What was already known
Erdős and Turán proved a logarithmic lower bound. Erdős asked whether the minimum grows faster than log n; the consecutive-integer construction gives an upper bound of order n/log n.
AI involvement
GPT-6 Astra found the argument and wrote Lean code in a network-isolated run without human mathematical steering. Adamczewski packaged the repository; Bloom later sketched three proof routes.
Benchmark and repository document no human guidance during the model proof search.
Importance
The compared theorem resolves the qualitative superlogarithmic question.
Limitations
Only the qualitative limit has a documented statement-matched Lean check; the stronger polynomial bounds are separate resolutions. Bloom’s sketch is not a documented complete check.
Technical verification
Source-reported Lean build and Comparator match to the trusted challenge in a pristine environment; no broader informal theorem is silently inferred.
Formal checking
compiled proof, trusted statement comparison, and standard-axiom check
- Checker
- Lean 4 kernel and FrontierMath Erdős Comparator
- Scope
- Problem #126 compared theorem
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The high-level mechanisms are described, but the distinct alternate proofs have not been fully digested in human accounts.
Correction and publication history
The compared theorem resolves the qualitative superlogarithmic question.
Read this published revision