Artin's conjecture on primitive roots

Remarks

The conjecture is open for every single value of $a$. The full conjecture also predicts a positive density for this set of primes; for many $a$ the predicted density is Artin's constant, about $0.3739558$. Christopher Hooley proved the conjecture in 1967 under the assumption of the generalized Riemann hypothesis. Roger Heath-Brown proved in 1986 that at most two primes $a$ fail the conjecture; so at least one of $2$, $3$, $5$ is a primitive root modulo infinitely many primes. The two hypotheses are needed: if $a = b^2$ is a square and $p \nmid a$, then $a^{(p-1)/2} \equiv b^{p-1} \equiv 1 \pmod p$, so $a$ is never a primitive root modulo an odd prime $p$. For $a = -1$, the multiplicative order of $-1$ is at most $2$, so it is a primitive root only modulo $2$ and $3$. For example, $2$ is a primitive root modulo $3$, $5$, $11$, $13$, $19$ and $29$.