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Squarefree quartics and power-free polynomial values

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

Proves the squarefree-values conjecture for irreducible integer quartics with no fixed prime-square divisor: squarefree values on positive integers have the predicted positive Euler-product density. More generally, establishes the \((d-2)\)-power-free density for irreducible integer polynomials of degrees four through eight under the necessary local condition; together with Browning’s higher-degree theorem, this covers every d ≥ 4. (OpenAI’s Lean note)

Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.

Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments

What the papers state

Theorem 1.1

In Squarefree values of quartics and power-free values of polynomials

The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.

Let f\in\Z[x] be irreducible over \Q, of degree d with 4\le d\le8, and put k=d-2. If (1.1) holds, then

S_{f,k}(X)=c_{f,k}X+o_f(X),\qquad c_{f,k}=\prod_p\left(1-\frac{\rho_f(p^k)}{p^k}\right)>0.

No primitivity or sign condition on the leading coefficient is required.

No Lean statement in the release is paired with this theorem.

Corollary 1.3 (Separable products and simultaneous values)

In Squarefree values of quartics and power-free values of polynomials

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

Fix an integer k\ge2. Let F\in\Z[x]\setminus\{0\} be separable over \Q, with every nonconstant irreducible factor of degree at most k+2. If \rho_F(p^k)<p^k for every prime p, then

S_{F,k}(N)=c_{F,k}N+o_{F,k}(N),\qquad c_{F,k}=\prod_p\left(1-\frac{\rho_F(p^k)}{p^k}\right)>0.

More generally, let \mathbf F=(F_1,\ldots,F_r), with r\ge1, be a fixed family of nonzero integer polynomials, each separable over \Q with all nonconstant irreducible factors of degree at most k+2. Factors may be shared by different F_i. Put

\Omega_p=\{a\in\Z/p^k\Z:p^k\mid F_i(a)\text{ for at least one }i\}.

Let S_{\mathbf F,k}(N) count the integers 1\le n\le N for which every F_i(n) is k-power-free. If \#\Omega_p<p^k for every prime p, then

S_{\mathbf F,k}(N)=c_{\mathbf F,k}N+o_{\mathbf F,k}(N),\qquad c_{\mathbf F,k}=\prod_p\left(1-\frac{\#\Omega_p}{p^k}\right)>0.

The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.

Two more possible matches

  • Corollary 1.2 in Squarefree values of quartics and power-free values of polynomials.
  • Proposition 1.4 in Squarefree values of quartics and power-free values of polynomials.
Reported proof verification
One Lean statement in the release; two independent groups report that Comparator accepted runs for this file at the version shown here.
Papers and explanations
One current paper, readable here as a page beside its PDF.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

Squarefree values of quartics and power-free values of polynomials

Undated, current edition.

Read it herePDF

We prove that every irreducible integer quartic with no fixed prime-square divisor takes squarefree values with the predicted positive Euler-product density. More generally, we obtain the corresponding \((d-2)\)-power-free density in degrees \(4\le d\le8\). The proof combines number-field factorization, determinant estimates with adaptive auxiliary primes, and explicit low-degree geometry. Together with Browning’s theorem for higher degrees, this gives the \((d-2)\)-power-free density for every d ≥ 4.

One Lean statement in this family

Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.

For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.

PowerFreeValues

The paper states

Corollary 1.3 (Separable products and simultaneous values)

In Squarefree values of quartics and power-free values of polynomials

Fix an integer k\ge2. Let F\in\Z[x]\setminus\{0\} be separable over \Q, with every nonconstant irreducible factor of degree at most k+2. If \rho_F(p^k)<p^k for every prime p, then

S_{F,k}(N)=c_{F,k}N+o_{F,k}(N),\qquad c_{F,k}=\prod_p\left(1-\frac{\rho_F(p^k)}{p^k}\right)>0.

More generally, let \mathbf F=(F_1,\ldots,F_r), with r\ge1, be a fixed family of nonzero integer polynomials, each separable over \Q with all nonconstant irreducible factors of degree at most k+2. Factors may be shared by different F_i. Put

\Omega_p=\{a\in\Z/p^k\Z:p^k\mid F_i(a)\text{ for at least one }i\}.

Let S_{\mathbf F,k}(N) count the integers 1\le n\le N for which every F_i(n) is k-power-free. If \#\Omega_p<p^k for every prime p, then

S_{\mathbf F,k}(N)=c_{\mathbf F,k}N+o_{\mathbf F,k}(N),\qquad c_{\mathbf F,k}=\prod_p\left(1-\frac{\#\Omega_p}{p^k}\right)>0.

The words they share: count, factor, free, power, rho, polynomials, integers, prime, irreducible, degree, products, values, put.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Positive density of power-free polynomial values

PowerFreeValues.lean, the release lists it as a main result.

theorem allDegrees (f : Polynomial ℤ)    (hirr : Irreducible (f.map (Int.castRingHom ℚ)))    (hdeglo : 4 ≤ f.natDegree)    (hlocal : LocallyAdmissible f (f.natDegree - 2)) :    DensityStatement f (f.natDegree - 2) := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

Other mathematical statements in OpenAI’s Lean library

Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.

MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).

Their note: None; statement is for every d >= 4 (stronger than the new range 4..8 alone).

“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.

Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express or imply the whole advertised result (their term: “MATCHES”).

“MATCHES”, in their words: The challenge statements state the headline as OpenAI wrote it.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

The formalized result proves positive-density power-free values for every integer polynomial $f$ irreducible over $\mathbb Q$ of degree $d\ge4$. Put $k=d-2$ and assume no prime $k$th power divides every value of $f$. Then the number of positive integers $n\le X$ for which $f(n)$ is $k$-free is $c_fX+o_f(X)$, where $c_f>0$ is the convergent product of local factors. Negative values are allowed and zero is excluded. No monicity, primitivity, or coefficient-height restriction is imposed.

AI-drafted comparisons, with no recorded human review

  • Positive density of power-free polynomial values PowerFreeValues

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the all-degree form of the power-free density theorem (degree d ≥ 4, k = d − 2), which contains the main theorem for degrees 4 to 8 as a case; the paper obtains degrees above 8 from Browning’s theorem.

Who ran the proof-verification program?

These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.

These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.

Positive density of power-free polynomial values PowerFreeValues

Two reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

Other proofs and exact calculations

These reports examine claims in the papers using exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.

  • Propositions 10.1 and 12.2 of Squarefree values of quartics and power-free values of polynomials

    Reported by
    cert-machine
    Method
    Exact computation
    Their verdict
    “CERTIFIED” Exact code written independently from the paper confirms that the published object has the claimed property.
    What it covers
    “a finite component: the exact parameter certificate of Proposition prop:parameter-certificate (all 3,989 cells for d = 4..8, Table tab:parameter-certificate, the 150 endpoint derivative inequalities), the printed exponent margins, and the enlarged quartic cell of Proposition prop:cyclotomic-central; not the density theorem, whose other steps are proved in prose”
    Original report
    Their report

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

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

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

This family’s papers cite no paper elsewhere in the release.

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

No paper elsewhere in the release cites 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-000811 lists the release’s claim as Solved with independent mathematical review pending; Lean proof column: reported complete, independent verification pending.

References outside the release

Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release

Explain a result, compare statements or propose a correction

Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.

Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.

Offer to write an explanationCompare the Lean statement with the paperSuggest a source attributionSuggest a change

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

  • Moritz Firsching, anxkhn, Paul Lezeau, Formal Conjectures: mark problems solved in the OpenAI math release and link formal proofs, 11 October 2026. Source

    A Formal Conjectures maintainer, after source inspection, links the power-free values result to Erdos problem 978.

Questions awaiting review

Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.

Read the Lean statement against the paper

  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form

Identify the theorem the catalogue describes

  • Family 020 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

Compare the proposed paper and Lean matches

  • Family 020 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form