# Pandigital number

In mathematics, a pandigital number is an integer that in a given base has among its significant digits each digit used in the base at least once. For example, 1234567890 (one billion two hundred thirty-four million five hundred sixty-seven thousand eight hundred ninety) is a pandigital number in base 10.

## Smallest pandigital numbers

The first few pandigital base 10 numbers are given by (sequence A171102 in the OEIS):
    1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689
The smallest pandigital number in a given base b is an integer of the form
    $b^{b-1}+\sum _{d=2}^{b-1}db^{b-1-d}={\frac {b^{b}-b}{(b-1)^{2}}}+(b-1)\times b^{b-2}-1$
The following table lists the smallest pandigital numbers of a few selected bases.
| Base | Smallest pandigital | Value in base 10 |
| 1 | 1 | 1 |
| 2 | 10 | 2 |
| 3 | 102 | 11 |
| 4 | 1023 | 75 |
| 5 | 10234 | 694 |
| 6 | 102345 | 8345 |
| 8 | 10234567 | 2177399 |
| 10 | 1023456789 | 1023456789 |
| 12 | 1023456789AB | 754777787027 |
| 16 | 1023456789ABCDEF | 1162849439785405935 |
| 36 | 1023456789ABCDEFGHIJKLMNOPQRSTUVWXYZ | 2959962226643665039859858867133882191922999717199870715 |
| Roman numerals | MCDXLIV | 1444 |
OEIS: A049363 gives the base 10 values for the first 18 bases.
In a trivial sense, all positive integers are pandigital in unary (or tallying). In binary, all integers are pandigital except for 0 and numbers of the form $2^{n}-1$ (the Mersenne numbers). The larger the base, the rarer pandigital numbers become, though one can always find runs of $b^{x}$ consecutive pandigital numbers with redundant digits by writing all the digits of the base together (but not putting the zero first as the most significant digit) and adding x + 1 zeroes at the end as least significant digits.
Conversely, the smaller the base, the fewer pandigital numbers without redundant digits there are. 2 is the only such pandigital number in base 2, while there are more of these in base 10.

## Variants and properties

Sometimes, the term is used to refer only to pandigital numbers with no redundant digits. In some cases, a number might be called pandigital even if it doesn't have a zero as a significant digit, for example, 923456781 (these are sometimes referred to as "zeroless pandigital numbers").
No base 10 pandigital number can be a prime number if it doesn't have redundant digits. The sum of the digits 0 to 9 is 45, passing the divisibility rule for both 3 and 9. The first base 10 pandigital prime is 10123457689; OEIS: A050288 lists more.
For different reasons, redundant digits are also required for a pandigital number (in any base except unary) to also be a palindromic number in that base. The smallest pandigital palindromic number in base 10 is 1023456789876543201.
The largest pandigital number without redundant digits to be also a square number is 9814072356 = 99066$^2$.
Two of the zeroless pandigital Friedman numbers are: 123456789 = ((86 + 2 × 7)$^5$ − 91) / 3$^4$, and 987654321 = (8 × (97 + 6/2)$^5$ + 1) / 3$^4$.
A pandigital Friedman number without redundant digits is the square: 2170348569 = 46587$^2$ + (0 × 139).
The concept of a "pandigital approximation" was introduced by Erich Friedman in 2004. With the digits from 1 to 9 (each used exactly once) and the mathematical symbols + – × / ( ) . and ^, Euler's number e can be approximated as $(1+9^{-4^{7\times 6}})^{3^{2^{85}}}$, which is correct to $1.8\cdot 10^{25}$ decimal places. The variant $(1+.2^{9^{7\times 6}})^{5^{3^{84}}}$ produces $8.3\cdot 10^{39}$ correct digits.
While much of what has been said does not apply to Roman numerals, there are pandigital numbers: MCDXLIV, MCDXLVI, MCDLXIV, MCDLXVI, MDCXLIV, MDCXLVI, MDCLXIV, MDCLXVI. These, listed in OEIS: A105416, use each of the digits just once, while OEIS: A105417 has pandigital Roman numerals with repeats.
Pandigital numbers are useful in fiction and in advertising. The Social Security number 987-65-4321 is a zeroless pandigital number reserved for use in advertising. Some credit card companies use pandigital numbers with redundant digits as fictitious credit card numbers (while others use strings of zeroes).

## Examples of base 10 pandigital numbers

* 123456789 = The first zeroless pandigital number.
* 381654729 = The only zeroless pandigital number where the first n digits are divisible by n.
* 987654321 = The largest zeroless pandigital number without redundant digits.
* 1023456789 = The first pandigital number.
* 1234567890 = The pandigital number with the digits in order.
* 3816547290 = The polydivisible pandigital number; the only pandigital number where the first n digits are divisible by n.
* 9814072356 = The largest pandigital square without redundant digits. It is the square of 99066.
* 9876543210 = The largest pandigital number without redundant digits.
* 12345678987654321 = A pandigital number with all the digits except zero in both ascending and descending order. It is the square of 111111111; see Demlo number. It is also a palindromic number.