{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R560",
  "content_hash": null,
  "slug": "pcpsg-artifact-pocklington-incumbent",
  "type": "artifact",
  "title": "Pocklington certificate for the incumbent pair",
  "summary": "Standard-library Python verifies complete n-1 factorizations, all witnesses, adjacency, and the gap.",
  "relevance": "For Closest prime square to the cube of a prime below one trillion, record pcpsg-artifact-pocklington-incumbent (“Pocklington certificate for the incumbent pair”) supplies evidence or a replay used to check the packet. The record states: Standard-library Python verifies complete n-1 factorizations, all witnesses, adjacency, and the gap.",
  "relevance_source": "recorded",
  "body": "The certificate uses a separate witness for each distinct prime divisor of \\(n-1\\). It verifies all small factor primes by trial division, then checks the Pocklington congruence and gcd conditions. The compact canonical report has SHA-256 digest `91d204497b47598411a8093cef855cd78b94d1725eae61cfb2f1b39ede71f35a`.",
  "status": "available",
  "evidence_grade": "executable",
  "scope": {
    "kind": "bounded",
    "statement": "Pocklington verification of p=1587809 and q=2000771023 and exact replay of their square-cube gap",
    "bounds": {
      "certified_integers": {
        "min": 2,
        "max": 2
      }
    },
    "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://arxiv.org/abs/math/0005139",
      "locator": "Inline standard-library Python computation executed by TheoremDB entry research on 2026-07-24"
    },
    "outputs": "{\"certified_prime_factors\":[2,3,19,29,59,337,2741],\"difference\":49600,\"p\":1587809,\"p_cube\":4003084686476516129,\"pocklington_checks\":[[1587809,2,3,1587808,1],[1587809,29,2,1496323,1],[1587809,59,2,583812,1],[2000771023,2,3,2000771022,1],[2000771023,3,11,1798115234,1],[2000771023,19,2,1835644527,1],[2000771023,337,2,1410779204,1],[2000771023,2741,2,1983323485,1]],\"q\":2000771023,\"q_square\":4003084686476466529}\n",
    "inline_source": [
      "import hashlib",
      "import json",
      "import math",
      "",
      "certificates = {",
      "    1587809: {",
      "        'factorization': {2: 5, 29: 2, 59: 1},",
      "        'witnesses': {2: 3, 29: 2, 59: 2},",
      "    },",
      "    2000771023: {",
      "        'factorization': {2: 1, 3: 1, 19: 2, 337: 1, 2741: 1},",
      "        'witnesses': {2: 3, 3: 11, 19: 2, 337: 2, 2741: 2},",
      "    },",
      "}",
      "",
      "def trial_prime(n):",
      "    if n < 2:",
      "        return False",
      "    if n % 2 == 0:",
      "        return n == 2",
      "    return all(n % d for d in range(3, math.isqrt(n) + 1, 2))",
      "",
      "checks = []",
      "for n, certificate in certificates.items():",
      "    product = 1",
      "    for prime, exponent in certificate['factorization'].items():",
      "        assert trial_prime(prime)",
      "        product *= prime ** exponent",
      "    assert product == n - 1",
      "    assert product * product > n",
      "    for prime, witness in certificate['witnesses'].items():",
      "        fermat = pow(witness, n - 1, n)",
      "        residue = pow(witness, (n - 1) // prime, n)",
      "        divisor_gcd = math.gcd(residue - 1, n)",
      "        assert fermat == 1 and divisor_gcd == 1",
      "        checks.append([n, prime, witness, residue, divisor_gcd])",
      "",
      "p = 1587809",
      "q = 2000771023",
      "cube = p ** 3",
      "square = q ** 2",
      "assert math.isqrt(cube) == q",
      "assert cube - square == 49600",
      "report = {",
      "    'p': p,",
      "    'q': q,",
      "    'p_cube': cube,",
      "    'q_square': square,",
      "    'difference': cube - square,",
      "    'pocklington_checks': checks,",
      "    'certified_prime_factors': sorted({",
      "        prime for certificate in certificates.values()",
      "        for prime in certificate['factorization']",
      "    }),",
      "}",
      "payload = json.dumps(report, sort_keys=True, separators=(',', ':'))",
      "assert hashlib.sha256(payload.encode()).hexdigest() == '91d204497b47598411a8093cef855cd78b94d1725eae61cfb2f1b39ede71f35a'",
      "print(payload)"
    ],
    "missing": [
      "command"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/math/0005139",
    "locator": "Inline standard-library Python computation executed by TheoremDB entry research on 2026-07-24"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R563",
      "title": "Pocklington certificates prove both numbers in the incumbent pair prime",
      "object_type": "claim",
      "relation": "verifies",
      "direction": "outgoing"
    },
    {
      "slug": "prime-cube-prime-square-gap-trillion",
      "title": "prime cube prime square gap trillion",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}