{
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  "source": "vibemath",
  "problem_id": "vibemathed:erdos-731-dyadic-regularity",
  "title": "Erdos Problem #731",
  "canonical_statement": "Let $A(n)$ be the least positive integer not dividing $\\binom{2n}{n}$. Erdos asked for the behaviour of $A(n)$ for reasonable $n$. Under an explicit dyadic-regularity formalization of reasonable, the distribution is determined on dyadic intervals against the scale $F_X = \\sqrt{2}(\\log 2)^{1/4} L^{1/4} \\exp\\sqrt{(\\log 2)L}$ with $L = \\log(2X)$.",
  "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "partial",
  "solution_events": [
    {
      "id": "vibemathed:erdos-731-dyadic-regularity:event",
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      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-06-27T00:00:00.000Z",
      "title": "arXiv:2606.29062 - A Resolution of Erdos Problem 731 under Dyadic Regularity",
      "summary": "resolved under an explicit formalization of 'reasonable', not in full generality",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:erdos-731-dyadic-regularity:attempt"
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      ],
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      "problem_id": "vibemathed:erdos-731-dyadic-regularity",
      "model": "ChatGPT, Aristotle",
      "provider": "OpenAI / Harmonic",
      "attempted_at": "2026-06-27T00:00:00.000Z",
      "human_collaborators": [
        "Eric Li"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The declaration says large language models, primarily ChatGPT, were used extensively throughout the research while the author originated the ideas, and that the accompanying Lean formalization was developed by the author with Aristotle.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv:2606.29062 - A Resolution of Erdos Problem 731 under Dyadic Regularity",
          "url": "https://arxiv.org/abs/2606.29062",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdos Problem #731",
          "url": "https://vibemathed.com/problem/erdos-731-dyadic-regularity",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:erdos-731-dyadic-regularity:verification",
      "problem_id": "vibemathed:erdos-731-dyadic-regularity",
      "solution_event_id": "vibemathed:erdos-731-dyadic-regularity:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "A Lean formalization accompanies the paper; we have not compiled it. The result is conditional on the paper's own formalization of Erdos's informal word 'reasonable', which the entry's resolution status reflects. arXiv preprint, not peer-reviewed.",
      "sources": [
        {
          "label": "VibeMathed: Erdos Problem #731",
          "url": "https://vibemathed.com/problem/erdos-731-dyadic-regularity",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2606.29062 - A Resolution of Erdos Problem 731 under Dyadic Regularity",
          "url": "https://arxiv.org/abs/2606.29062",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
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  "sources": [
    {
      "label": "arXiv:2606.29062 - A Resolution of Erdos Problem 731 under Dyadic Regularity",
      "url": "https://arxiv.org/abs/2606.29062",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdos Problem #731",
      "url": "https://vibemathed.com/problem/erdos-731-dyadic-regularity",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "expand",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-731-dyadic-regularity:signal:recent_partial_result",
      "problem_id": "vibemathed:erdos-731-dyadic-regularity",
      "kind": "recent_partial_result",
      "reason": "A source-reported partial result may support bounded expansion.",
      "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"
}