What was already known
The individual Dickman law was classical. Erdős and Pomerance conjectured joint independence; Teräväinen established the product in logarithmic density, while ordinary natural density remained open.
AI involvement
OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.
OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.
Importance
The reported natural-density result resolves the Erdős–Pomerance joint independence conjecture and implies that either ordering of the largest prime factors has density one half.
Limitations
The statement concerns two consecutive integers and ordinary natural density; it does not establish analogous laws for longer runs. The Lean check is reported by OpenAI and was not reproduced here.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned Lean scope note identifies the joint natural-density theorem and equal ordering densities.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- For every fixed 0<a,b<1, the natural density of integers n with P⁺(n)≤nᵃ and P⁺(n+1)≤nᵇ is ρ(1/a)ρ(1/b), where P⁺ is the largest prime factor and ρ is the Dickman function.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.
Correction and publication history
Primary manuscript, attribution, prior-work, and scoped checking review for non-notifying local inclusion.
Read this published revision