Not evaluated
Primitive roots (Artin's conjecture): announced proof that every non-square integer other than −1 is a primitive root for a positive proportion of primes (Bhargava, with GPT-6)
The gist
On October 14, 2026, Manjul Bhargava posted on Hexagon a proof, written up and formalised in Lean with GPT-6, that every non-square integer other than −1 is a primitive root modulo a positive proportion of primes.
DiscoverIA status on October 24, 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 24, 2026: a citation always points to a dated snapshot.
Cite snapshot 1, Oct 24, 2026
APA citation with the dated link
BibTeX @misc, key discoveria-bhargava-artin-densite-positive-2026-10-14-r1
@misc{discoveria-bhargava-artin-densite-positive-2026-10-14-r1,
author = {{DiscoverIA}},
title = {Primitive roots (Artin's conjecture): announced proof that every non-square integer other than −1 is a primitive root for a positive proportion of primes (Bhargava, with GPT-6)},
howpublished = {DiscoverIA registry, entry bhargava-artin-densite-positive-2026-10-14},
note = {Status “Not evaluated”, snapshot 1, recorded on October 24, 2026. New result},
year = {2026},
url = {https://discoveria.ai/entry/bhargava-artin-densite-positive-2026-10-14#releve-1},
urldate = {2026-10-24}
}Dated link opens this snapshot in the history
To understand ≈ 40 s reading time, estimated
What it changes
An integer a is a primitive root modulo a prime p when its powers run through every non-zero remainder modulo p. If the proof held, every non-square integer other than −1 would be one for a positive proportion of primes, with no assumption beyond recent theorems by Cureton and Thorner and by OpenAI. According to the paper, an OpenAI preprint had already shown that there are infinitely many such primes; the gain would be to reach a positive proportion, short of the exact density predicted by Artin's conjecture.
What remains unclear
The proof rests on two recent results that have not been peer-reviewed: Cureton and Thorner's zero-free region and estimates by OpenAI. The Lean code checks only the deductions, with both results left as assumptions. According to the page, some details added by GPT-6 were checked by Lean but not yet by any human.
For experts
Sources, method, key facts, status history, report an error
Status
Not evaluated
Recorded on · hexagonmath.org · 2610.00228v1
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 sourcehexagonmath.org · 2610.00228v1page read word for word · consulted on Oct 24, 2026 ·
we prove that a positive lower density of primes have a as a primitive root
- Paperhexagonmath.org · pdf/2610.00228v1page read word for word · consulted on Oct 24, 2026 ·
Some details added by GPT-6 were checked by Lean but not (yet) by any human.
Method
Cureton and Thorner's announcement led to handwritten notes by Manjul Bhargava and a conversation with Peter Sarnak; the notes were converted into LaTeX and then completed by the author and GPT-6, which also helped with the Lean formalisation. The text links the width of a zero-free region shared by all Dedekind zeta functions to the proportion obtained, and uses N = 6000 from version 1 of Cureton and Thorner. DiscoverIA status: not evaluated, meaning that no third-party evaluation has been found in the sources read. Set on October 24, 2026 from hexagonmath.org (page read, quotation checked word for word: “we prove that a positive lower density of primes have a as a primitive root”).
Limits
The claimant supplies Lean proofs for the deductions only (the cited theorems remain assumptions); DiscoverIA has found no independent verification of them.
Key facts
- Claimed by
- Manjul Bhargava, with GPT-6 (OpenAI)
announced on Oct 14, 2026 - Nature
- new result
- Field
- Mathematics (number theory / primitive roots)
- Checked by
- no verifier identified
- Sources
- 2 : 2 read word for word
- This entry
- AI draft, not reviewed
Status history
Only one observation so far.
- 01
First observation:Not evaluatedCurrent status
preprint on Hexagon, a platform without peer review; Lean covers the deductions only, with the Cureton-Thorner and OpenAI theorems left as assumptions; no third-party evaluation…
Read the full reason
preprint on Hexagon, a platform without peer review; Lean covers the deductions only, with the Cureton-Thorner and OpenAI theorems left as assumptions; no third-party evaluation found
Source: hexagonmath.orgread word for word
An observation is added; none is ever erased.
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