ArmstrongNumber 2006-07-06 17:01:03 2006-07-06 17:01:03 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Given a base $b$ integer $$n = \sum_{i = 1}^k d_ib^{i - 1}$$ where $d_1$ is the least significant digit and $d_k$ is the most significant, if it's also the case that for some power $m$ the equality $$n = \sum_{i = 1}^k {d_i}^m$$ also holds, then $n$ is an {\em Armstrong number} or {\em narcissistic number} or {\em plus perfect number} or {\em perfect digital invariant}. In any given base $b$ there is a finite amount of Armstrong numbers, since the inequality $k(b - 1)^m > b^{k - 1}$ is false after a certain threshold.