On power values of pyramidal numbers, II

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.