{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-728-factorial-divisibility",
  "title": "Erdős Problem #728: Factorial Divisibility",
  "canonical_statement": "Whether there are infinitely many integers $a, b, n$ with $a, b \\ge \\varepsilon n$ such that $a!\\cdot b!$ divides $n!\\cdot(a+b-n)!$ while $a+b$ exceeds $n$ by more than $C\\cdot\\log n$.",
  "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "resolved",
  "solution_events": [
    {
      "id": "vibemathed:erdos-728-factorial-divisibility:event",
      "problem_id": "vibemathed:erdos-728-factorial-divisibility",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-01-06T00:00:00.000Z",
      "title": "erdosproblems.com",
      "summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:erdos-728-factorial-divisibility:attempt"
      ],
      "method_family_ids": [
        "vibemathed:erdos-728-factorial-divisibility:method"
      ],
      "source_assertion_ids": [
        "vibemathed:erdos-728-factorial-divisibility:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:erdos-728-factorial-divisibility:attempt",
      "problem_id": "vibemathed:erdos-728-factorial-divisibility",
      "model": "Aristotle (Harmonic) + GPT-5.2 Pro",
      "provider": "Harmonic / OpenAI",
      "attempted_at": "2026-01-06T00:00:00.000Z",
      "human_collaborators": [
        "Boris Alexeev",
        "Kevin Barreto",
        "Liam Price",
        "Nat Sothanaphan"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Aristotle (Harmonic's Lean-based prover) and GPT-5.2 Pro produced a fully autonomous, Lean-verified proof. Regarded by the erdosproblems.com maintainers as the first Erdős problem resolved autonomously by AI systems, about three months before the more widely covered #1196 result.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Writeup of Aristotle's autonomous Lean proof (Barreto)",
          "url": "https://arxiv.org/abs/2601.07421",
          "kind": "paper",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "erdosproblems.com",
          "url": "https://www.erdosproblems.com/728",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdős Problem #728: Factorial Divisibility",
          "url": "https://vibemathed.com/problem/erdos-728-factorial-divisibility",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:erdos-728-factorial-divisibility:verification",
      "problem_id": "vibemathed:erdos-728-factorial-divisibility",
      "solution_event_id": "vibemathed:erdos-728-factorial-divisibility:event",
      "level": "lean_verified_statement_audited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "audited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Machine-checked end to end in the Lean proof assistant - every logical step formally verified, not just human-reviewed. Written up formally by Nat Sothanaphan.",
      "sources": [
        {
          "label": "VibeMathed: Erdős Problem #728: Factorial Divisibility",
          "url": "https://vibemathed.com/problem/erdos-728-factorial-divisibility",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "erdosproblems.com",
          "url": "https://www.erdosproblems.com/728",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "erdosproblems.com",
      "url": "https://www.erdosproblems.com/728",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős Problem #728: Factorial Divisibility",
      "url": "https://vibemathed.com/problem/erdos-728-factorial-divisibility",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "replay",
  "recommended_exposure": "method_aware",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-728-factorial-divisibility:signal:replay_ready",
      "problem_id": "vibemathed:erdos-728-factorial-divisibility",
      "kind": "replay_ready",
      "reason": "A public source and a non-unreviewed verification label support replay triage.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}