Reducing the Erdos-Moser equation 1^n + 2^n + . . . + k^n = (k+1)^n modulo k and k^2
Number Theory
2011-07-13 v1
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.
Keywords
Cite
@article{arxiv.1011.2154,
title = {Reducing the Erdos-Moser equation 1^n + 2^n + . . . + k^n = (k+1)^n modulo k and k^2},
author = {Jonathan Sondow and Kieren MacMillan},
journal= {arXiv preprint arXiv:1011.2154},
year = {2011}
}
Comments
10 pages, 2 tables, submitted for publication