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| {{redirect|Pandigital|the consumer electronics company|Pandigital (company)}}
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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}}:
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| 1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689
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| The smallest pandigital number in a given base ''b'' is an integer of the form
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| <math>b^{b - 1} + \sum_{d = 2}^{b - 1} db^{b - 1 - d}</math>
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| The following table lists the smallest pandigital numbers of a few selected bases:
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| {| class="wikitable"
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| |-
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| !Base
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| !Smallest pandigital
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| !Values in base 10
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| |-
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| |2
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| |10
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| |[[2 (number)|2]]
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| |-
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| |3
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| |102
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| |[[11 (number)|11]]
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| |-
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| |4
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| |1023
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| |[[75 (number)|75]]
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| |-
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| |10
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| |1023456789
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| |1023456789
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| |-
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| |16
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| |1023456789ABCDEF
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| |1162849439785405935
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| |-
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| |Roman<br>numerals
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| |MCDXLIV
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| |1444
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| |}
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| {{OEIS2C|id=A049363}} gives the base 10 values for the first 18 bases.
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| 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.
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| 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.
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| 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").
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| 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.
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| 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.
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| The largest pandigital number without redundant digits to be also a [[square number]] is [[9814072356 (number)|9814072356]].
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| 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>.
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| A pandigital [[Friedman number]] without redundant digits is the square: 2170348569 = 46587^2 + (0 × 139).
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| 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.
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| 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).
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| ==Examples of pandigital numbers==
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| *123456789 = The first zeroless pandigital number.
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| *381654729 = The only zeroless pandigital number where the first ''n'' digits are divisible by ''n''.
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| *987654321 = The largest zeroless pandigital number without redundant digits.
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| *1023456789 = The first pandigital number.
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| *1234567890 = The first pandigital number with the digits in order.
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| *3816547290 = The only pandigital number without redundant digits, where the first ''n'' digits are divisible by ''n''.
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| *9876543210 = The largest pandigital number without redundant digits.
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| *9814072356 = The largest pandigital square without redundant digits. It is the square of 99,066.
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| *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]].
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| ==See also==
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| * [[Champernowne constant]]
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| == References==
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| * {{mathworld | urlname = PandigitalNumber| title = Pandigital number}}
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| * De Geest, P. ''The Nine Digits Page'' [http://www.worldofnumbers.com/ninedigits.htm]
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| * 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.
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| {{Classes of natural numbers}}
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| [[Category:Pandigital numbers| ]]
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| [[Category:Base-dependent integer sequences]]
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