# Artin’s conjecture on primitive roots

Let $m$ be a number in the list $2,4,{p}^{k}$ or $2{p}^{k}$ for some $k\ge 1$. Then we know that $m$ has a primitive root^{}, but finding one can be a rather challenging problem (theoretically and computationally).

Gauss conjectured that the number $10$ is a primitive root for infinitely many primes $p$. Much later, in $1927$, Emil Artin made the following conjecture:

###### Artin’s Conjecture.

Let $n$ be an integer not equal to $\mathrm{-}\mathrm{1}$ or a square. Then $n$ is a primitive root for infinitely many primes $p$.

However, up to now, nobody has been able to show that a single integer $n$ is a primitive root for infinitely many primes. It can be shown that the number $3$ is a primitive root for every Fermat prime^{} but, unfortunately, the existence of infinitely many Fermat primes is far from obvious, and in fact it is quite dubious (only five Fermat primes are known!).

Title | Artin’s conjecture on primitive roots |
---|---|

Canonical name | ArtinsConjectureOnPrimitiveRoots |

Date of creation | 2013-03-22 16:21:04 |

Last modified on | 2013-03-22 16:21:04 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 5 |

Author | alozano (2414) |

Entry type | Conjecture |

Classification | msc 11-00 |

Synonym | Artin’s conjecture |