  For $m \geq 3$, we define the $m$th order pyramidal number by \[
\mathrm{Pyr}_m(x) = \frac{1}{6} x(x+1)((m-2)x+5-m). \] In a previous paper,
written by the first-, second-, and fourth-named authors, all solutions to the
equation $\mathrm{Pyr}_m(x) = y^2$ are found in positive integers $x$ and $y$,
for $6 \leq m \leq 100$. In this paper, we consider the question of higher
powers, and find all solutions to the equation $\mathrm{Pyr}_m(x) = y^n$ in
positive integers $x$, $y$, and $n$, with $n \geq 3$, and $5 \leq m \leq 50$.
We reduce the problem to a study of systems of binomial Thue equations, and use
a combination of local arguments, the modular method via Frey curves, and
bounds arising from linear forms in logarithms to solve the problem.
