Artin's Primitive Root Conjecture: Infinitude for Every Admissible Base

Artin conjectured in 1927 that every integer $a\ne-1$ that is not a perfect square is a primitive root modulo infinitely many primes $p$ , and more precisely that the number of such $p\le x$ is asymptotic to $A(a)\,x/\log x$ for an explicit constant $A(a)>0$ . Hooley (1967) proved this under the generalized Riemann hypothesis for Kummer fields. Unconditionally, Gupta-Murty and Heath-Brown (1986) showed that at most two prime bases fail the infinitude assertion, so one of $2,3,5$ works, but no single prescribed base such as $a=2$ was known to be a primitive root for infinitely many primes. Is every admissible integer $a$ a primitive root modulo infinitely many primes, with the predicted density?

Result: Proved(see note)
Status: Partial result
AI contribution: AI-discovered
Method: Argument
Field: Analytic number theory, primitive roots
Posed by: Emil Artin (1927), recorded in the Artin-Hasse correspondence and Hasse's diaries
Year posed: 1927
Years open: 99y
Solved: 2026-10-04
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 63 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for every integer $a$ that is neither $-1$ nor a square there are $c_a>0$ and $x_a$ such that for $x\ge x_a$ at least $c_a\,x/(\log x)^2$ primes $p\in(x,2x)$ have $a$ as a primitive root. This gives the infinitude part of Artin's conjecture for every admissible base, including $a=2$ . It does not give the predicted asymptotic $A(a)\,x/\log x$ : the count is short by a factor $\log x$ and constants depend on $a$ . The analytic input is Theorem 1.2: for every cyclotomic field $F\supseteq\mu_{12}$ and every finite-order Hecke character $\eta$ of $F$ , $L_F(s,\eta)$ has no zeros with $\mathrm{Re}\,s>1-10^{-6}$ . The companion on simultaneous primitive roots gives the same lower bound for any finite set of prime bases, assuming four inputs from this paper.

What the AI did

Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorems 1.1 and 1.2, read against Artin's conjecture as cited (Artin-Hasse correspondence, Hooley 1967, Heath-Brown 1986). The proof was not refereed. No Lean main result. The proof imports a marked Type II estimate from the companion manuscript on the Poisson-Dirichlet law for prime predecessors (Theorem 3.1), which is itself an unchecked manuscript in the same release, and adapts mechanisms from the release's quasi-Riemann hypothesis manuscript. The analytic core, a uniform zero-free strip for Hecke L-functions, is itself a very strong claim.

Artin's Primitive Root Conjecture: Infinitude for Every Admissible Base

What was actually shown

What the AI did

Verification

Sources

Discussion