On power values of pyramidal numbers, II

Abstract

For $m\ge 3$, 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 $6\le m\le 100$. 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 $n\ge 3$, and $5\le m\le 50$. 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.

Minor corrections, to appear in Acta Arithmetica

References in corpus (1)

• A Unique Perfect Power Decagonal Number