Reducing the Erdos-Moser equation 1^n + 2^n + . . . + k^n = (k+1)^n modulo k and k^2

TL;DR

This paper investigates the Erdos-Moser conjecture by analyzing the equation modulo k and k^2, providing new conditions, proofs for odd exponents, and connections to number theory concepts.

Contribution

It introduces necessary and sufficient conditions for solutions to the congruences, offering a new proof for the conjecture for odd exponents and linking solutions to pseudoperfect numbers and Zagier's work.

Findings

Confirmed the conjecture for odd exponents n.

Established conditions for solutions to the congruences.

Connected solutions to pseudoperfect numbers and Zagier's results.

Abstract

An open conjecture of Erdos and Moser is that the only solution of the Diophantine equation in the title is the trivial solution 1+2=3. Reducing the equation modulo k and k^2, we give necessary and sufficient conditions on solutions to the resulting congruence and supercongruence. A corollary is a new proof of Moser's result that the conjecture is true for odd exponents n. We also connect solutions k of the congruence to primary pseudoperfect numbers and to a result of Zagier. The proofs use divisibility properties of power sums as well as Lerch's relation between Fermat and Wilson quotients.