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

Abstract

The Erdös-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.

22 pages, comments are welcome

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• Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)
• The Equivalence of Giuga's and Agoh's Conjectures
• Giuga Numbers and the arithmetic derivative
• Generalizations of Carmichael numbers I

Cited by in corpus (1)

• On the Diophantine equation $(x+1{)}^k+(x+2{)}^k+...+(2x{)}^k=y^n$