{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R648",
  "content_hash": null,
  "slug": "rdcr-claim-factorized-run",
  "type": "claim",
  "title": "Exact factorizations certify the length-fourteen witness",
  "summary": "Trial division through the square root gives complete prime factorizations and reproduces all fourteen divisor counts.",
  "relevance": "For Longest rainbow divisor-count interval below 10^12, record rdcr-claim-factorized-run (“Exact factorizations certify the length-fourteen witness”) records a bound, answer, status fact, or structural consequence. The record states: Trial division through the square root gives complete prime factorizations and reproduces all fourteen divisor counts.",
  "relevance_source": "recorded",
  "body": "The complete factorizations are\n\\[\n\\begin{array}{rcl}\n1745175039&=&3\\cdot31^2\\cdot605333,\\\\\n1745175040&=&2^9\\cdot5\\cdot7\\cdot97387,\\\\\n1745175041&=&37\\cdot103\\cdot173\\cdot2647,\\\\\n1745175042&=&2\\cdot3^2\\cdot13\\cdot19\\cdot59\\cdot6653,\\\\\n1745175043&=&53\\cdot977\\cdot33703,\\\\\n1745175044&=&2^2\\cdot436293761,\\\\\n1745175045&=&3\\cdot5\\cdot41\\cdot773\\cdot3671,\\\\\n1745175046&=&2\\cdot872587523,\\\\\n1745175047&=&7\\cdot11\\cdot61^2\\cdot6091,\\\\\n1745175048&=&2^3\\cdot3\\cdot23\\cdot47\\cdot137\\cdot491,\\\\\n1745175049&=&1745175049,\\\\\n1745175050&=&2\\cdot5^2\\cdot29\\cdot167\\cdot7207,\\\\\n1745175051&=&3^4\\cdot113\\cdot190667,\\\\\n1745175052&=&2^2\\cdot17^2\\cdot1039\\cdot1453.\n\\end{array}\n\\]\nTrial division by every prime through the square root leaves either 1 or the displayed final prime, which proves each factorization complete. Applying \\(\\prod(e+1)\\) gives, in the same order,\n\\[\n12,80,16,96,8,6,32,4,24,128,2,48,20,36.\n\\]",
  "status": "established",
  "evidence_grade": "mathematical_identity",
  "scope": {
    "kind": "bounded",
    "statement": "the fourteen integers from 1745175039 through 1745175052",
    "bounds": {
      "n": {
        "min": 1745175039,
        "max": 1745175052
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1510.07081",
      "locator": "Independent trial-division replay on 2026-07-25 and factor output from rdcr-artifact-segmented-prefix-sweep"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1510.07081",
    "locator": "Independent trial-division replay on 2026-07-25 and factor output from rdcr-artifact-segmented-prefix-sweep"
  },
  "relations": [
    {
      "slug": "R646",
      "title": "Segmented factor sieve and cross-segment rainbow search",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R649",
      "title": "The requested maximum is at least fourteen",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "rainbow-divisor-count-run-1e12",
      "title": "rainbow divisor count run 1e12",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}