A top hat for Moser's four mathemagical rabbits

TL;DR

This paper revisits Moser's 1953 result on the non-existence of small solutions to a specific exponential sum equation, providing a new proof based on a von Staudt-Clausen type theorem that unifies the original identities.

Contribution

The paper introduces a novel proof of Moser's theorem using a von Staudt-Clausen type theorem, simplifying and unifying the derivation of the four identities involved.

Findings

Moser's theorem holds for m > 10^{10^6}.

A new proof based on classical number theory is provided.

The identities used by Moser can be derived uniformly from the theorem.

Abstract

If the equation 1^k+2^k+...+(m-2)^k+(m-1)^k=m^k has an integer solution with k>1, then m>10^{10^6}. Leo Moser showed this in 1953 by remarkably elementary methods. His proof rests on four identities he derives separately. It is shown here that Moser's result can be derived from a von Staudt-Clausen type theorem (an easy proof of which is also presented here). In this approach the four identities can be derived uniformly. The mathematical arguments used in the proofs were already available during the lifetime of Lagrange (1736-1813).