{
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  "source": "vibemath",
  "problem_id": "vibemathed:erdos-consecutive-integers-prime-factors",
  "title": "Erdos's Conjecture on Consecutive Integers Free of Certain Prime Factors",
  "canonical_statement": "Let $n_k$ be the least $n > 2k$ such that $(n-k)(n-k+1)\\cdots(n-1)$ has no prime factor in $(k, 2k)$. Erdos conjectured a superpolynomial lower bound; for all large $k$, $n_k > e^{\\log^2 k / (20 \\log\\log k)}$.",
  "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-consecutive-integers-prime-factors:event",
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      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-06-18T00:00:00.000Z",
      "title": "arXiv:2606.19863 - Consecutive integers free of certain prime factors",
      "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-consecutive-integers-prime-factors:attempt"
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      "id": "vibemathed:erdos-consecutive-integers-prime-factors:attempt",
      "problem_id": "vibemathed:erdos-consecutive-integers-prime-factors",
      "model": "ChatGPT 5.5 Pro, Aristotle",
      "provider": "OpenAI / Harmonic",
      "attempted_at": "2026-06-18T00:00:00.000Z",
      "human_collaborators": [
        "Wouter van Doorn",
        "Quanyu Tang"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The paper says the bound was conceived of by ChatGPT 5.5 Pro and that the original paper the model wrote remains publicly available, so the provenance is checkable rather than asserted. The supporting proofs were obtained by Harmonic's Aristotle and are stated to be fully self-contained.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
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          "label": "arXiv:2606.19863 - Consecutive integers free of certain prime factors",
          "url": "https://arxiv.org/abs/2606.19863",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdos's Conjecture on Consecutive Integers Free of Certain Prime Factors",
          "url": "https://vibemathed.com/problem/erdos-consecutive-integers-prime-factors",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
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      "independence": "unknown"
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      "solution_event_id": "vibemathed:erdos-consecutive-integers-prime-factors:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Proofs produced by an automated theorem prover and described as fully self-contained; we have not independently checked them. arXiv preprint, not peer-reviewed.",
      "sources": [
        {
          "label": "VibeMathed: Erdos's Conjecture on Consecutive Integers Free of Certain Prime Factors",
          "url": "https://vibemathed.com/problem/erdos-consecutive-integers-prime-factors",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
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          "label": "arXiv:2606.19863 - Consecutive integers free of certain prime factors",
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          "kind": "primary-mathematical-source",
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        }
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  "sources": [
    {
      "label": "arXiv:2606.19863 - Consecutive integers free of certain prime factors",
      "url": "https://arxiv.org/abs/2606.19863",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdos's Conjecture on Consecutive Integers Free of Certain Prime Factors",
      "url": "https://vibemathed.com/problem/erdos-consecutive-integers-prime-factors",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
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  "recommended_exposure": "result_only",
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    {
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      "problem_id": "vibemathed:erdos-consecutive-integers-prime-factors",
      "kind": "watch_only",
      "reason": "Current public evidence does not meet the default replay threshold.",
      "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"
}