Not evaluated
Artin's primitive root conjecture: infinitude for every base (family 029)
The gist
On October 6, 2026, OpenAI announced, from an unreleased internal model, a manuscript on Artin's primitive root conjecture, for every admissible integer base.
DiscoverIA status on October 12, 2026: not evaluated, meaning no third-party evaluation found.
AI draft, not reviewedWhat has not been reviewed?
Text written by an AI (DiscoverIA pipeline), on . The status and its sources come from the registry; only the explanatory text has not been reviewed. A reported error is corrected, then the text is reviewed.
Cite this entry
A single status snapshot since Oct 12, 2026: a citation always points to a dated snapshot.
Cite snapshot 1, Oct 12, 2026
APA citation with the dated link
BibTeX @misc, key discoveria-oai-fam-029-artins-primitive-root-conjecture-infinitude-r1
@misc{discoveria-oai-fam-029-artins-primitive-root-conjecture-infinitude-r1,
author = {{DiscoverIA}},
title = {Artin's primitive root conjecture: infinitude for every base (family 029)},
howpublished = {DiscoverIA registry, entry oai-fam-029-artins-primitive-root-conjecture-infinitude},
note = {Status “Not evaluated”, snapshot 1, recorded on October 12, 2026. New result},
year = {2026},
url = {https://discoveria.ai/entry/oai-fam-029-artins-primitive-root-conjecture-infinitude#releve-1},
urldate = {2026-10-12}
}Dated link opens this snapshot in the history
To understand ≈ 20 s reading time, estimated
What it changes
If the manuscript were correct, for every admissible integer base, infinitely many primes p would have that base as a primitive root, that is, a number whose successive powers, modulo p, run through all the non-zero remainders. The sources read do not give its precise statement.
What remains unclear
No Lean proof (checkable by computer) is supplied for this family, and no third-party assessment is recorded. The sources read do not say where the question stood before the announcement.
For experts
Sources, method, key facts, status history, report an error
Status
Not evaluated
Recorded on · github.com · README.md
Step 1 of 4 on the scale: not evaluated. Next step: under review.
- Announced
- Recordednot evaluated
- nextTo follow
What this status means
Source reading : Source read word for word (level 3 of 3)
The result is declared neither right nor wrong.
To move to “Under review”: a public review under way. To “Peer reviewed”: a journal publication or a review by named experts.
Sources
- Main sourcegithub.com · CONTENTS.mdlink only, reading not recorded
- Repositorygithub.com · Primitive-roots-for-every…link only, reading not recorded
- Repositorygithub.com · Simultaneous-primitive…link only, reading not recorded
Method
In the openai/math repository, family 029 comprises two manuscripts dated October 4, 2026: “Primitive roots for every admissible integer base” and “Simultaneous primitive roots: a conditional lower bound for prime bases”, whose title announces a conditional lower bound for prime bases. There is no file lean/docs/029.md, and history.md reports no revision of these manuscripts. DiscoverIA status: not evaluated, meaning that no third-party evaluation has been found in the sources read. Set on October 12, 2026 from github.com (page read, quotation checked word for word: “Not all have accompanying Lean formalizations.”).
Limits
DiscoverIA has not checked this result itself.
Key facts
- Claimed by
- OpenAI (unreleased internal model)
announced on Oct 6, 2026 - Nature
- new result
- Field
- Mathematics (Number theory)
- In openai/math
- family 029, section “Number theory”, 2 manuscripts dated Oct 4, 2026
- Lean proof
- none supplied
- Revisions
- none reported by the claimant
- Checked by
- no verifier identified
- Sources
- 3 : 3 links only
- This entry
- AI draft, not reviewed
Status history
Only one observation so far.
- 01
First observation:Not evaluatedCurrent status
Family with no Lean proof supplied. No third-party evaluation read.
Source: github.comread word for word
An observation is added; none is ever erased.
In the same set
openai/math: 719 manuscripts in 372 families from an unreleased internal model · entry 29 of 372, by family number
- Not evaluated : Positive lower density of large prime gaps (family 026)
- Not evaluated : Integral density on curve character varieties (family 027)
- Not evaluated : The Gaussian moat conjecture (family 028)
- Not evaluated : Artin's primitive root conjecture: infinitude for every base (family 029)
- Not evaluated : Modularity over imaginary quadratic fields (family 030)
- Not evaluated : Uchida's conjecture for open Galois homomorphisms (family 031)
- Not evaluated : Rational Hodge conjecture for CM abelian varieties (family 032)
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