Artin's conjecture on primitive roots for two

Statement

This is a special case of Artin's conjecture on primitive roots for the number $2$. It is still open.

Infinitude version

There are infinitely many primes $p$ such that $2$ is a primitive root modulo $p$.

Density version

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Relation with other facts and conjectures

Stronger conjectures

• There are infinitely many Fermat primes. Note that this conjecture is not generally believed to be true; in fact, many believe the opposite: finitude conjecture for Fermat primes
• There are infinitely many safe primes that are congruent to $3$ modulo $8$, or equivalently, infinitely many Sophie Germain primes that are congruent to $1$ modulo $4$: For full proof, refer: Safe prime has plus or minus two as a primitive root

Weaker facts

• Limit inferior of least primitive root is finite: This statement would certainly follow if there are infinitely many primes with 2 as a primitive root. However, it has independently been shown to be true using the Heath-Brown theorem on Artin's conjecture.

Related facts

• Heath-Brown theorem on Artin's conjecture states that Artin's conjecture holds for all but at most two exceptional numbers. However, it gives no explicit information about what those two numbers may be, and hence $2$ may be one of those exceptional numbers.