# 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 $k\geq b$, if for each $-1 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

 $b^{b-1}+\sum_{d=2}^{b-1}db^{(b-1)-d},$

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 zeroless pandigital number.

Title pandigital number PandigitalNumber 2013-03-22 16:04:28 2013-03-22 16:04:28 PrimeFan (13766) PrimeFan (13766) 4 PrimeFan (13766) Definition msc 11A63 zeroless pandigital number