pandigital number

Given a base $b$ integer

$$n=\sum _{i=1}^k{d}_i{b}^{i-1}$$

where $d_1$ is the least significant digit and $d_k$ is the most significant, and $k\ge b$, if for each there is at least one $d_x=m$ among the digits of $n$, then $n$ is a pandigital number in base $b$.

The smallest pandigital number in base $b$ is

$$b^{b-1}+\sum _{d=2}^{b-1}d{b}^{(b-1)-d},$$

while the largest (with only one instance of each digit) is

$$\sum _{d=1}^{b-1}d{b}^d.$$

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 zeroless pandigital number.

Title	pandigital number
Canonical name	PandigitalNumber
Date of creation	2013-03-22 16:04:28
Last modified on	2013-03-22 16:04:28
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	4
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11A63
Defines	zeroless pandigital number