Claimed AI-led

Artin's primitive root conjecture: infinitely many primes for every admissible base

An OpenAI manuscript claims that every integer other than $-1$ and the perfect squares is a primitive root modulo infinitely many primes, the qualitative half of Artin's 1927 conjecture, which until now was known unconditionally for no single prescribed base.

Lab: OpenAI
Model: Unreleased internal OpenAI model, not named in the release
Field: Mathematics
Date: 2026-10-06
Problem posed: 1927 · open 99 yrs

Sources

What was found

For each admissible $a$ the manuscript proves at least $c_a x/(\log x)^2$ primes in $(x, 2x)$ with primitive root $a$ , against the predicted $x/\log x$ . Hooley proved the full predicted asymptotic in 1967 under the generalized Riemann hypothesis; unconditionally, Heath-Brown showed in 1986 that at most two prime bases fail, so one of 2, 3 and 5 works without saying which. The analytic core is a separate theorem: for every cyclotomic field containing the twelfth roots of unity and every finite-order Hecke character, the $L$ -function has no zeros with $\Re s > 1 - 10^{-6}$ , a strip of uniform width with no exceptional zero. It is built from the cubic-theta and coefficient-reflection machinery of the release's quasi-Riemann hypothesis manuscript, extended from $\mathbb{Q}(\sqrt{-3})$ to these fields. A companion manuscript gives a conditional result for simultaneous primitive roots.

Novelty check

Read the introduction at commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb on 2026-10-12. Its account of prior work (Hooley, Gupta and Murty, Heath-Brown, and the averaged results of Goldfeld, Stephens and of Klurman, Shparlinski and Teräväinen) matches the standard one, in which no specific base such as 2 is known unconditionally. No earlier unconditional claim for a prescribed base was found in the registry.

Caveats and known objections

The uniform zero-free strip is itself a statement of the same kind as the quasi-Riemann hypothesis and depends on the methods of that companion manuscript, so this entry stands or falls with 2026-10-06-quasi-riemann-hypothesis, which is also unchecked. No Lean formalization accompanies the family. The bound is a lower bound of the wrong order for the density conjecture; the quantitative half of Artin's conjecture is untouched. Nobody outside OpenAI is on record as having read the argument. Autonomy is graded ai-led for the same reasons as the other families split from this release: humans chose the problems and filtered the output, and the single-agent account rests on OpenAI alone.

The open problem

MathDBArtin's primitive root conjecture

370026

Nothing mechanically. It records what a problem says, what is known about it, and the standing of any claimed solution.

Also recorded at

vibemathedartin-primitive-root-infinitude

Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.

Nobody outside the lab has checked this yet.

Reading the primary source closely enough to say whether it supports the claim counts as a check, and you are credited on the entry.

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History of this entry

2026-10-16CorrectedRecorded the vibemathed record for this result and the MathDB record for the conjecture as registrations. vibemathed cites the same manuscript and grades it partial, as this entry does. No grade moved.
2026-10-12AddedEntered the registry graded Claimed and AI-led.

Entries are never deleted. A grade that does not hold up is downgraded on the record, with the reason beside it.

Community discussion

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Cite this entry

Plain text

whataifound.org. (2026). Artin's primitive root conjecture: infinitely many primes for every admissible base. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-10-06-artin-primitive-root

BibTeX

@misc{whataifound-openai-2026-root,
  title        = {Artin's primitive root conjecture: infinitely many primes for every admissible base},
  author       = {{whataifound.org}},
  year         = {2026},
  howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
  note         = {Result by OpenAI. Verification: Claimed. Autonomy: AI-led.},
  url          = {https://whataifound.org/finding/2026-10-06-artin-primitive-root}
}

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