Math Conjectures

Artin’s Primitive Root Conjecture

OPENMajorConjectureProposed 1927 · Standard version

Canonical statement

Let g∈Z∖{−1,0,1}g\in\mathbb Z\setminus\{-1,0,1\} be not a perfect square. For n≥1n\ge1, let ζn=e2πi/n\zeta_n=e^{2\pi i/n}, let μ(n)\mu(n) be the Möbius function (zero if a prime square divides nn, and (−1)r(-1)^r if nn is a product of rr distinct primes), and put
δ(g)=∑n=1∞μ(n)[Q(ζn,g1/n):Q]. \delta(g)=\sum_{n=1}^{\infty} \frac{\mu(n)} {[\mathbb Q(\zeta_n,g^{1/n}):\mathbb Q]}.
Then δ(g)>0\delta(g)>0 and, as x→∞x\to\infty,
#{p≤x:p prime, p∤g, g generates (Z/pZ)×}∼δ(g)Li⁡(x), \#\{p\le x:p\text{ prime},\ p\nmid g,\ g\text{ generates }(\mathbb Z/p\mathbb Z)^\times\} \sim \delta(g)\operatorname{Li}(x),
where Li⁡(x)=∫2xdt/log⁡t\operatorname{Li}(x)=\int_2^xdt/\log t; the field in the denominator is independent of the choice of the nn-th root of gg.
View source LaTeX
Let \(g\in\mathbb Z\setminus\{-1,0,1\}\) be not a perfect square. For \(n\ge1\), let \(\zeta_n=e^{2\pi i/n}\), let \(\mu(n)\) be the Möbius function (zero if a prime square divides \(n\), and \((-1)^r\) if \(n\) is a product of \(r\) distinct primes), and put
\[
  \delta(g)=\sum_{n=1}^{\infty}
    \frac{\mu(n)}
    {[\mathbb Q(\zeta_n,g^{1/n}):\mathbb Q]}.
\] Then \(\delta(g)>0\) and, as \(x\to\infty\),
\[
  \#\{p\le x:p\text{ prime},\ p\nmid g,\
     g\text{ generates }(\mathbb Z/p\mathbb Z)^\times\}
    \sim \delta(g)\operatorname{Li}(x),
\] where \(\operatorname{Li}(x)=\int_2^xdt/\log t\); the field in the denominator is independent of the choice of the \(n\)-th root of \(g\).
NT-PRIME-010:[Artin1927Primitive][Hooley1967Artin][Moree2012Artin][GoldmakherEtAl2025Artin]
Number theoryAlgebra and representation theory

Notes

In a letter to Hasse of 27 September 1927, Artin conjectured that any integer gg other than −1-1, 00, 11 and not a perfect square is a primitive root modulo pp for infinitely many primes pp, indeed for a set of primes of positive density [Artin1927Primitive]. The quantitative form predicts that the count of such primes up to xx is asymptotic to δ(g)Li⁡(x)\delta(g)\operatorname{Li}(x), where δ(g)\delta(g) is a Möbius-weighted sum over the degrees of the fields Q(ζn,g1/n)\mathbb Q(\zeta_n,g^{1/n}); this expression incorporates the correction, needed for special gg, to Artin's originally proposed universal constant.

Hooley proved the full asymptotic conditionally, assuming generalized Riemann hypotheses for the relevant Dedekind zeta functions [Hooley1967Artin]. Unconditionally the situation is curious: results are known for restricted alternatives, valid outside small exceptional sets of candidate bases, yet not a single specific admissible gg has been proved unconditionally to satisfy the conjecture. The literature is surveyed by Moree [Moree2012Artin], and refinements continue to be studied [GoldmakherEtAl2025Artin].

A resolution requires either removing the Riemann hypotheses from Hooley's argument or a different approach entirely; the conjecture is open.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.