# Sierpiński conjecture

In 1960 Wacław Sierpiński (1882-1969) proved the following interesting result:

Theorem: There exist infinitely many odd integers $k$ such that $k{2}^{n}+1$ is composite for every integer $n\ge 1$.

A multiplier $k$ with this property is called a Sierpiński number (http://planetmath.org/SierpinskiNumbers). The Sierpiński problem consists in determining the smallest Sierpiński number. In 1962, John Selfridge discovered the Sierpiński number $k=78557$, which is now believed to be in fact the smallest such number.

Conjecture: The integer $k=78557$ is the smallest Sierpiński number.

To prove the conjecture, it would be sufficient to exhibit a prime $k{2}^{n}+1$ for each $$.

Title | Sierpiński conjecture |
---|---|

Canonical name | SierpinskiConjecture |

Date of creation | 2013-03-22 13:34:16 |

Last modified on | 2013-03-22 13:34:16 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 12 |

Author | yark (2760) |

Entry type | Conjecture |

Classification | msc 11B83 |

Synonym | Sierpinski conjecture^{} |