On power values of pyramidal numbers, II

TL;DR

This paper investigates solutions to pyramidal number equations involving higher powers, extending previous work by solving for all positive integer solutions where the pyramidal number equals a perfect power with exponent at least 3.

Contribution

It generalizes earlier results by determining all solutions to  pyr_m(x) = y^n for higher powers and a broader range of m, using advanced number theoretic techniques.

Findings

All solutions to  pyr_m(x) = y^n for n \u2265 3 and 5 q m q 50 are found.

The problem reduces to systems of binomial Thue equations, which are solved using modern methods.

The approach combines local arguments, modular method, and bounds from linear forms in logarithms.

Abstract

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.