{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R563",
  "content_hash": null,
  "slug": "pcpsg-claim-incumbent-certified",
  "type": "claim",
  "title": "Pocklington certificates prove both numbers in the incumbent pair prime",
  "summary": "Fully factored n-1 certificates establish primality, and q=floor(sqrt(p^3)) proves lower adjacency.",
  "relevance": "For Closest prime square to the cube of a prime below one trillion, record pcpsg-claim-incumbent-certified (“Pocklington certificates prove both numbers in the incumbent pair prime”) records a bound, answer, status fact, or structural consequence. The record states: Fully factored n-1 certificates establish primality, and q=floor(sqrt(p^3)) proves lower adjacency.",
  "relevance_source": "recorded",
  "body": "The factorizations\n\\[\n1587809-1=2^5\\cdot29^2\\cdot59\n\\]\nand\n\\[\n2000771023-1=2\\cdot3\\cdot19^2\\cdot337\\cdot2741\n\\]\nare complete. Trial division certifies the seven prime factors. For every distinct prime divisor \\(s\\) of \\(n-1\\), the replay gives a witness \\(a\\) satisfying\n\\[\na^{n-1}\\equiv1\\pmod n,\n\\qquad\n\\gcd(a^{(n-1)/s}-1,n)=1.\n\\]\nSince the factored part is all of \\(n-1\\), Pocklington's criterion proves each \\(n\\) prime.\n\nExact integer square root gives\n\\[\n\\lfloor\\sqrt{1587809^3}\\rfloor=2000771023.\n\\]\nThe certified prime \\(q\\) is therefore the largest integer below \\(p^{3/2}\\), so it is automatically the adjacent lower prime.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "the pair p=1587809 and q=2000771023",
    "bounds": {
      "p": {
        "min": 1587809,
        "max": 1587809
      },
      "q": {
        "min": 2000771023,
        "max": 2000771023
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2203.16341",
      "locator": "Pocklington criterion and executable verification in pcpsg-artifact-pocklington-incumbent"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2203.16341",
    "locator": "Pocklington criterion and executable verification in pcpsg-artifact-pocklington-incumbent"
  },
  "relations": [
    {
      "slug": "R560",
      "title": "Pocklington certificate for the incumbent pair",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "R565",
      "title": "The minimum gap through p=100 billion is 49,600",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "prime-cube-prime-square-gap-trillion",
      "title": "prime cube prime square gap trillion",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}