# pandigital number

Given a base $b$ integer

$$n=\sum _{i=1}^{k}{d}_{i}{b}^{i-1}$$ |

where ${d}_{1}$ is the least significant digit and ${d}_{k}$ is the most significant, and $k\ge b$, if for each $$ there is at least one ${d}_{x}=m$ among the digits of $n$, then $n$ is a pandigital number in base $b$.

The smallest pandigital number in base $b$ is

$${b}^{b-1}+\sum _{d=2}^{b-1}d{b}^{(b-1)-d},$$ |

while the largest (with only one instance of each digit) is

$$\sum _{d=1}^{b-1}d{b}^{d}.$$ |

There are infinitely many pandigital numbers with more than one instance of one or more digits.

If $b$ is not prime, a pandigital number must have at least $b+1$ digits to be prime. With $k=b$ for the length of digits of a pandigital number $n$, it follows from the divisibility rules^{} in that base that $(b-1)|n$.

Sometimes a number with at least one instance each of the digits 1 through $b-1$ but no instances of 0 is called a zeroless pandigital number.

Title | pandigital number |
---|---|

Canonical name | PandigitalNumber |

Date of creation | 2013-03-22 16:04:28 |

Last modified on | 2013-03-22 16:04:28 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 4 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 11A63 |

Defines | zeroless pandigital number |