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 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.