# Beatty’s theorem

If $p$ and $q$ are positive irrationals such that

$$\frac{1}{p}+\frac{1}{q}=1$$ |

then the sequences

$\mathrm{\{}\lfloor np\rfloor \}{}_{n=1}{}^{\mathrm{\infty}}$ | $=$ | $\lfloor p\rfloor ,\lfloor 2p\rfloor ,\lfloor 3p\rfloor ,\mathrm{\dots}$ | ||

$\mathrm{\{}\lfloor nq\rfloor \}{}_{n=1}{}^{\mathrm{\infty}}$ | $=$ | $\lfloor q\rfloor ,\lfloor 2q\rfloor ,\lfloor 3q\rfloor ,\mathrm{\dots}$ |

where $\lfloor x\rfloor $ denotes the floor (or greatest integer function) of $x$, constitute a partition^{} of the set of positive integers.

That is, every positive integer is a member exactly once of one of the two sequences and the two sequences have no common terms.

Title | Beatty’s theorem |

Canonical name | BeattysTheorem |

Date of creation | 2013-03-22 11:56:34 |

Last modified on | 2013-03-22 11:56:34 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 6 |

Author | drini (3) |

Entry type | Theorem |

Classification | msc 11B83 |

Related topic | Sequence |

Related topic | Irrational |

Related topic | Partition |

Related topic | Floor |

Related topic | Ceiling |

Related topic | BeattySequence |

Related topic | FraenkelsPartitionTheorem |

Related topic | FraenkelsPartitionTheorem2 |

Related topic | ConjugateIndex |