We compute the rainbow numbers of $\mathbb{Z}_p$ for the
equation $x-y=z^k$, where $p$ is prime and $k\geq 2$.
