{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-387-binomial-divisors",
  "title": "Erdős Problem #387",
  "canonical_statement": "Erdős and Graham asked whether $\\binom{n}{k}$ with $1 \\le k \\le n/2$ must always have a divisor $\\le n$ that is close to $n$, meaning bigger than a fixed constant times $n$. Settled in both directions: true when $k$ is large enough as a function of $n$, but false in general, since there are $\\binom{n}{k}$ with $k$ small compared to $n$ having no such divisor.",
  "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-387-binomial-divisors:event",
      "problem_id": "vibemathed:erdos-387-binomial-divisors",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-05-20T00:00:00.000Z",
      "title": "arXiv:2605.21221 - Binomial coefficients with divisors avoiding an interval",
      "summary": "false in general; the positive direction holds for k large relative to n",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:erdos-387-binomial-divisors:attempt"
      ],
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      ],
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    {
      "id": "vibemathed:erdos-387-binomial-divisors:attempt",
      "problem_id": "vibemathed:erdos-387-binomial-divisors",
      "model": "ChatGPT 5.5 Pro",
      "provider": "OpenAI",
      "attempted_at": "2026-05-20T00:00:00.000Z",
      "human_collaborators": [
        "Hung M. Bui",
        "Slava Naprienko",
        "Kyle Pratt",
        "Alexandru Zaharescu"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The disclosure is carefully scoped rather than blanket. The main ideas in the proof of Theorem 5.1 were developed in interactive sessions between the authors and ChatGPT 5.5 Pro, and some documents and code in the accompanying repository were generated with AI assistance. The authors separately used ChatGPT for literature searches and for spotting typos, and they state that all text in the paper is human-generated. The heavier half of the paper, a restricted covering problem attacked with sieve methods and exponential sum estimates, is presented as the authors' own.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
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          "label": "arXiv:2605.21221 - Binomial coefficients with divisors avoiding an interval",
          "url": "https://arxiv.org/abs/2605.21221",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdős Problem #387",
          "url": "https://vibemathed.com/problem/erdos-387-binomial-divisors",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
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      "cost_usd": null,
      "independence": "unknown"
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      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "arXiv preprint, not peer-reviewed. The authors credit the model with the main ideas of one theorem rather than the paper, so the bulk of the argument rests on ordinary refereeing.",
      "sources": [
        {
          "label": "VibeMathed: Erdős Problem #387",
          "url": "https://vibemathed.com/problem/erdos-387-binomial-divisors",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2605.21221 - Binomial coefficients with divisors avoiding an interval",
          "url": "https://arxiv.org/abs/2605.21221",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
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  "sources": [
    {
      "label": "arXiv:2605.21221 - Binomial coefficients with divisors avoiding an interval",
      "url": "https://arxiv.org/abs/2605.21221",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős Problem #387",
      "url": "https://vibemathed.com/problem/erdos-387-binomial-divisors",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
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  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-387-binomial-divisors:signal:watch_only",
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      "generated_at": "2026-09-13T16:28:40.178Z"
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  "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"
}