{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R682",
  "content_hash": null,
  "slug": "rsg-claim-prefix-exact-five-million",
  "type": "claim",
  "title": "The exact rainbow maximum through five million is 7",
  "summary": "An exhaustive square-divisor sieve and exact matching test find maximum endpoint distance 7, first attained at 30,922 and 30,929.",
  "relevance": "For Largest rainbow squarefree gap below 10^12, record rsg-claim-prefix-exact-five-million (“The exact rainbow maximum through five million is 7”) records a bound, answer, status fact, or structural consequence. The record states: An exhaustive square-divisor sieve and exact matching test find maximum endpoint distance 7, first attained at 30,922 and 30,929.",
  "relevance_source": "recorded",
  "body": "For consecutive squarefree endpoints \\(a<b\\leq5{,}000{,}000\\), the largest distance admitting distinct square-prime labels on every interior integer is\n\\[\n\\boxed{b-a=7}.\n\\]\nThe first attaining pair is\n\\[\n(a,b)=(30922,30929).\n\\]\n\nThe replay marks every multiple of \\(p^2\\) for every prime \\(p\\leq\\sqrt{5{,}000{,}000}\\). It then visits every consecutive squarefree pair. Once the displayed distance-7 witness is checked, only gaps of distance at least 7 need a matching test. There are 344 such ordinary squarefree gaps and 33 admit a full distinct-prime matching. None has distance above 7.",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "every pair of consecutive squarefree integers a < b with b at most 5000000",
    "bounds": {
      "b": {
        "min": 2,
        "max": 5000000
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1912.04972",
      "locator": "Exact replay in rsg-artifact-prefix-sieve-five-million"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1912.04972",
    "locator": "Exact replay in rsg-artifact-prefix-sieve-five-million"
  },
  "relations": [
    {
      "slug": "R679",
      "title": "Exact square-divisor sieve and matching replay through five million",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "R683",
      "title": "The interval 30,922 to 30,929 has a six-color square-prime certificate",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "rainbow-squarefree-gap-1e12",
      "title": "rainbow squarefree gap 1e12",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}