# Armstrong 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, if it’s also the case that for some power $m$ the equality

$$n=\sum _{i=1}^{k}d_{i}{}^{m}$$ |

also holds, then $n$ is an Armstrong number or narcissistic number or plus perfect number or perfect digital invariant.

In any given base $b$ there is a finite amount of Armstrong numbers, since the inequality $k{(b-1)}^{m}>{b}^{k-1}$ is false after a certain threshold.

Title | Armstrong number |
---|---|

Canonical name | ArmstrongNumber |

Date of creation | 2013-03-22 16:04:06 |

Last modified on | 2013-03-22 16:04:06 |

Owner | CompositeFan (12809) |

Last modified by | CompositeFan (12809) |

Numerical id | 4 |

Author | CompositeFan (12809) |

Entry type | Definition |

Classification | msc 11A63 |

Synonym | narcissistic number |

Synonym | plus perfect number |

Synonym | perfect digital invariant |