{
  "vercy": "1.0-draft",
  "publication": {
    "status": "research-draft",
    "adjudicationStatus": "unreviewed",
    "publishableCanonical": false,
    "generatedAt": "2026-09-12T21:03:38Z",
    "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": "2026-10-06T06:57:30+00:00",
      "fields": [
        "agent_conduct",
        "ethics",
        "owners"
      ]
    },
    "gate": {
      "passed": true,
      "version": "1.0",
      "at": "2026-10-06T19:51:58+00:00",
      "evidence": [
        "completeness 1.00",
        "sense: Wikidata description",
        "2 live sources",
        "review by codex: pass"
      ]
    }
  },
  "metaModel": {
    "id": "THING-Q2661866",
    "registryId": "vr.tr.leyland-number",
    "name": "Leyland number",
    "version": "0.2.0-wave.3",
    "entryKind": "thing",
    "family": "Thing Registry",
    "domain": [
      "XCT.QTY"
    ],
    "status": "research-draft"
  },
  "canonicalUrl": "https://ver.cy/models/thing/q2661866/",
  "model": {
    "registry_id": "vr.tr.leyland-number",
    "name": "Leyland number",
    "purpose": "Let an agent define Leyland numbers precisely, compute and test them, relay known Leyland primes and their role in primality testing, and point to sequence references.",
    "definition": "A number of the form x to the power y plus y to the power x, where x and y are integers greater than 1, named after Paul Leyland; the first are 8, 17, 32, 54, 57, 100 and 145, and Leyland primes such as 17, 593 and 32993 are used as test cases for primality proving because they have no special algebraic form.",
    "what_it_is_for": "Not for anything; test cases in computational number theory.",
    "affordances": [
      "compute Leyland numbers for given x and y",
      "list examples and known primes",
      "explain their use in primality proving",
      "point to sequence references"
    ],
    "distinguishing_features": [
      "Symmetric exponential form",
      "No special algebraic structure aiding primality proofs",
      "Named after Paul Leyland",
      "Second kind with subtraction"
    ],
    "appearance": "Not visible; a property of integers.",
    "visual_identification": [
      "Form x to the y plus y to the x with x, y greater than 1",
      "First terms 8, 17, 32, 54, 57, 100",
      "Leyland numbers of the second kind subtract instead of add"
    ],
    "physical_properties": [
      {
        "quantity": "first terms",
        "typical_range": "8, 17, 32, 54, 57, 100, 145, 177, 320",
        "unit": "integers",
        "note": ""
      },
      {
        "quantity": "condition",
        "typical_range": "x and y greater than 1",
        "unit": "condition",
        "note": "to exclude trivial cases"
      }
    ],
    "families_and_kinds": [
      "Leyland numbers",
      "Leyland primes",
      "Leyland numbers of the second kind",
      "generalisations with other bases"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "positive integer",
        "why": "a property of integers"
      },
      {
        "relation": "is used in",
        "target": "primality test",
        "why": "as test cases"
      },
      {
        "relation": "is listed in",
        "target": "On-Line Encyclopedia of Integer Sequences",
        "why": "sequence A076980"
      },
      {
        "relation": "is related to",
        "target": "prime number",
        "why": "Leyland primes"
      }
    ],
    "identifiers": [
      {
        "scheme": "OEIS",
        "value_or_pattern": "A076980",
        "note": "Leyland numbers"
      },
      {
        "scheme": "OEIS",
        "value_or_pattern": "A094133",
        "note": "Leyland primes"
      }
    ],
    "standards_and_regulation": [
      "No regulation; definitions follow the literature and the OEIS"
    ],
    "failure_modes_and_hazards": [
      "Including trivial cases with x or y equal to 1",
      "Confusing the two kinds",
      "Quoting outdated lists of known primes"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": [],
    "agent_conduct": {
      "may": [
        "Define Leyland numbers and compute them for given values of x and y.",
        "Relay known Leyland primes and their use as test cases for primality proving, with sources.",
        "Test whether a given number is a Leyland number."
      ],
      "must_not": [
        "Include cases with x or y equal to 1 as Leyland numbers.",
        "Quote outdated lists of the largest known Leyland primes as current.",
        "Claim a probable prime is a proven prime without a primality certificate.",
        "Confuse Leyland numbers of the first kind with those of the second kind."
      ],
      "requires_human": [
        "Announcing a new record Leyland prime."
      ]
    },
    "ethics": {
      "considerations": [
        "Credit for record primes belongs to the people and projects that found and proved them.",
        "Errors in published primes can propagate into software tests and benchmarks."
      ],
      "affected_parties": [
        "Mathematicians",
        "Prime search volunteers",
        "Students and learners"
      ]
    },
    "owners": {
      "steward": "Nobody owns the concept; the mathematical community and prime record lists keep the record.",
      "master_systems": []
    }
  },
  "sources": [
    {
      "id": "wikidata",
      "title": "Wikidata item Q2661866: Leyland number",
      "url": "https://www.wikidata.org/wiki/Q2661866",
      "url_status": "live",
      "what_it_supports": "identity and sense of the item",
      "checked_by": "wikidata-api"
    },
    {
      "id": "wikipedia-en",
      "title": "Wikipedia: Leyland number",
      "url": "https://en.wikipedia.org/wiki/Leyland_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 Leyland number, and why are x and y required to exceed 1?",
                    "kind": "definition"
                  },
                  {
                    "text": "Is the question about the first or second kind?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          },
          {
            "id": "examples",
            "name": "Examples",
            "description": "Examples.",
            "findings": [
              {
                "id": "examples-finding",
                "name": "Examples",
                "description": "Examples.",
                "questions": [
                  {
                    "text": "What are the Leyland numbers below a given bound, and which pairs produce them?",
                    "kind": "measurement"
                  },
                  {
                    "text": "Which entry fits integer sequences?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "primes",
        "name": "Primes",
        "description": "Leyland primes.",
        "rationale": "Primality.",
        "layers": [
          {
            "id": "known",
            "name": "Known",
            "description": "Known primes.",
            "findings": [
              {
                "id": "known-finding",
                "name": "Known",
                "description": "Known.",
                "questions": [
                  {
                    "text": "Which Leyland primes are known, and where are records kept?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Is the list current?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          },
          {
            "id": "testing",
            "name": "Testing",
            "description": "Role in primality proving.",
            "findings": [
              {
                "id": "testing-finding",
                "name": "Testing",
                "description": "Testing.",
                "questions": [
                  {
                    "text": "Why are Leyland numbers useful test cases for general primality proving algorithms?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits primality proving?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "compute",
        "name": "Compute",
        "description": "Computation.",
        "rationale": "Computation.",
        "layers": [
          {
            "id": "generate",
            "name": "Generate",
            "description": "Generating.",
            "findings": [
              {
                "id": "generate-finding",
                "name": "Generate",
                "description": "Generate.",
                "questions": [
                  {
                    "text": "How are Leyland numbers generated in order without duplicates?",
                    "kind": "action"
                  },
                  {
                    "text": "What are the computational limits?",
                    "kind": "measurement"
                  }
                ]
              }
            ]
          },
          {
            "id": "second",
            "name": "Second kind",
            "description": "Second kind.",
            "findings": [
              {
                "id": "second-finding",
                "name": "Second",
                "description": "Second.",
                "questions": [
                  {
                    "text": "What are Leyland numbers of the second kind, and what is known about them?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which references are standard?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "context",
        "name": "Context",
        "description": "History and teaching.",
        "rationale": "Context.",
        "layers": [
          {
            "id": "history",
            "name": "History",
            "description": "Origin.",
            "findings": [
              {
                "id": "history-finding",
                "name": "History",
                "description": "History.",
                "questions": [
                  {
                    "text": "Who is Paul Leyland, 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 Leyland numbers used in teaching and computation exercises?",
                    "kind": "action"
                  },
                  {
                    "text": "Which misconceptions arise?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should Leyland primes be a separate entry?",
    "How should prime records be linked?",
    "Which conjectures should be recorded?"
  ],
  "statistics": {
    "bundles": 4,
    "layers": 8,
    "findings": 8,
    "questions": 16
  }
}