Breakthrough Tracker record
Erdős Problem #728: a Lean-formalized resolution of the reconstructed intended statement
The proof establishes an infinite family of middle-range factorial-divisibility triples whose logarithmic gap lies between any prescribed positive constant multiples of log n. A Lean proof is translated into conventional mathematical prose.
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- Stable ID
math-erdos-728-factorial-divisibility-2026- Revision
math-erdos-728-factorial-divisibility-2026.v1- Field
- Mathematics · Number theory and formal mathematics
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Current · Formally verified resolution of a reconstructed formulation
- Last checked
AI role
GPT-5.2 Pro and Harmonic's Aristotle performed the formal proof search, with Kevin Barreto operating the system and Sothanaphan writing the exposition.
Record details
- Problem or result
- Erdős Problem #728
- Authors
- Nat Sothanaphan; GPT-5.2 Pro and Aristotle operated by Kevin Barreto
- Institutions
- Harmonic for Aristotle; see the paper for author affiliation
- Result date
- Submitted January 12, 2026; revised through January 26, 2026
Why it matters
The authors describe it as the first problem in the Erdős Problems database regarded as fully resolved autonomously by an AI system.
Limits
The historical wording of #728 was ambiguous, so the project formalized a reconstructed intended statement. Related older literature also complicates an unqualified novelty claim.
Sources
- Primary: Nat Sothanaphan, Resolution of Erdős Problem #728
- Independent: Erdős Problems project blog
- Independent: Problem #728 discussion and formulation notes
Correction and revision history
- Added on 2026-07-22 with an explicit formulation and prior-literature warning rather than an unqualified autonomous-solution claim.
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