PandigitalNumber 2006-07-10 13:20:10 2006-07-10 13:20:10 Definition zeroless pandigital number % 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, and $k \ge b$, if for each $-1 < m < b$ there is at least one $d_x = m$ among the digits of $n$, then $n$ is a {\em pandigital number} in base $b$. The smallest pandigital number in base $b$ is $$b^{b - 1} + \sum_{d = 2}^{b - 1} db^{(b - 1) - d},$$ while the largest (with only one instance of each digit) is $$\sum_{d = 1}^{b - 1} db^d.$$ There are infinitely many pandigital numbers with more than one instance of one or more digits. If $b$ is not prime, a pandigital number must have at least $b + 1$ digits to be prime. With $k = b$ for the length of digits of a pandigital number $n$, it follows from the divisibility rules in that base that $(b - 1)|n$. Sometimes a number with at least one instance each of the digits 1 through $b - 1$ but no instances of 0 is called a {\em zeroless pandigital number}.