---
id: "2026-10-06-openai-artin-primitive-root-conjecture-infinitude"
url: "https://postcutoff.com/e/2026-10-06-openai-artin-primitive-root-conjecture-infinitude/"
as_of: "2026-10-30T13:42:00+01:00"
date: "2026-10-06"
date_precision: day
category: science
importance: 4
confidence: medium
status: [Awaiting review]
verification: Preprint only
sources: 7
editor: human editor
human_review: null
version: "2026-10-29"
---

As of: 2026-10-30 13:42 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-10-06-openai-artin-primitive-root-conjecture-infinitude/

# OpenAI claims Artin's primitive root conjecture (infinitude)

Full title: OpenAI's release claims the infinitude part of Artin's primitive root conjecture (1927) for every admissible base, including 2; Bhargava and others build on it

"Primitive roots for every admissible integer base" (OpenAI, 4 Oct 2026, family 029) proves, for every integer a that is neither −1 nor a square, that at least c_a·x/(log x)² primes in (x, 2x) have a as a primitive root. This gives the infinitude assertion of Artin's 1927 conjecture for every admissible base, including 2, though not the conjectured density. The proof combines a zero-free half-plane for Hecke L-functions, of the kind behind the release's quasi-Riemann hypothesis, with sieve estimates for primes p whose p − 1 has a controlled shape. It is unrefereed and not formalized. Follow-ups by Cureton–Thorner and by Manjul Bhargava (Hexagon, partly AI-written) extend the zero-free region and upgrade the count to a positive lower density.

## Key facts
- Theorem 1.1 (verbatim): 'For every admissible integer a, there are constants ca > 0 and xa ≥ 2 such that, for every real x ≥ xa, #{p prime : x < p < 2x, p ∤ a, ordp(a) = p − 1} ≥ ca x/(log x)².' The paper: 'The predicted counting scale is x/ log x … In particular, it proves the infinitude assertion for each fixed admissible integer, including the prescribed base 2.' 92 pages, dated 4 Oct 2026; companion 'Simultaneous primitive roots: a conditional lower bound for prime bases' (21 pp.)
- Prior state (paper, §1.1): Hooley proved Artin's predicted asymptotic assuming the Riemann hypothesis for the relevant Dedekind zeta functions (1967); unconditionally, Gupta and Murty gave a set of thirteen integers one of which works, and Heath-Brown 'proved that at most two prime bases fail the infinitude assertion, so at least one of 2, 3, 5 satisfies it' (verbatim), without identifying which
- Method: a zero-free half-plane for Hecke L-functions over cyclotomic fields from a cubic theta input, a quadratic large sieve, and 'marked Type II estimates' to count primes with p − 1 equal to a small cofactor times one large prime. A blog commenter ('DH', 9 Oct) noted that the write-up contains a zero-free region Re s > 999999/1000000 for finite-order Hecke characters of cyclotomic fields containing ζ12. No Lean scope note
- Frank Calegari (Persiflage, 8 Oct, verbatim): 'the argument is a little bit different, but also not at all surprising'; 'this is an interesting type of argument which I don’t find particularly surprising. But no one was going to write this paper, which depends on the quasi-Riemann hypothesis for Dirichlet L-functions, without knowing anything about general Dedekind L-functions.'
- Follow-ups: Dean Cureton and Jesse Thorner (Hexagon, 12 Oct) prove a quasi-Riemann hypothesis for every Dedekind zeta function (Re s > 31/32), building on the zero-free argument of this family (see [2026-10-12-cureton-thorner-dedekind-zeta-zero-free-axiomprover](https://postcutoff.com/e/2026-10-12-cureton-thorner-dedekind-zeta-zero-free-axiomprover/index.md)). Manjul Bhargava (Hexagon 2610.00228, 14 Oct, mixed human and AI text): 'a positive lower density of primes have a as a primitive root', using Cureton–Thorner's half-plane and 'the marked Type II estimates of OpenAI'; 'The argument gives a lower bound rather than the exact Artin density.' His provenance note mentions handwritten notes, 'a conversation with Peter Sarnak', GPT-6 and a Lean check of the deductions only
- Scientific American (8 Oct) listed Artin's conjecture among the release's other important number-theory results, without expert comment

## What happened

A number a is a primitive root modulo a prime p when its powers run through every nonzero residue. Emil Artin conjectured in 1927
that every integer other than −1 and perfect squares is a primitive root for infinitely many primes, with a predicted density.
Christopher Hooley proved this in 1967 assuming the generalized Riemann hypothesis. Without that assumption, Gupta, Murty and
Heath-Brown showed in the 1980s that at most two prime bases can fail, so one of 2, 3 and 5 works, but nobody could name a single
base, not even 2, that provably works.

OpenAI's manuscript claims the infinitude statement for every admissible base, with a lower bound of order x/(log x)² primes in each
dyadic range rather than the predicted x/log x. As Calegari reconstructs it, the idea is to find primes p for which p − 1 has a
special shape, a small cofactor times a large prime, so that only a few cases of Hooley's argument need a zero-free region. That
region comes from the same automorphic machinery as the release's quasi-Riemann hypothesis.

## Why it matters

Artin's conjecture is one of the classical problems of number theory. A proof of its qualitative part for every base, without the
Riemann hypothesis, would settle what has been open for almost a century, although the exact Artin density remains conjectural.

## What is disputed or not yet verified

- Verification: Unrefereed manuscript, no Lean formalization; follow-ups take its estimates as inputs

## Reactions

### Reactions and follow-ups

Calegari found the argument natural once quasi-RH is available, but noted that nobody would have written it without that input. Dean
Cureton and Jesse Thorner extended the zero-free half-plane to every Dedekind zeta function within a week, and Manjul Bhargava posted a
note (written with GPT-6, deductions checked in Lean) upgrading the count to a positive proportion of primes. No independent check of
the 92-page proof itself has been published.

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 159 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 98 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 189 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 128 days after its cutoff

## Sources

1. [OpenAI: Primitive roots for every admissible integer base (4 Oct)](https://github.com/openai/math/blob/main/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/primitive-roots-all-integer-bases.pdf) (github.com, paper)
2. [OpenAI: Simultaneous primitive roots, a conditional lower bound for prime bases](https://github.com/openai/math/blob/main/preprints/Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases-October-4-2026/simultaneous-primitive-roots-conditional-lower-bound-prime-bases.pdf) (github.com, paper)
3. [Hexagon 2610.00228: Every nonsquare integer not equal to −1 is a primitive root modulo a positive lower density of primes (contributed by Manjul Bhargava)](https://hexagonmath.org/abs/2610.00228v1) (hexagonmath.org, paper)
4. [Hexagon 2610.00200: Cureton and Thorner, A quasi-Riemann hypothesis for Dedekind zeta functions](https://hexagonmath.org/abs/2610.00200v2) (hexagonmath.org, paper)
5. [Scientific American: The most exciting claims from OpenAI's heap of new proofs](https://www.scientificamerican.com/article/the-most-exciting-claims-from-openais-heap-of-new-proofs/) (scientificamerican.com, press)
6. [Persiflage (Frank Calegari): A Brief Digest of Some New OpenAI Results (8 Oct, item 4 and comments)](https://galoisrepresentations.org/2026/10/08/a-brief-digest-of-some-new-openai-results/) (galoisrepresentations.org, discussion)
7. [oai-math-revised: community notes on the primitive-root manuscript](https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/AGENT_NOTES.md) (github.com, discussion)

## Changes

- 2026-10-29 (filed): Created (families 001–125; manuscript introduction, Persiflage digest and comments, Bhargava's and Cureton–Thorner's Hexagon pages read)

## Related

- 2026-10-12: [Cureton and Thorner, with AxiomProver, extend the quasi-Riemann zero-free half-plane to every Dedekind zeta function (Re s > 31/32, Lean-formalized)](https://postcutoff.com/e/2026-10-12-cureton-thorner-dedekind-zeta-zero-free-axiomprover/index.md)
- 2026-10-06: [OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems](https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/index.md)
- 2026-10-06: [OpenAI model claims the quasi-Riemann hypothesis](https://postcutoff.com/e/2026-10-06-quasi-riemann-hypothesis-openai/index.md)