# Limit inferior of least primitive root is finite

From Number

## Statement

There is a finite number (at most ) that equals the limit inferior of the sequence of least primitive roots modulo prime numbers.

## Facts used

## Proof

By the Heath-Brown theorem on Artin's conjecture, there are at most two exceptional positive values of that are not perfect squares and are primitive roots for only finitely many primes. Thus, in the set there is at least one value that is a primitive root for infinitely many primes. The limit inferior of the least primitive root is bounded by that element, and hence by .