{
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  "source": "vibemath",
  "problem_id": "vibemathed:distinct-multiples-in-intervals",
  "title": "Improved Bounds for Distinct Multiples in Intervals",
  "canonical_statement": "For the Erdős–Pomerance functions $F(n)$ and $h_{\\mathbb{P}}(n)$ counting how many consecutive integers are needed to contain a distinct multiple of each integer, respectively prime, up to $n$, the paper proves $F(n) \\ge h_{\\mathbb{P}}(n) \\ge n\\exp\\left(\\left(\\frac{\\log 2}{2} - o(1)\\right)\\frac{\\log n}{\\log\\log n}\\right)$, disproving Kominers' conjecture that $F(n) \\ll n\\log n$. The paper also significantly  improves known upper bounds (which were on the order of $n^{3/2}$) to $F(n) \\le n^{4/3 + o(1)}$ and $h_{\\mathbb{P}}(n) \\le n^{4/3 - o(1)}$.",
  "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",
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    {
      "id": "vibemathed:distinct-multiples-in-intervals:event",
      "problem_id": "vibemathed:distinct-multiples-in-intervals",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-07-29T00:00:00.000Z",
      "title": "arXiv",
      "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_co_developed",
      "attempt_ids": [
        "vibemathed:distinct-multiples-in-intervals:attempt"
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      "id": "vibemathed:distinct-multiples-in-intervals:attempt",
      "problem_id": "vibemathed:distinct-multiples-in-intervals",
      "model": "ChatGPT 5.x",
      "provider": "OpenAI",
      "attempted_at": "2026-07-29T00:00:00.000Z",
      "human_collaborators": [
        "Kaizhe Chen",
        "Samuel Korsky"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The note carries a dedicated Statement on AI saying that the main proofs in it were developed with the assistance of ChatGPT 5.x. That is a claim about the mathematics rather than the exposition, which is why this sits a tier above the rest of its batch.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2607.26450",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Improved Bounds for Distinct Multiples in Intervals",
          "url": "https://vibemathed.com/problem/distinct-multiples-in-intervals",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
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      "independence": "unknown"
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      "solution_event_id": "vibemathed:distinct-multiples-in-intervals:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "No independent check. The disclosure credits the model with the main proofs, so the result rests entirely on the author's own verification. Preprint, not refereed.",
      "sources": [
        {
          "label": "VibeMathed: Improved Bounds for Distinct Multiples in Intervals",
          "url": "https://vibemathed.com/problem/distinct-multiples-in-intervals",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2607.26450",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
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      "label": "arXiv",
      "url": "https://arxiv.org/abs/2607.26450",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Improved Bounds for Distinct Multiples in Intervals",
      "url": "https://vibemathed.com/problem/distinct-multiples-in-intervals",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
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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"
}