Divisibility of Power Sums and the Generalized Erdos-Moser Equation

TL;DR

This paper investigates the divisibility properties of power sums and provides a simple proof that solutions to a generalized Erdos-Moser equation must have an odd m, extending previous formulas and results.

Contribution

It generalizes Lengyel's formula for power sums when m is a power of 2 and proves that solutions to the generalized Erdos-Moser equation have an odd m using elementary methods.

Findings

Determined the highest power of 2 dividing power sums.

Extended Lengyel's formula to broader cases.

Proved that solutions to the generalized Erdos-Moser equation have odd m.

Abstract

Using elementary methods, we determine the highest power of 2 dividing a power sum 1^n + 2^n + . . . + m^n, generalizing Lengyel's formula for the case where m is itself a power of 2. An application is a simple proof of Moree's result that, if (a,m,n) is any solution of the generalized Erdos-Moser Diophantine equation 1^n + 2^n + . . . + (m-1)^n = am^n, then m is odd.