Pandigital number

A pandigital number (from the Greek παν: “each” and digital in the sense of being represented by digits) is a decimal whole number that contains each of the ten digits from 0 to 9 exactly once. The first digit must not be 0.

Pandigital numbers have no real meaning in mathematics or in any other field of application. They are mostly used as a curiosity in mathematical puzzles like Latin squares or Sudokus .

An example is the number 1748592603.

Every pandigital number has the checksum 45 and is therefore divisible by 9 :

0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45

There are 9 9 in total! = 3265920 pandigital numbers: There are 9 possibilities for the first digit (because the zero is excluded), 9 for the second (because the first digit is excluded), 8 for the third (the first two digits must not be used again) , 7 for the fourth, etc.

The first pandigital numbers are 1023456789, 1023456798, 1023456879, 1023456897, 1023456978 (sequence A050278 in OEIS ).

Multi-digital

A more general definition of pandigital numbers is as follows: A number or a mathematical expression that contains each digit on a base exactly once. In France, those numbers are also multi digitally called the numbers to base 10 decadigital . ${\displaystyle n}$

For base 4, 1320 is a pandigital number and 2 + 1 = 3 + 0 is a pandigital sum.

Pandigital breaks

Pandigital fractions are fractions that contain the digits 1 to 9 exactly once.

Examples:

${\displaystyle {\frac {1}{2}}={\frac {7269}{14538}}}$or or ${\displaystyle {\frac {1}{3}}={\frac {5832}{17496}}}$ ${\displaystyle {\frac {1}{8}}={\frac {8174}{65392}}}$

Pandigital formulas

Pandigital formulas are formulas that contain the digits 1 to 9 exactly once.

Examples:

${\displaystyle (1+9^{-4^{6\cdot 7}})^{3^{2^{85}}}}$

approximates Euler's number e exactly to 18457734525360901453873570 decimal places.

Special pandigital numbers

3,816,547,290 is the only pandigital number in which the first n digits (read as numbers) are each divisible by n ; the first digit through 1, the first two digits through 2, the first three digits through 3, etc .:

3	→ divisible by 1
38	→ divisible by 2
381	→ divisible by 3
3816	→ divisible by 4
38165	→ divisible by 5
381654	→ divisible by 6
3816547	→ divisible by 7
38165472	→ divisible by 8
381654729	→ divisible by 9
3816547290	→ divisible by 10

9,814,072,356 is the largest pandigital square number. Its square root is the “rotatable” number 99066.

See also

• List of special numbers

Web links

• Eric W. Weisstein : Pandigital Number . In: MathWorld (English).
• Eric W. Weisstein : Pandigital Fraction . In: MathWorld (English).

Individual evidence

1. ↑ immeron.edu
2. ↑ johndcook.com
3. ^ The Nine Digits Page with some Ten Digits (pandigital) exceptions . World! Of Numbers; accessed on March 2, 2014.
4. ↑ Gleick: The information . 1st edition. Redlineverlag, 2011, ISBN 978-3-86881-312-8 , p. 366