{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R68",
  "content_hash": null,
  "slug": "bdr1m-claim-exact-prefix-20000",
  "type": "claim",
  "title": "The exact record through n=20,000 occurs at (19,971, 9,949)",
  "summary": "An exact 100-million-state recurrence sweep gives a 469-digit divisor count.",
  "relevance": "For Most divisors of a binomial coefficient with top at most 10^6, record bdr1m-claim-exact-prefix-20000 (“The exact record through n=20,000 occurs at (19,971, 9,949)”) records a bound, answer, status fact, or structural consequence. The record states: An exact 100-million-state recurrence sweep gives a 469-digit divisor count.",
  "relevance_source": "recorded",
  "body": "Using \\(k\\leq n/2\\) by symmetry, the exact prefix record is attained at\n\\[\n(n,k)=(19971,9949).\n\\]\nIts divisor count is\n\\[\n\\boxed{2348387885204790879830610540244537680880212204266092283665817947799112331315722388739095226210492168628520272301982570612855049692984247653163656175812378814497351725872902200308583530162859385427897865865873620878587583646796243101440659894852088456115074260792754959770155740574399754249752554204207499681748634739352025413649254209837693841360334377369629570834866128375706364256515169543130470863765460450748221403831366558257535620632378709575168022539059180077056}.\n\\]\nThe prime-exponent histogram of \\(\\binom{19971}{9949}\\) is\n\\[\n\\#\\{p:v_p=1\\}=1527,\\quad \\#\\{p:v_p=2\\}=9,\\quad \\#\\{p:v_p=3\\}=4,\n\\]\nwith one exponent 8 and one exponent 10. The factors having exponent above one are\n\\[\n2^{10}3^8 5^3 7^2 11^3 13^3 19^2 23^3 41^2 59^2 79^2 107^2 109^2 127^2 131^2.\n\\]\nThus the displayed value also equals \\(2^{1527}3^9 4^4 9\\cdot11\\). The ordered full factorization has SHA-256 digest `11d4dc8a9b2dc54225e841546f236a3a9cd5c2f61fe0db6ab2455f28749b9c6c` under the `p^e` convention documented in the certificate artifact.",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "every integer pair (n,k) with 1 <= k < n <= 20000",
    "bounds": {
      "n": {
        "min": 2,
        "max": 20000
      },
      "pairs_checked_using_symmetry": {
        "min": 100000000,
        "max": 100000000
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1134/S0001434613010331",
      "locator": "Exact recurrence sweep in bdr1m-artifact-exact-prefix-sweep and Legendre certificate in bdr1m-artifact-factorization-replay"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1134/S0001434613010331",
    "locator": "Exact recurrence sweep in bdr1m-artifact-exact-prefix-sweep and Legendre certificate in bdr1m-artifact-factorization-replay"
  },
  "models": [],
  "relations": [
    {
      "slug": "R64",
      "title": "Exact 100-million-state binomial recurrence sweep",
      "object_type": "artifact",
      "relation": "reproduces",
      "direction": "incoming"
    },
    {
      "slug": "R65",
      "title": "Legendre factorization and divisor-count replay",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R66",
      "title": "The literature audit found asymptotic and central-coefficient results",
      "object_type": "attempt",
      "relation": "contextualizes",
      "direction": "incoming"
    },
    {
      "slug": "binomial-divisor-record-1e6",
      "title": "binomial divisor record 1e6",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}