Moser's mathemagical work on the equation 1^k+2^k+...+(m-1)^k=m^k
Number Theory
2020-08-28 v2
Abstract
If the equation of the title has an integer solution with k>=2, then m>10^{10^6}. Leo Moser showed this in 1953 by amazingly elementary methods. With the hindsight of more than 50 years his proof can be somewhat simplified. We give a further proof showing that Moser's result can be derived from a von Staudt-Clausen type theorem. Based on more recent developments concerning this equation, we derive a new result using the divisibility properties of numbers in the sequence 2^{2e+1}+1, e=0,1,2,..... In the final section we show that certain Erdos-Moser type equations arising in a recent paper of Kellner can be solved completely.
Keywords
Cite
@article{arxiv.1011.2940,
title = {Moser's mathemagical work on the equation 1^k+2^k+...+(m-1)^k=m^k},
author = {Pieter Moree},
journal= {arXiv preprint arXiv:1011.2940},
year = {2020}
}
Comments
22 pages, Rocky Mountain Journal of Mathematics, to appear