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pandigital number (Definition)

Given a base $ b$ integer

$\displaystyle n = \sum_{i = 1}^k d_ib^{i - 1}$
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

$\displaystyle b^{b - 1} + \sum_{d = 2}^{b - 1} db^{(b - 1) - d},$
while the largest (with only one instance of each digit) is
$\displaystyle \sum_{d = 1}^{b - 1} db^d.$

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)\vert 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.



"pandigital number" is owned by CompositeFan. [ owner history (1) ]
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Also defines:  zeroless pandigital number
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Cross-references: number, divisibility rules, length, prime, digits, least significant digit, integer, base
There are 5 references to this entry.

This is version 1 of pandigital number, born on 2006-07-10.
Object id is 8131, canonical name is PandigitalNumber.
Accessed 1441 times total.

Classification:
AMS MSC: 11A63 (Number theory :: Elementary number theory :: Radix representation; digital problems)
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