# Sierpinski number

An integer $k$ is a *Sierpinski number* if for every positive integer $n$, the number $k{2}^{n}+1$ is composite.

That such numbers exist is amazing, and even more surprising is that there are infinitely many of them (in fact, infinitely many odd ones). The smallest *known* Sierpinski number is 78557, but it is not known whether or not this is the smallest one. The smallest number $m$ for which it
is unknown whether or not $m$ is a Sierpinski number is 10223.

A process for generating Sierpinski numbers using covering sets of primes can be found at

Visit

for the distributed computing effort to show that 78557 is indeed the smallest Sierpinski number (or find a smaller one).

Similarly, a *Riesel number* is a number $k$ such that for every positive integer $n$, the number $k{2}^{n}-1$ is composite. The smallest known Riesel number is 509203, but again, it is not known for sure that this is the smallest.

Title | Sierpinski number |
---|---|

Canonical name | SierpinskiNumber |

Date of creation | 2013-03-22 13:55:33 |

Last modified on | 2013-03-22 13:55:33 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 7 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 11B83 |

Defines | Riesel number |

Defines | Sierpiński number |