{
  "vercy": "1.0-draft",
  "publication": {
    "status": "research-draft",
    "adjudicationStatus": "unreviewed",
    "publishableCanonical": false,
    "generatedAt": "2026-09-12T20:13:51Z",
    "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": "2026-10-06T07:00:12+00:00",
      "fields": [
        "agent_conduct",
        "ethics",
        "owners"
      ]
    },
    "gate": {
      "passed": true,
      "version": "1.0",
      "at": "2026-10-06T19:52:00+00:00",
      "evidence": [
        "completeness 1.00",
        "sense: Wikidata description",
        "2 live sources",
        "review by codex: pass"
      ]
    }
  },
  "metaModel": {
    "id": "THING-Q532389",
    "registryId": "vr.tr.cullen-number",
    "name": "Cullen number",
    "version": "0.2.0-wave.3",
    "entryKind": "thing",
    "family": "Thing Registry",
    "domain": [
      "XCT.QTY"
    ],
    "status": "research-draft"
  },
  "canonicalUrl": "https://ver.cy/models/thing/q532389/",
  "model": {
    "registry_id": "vr.tr.cullen-number",
    "name": "Cullen number",
    "purpose": "Let an agent define Cullen numbers precisely, list small examples and the known Cullen primes, relate them to Woodall and generalised Cullen numbers, and point to sequence references and open problems.",
    "definition": "A natural number of the form n times 2 to the power n, plus 1, named after James Cullen who studied them in 1905; almost all Cullen numbers are composite, only a few Cullen primes are known, and the related Woodall numbers have the form n times 2 to the power n, minus 1.",
    "what_it_is_for": "Not for anything; an object of number theory.",
    "affordances": [
      "compute Cullen numbers for given n",
      "list known Cullen primes",
      "distinguish Cullen from Woodall and generalised forms",
      "point to sequence references"
    ],
    "distinguishing_features": [
      "Defined by n times 2 to the n, plus 1",
      "Almost all composite",
      "Few known primes",
      "Paired with Woodall numbers"
    ],
    "appearance": "Not visible; a property of integers.",
    "visual_identification": [
      "Form n times 2 to the n, plus 1",
      "First terms 1, 3, 9, 25, 65, 161",
      "Woodall numbers subtract 1 instead of adding"
    ],
    "physical_properties": [
      {
        "quantity": "first terms",
        "typical_range": "1, 3, 9, 25, 65, 161, 385",
        "unit": "integers",
        "note": "for n from 0"
      },
      {
        "quantity": "smallest Cullen prime beyond trivial",
        "typical_range": "n equal to 1 and n equal to 141",
        "unit": "indices",
        "note": "Cullen primes are rare"
      }
    ],
    "families_and_kinds": [
      "Cullen numbers",
      "Cullen primes",
      "generalised Cullen numbers with other bases",
      "Woodall numbers as the related family"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "positive integer",
        "why": "a property of integers"
      },
      {
        "relation": "is related to",
        "target": "Woodall number",
        "why": "the minus-one counterpart"
      },
      {
        "relation": "is listed in",
        "target": "On-Line Encyclopedia of Integer Sequences",
        "why": "sequence A002064"
      },
      {
        "relation": "is studied by",
        "target": "number theory",
        "why": "the field"
      }
    ],
    "identifiers": [
      {
        "scheme": "OEIS",
        "value_or_pattern": "A002064",
        "note": "Cullen numbers"
      },
      {
        "scheme": "OEIS",
        "value_or_pattern": "A005849",
        "note": "indices of Cullen primes"
      }
    ],
    "standards_and_regulation": [
      "No regulation; definitions follow the literature and the OEIS"
    ],
    "failure_modes_and_hazards": [
      "Confusing Cullen with Woodall numbers",
      "Quoting outdated lists of known primes",
      "Presenting conjectures as theorems"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": [],
    "agent_conduct": {
      "may": [
        "Compute Cullen numbers and report known Cullen primes from published records.",
        "Explain their relation to Woodall numbers."
      ],
      "must_not": [
        "Announce a new Cullen prime without independent verification.",
        "Confuse Cullen numbers with Woodall numbers or with generalised forms.",
        "Use other people's computing resources for prime searches without permission."
      ],
      "requires_human": [
        "Announcing a new record to the mathematical community."
      ]
    },
    "ethics": {
      "considerations": [
        "Credit for discoveries in distributed searches belongs to the people who ran them.",
        "False record claims mislead others."
      ],
      "affected_parties": [
        "Researchers and volunteers in prime searches",
        "Students"
      ]
    },
    "owners": {
      "steward": "Nobody owns the concept; it belongs to mathematics.",
      "master_systems": []
    }
  },
  "sources": [
    {
      "id": "wikidata",
      "title": "Wikidata item Q532389: Cullen number",
      "url": "https://www.wikidata.org/wiki/Q532389",
      "url_status": "live",
      "what_it_supports": "identity and sense of the item",
      "checked_by": "wikidata-api"
    },
    {
      "id": "wikipedia-en",
      "title": "Wikipedia: Cullen number",
      "url": "https://en.wikipedia.org/wiki/Cullen_number",
      "url_status": "live",
      "what_it_supports": "general description of the item",
      "checked_by": "wikidata-api sitelink"
    }
  ],
  "structure": {
    "bundles": [
      {
        "id": "define",
        "name": "Define",
        "description": "The definition.",
        "rationale": "Precision.",
        "layers": [
          {
            "id": "definition",
            "name": "Definition",
            "description": "Definition.",
            "findings": [
              {
                "id": "definition-finding",
                "name": "Definition",
                "description": "Definition.",
                "questions": [
                  {
                    "text": "What is a Cullen number, and how does it differ from a Woodall number?",
                    "kind": "definition"
                  },
                  {
                    "text": "Is the question about Cullen, Woodall or generalised Cullen numbers?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          },
          {
            "id": "examples",
            "name": "Examples",
            "description": "Examples.",
            "findings": [
              {
                "id": "examples-finding",
                "name": "Examples",
                "description": "Examples.",
                "questions": [
                  {
                    "text": "What are the first Cullen numbers, and how are they computed?",
                    "kind": "measurement"
                  },
                  {
                    "text": "Which entry fits integer sequences?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "primes",
        "name": "Primes",
        "description": "Cullen primes.",
        "rationale": "Primality.",
        "layers": [
          {
            "id": "known",
            "name": "Known",
            "description": "Known primes.",
            "findings": [
              {
                "id": "known-finding",
                "name": "Known",
                "description": "Known.",
                "questions": [
                  {
                    "text": "Which Cullen primes are known, and where is the current list maintained?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Is the list current?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          },
          {
            "id": "divisibility",
            "name": "Divisibility",
            "description": "Divisibility results.",
            "findings": [
              {
                "id": "divisibility-finding",
                "name": "Divisibility",
                "description": "Divisibility.",
                "questions": [
                  {
                    "text": "What divisibility properties make almost all Cullen numbers composite?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which references are standard?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "compute",
        "name": "Compute",
        "description": "Computation and search.",
        "rationale": "Computation.",
        "layers": [
          {
            "id": "search",
            "name": "Search",
            "description": "Prime searches.",
            "findings": [
              {
                "id": "search-finding",
                "name": "Search",
                "description": "Search.",
                "questions": [
                  {
                    "text": "How are Cullen primes searched for, and which projects do it?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits distributed computing projects?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "generalise",
            "name": "Generalise",
            "description": "Generalisations.",
            "findings": [
              {
                "id": "generalise-finding",
                "name": "Generalise",
                "description": "Generalise.",
                "questions": [
                  {
                    "text": "What generalised Cullen numbers exist for other bases?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Should a generalisation be a separate entry?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "context",
        "name": "Context",
        "description": "History and teaching.",
        "rationale": "Context.",
        "layers": [
          {
            "id": "history",
            "name": "History",
            "description": "Cullen and Cunningham.",
            "findings": [
              {
                "id": "history-finding",
                "name": "History",
                "description": "History.",
                "questions": [
                  {
                    "text": "Who were Cullen and Cunningham, and how did the numbers get their name?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits the person?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "teach",
            "name": "Teach",
            "description": "Teaching.",
            "findings": [
              {
                "id": "teach-finding",
                "name": "Teach",
                "description": "Teaching.",
                "questions": [
                  {
                    "text": "How are Cullen numbers used in teaching and puzzles?",
                    "kind": "action"
                  },
                  {
                    "text": "Which misconceptions arise?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should Cullen primes be a separate entry?",
    "How should prime search projects be linked?",
    "Which conjectures, such as infinitude of Cullen primes, should be recorded?"
  ],
  "statistics": {
    "bundles": 4,
    "layers": 8,
    "findings": 8,
    "questions": 16
  }
}