The $p$-adic Order of Power Sums, the Erdös-Moser Equation, and Bernoulli Numbers

Jonathan Sondow, Emmanuel Tsukerman

Published 2014-01-01Version 1

The Erd\"os-Moser equation is a Diophantine equation proposed more than 60 years ago which remains unresolved to this day. In this paper, we consider the problem in terms of divisibility of power sums and in terms of certain Egyptian fraction equations. As a consequence, we show that solutions must satisfy strong divisibility properties and a restrictive Egyptian fraction equation. Our studies lead us to results on the Bernoulli numbers and allow us to motivate Moser's original approach to the problem.