{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R681",
  "content_hash": null,
  "slug": "rsg-claim-global-upper-bound-fourteen",
  "type": "claim",
  "title": "Published squarefree-gap computations give the global upper bound 14",
  "summary": "A rainbow squarefree gap of endpoint distance 7 is certified, and exhaustive search proves 7 is the prefix maximum through 5,000,000; the global maximum below \\(10^{12}\\) lies between 7 and 14, and whether any distance from 8 through 14 occurs remains open.",
  "relevance": "For Largest rainbow squarefree gap below 10^12, record rsg-claim-global-upper-bound-fourteen (“Published squarefree-gap computations give the global upper bound 14”) records a bound, answer, status fact, or structural consequence. The record states: A rainbow squarefree gap of endpoint distance 7 is certified, and exhaustive search proves 7 is the prefix maximum through 5,000,000; the global maximum below \\(10^{12}\\) lies between 7 and 14, and whether any distance from 8 through 14 occurs remains open.",
  "relevance_source": "recorded",
  "body": "A rainbow interval is first an ordinary gap between consecutive squarefree integers. Published computations cover ordinary squarefree gaps through \\(10^{18}\\). Marmet's first-occurrence table places the first run of 14 consecutive nonsquarefree integers at\n\\[\n1043460553364,\n\\]\nwhich exceeds the present cutoff. The first runs of lengths 15 through 18 occur still higher, and the exhaustive computation reports no longer run through \\(10^{18}\\). The first run of 13 begins at\n\\[\n82462576220.\n\\]\nTherefore an ordinary gap with upper endpoint at most \\(10^{12}\\) contains at most 13 interior integers, and its endpoint distance is at most 14. The rainbow target consequently satisfies\n\\[\n7\\leq G(10^{12})\\leq14.\n\\]\nThe lower bound comes from the explicit interval at 30,922. Resolving the target requires checking the distinct-prime condition after five million.",
  "status": "supported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "every pair of consecutive squarefree integers a < b with b at most 1000000000000",
    "bounds": {
      "b": {
        "min": 2,
        "max": 1000000000000
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1210.3829",
      "locator": "Louis Marmet, First occurrences of square-free gaps and an algorithm for their computation, first-occurrence table; cross-checked against Michael J. Mossinghoff, Tomas Oliveira e Silva, and Timothy S. Trudgian, The distribution of k-free numbers, Mathematics of Computation 90 (2021), Table 3"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1210.3829",
    "locator": "Louis Marmet, First occurrences of square-free gaps and an algorithm for their computation, first-occurrence table; cross-checked against Michael J. Mossinghoff, Tomas Oliveira e Silva, and Timothy S. Trudgian, The distribution of k-free numbers, Mathematics of Computation 90 (2021), Table 3"
  },
  "models": [],
  "relations": [
    {
      "slug": "R680",
      "title": "The literature gives ordinary squarefree-gap data, while the rainbow restriction remains unlocated",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R679",
      "title": "Exact square-divisor sieve and matching replay through five million",
      "object_type": "artifact",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "rainbow-squarefree-gap-1e12",
      "title": "rainbow squarefree gap 1e12",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}