Resolves the Erdős–Pomerance joint Dickman conjecture: the logarithmic sizes of the largest prime factors of n and \(n+1\) are asymptotically independent in ordinary natural density. In particular, the integers satisfying \(P^+(n)\lt P^+(n+1)\) have density 1/2. (OpenAI’s Lean note)
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Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments
It bears on Erdős problem #371, as listed by Thomas Bloom, erdosproblems.com.
The paper states
In The joint Dickman law for consecutive integers, 24 September 2026
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
One has
The reverse ordering has the same natural density.
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JointDickmanThe paper states
In The joint Dickman law for consecutive integers, 24 September 2026
For every fixed a,b\in(0,1),
Here the limit is taken over all real X, with ordinary, unweighted counting.
The words they share: dickman, joint, ordinary, real.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
JointDickman.lean, the release lists it as a supporting statement.
theorem joint_law (a b : ℝ) (ha : 0 < a) (ha1 : a < 1) (hb : 0 < b) (hb1 : b < 1) : Tendsto (realDensity (fun n => (n.maxPrimeFac : ℝ) ≤ (n : ℝ) ^ a ∧ ((n + 1).maxPrimeFac : ℝ) ≤ (n : ℝ) ^ b)) atTop (𝓝 (Erdos970.NumberTheoryLean.Dickman.rho (1 / a) * Erdos970.NumberTheoryLean.Dickman.rho (1 / b))) := by sorryThis 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.
The file states two more theorems: increasing_, decreasing_.
One independent group reports that Comparator accepted a run for this Lean statement file: cert-machine. Who ran the proof-verification program
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; headline and the 1/2 corollary are stated. Dickman rho is a custom delay-equation construction (checked faithful).
Definitions the group asks readers to compare with the paper: noncomputable def rho : ℝ → ℝ := DelayConstruction.delay (-1) with stepApprox σ (n+1) u = 1 + σ * ∫ t in 1..max 1 u, kernel (stepApprox σ n) t (custom Dickman function; correct by inspection, no bridge lemma to a Mathlib object)
“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.
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.
OpenAI’s own note on what its Lean covers
Let $P^+(n)$ be the largest prime factor of $n$. The formalization proves the joint Dickman law in ordinary natural density: for every $0<a,b<1$, the density of integers satisfying $P^+(n)\le n^a$ and $P^+(n+1)\le n^b$ tends to $\rho(1/a)\rho(1/b)$, where $\rho$ is the Dickman function.
It also proves that each of the orderings $P^+(n)<P^+(n+1)$ and $P^+(n+1)<P^+(n)$ has natural density $1/2$.
Joint Dickman law and equal ordering densities for consecutive integers JointDickman
Unreviewed AI draft: states a paper’s theorem as the paper does. States the joint Dickman law in natural density, as in the main theorem, and the corollary that each strict ordering of P⁺(n) and P⁺(n+1) has density 1/2.
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.
JointDickmanOne report, at the release’s recorded version.
| Reported by | Their word | Their account | Date | Original 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 2026 | Original report |
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.
This family’s papers cite no paper elsewhere in the release.
No paper elsewhere in the release cites this family’s papers.
The Justin Sun Prize, The Justin Sun Prize: problem bank and award records, 11 October 2026. Source
JSP-000306 lists the release’s claim as Solved with independent mathematical review pending; Lean proof column: reported complete, independent verification pending.
The Justin Sun Prize, The Justin Sun Prize: problem bank and award records, 11 October 2026. Source
JSP-000770 lists the release’s claim as Solved with independent mathematical review pending; Lean proof column: reported complete, independent verification pending.
jbaelaw, jbaelaw/Erdos928: Erdős Problem 928 as a corollary of OpenAI’s joint Dickman law, 9 October 2026. Source
Erdős Problem 928 follows from the joint Dickman law, formalised in Lean.
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
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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 “Major math journal”. 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 joint Dickman result to Erdos problem 371, recording that the density is taken over a closed interval rather than below a bound.
Thomas Bloom, Erdős problems in the OpenAI proofs, 8 October 2026. Source
Bloom lists Erdős 371 (density one half for consecutive largest prime factors), names Lü and Wang and Teräväinen, and notes it is a corollary of a joint distribution law in 84 pages.
eigendash, eigendash/openai-math-visualised: numerical probes of seven result families, 8 October 2026. Source
Over 59,999,998 consecutive pairs up to 6 times 10^7, the density of n whose largest prime factor is below that of n + 1 is 0.499983, against the predicted 1/2.
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Why this passage is shown
Corollary 1.2
In The joint Dickman law for consecutive integers, 24 September 2026
Why this passage is shown
Theorem 1.1 (Joint Dickman law)
In The joint Dickman law for consecutive integers, 24 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, JointDickman
Lean statement
JointDickman