{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-prime-number",
  "semantic_family_id": "sf-prime-number",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "自然數質數"
  },
  "summary": {
    "zh-Hant": "自然數質數是大於或等於 2，且正因數只有 1 與自身的自然數。"
  },
  "statement": {
    "zh-Hant": "自然數 p 為質數，當且僅當 2≤p，且每個自然數 d 若 d∣p，則 d=1 或 d=p。0 與 1 都不是質數。"
  },
  "formula": {
    "tex": "p\\in N",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>p</mi><mo>∈</mo><mi>N</mi></mrow></math>",
    "semantic_ast": {
      "type": "membership",
      "element": {
        "type": "symbol",
        "name": "p",
        "glyph": "p"
      },
      "set": {
        "type": "symbol",
        "name": "N",
        "glyph": "N"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "6ce96954967e6810bc789d634c22f7051c3677c228e6203e49a65f163b38372e"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "p、d 是包含 0 的自然數；質數定義另要求 2≤p。"
    },
    {
      "id": "a2",
      "zh-Hant": "因數判定使用自然數整除，不以近似平方根或浮點除法判斷。"
    }
  ],
  "symbols": [
    {
      "token": "p",
      "role": "prime_candidate",
      "role_zh": "候選質數",
      "scope": "representative_formula"
    },
    {
      "token": "N",
      "role": "natural_domain",
      "role_zh": "自然數論域",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-natural-divisibility",
      "title_zh": "自然數整除",
      "reason_zh": "質數由自然數因數結構定義。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "判定有限候選 p 時可列出全部正因數；或只檢查 2 到 floor(sqrt(p)) 的候選因數，因為合數必有不超過平方根的非平凡因數。",
      "0、1 的因數行為不符合質數定義；2 是最小質數，也是唯一偶質數。",
      "找到非平凡因數即可給出合數見證；沒在不完整候選列表中找到因數，不能單獨當成質數證明。"
    ],
    "common_misconception_zh": "把 1 列為質數，或只測少數小因數就宣稱大數是質數。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪只涵蓋安全整數範圍內的完整因數或平方根界檢查；不構成任意大數的形式質數證明。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "綁定 Nat.Prime、Nat.dvd_prime、最小因數與可判定性；另建立大整數證書途徑。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-prime-number.md 整理；2026-10-13 脈衝 R04（使用者手動接續 2026-10-18 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}