Artin’s conjecture on primitive roots

Let $m$ be a number in the list $2,4,p^k$ or $2p^k$ 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