{
  "vercy": "1.0-draft",
  "publication": {
    "status": "research-draft",
    "adjudicationStatus": "unreviewed",
    "publishableCanonical": false,
    "generatedAt": "2026-09-13T02:44:35Z",
    "providers": [
      "Claude"
    ],
    "breadth": "written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify",
    "pass": 2,
    "wave": 3,
    "engine": "claude",
    "enrichment": {
      "pass": 3,
      "engine": "claude",
      "producedAt": "2026-10-06T10:00:00Z",
      "ingestedAt": "2026-10-06T07:11:28+00:00",
      "fields": [
        "agent_conduct",
        "ethics",
        "owners"
      ]
    },
    "revisions": [
      {
        "at": "2026-10-06T22:32:47+00:00",
        "engine": "codex",
        "fields": [
          "definition",
          "distinguishing_features"
        ],
        "answering_review": "4f2bd9b8cfaa499badd2060a5f3e4e4ce80cc3491c806fa65d956d4a68158592"
      }
    ],
    "gate": {
      "passed": true,
      "version": "1.0",
      "at": "2026-10-07T10:10:25+00:00",
      "evidence": [
        "completeness 1.00",
        "sense: Wikidata description",
        "2 live sources",
        "review by codex: pass"
      ]
    }
  },
  "metaModel": {
    "id": "THING-Q1318413",
    "registryId": "vr.tr.double-mersenne-number",
    "name": "double Mersenne number",
    "version": "0.2.0-wave.3",
    "entryKind": "thing",
    "family": "Thing Registry",
    "domain": [
      "XCT.QTY"
    ],
    "status": "research-draft"
  },
  "canonicalUrl": "https://ver.cy/models/thing/q1318413/",
  "model": {
    "registry_id": "vr.tr.double-mersenne-number",
    "name": "double Mersenne number",
    "purpose": "Let an agent explain double Mersenne numbers, relay known results and open problems from mathematical references, describe their relation to Mersenne primes and the Catalan-Mersenne conjecture, and distinguish them from ordinary Mersenne numbers.",
    "definition": "A number of the form 2^(2^p - 1) - 1, denoted MM(p), where both p and 2^p - 1 are prime, making it a Mersenne number whose exponent is itself a Mersenne prime; the first four cases MM(2), MM(3), MM(5) and MM(7) are prime, while MM(13) is known to be composite. The related Catalan-Mersenne sequence consists of iterated Mersenne numbers whose primality beyond the fifth term is unknown; double Mersenne numbers are among the largest numbers studied for primality.",
    "what_it_is_for": "Not applicable; a mathematical concept.",
    "affordances": [
      "explain the definition",
      "relay known results",
      "describe open problems",
      "distinguish from Mersenne numbers"
    ],
    "distinguishing_features": [
      "The nested Mersenne form requires both p and 2^p - 1 to be prime in this sense.",
      "Four double Mersenne primes are known.",
      "These numbers are related to the Catalan-Mersenne sequence.",
      "Some cases beyond MM(7), including MM(13), are known to be composite, while other cases remain unresolved."
    ],
    "appearance": "Not a visible object; enormous integers.",
    "visual_identification": [
      "Mersenne number with Mersenne exponent",
      "MM(2) = 7, MM(3) = 127, MM(5), MM(7) prime; MM(13) composite",
      "Ordinary Mersenne numbers are 2^n - 1; Fermat numbers are 2^(2^n) + 1"
    ],
    "physical_properties": [
      {
        "quantity": "known double Mersenne primes",
        "typical_range": "4",
        "unit": "count",
        "note": "MM(2), MM(3), MM(5), MM(7)"
      },
      {
        "quantity": "MM(7)",
        "typical_range": "2^127 - 1",
        "unit": "value",
        "note": "about 1.7e38"
      },
      {
        "quantity": "MM(13) digits",
        "typical_range": "about 2466",
        "unit": "count",
        "note": "known composite"
      }
    ],
    "families_and_kinds": [
      "double Mersenne primes",
      "composite double Mersenne numbers such as MM(13), MM(17), MM(19), MM(31)",
      "the Catalan-Mersenne sequence",
      "candidates such as MM(61) with unknown status"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "Mersenne number",
        "why": "in registry terms"
      },
      {
        "relation": "is prime for",
        "target": "Mersenne prime",
        "why": "exponents in known cases"
      },
      {
        "relation": "is the subject of",
        "target": "Catalan-Mersenne conjecture",
        "why": "about iterated Mersenne numbers"
      },
      {
        "relation": "is studied in",
        "target": "primality testing",
        "why": "and factor searches"
      }
    ],
    "identifiers": [
      {
        "scheme": "OEIS",
        "value_or_pattern": "A077586",
        "note": "double Mersenne numbers"
      }
    ],
    "standards_and_regulation": [
      "No regulation; standard mathematical definitions"
    ],
    "failure_modes_and_hazards": [
      "Assuming MM(p) is prime whenever M(p) is",
      "Confusing with Mersenne numbers or Fermat numbers",
      "Overstating what is proved"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": [],
    "agent_conduct": {
      "may": [
        "Explain what double Mersenne numbers are and report the known primality status of the small cases.",
        "Distinguish proven results from open conjectures about these numbers."
      ],
      "must_not": [
        "Present the conjecture about further primes as settled.",
        "Claim a primality result for a large case without a verifiable proof or certificate.",
        "Misattribute a factorisation or search result to the wrong contributor.",
        "Present distributed search progress as a completed proof."
      ],
      "requires_human": [
        "Publishing a claimed primality or factorisation result under a person's name."
      ]
    },
    "ethics": {
      "considerations": [
        "Claims in computational number theory must be verifiable, or the record is corrupted.",
        "Volunteer searchers deserve credit for the results their computation produced."
      ],
      "affected_parties": [
        "Mathematicians and students",
        "Volunteer distributed computing participants"
      ]
    },
    "owners": {
      "steward": "Nobody owns them; mathematical and integer sequence databases record the known status.",
      "master_systems": [
        "Published integer sequence and factorisation databases"
      ]
    }
  },
  "sources": [
    {
      "id": "wikidata",
      "title": "Wikidata item Q1318413: double Mersenne number",
      "url": "https://www.wikidata.org/wiki/Q1318413",
      "url_status": "live",
      "what_it_supports": "identity and sense of the item",
      "checked_by": "wikidata-api"
    },
    {
      "id": "wikipedia-en",
      "title": "Wikipedia: Double Mersenne number",
      "url": "https://en.wikipedia.org/wiki/Double_Mersenne_number",
      "url_status": "live",
      "what_it_supports": "general description of the item",
      "checked_by": "wikidata-api sitelink"
    }
  ],
  "structure": {
    "bundles": [
      {
        "id": "understand",
        "name": "Understand",
        "description": "What a double Mersenne number is.",
        "rationale": "Mathematics.",
        "layers": [
          {
            "id": "definition",
            "name": "Definition",
            "description": "Definition.",
            "findings": [
              {
                "id": "definition-finding",
                "name": "Definition",
                "description": "Definition.",
                "questions": [
                  {
                    "text": "What is a double Mersenne number, and how does it relate to Mersenne numbers and primes?",
                    "kind": "definition"
                  },
                  {
                    "text": "Is the question about double Mersenne numbers, Mersenne numbers or Fermat numbers?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          },
          {
            "id": "known",
            "name": "Known",
            "description": "Known results.",
            "findings": [
              {
                "id": "known-finding",
                "name": "Known",
                "description": "Known.",
                "questions": [
                  {
                    "text": "Which double Mersenne numbers are known to be prime or composite, and how were factors found?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits Mersenne prime?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "problems",
        "name": "Problems",
        "description": "Open problems.",
        "rationale": "Mathematics.",
        "layers": [
          {
            "id": "catalan",
            "name": "Catalan",
            "description": "Catalan-Mersenne conjecture.",
            "findings": [
              {
                "id": "catalan-finding",
                "name": "Catalan",
                "description": "Catalan.",
                "questions": [
                  {
                    "text": "What is the Catalan-Mersenne sequence, and why is the primality of its fifth term unknown?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which references are standard?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          },
          {
            "id": "mm61",
            "name": "MM(61)",
            "description": "Candidates.",
            "findings": [
              {
                "id": "mm61-finding",
                "name": "MM(61)",
                "description": "MM(61).",
                "questions": [
                  {
                    "text": "Why are MM(61) and larger candidates beyond current primality testing?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which sources are cited?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "compute",
        "name": "Compute",
        "description": "Computation.",
        "rationale": "Mathematics.",
        "layers": [
          {
            "id": "tests",
            "name": "Tests",
            "description": "Primality tests.",
            "findings": [
              {
                "id": "tests-finding",
                "name": "Tests",
                "description": "Tests.",
                "questions": [
                  {
                    "text": "How does the Lucas-Lehmer test apply, and why is it infeasible for huge double Mersenne numbers?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits Lucas-Lehmer test?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "factors",
            "name": "Factors",
            "description": "Factor searches.",
            "findings": [
              {
                "id": "factors-finding",
                "name": "Factors",
                "description": "Factors.",
                "questions": [
                  {
                    "text": "How are factors of double Mersenne numbers searched for?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits distributed computing in number theory?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "context",
        "name": "Context",
        "description": "History and culture.",
        "rationale": "Context.",
        "layers": [
          {
            "id": "history",
            "name": "History",
            "description": "History.",
            "findings": [
              {
                "id": "history-finding",
                "name": "History",
                "description": "History.",
                "questions": [
                  {
                    "text": "How did Catalan, Lucas and later mathematicians study these numbers?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits the history of number theory?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "culture",
            "name": "Culture",
            "description": "Culture.",
            "findings": [
              {
                "id": "culture-finding",
                "name": "Culture",
                "description": "Culture.",
                "questions": [
                  {
                    "text": "How have double Mersenne numbers appeared in popular mathematics and fiction?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits mathematics in popular culture?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should the Catalan-Mersenne conjecture be a separate entry?",
    "How should mathematical references be linked?",
    "How should the status of MM(61) be tracked?"
  ],
  "statistics": {
    "bundles": 4,
    "layers": 8,
    "findings": 8,
    "questions": 16
  }
}