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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, 1223334444555567890 is a pandigital number in base 10. The first few pandigital base 10 numbers are given by {{OEIS|id=A050278}}:
 
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}</math>
 
The following table lists the smallest pandigital numbers of a few selected bases:
 
{| class="wikitable"
|-
!Base
!Smallest pandigital
!Values in base 10
|-
|2
|10
|[[2 (number)|2]]
|-
|3
|102
|[[11 (number)|11]]
|-
|4
|1023
|[[75 (number)|75]]
|-
|10
|1023456789
|1023456789
|-
|16
|1023456789ABCDEF
|1162849439785405935
|-
|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 (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.
 
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]].
 
Two of the zeroless pandigital [[Friedman number]]s are: 123456789 = ((86 + 2 × 7)<sup>5</sup> - 91) / 3<sup>4</sub>, 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 = 46587^2 + (0 × 139).
 
While much of what has been said does not apply to [[Roman numeral]]s, 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 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 first pandigital number with the digits in order.
*3816547290 = The only pandigital number without redundant digits, where the first ''n'' digits are divisible by ''n''.
*9876543210 = The largest pandigital number without redundant digits.
*9814072356 = The largest pandigital  square without redundant digits. It is the square of 99,066.
*12345678987654321 = A pandigital number with all the digits except zero in both ascending and descending order. It is the [[square (algebra)|square]] of 111111111. It is also a [[palindrome number]].
 
==See also==
* [[Champernowne constant]]
 
== References==
* {{mathworld | urlname = PandigitalNumber| title = Pandigital number}}
* De Geest, P. ''The Nine Digits Page'' [http://www.worldofnumbers.com/ninedigits.htm]
* Sloane, N. J. A. ''Sequences'' [http://oeis.org/A050278], [http://oeis.org/A050288], [http://oeis.org/A050289], and [http://oeis.org/A050290] in "The On-Line Encyclopedia of Integer Sequences.
 
{{Classes of natural numbers}}
[[Category:Pandigital numbers| ]]
[[Category:Base-dependent integer sequences]]

Latest revision as of 15:46, 10 September 2014

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