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Primitive roots for every admissible integer base

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Proves the infinitude assertion in Artin’s primitive root conjecture for every integer a that is neither −1 nor a square. For each such base, at least \(c_a x/(\log x)^2\) primes in every sufficiently large interval \((x,2x)\) have primitive root a, with \(c_a\gt 0\).

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

The paper states

Theorem 1.1

In Primitive roots for every admissible integer base, 4 October 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

The other possible matches are Theorem 1.1 in Simultaneous primitive roots () and Theorem 1.2 in the same paper ().

For every admissible integer a, there are constants c_a>0 and x_a\ge2 such that, for every real x\ge x_a,

\#\{p\text{ prime}:x<p<2x,\ p\nmid a,\ \ord_p(a)=p-1\} \ge c_a\frac{x}{(\log x)^2}.
Reported proof verification
The release has no Lean statement for this family. The papers are available to read and assess.
Papers and explanations
Two current papers, each readable here as a page beside its PDF. One explanation by others.

Papers

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Primitive roots for every admissible integer base

4 October 2026, current edition.

Read it herePDF

We prove the infinitude assertion in Artin’s primitive root conjecture: for every integer a that is neither −1 nor a square, there are at least \(c_a x/(\log x)^2\) primes in \((x,2x)\) with primitive root a, for some \(c_a\gt 0\) and every sufficiently large x.

Simultaneous primitive roots: a conditional lower bound for prime bases

4 October 2026, current edition.

Read it herePDF

For every fixed finite set of distinct positive primes, we prove that at least \(cx/(\log x)^2\) primes in \((x,2x)\) have every member of the set as a primitive root, for some c > 0 and all sufficiently large x. The result assumes four explicitly stated analytic and sieve inputs from the companion paper on primitive roots for admissible integer bases.

Lean statements

The release has no Lean statement for this family. Its papers remain available to read, and contributors can propose a formal statement of a result. Suggest a change

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

The release contains no Lean statement for this family, so there is no formal statement to compare with OpenAI’s description.

Explanations by others

Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.

  • Frank Calegari, A Brief Digest of Some New OpenAI Results, 8 October 2026. Source

    Artin’s conjecture follows from quasi-RH for Dirichlet L-functions via primes p = 2aq + 1 with large prime factors in a; they call the argument unsurprising but one nobody would have written without quasi-RH.

What these papers cite in the release, and what cites them

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Within this family

Papers elsewhere in the release that this family’s papers cite

Papers elsewhere in the release that cite this family’s papers

Follow-up work by others

  • The Justin Sun Prize, The Justin Sun Prize: problem bank and award records, 11 October 2026. Source

    JSP-000027 lists the release’s claim as Solved with independent mathematical review pending; Lean proof column: No.

References outside the release

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What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.

    Author not stated. Source

Questions awaiting review

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Identify the theorem the catalogue describes

  • Family 029 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

State the result in Lean

  • Family 029 The release has no Lean statement for this family. Open the form