Pandigital number

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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, 1234567890 (one billion two hundred thirty-four million five hundred sixty-seven thousand eight hundred ninety) is a pandigital number in base 10.

Contents

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

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

BaseSmallest pandigitalValue in base 10
1 1 1
2 10 2
3 102 11
4 1023 75
5 10234694
6 1023458345
8 102345672177399
10 10234567891023456789
12 1023456789AB754777787027
16 1023456789ABCDEF1162849439785405935
361023456789ABCDEFGHIJKLMNOPQRSTUVWXYZ2959962226643665039859858867133882191922999717199870715
Roman
numerals
MCDXLIV1444

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 (the Mersenne numbers). The larger the base, the rarer pandigital numbers become, though one can always find runs of 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 = 990662.

Two of the zeroless pandigital Friedman numbers are: 123456789 = ((86 + 2 × 7)5 − 91) / 34, and 987654321 = (8 × (97 + 6/2)5 + 1) / 34.

A pandigital Friedman number without redundant digits is the square: 2170348569 = 465872 + (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 , which is correct to decimal places. [1] The variant produces correct digits. [2]

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

See also

Related Research Articles

A palindromic number is a number that remains the same when its digits are reversed. In other words, it has reflectional symmetry across a vertical axis. The term palindromic is derived from palindrome, which refers to a word whose spelling is unchanged when its letters are reversed. The first 30 palindromic numbers are:

19 (nineteen) is the natural number following 18 and preceding 20. It is a prime number.

73 (seventy-three) is the natural number following 72 and preceding 74. In English, it is the smallest natural number with twelve letters in its spelled out name.

<span class="mw-page-title-main">37 (number)</span> Natural number

37 (thirty-seven) is the natural number following 36 and preceding 38.

101 is the natural number following 100 and preceding 102.

1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000.

300 is the natural number following 299 and preceding 301.

<span class="mw-page-title-main">360 (number)</span> Natural number

360 is the natural number following 359 and preceding 361.

400 is the natural number following 399 and preceding 401.

500 is the natural number following 499 and preceding 501.

It is:

10,000 is the natural number following 9,999 and preceding 10,001.

<span class="mw-page-title-main">1,000,000</span> Natural number

1,000,000, or one thousand thousand, is the natural number following 999,999 and preceding 1,000,001. The word is derived from the early Italian millione, from mille, "thousand", plus the augmentative suffix -one.

<span class="mw-page-title-main">1,000,000,000</span> Natural number

1,000,000,000 is the natural number following 999,999,999 and preceding 1,000,000,001. With a number, "billion" can be abbreviated as b, bil or bn.

100,000 (one hundred thousand) is the natural number following 99,999 and preceding 100,001. In scientific notation, it is written as 105.

138 is the natural number following 137 and preceding 139.

181 is the natural number following 180 and preceding 182.

100,000,000 is the natural number following 99,999,999 and preceding 100,000,001.

20,000 is the natural number that comes after 19,999 and before 20,001.

60,000 is the natural number that comes after 59,999 and before 60,001. It is a round number. It is the value of (75025).

References

  1. Weisstein, Eric W. "e Approximations". Wolfram MathWorld. Archived from the original on 26 May 2024. Retrieved 21 August 2024.
  2. Friedman, Erich (2004). "Problem of the Month (August 2004)". Archived from the original on 4 June 2024. Retrieved 21 August 2024.