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{{short description|Integer whose representation contains every digit in its number base}}
{{redirect|Pandigital|the consumer electronics company|Pandigital (company)}}
{{No footnotes|date=January 2017}}

In [[mathematics]], a '''pandigital number''' is an [[integer]] that in a given [[Radix|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 {{OEIS|id=A171102 }}:

: 1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689

The smallest pandigital number in a given base ''b'' is an integer of the form

: <math>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</math> 

The following table lists the smallest pandigital numbers of a few selected bases.

{| class="wikitable"
|-
!Base
!Smallest pandigital
!Value in base 10
|-
|[[Unary numeral system|1]]
|1
|[[1 (number)|1]]
|-
|[[Binary number|2]]
|10
|[[2 (number)|2]]
|-
|[[Ternary numeral system|3]]
|102
|[[11 (number)|11]]
|-
|[[Quaternary numeral system|4]]
|1023
|[[75 (number)|75]]
|-
|[[Quinary|5]]
|10234
|694
|-
|[[Senary|6]]
|102345
|8345
|-
|[[Octal|8]]
|10234567
|2177399
|-
|[[Decimal|10]]
|1023456789
|1023456789
|-
|[[Duodecimal|12]]
|1023456789AB
|754777787027
|-
|[[Hexadecimal|16]]
|1023456789ABCDEF
|1162849439785405935
|-
|36
|1023456789ABCDEFGHIJKLMNOPQRSTUVWXYZ
|2959962226643665039859858867133882191922999717199870715
|-
|[[Roman numerals|Roman<br>numerals]]
|MCDXLIV
|1444
|}

{{OEIS2C|id=A049363}} gives the base 10 values for the first 18 bases.

In a trivial sense, all positive integers are pandigital in [[Unary numeral system|unary]] (or tallying). In binary, all integers are pandigital except for 0 and numbers of the form <math>2^n - 1</math> (the [[Mersenne prime|Mersenne number]]s). The larger the base, the rarer pandigital numbers become, though one can always find runs of <math>b^x</math> 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; {{OEIS2C|id=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 (number)|9814072356]] = 99066<sup>2</sup>.

Two of the zeroless pandigital [[Friedman number]]s are: 123456789 = ((86 + 2 × 7)<sup>5</sup> − 91) / 3<sup>4</sup>, and 987654321 = (8 × (97 + 6/2)<sup>5</sup> + 1) / 3<sup>4</sup>.

A pandigital [[Friedman number]] without redundant digits is the square: 2170348569 = 46587<sup>2</sup> + (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]] {{mvar|e}} can be approximated as <math>(1+9^{-4^{7\times 6}})^{3^{2^{85}}}</math>, which is correct to <math>1.8\cdot10^{25}</math> decimal places.<ref>{{Cite web|url=https://mathworld.wolfram.com/eApproximations.html|title=e Approximations|work=Wolfram MathWorld|first=Eric W.|last=Weisstein|access-date=21 August 2024|archive-url=https://web.archive.org/web/20240526055255/https://mathworld.wolfram.com/eApproximations.html|archive-date=26 May 2024|url-status=live}}</ref> The variant <math>(1+.2^{9^{7\times 6}})^{5^{3^{84}}}</math> produces <math>8.3\cdot10^{39}</math> correct digits.<ref>{{Cite web|url=https://erich-friedman.github.io/mathmagic/0804.html|first=Erich|last=Friedman|title=Problem of the Month (August 2004)|year=2004|access-date=21 August 2024|archive-url=https://web.archive.org/web/20240604202602/https://erich-friedman.github.io/mathmagic/0804.html|archive-date=4 June 2024|url-status=live}}</ref>

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 {{OEIS2C|id=A105416}}, use each of the digits just once, while {{OEIS2C|id=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 {{mvar|n}} digits are divisible by {{mvar|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 number#Related problems|polydivisible pandigital number]]; the only pandigital number where the first {{mvar|n}} digits are divisible by {{mvar|n}}.
*9814072356 = The largest pandigital square without redundant digits. It is the [[square (algebra)|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 (algebra)|square]] of 111111111; see [[Demlo number]]. It is also a [[palindromic number]].

==See also==
* [[Champernowne constant]]
* [[Pangram]], a word (or a sentence) using every letters A to Z at least once, a pandigital number in [[base 36]] will become a pangram

== References==
{{Reflist}}

== External links==
* {{mathworld | urlname = PandigitalNumber| title = Pandigital number}}
* De Geest, P. ''The Nine Digits Page'' [http://www.worldofnumbers.com/ninedigits.htm]
* {{Cite OEIS|1=A050278|2=Pandigital numbers: numbers containing the digits 0-9. Version 1: each digit appears exactly once}}
* {{Cite OEIS|1=A050288|2=Pandigital primes}}
* {{Cite OEIS|1=A050289|2=Zeroless pandigital numbers: numbers containing the digits 1-9 and no 0's}}
* {{Cite OEIS|1=A050290|2=Zeroless pandigital primes}}

{{Classes of natural numbers}}

[[Category:Base-dependent integer sequences]]