# Thabit number

An integer of the form $3\cdot {2}^{n}-1$, or ${2}^{n+1}+{2}^{n}-1$. They are listed in A055010 of Sloane’s OEIS. The Thabit numbers are a subset of the Proth numbers.

The mathematician and astronomer Thabit ibn Qurra studied these numbers in search of a formula for amicable pairs. He found that when two consecutive Thabit numbers are also prime numbers^{} (corresponding to indices $n$ and $n-1$) and $9\cdot {2}^{2n-1}-1$ is a prime number, too, then these numbers multiplied by ${2}^{n}$ will reveal an amicable pair. The only $n$ known to fit these criteria are 2, 4 and 7. The largest Thabit number known to be prime corresponds to index 2312734, its immediate lower neighbor is composite.

It is conjectured that the nimfactorial of a Thabit number is always 2.

Title | Thabit number |

Canonical name | ThabitNumber |

Date of creation | 2013-03-22 15:52:58 |

Last modified on | 2013-03-22 15:52:58 |

Owner | Mravinci (12996) |

Last modified by | Mravinci (12996) |

Numerical id | 5 |

Author | Mravinci (12996) |

Entry type | Definition |

Classification | msc 11A05 |

Synonym | Thabit ibn Kurra number |

Synonym | Thabit ibn Kurrah number |

Synonym | Thabit ibn Qurra number |

Synonym | Thabit ibn Qurrah number |

Synonym | Thabit bin Kurra number |

Synonym | Thabit bin Kurrah number |

Synonym | Thabit bin Qurra number |

Synonym | Thabit bin Qurrah number |

Related topic | AFormulaForAmicablePairs |