{
  "vercy": "1.0-draft",
  "publication": {
    "status": "research-draft",
    "adjudicationStatus": "unreviewed",
    "publishableCanonical": false,
    "generatedAt": "2026-09-12T23:41:50Z",
    "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": "2026-10-06T07:00:04+00:00",
      "fields": [
        "agent_conduct",
        "ethics",
        "owners"
      ]
    },
    "gate": {
      "passed": true,
      "version": "1.0",
      "at": "2026-10-06T19:52:01+00:00",
      "evidence": [
        "completeness 1.00",
        "sense: Wikidata description",
        "2 live sources",
        "review by codex: pass"
      ]
    }
  },
  "metaModel": {
    "id": "THING-Q593418",
    "registryId": "vr.tr.proth-number",
    "name": "Proth number",
    "version": "0.2.0-wave.3",
    "entryKind": "thing",
    "family": "Thing Registry",
    "domain": [
      "XCT.QTY"
    ],
    "status": "research-draft"
  },
  "canonicalUrl": "https://ver.cy/models/thing/q593418/",
  "model": {
    "registry_id": "vr.tr.proth-number",
    "name": "Proth number",
    "purpose": "Let an agent define Proth numbers and Proth primes, explain Proth theorem and its use as a primality test, relate them to Fermat and Cullen numbers, and help test or list examples.",
    "definition": "A positive integer of the form k times 2 to the power n plus 1, where k is odd and k is less than 2 to the power n, such as 3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81 and 97; Proth numbers are named after Francois Proth, whose 1878 theorem gives a simple primality test for them, and Proth primes include many of the largest known primes found by distributed computing.",
    "what_it_is_for": "Primality testing and prime searching.",
    "affordances": [
      "define the numbers and primes",
      "explain Proth theorem",
      "relate to other special forms",
      "help test or list examples"
    ],
    "distinguishing_features": [
      "Special form",
      "Simple primality test",
      "Source of large primes",
      "Connection to Fermat numbers"
    ],
    "appearance": "Not a visible object; integers of a special form.",
    "visual_identification": [
      "k times 2 to the n plus 1 with odd k less than 2 to the n",
      "Sequence 3, 5, 9, 13, 17, 25, 33",
      "Fermat numbers are a special case; Sierpinski numbers are related"
    ],
    "physical_properties": [
      {
        "quantity": "first terms",
        "typical_range": "3, 5, 9, 13, 17, 25, 33, 41, 49, 57",
        "unit": "sequence",
        "note": ""
      },
      {
        "quantity": "Proth theorem",
        "typical_range": "1878",
        "unit": "year",
        "note": ""
      }
    ],
    "families_and_kinds": [
      "Proth numbers",
      "Proth primes",
      "Fermat numbers as Proth numbers with k equal to 1",
      "related forms such as Cullen and Sierpinski numbers"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "positive integer",
        "why": "in registry terms"
      },
      {
        "relation": "is tested by",
        "target": "Proth theorem",
        "why": "the primality test"
      },
      {
        "relation": "includes",
        "target": "Fermat number",
        "why": "as a special case"
      },
      {
        "relation": "is named after",
        "target": "Francois Proth",
        "why": "the mathematician"
      }
    ],
    "identifiers": [
      {
        "scheme": "OEIS",
        "value_or_pattern": "A080075",
        "note": "Proth numbers"
      }
    ],
    "standards_and_regulation": [
      "No regulation; the OEIS and prime databases are references"
    ],
    "failure_modes_and_hazards": [
      "Omitting the condition k less than 2 to the n",
      "Confusing Proth numbers with Proth primes",
      "Computation errors for large cases"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": [],
    "agent_conduct": {
      "may": [
        "State the definition of a Proth number including the condition that k is less than 2 to the n.",
        "Check whether a number is a Proth number and show the working.",
        "Explain Proth's theorem and its use for testing primes."
      ],
      "must_not": [
        "Omit the condition that k is odd and less than 2 to the n.",
        "Confuse Proth numbers with Proth primes.",
        "Report unverified results for large cases as confirmed.",
        "Claim new prime discoveries without verification."
      ],
      "requires_human": [
        "Announcing a newly found large Proth prime."
      ]
    },
    "ethics": {
      "considerations": [
        "Correct credit for discoveries matters to mathematicians and volunteer projects.",
        "Accurate definitions keep students from learning errors."
      ],
      "affected_parties": [
        "Students",
        "Mathematicians",
        "Volunteer computing participants"
      ]
    },
    "owners": {
      "steward": "Nobody: a mathematical concept held in common.",
      "master_systems": []
    }
  },
  "sources": [
    {
      "id": "wikidata",
      "title": "Wikidata item Q593418: Proth number",
      "url": "https://www.wikidata.org/wiki/Q593418",
      "url_status": "live",
      "what_it_supports": "identity and sense of the item",
      "checked_by": "wikidata-api"
    },
    {
      "id": "wikipedia-en",
      "title": "Wikipedia: Proth number",
      "url": "https://en.wikipedia.org/wiki/Proth_number",
      "url_status": "live",
      "what_it_supports": "general description of the item",
      "checked_by": "wikidata-api sitelink"
    }
  ],
  "structure": {
    "bundles": [
      {
        "id": "define",
        "name": "Define",
        "description": "What Proth numbers are.",
        "rationale": "Definition.",
        "layers": [
          {
            "id": "definition",
            "name": "Definition",
            "description": "Definition and examples.",
            "findings": [
              {
                "id": "definition-finding",
                "name": "Definition",
                "description": "Definition.",
                "questions": [
                  {
                    "text": "What is a Proth number, and what are the first examples?",
                    "kind": "definition"
                  },
                  {
                    "text": "Does the number in question satisfy the form and the condition on k?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          },
          {
            "id": "primes",
            "name": "Primes",
            "description": "Proth primes.",
            "findings": [
              {
                "id": "primes-finding",
                "name": "Primes",
                "description": "Primes.",
                "questions": [
                  {
                    "text": "What are Proth primes, and which large primes have this form?",
                    "kind": "definition"
                  },
                  {
                    "text": "Which entry fits Proth prime?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "test",
        "name": "Test",
        "description": "Primality testing.",
        "rationale": "Computation.",
        "layers": [
          {
            "id": "theorem",
            "name": "Theorem",
            "description": "Proth theorem.",
            "findings": [
              {
                "id": "theorem-finding",
                "name": "Theorem",
                "description": "Theorem.",
                "questions": [
                  {
                    "text": "What does Proth theorem state, and how is it applied as a test?",
                    "kind": "action"
                  },
                  {
                    "text": "Is the Proth number in question prime?",
                    "kind": "measurement"
                  }
                ]
              }
            ]
          },
          {
            "id": "algorithms",
            "name": "Algorithms",
            "description": "Algorithms.",
            "findings": [
              {
                "id": "algorithms-finding",
                "name": "Algorithms",
                "description": "Algorithms.",
                "questions": [
                  {
                    "text": "How is the test implemented efficiently for large numbers?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which references are standard?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "theory",
        "name": "Theory",
        "description": "Related forms.",
        "rationale": "Theory.",
        "layers": [
          {
            "id": "fermat",
            "name": "Fermat",
            "description": "Fermat numbers.",
            "findings": [
              {
                "id": "fermat-finding",
                "name": "Fermat",
                "description": "Fermat.",
                "questions": [
                  {
                    "text": "How are Fermat numbers Proth numbers, and what does Pepin test have to do with Proth theorem?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits Fermat number?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "sierpinski",
            "name": "Sierpinski",
            "description": "Sierpinski and related problems.",
            "findings": [
              {
                "id": "sierpinski-finding",
                "name": "Sierpinski",
                "description": "Sierpinski.",
                "questions": [
                  {
                    "text": "How do Proth numbers relate to Sierpinski numbers and the Seventeen or Bust project?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which sources are cited?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "context",
        "name": "Context",
        "description": "History and computing.",
        "rationale": "Context.",
        "layers": [
          {
            "id": "history",
            "name": "History",
            "description": "History.",
            "findings": [
              {
                "id": "history-finding",
                "name": "History",
                "description": "History.",
                "questions": [
                  {
                    "text": "Who was Francois Proth, and how was his theorem received?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits the history of number theory?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "computing",
            "name": "Computing",
            "description": "Distributed prime searches.",
            "findings": [
              {
                "id": "computing-finding",
                "name": "Computing",
                "description": "Computing.",
                "questions": [
                  {
                    "text": "How do distributed projects search for Proth primes?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits distributed computing?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should Proth prime be a separate entry?",
    "How should prime databases be linked?",
    "How should records of largest Proth primes be kept current?"
  ],
  "statistics": {
    "bundles": 4,
    "layers": 8,
    "findings": 8,
    "questions": 16
  }
}