{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R65",
  "content_hash": null,
  "slug": "bdr1m-artifact-factorization-replay",
  "type": "artifact",
  "title": "Legendre factorization and divisor-count replay",
  "summary": "Standard-library Python regenerates both full factorizations and checks their SHA-256 digests.",
  "relevance": "For Most divisors of a binomial coefficient with top at most 10^6, record bdr1m-artifact-factorization-replay (“Legendre factorization and divisor-count replay”) supplies evidence or a replay used to check the packet. The record states: Standard-library Python regenerates both full factorizations and checks their SHA-256 digests.",
  "relevance_source": "recorded",
  "body": "The verifier sieves every prime through each top parameter. Legendre's factorial valuation gives every binomial exponent as an integer. The code joins the nonzero pairs in ascending prime order using `p^e` and hashes the resulting factorization. It then multiplies every \\(e+1\\), hashes the exact decimal divisor count, and reports each exponent histogram.\n\nThe prefix factorization contains 1,542 terms. The cutoff-scale factorization contains 53,478 terms. Their report payload has SHA-256 digest `fea7d94ea1b9b59e253076f13b8caf82c47db0d875c7e0b71d351554ee7eb723`.",
  "status": "available",
  "evidence_grade": "executable",
  "scope": {
    "kind": "bounded",
    "statement": "the complete prime-exponent factorizations at (19971,9949) and (1000000,499985)",
    "bounds": {
      "pairs": {
        "min": 2,
        "max": 2
      },
      "largest_n": {
        "min": 1000000,
        "max": 1000000
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "partial",
    "kind": "inline_python_computation",
    "entrypoint": "join source_lines with newline and run with python3",
    "runtime": "CPython 3.9 or later, standard library only",
    "citation": {
      "url": "https://doi.org/10.1134/S0001434613010331",
      "locator": "Inline CPython standard-library computation reproduced on 2026-07-25"
    },
    "inline_source": [
      "from hashlib import sha256",
      "from json import dumps",
      "from math import isqrt",
      "",
      "CASES = [",
      "    (19971, 9949, 1542, \"11d4dc8a9b2dc54225e841546f236a3a9cd5c2f61fe0db6ab2455f28749b9c6c\",",
      "     \"87551dd1380ee16fd2443feb0ea48da980013632cae016af476f8981a8f14683\"),",
      "    (1000000, 499985, 53478, \"fb3db7328c0c940f68d88e873c6554c9ec65616c825f5b933fd2e55e492e1be2\",",
      "     \"37e6b0aec8c146fa82e6e8d0eb776dbb1504fb2fff80e5fa74bff8eaddfee951\"),",
      "]",
      "",
      "def primes_through(limit):",
      "    sieve = bytearray(b\"\\x01\") * (limit + 1)",
      "    sieve[:2] = b\"\\x00\\x00\"",
      "    for p in range(2, isqrt(limit) + 1):",
      "        if sieve[p]:",
      "            sieve[p*p:limit+1:p] = b\"\\x00\" * ((limit-p*p)//p+1)",
      "    return [p for p in range(2, limit + 1) if sieve[p]]",
      "",
      "def factorial_valuation(n, p):",
      "    value = 0",
      "    while n:",
      "        n //= p",
      "        value += n",
      "    return value",
      "",
      "def certify(n, k, expected_terms, expected_factor_sha, expected_tau_sha):",
      "    factors = []",
      "    tau = 1",
      "    histogram = {}",
      "    for p in primes_through(n):",
      "        exponent = factorial_valuation(n,p)-factorial_valuation(k,p)-factorial_valuation(n-k,p)",
      "        assert exponent >= 0",
      "        if exponent:",
      "            factors.append((p,exponent))",
      "            tau *= exponent+1",
      "            histogram[exponent] = histogram.get(exponent,0)+1",
      "    factor_data = \"*\".join(f\"{p}^{e}\" for p,e in factors)",
      "    tau_text = str(tau)",
      "    assert len(factors) == expected_terms",
      "    assert sha256(factor_data.encode()).hexdigest() == expected_factor_sha",
      "    assert sha256(tau_text.encode()).hexdigest() == expected_tau_sha",
      "    return {",
      "        \"n\":n, \"k\":k, \"factor_terms\":len(factors),",
      "        \"exponent_histogram\":sorted(histogram.items()),",
      "        \"factor_sha256\":expected_factor_sha, \"tau_digits\":len(tau_text),",
      "        \"tau_prefix\":tau_text[:32], \"tau_suffix\":tau_text[-32:],",
      "        \"tau_sha256\":expected_tau_sha,",
      "    }",
      "",
      "report = [certify(*case) for case in CASES]",
      "payload = dumps(report,sort_keys=True,separators=(\",\",\":\"))",
      "assert sha256(payload.encode()).hexdigest() == \"fea7d94ea1b9b59e253076f13b8caf82c47db0d875c7e0b71d351554ee7eb723\"",
      "print(payload)"
    ],
    "missing": [
      "command",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1134/S0001434613010331",
    "locator": "Inline CPython standard-library computation reproduced on 2026-07-25"
  },
  "relations": [
    {
      "slug": "R68",
      "title": "The exact record through n=20,000 occurs at (19,971, 9,949)",
      "object_type": "claim",
      "relation": "evidences",
      "direction": "outgoing"
    },
    {
      "slug": "R67",
      "title": "A certified cutoff-scale coefficient gives a 16,113-digit lower bound",
      "object_type": "claim",
      "relation": "evidences",
      "direction": "outgoing"
    },
    {
      "slug": "binomial-divisor-record-1e6",
      "title": "binomial divisor record 1e6",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}