Open problem, Arithmetic and number theory, Algebra, posed 1927

Artin's conjecture on primitive roots

Open

Every integer $a$ that is neither $-1$ nor a perfect square is a primitive root modulo $p$ for infinitely many primes $p$ , and when $a$ is not a perfect power and its square-free part is not congruent to $1 \pmod{4}$ , the set of such primes has asymptotic density $C_{\mathrm{Artin}} = \prod_{p} \left(1 - \frac{1}{p(p-1)}\right) \approx 0.3739558136$ inside the set of all primes.

Research frontier as of 2026

As of 2026, Artin's conjecture on primitive roots remains open unconditionally: there is not a single specific integer $a$ (such as $a=2$ or $a=10$ ) for which it is proved that $a$ is a primitive root modulo infinitely many primes. Conditionally on the Generalized Riemann Hypothesis for Kummer fields $\mathbb{Q}(\zeta_q, a^{1/q})$ , Hooley's 1967 theorem gives the full asymptotic formula. Unconditionally, combining the multiplicative structure of primitive roots with Chen-type linear sieves and Bombieri–Vinogradov-style distribution estimates shows that the exceptional set of prime bases for which the conjecture fails has size at most $2$ (Heath-Brown, 1986), and that among any three multiplicatively independent integers (with mild sign and square conditions), at least one is a primitive root infinitely often.

Best known results

Conditional on the Generalized Riemann Hypothesis, Artin's conjecture holds with the exact asymptotic density for every non-square integer $a \ne -1$ (Hooley, 1967).
Unconditionally, there are at most two prime numbers $q$ for which Artin's conjecture fails; hence at least one of $2$ , $3$ , or $5$ is a primitive root modulo infinitely many primes (Heath-Brown, 1986).

Tools and where they stop

Tool	Achieved	Where it stops
Chebotarev density theorem in Kummer extensions	Characterizes primes $p$ for which the index of $\langle a \rangle$ is divisible by $q$ via splitting of $p$ in $\mathbb{Q}(\zeta_q, a^{1/q})$ , yielding the exact conditional density under GRH.	Unconditional error terms in the Chebotarev density theorem are too weak when the degree $[\mathbb{Q}(\zeta_q, a^{1/q}):\mathbb{Q}] \approx q(q-1)$ grows with $x$ .
Linear sieve and multiplicative dependence arguments (Gupta–Murty, Heath-Brown)	Constructs primes $p$ where $p-1$ has only boundedly many prime factors (such as $p-1 = 2q$ or $2q_1 q_2$ ), limiting the number of possible prime counterexamples to at most $2$ .	Non-constructive: character sum bounds require averaging over multiple independent generators, leaving the status of any single integer $a$ unresolved.

Open questions

Can it be proved unconditionally that $2$ (or $10$ ) is a primitive root modulo infinitely many primes?
Can the size of the possible exceptional set of prime bases be reduced unconditionally from $2$ to $1$ or $0$ ?

References

Christopher Hooley (1967). On Artin's conjecture · DOI:10.1515/crll.1967.225.209
D. R. Heath-Brown (1986). Artin's Conjecture for Primitive Roots · DOI:10.1093/qmath/37.1.27
M. Ram Murty (1988). Artin's conjecture on primitive roots: an update · DOI:10.1007/BF03023749