{ "schema_version": "ocme-derivation-reader-v1", "guide_id": "derivation-mko-unique-prime-factorization-01", "mko_id": "mko-unique-prime-factorization", "title": { "zh-Hant": "存在、Euclid引理與唯一質因數", "en": "Existence, Euclid's lemma and unique factors" }, "contract": { "schema_version": "ocme-derivation-guide-v0.1", "guide_id": "derivation-mko-unique-prime-factorization-01", "source": { "mko_id": "mko-unique-prime-factorization", "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f", "sha256": "0f01853edccbfb8cdf298587be277da27032f1f31ab87712ee5745644aa66cff", "english_sha256": "daa563be289db97eeabc811b523c607b1c88cb0a7c4b7b320a62dee46bcc27ac" }, "mode": "theorem_proof", "audience_prerequisites": [ "calc" ], "goal": { "zh-Hant": "正整數n>1存在唯一質因數多重集合；360=2³·3²·5，負號與units另列。", "en": "Every positive integer n>1 has a unique prime multiset; 360=2^3*3^2*5, with signs and units separate." }, "givens": [ { "id": "G1", "statement": { "zh-Hant": "主定理n∈Z,n>1；實例n=360；分解忽略因數順序但保留重數。", "en": "The theorem domain is integers n>1; the example is 360; factor order is ignored but multiplicity retained." } } ], "steps": [ { "id": "S01", "uses": [ "G1" ], "before_tex": "n>1,\\quad d=\\min\\{m>1:m\\mid n\\}", "after_tex": "d\\text{ prime}", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "若d=uv且11\\Rightarrow q\\text{ has prime factors by induction}", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "商比n小，強歸納提供q分解，乘回d證存在；n=2的基礎亦落q=1分支。", "en": "The quotient is smaller than n; strong induction factors q and adjoining d proves existence, with n=2 covered by q=1." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S01" ], "explanation": { "zh-Hant": "歸納只用更小正整數，不用待證同一n。", "en": "Induction uses only smaller positive integers, not the same n." } }, "leaf_basis": "calc" }, { "id": "S03", "uses": [ "G1" ], "before_tex": "p\\text{ prime},\\quad p\\mid ab,\\quad p\\nmid a", "after_tex": "\\gcd(p,a)=1\\Rightarrow\\exists u,v:\\ up+va=1", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "p因數只有1、p，且p不整除a，故互質，可用明示Bezout前提。", "en": "The only positive divisors of p are 1,p; since p does not divide a, they are coprime and the stated Bezout prerequisite applies." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1" ], "explanation": { "zh-Hant": "這一步標示實際用到的Bezout，不把Euclid引理當自證。", "en": "This identifies the actual Bezout dependency rather than assuming the Euclid lemma being proved." } }, "leaf_basis": "calc" }, { "id": "S04", "uses": [ "G1", "S03" ], "before_tex": "up+va=1,\\quad p\\mid ab", "after_tex": "b=upb+vab\\Rightarrow p\\mid b", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "乘b後兩項皆被p整除，證p|ab則p|a或p|b。", "en": "After multiplication by b, both terms are divisible by p, proving Euclid's lemma." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S03" ], "explanation": { "zh-Hant": "若p|a則直接在另一分支；合數p不保證前一步互質。", "en": "The case p dividing a is immediate; composite p does not ensure the preceding coprimality." } }, "leaf_basis": "calc" }, { "id": "S05", "uses": [ "G1", "S04" ], "before_tex": "\\prod_{i=1}^{r}p_i=\\prod_{j=1}^{s}q_j,\\quad p_i,q_j\\text{ prime}", "after_tex": "p_1\\mid\\prod_{j=1}^{s}q_j\\Rightarrow\\exists j:\\ p_1\\mid q_j\\Rightarrow p_1=q_j", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "反覆對有限乘積使用Euclid引理，p₁必等於右側某個prime。", "en": "Finite-product induction using Euclid's lemma makes p1 equal to one prime on the right." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S04" ], "explanation": { "zh-Hant": "不是只比兩個已算答案；對任意兩個有限prime分解適用。", "en": "This handles arbitrary finite prime factorizations, not merely two computed outputs." } }, "leaf_basis": "calc" }, { "id": "S06", "uses": [ "G1", "S02", "S05" ], "before_tex": "p_1=q_j\\ne0", "after_tex": "\\text{cancel }p_1\\text{ and induct on }r+s\\Rightarrow\\{p_i\\}_{\\rm multiset}=\\{q_j\\}_{\\rm multiset}", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "消去非零整數prime使總因數數下降，直到空乘積1；非空prime乘積≥2，不能等1。", "en": "Cancel the nonzero prime and decrease the total factor count until empty product 1; a nonempty prime product is at least 2 and cannot equal 1." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S02", "S05" ], "explanation": { "zh-Hant": "唯一性保留重數，只忽略次序，空與非空終點亦已區分。", "en": "Uniqueness retains multiplicities while ignoring order, and distinguishes empty from nonempty endpoints." } }, "leaf_basis": "calc" }, { "id": "S07", "uses": [ "G1", "S01", "S02" ], "before_tex": "360", "after_tex": "360=2\\cdot180=2^2\\cdot90=2^3\\cdot45=2^3\\cdot3\\cdot15=2^3\\cdot3^2\\cdot5", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "完整抽取2三次與3兩次，餘5；2、3皆prime，餘5只需試2因3²>5，故5prime。", "en": "Extract 2 three times and 3 twice, leaving 5. Primes 2,3 qualify, and testing 2 suffices for remaining 5 since 3^2>5." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S01", "S02" ], "explanation": { "zh-Hant": "剩餘5不是未處理的合數；已完整核到平方根界。", "en": "The remaining 5 is not an unchecked composite; trial division reaches the square-root boundary." } }, "leaf_basis": "calc" }, { "id": "S08", "uses": [ "G1", "S06", "S07" ], "before_tex": "2^3\\cdot3^2\\cdot5=360", "after_tex": "360\\text{ has multiset }\\{2,2,2,3,3,5\\}\\text{ uniquely}", "relation": "implies", "rule": "mko_definition", "conditions": [], "explanation": { "zh-Hant": "乘積精確重驗，並由一般唯一性證其他prime列表只有排列不同。", "en": "Recheck the exact product; the uniqueness argument ensures any other prime list differs only in order." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S06", "S07" ], "explanation": { "zh-Hant": "不能只寫360=p^k，因這個例子有三種互異prime。", "en": "A single prime-power expression is insufficient because this example has three distinct primes." } }, "leaf_basis": "mko_definition" }, { "id": "S09", "uses": [ "G1", "S06", "S08" ], "before_tex": "-360,\\quad1,\\quad-1,\\quad0", "after_tex": "-360=-(2^3 3^2 5),\\quad\\pm1\\text{ use empty prime product};\\quad0\\text{ excluded}", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "負號是單位因子；±1無prime因數而用空乘積1；0不屬n>1分解域。", "en": "A negative sign is a unit factor; ±1 use empty prime product 1, and zero is outside the n>1 factorization domain." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S06", "S08" ], "explanation": { "zh-Hant": "0可被任意prime整除，不能給有限唯一prime多重集合。", "en": "Every prime divides zero, so it cannot have a finite unique prime multiset." } }, "leaf_basis": "calc" } ], "result": { "zh-Hant": "正整數n>1存在唯一質因數多重集合；360=2³·3²·5，負號與units另列。", "en": "Every positive integer n>1 has a unique prime multiset; 360=2^3*3^2*5, with signs and units separate." }, "negative_controls": [ { "claim": { "zh-Hant": "360=2³·3·5。", "en": "360=2^3*3*5." }, "expected": "reject", "reason": { "zh-Hant": "右側120，漏掉一個3。", "en": "The right side is 120, missing a factor 3." } }, { "claim": { "zh-Hant": "Euclid引理對合數4也成立。", "en": "Euclid's lemma also holds for composite 4." }, "expected": "reject", "reason": { "zh-Hant": "4|2·2但4不整除任一2。", "en": "4 divides 2*2 but neither factor 2." } }, { "claim": { "zh-Hant": "0的prime factorization是空乘積。", "en": "Zero's prime factorization is the empty product." }, "expected": "reject", "reason": { "zh-Hant": "空乘積是1，不是0。", "en": "The empty product is 1, not zero." } } ], "status": "candidate", "gaps": [], "verification": { "structure": "checked_local_script", "mathematics": "candidate_argument", "formalization": "not_formalized" }, "rule_glossary": [ { "id": "calc", "kind": "prerequisite", "statement": { "zh-Hant": "假設精確整數算術、帶餘除法、整除與gcd的定義；同餘等式始終綁定明示模數。另假設正整數良基／強歸納原理，以及互質整數的Bezout恆等式。", "en": "Assume exact integer arithmetic, division with remainder, and definitions of divisibility and gcd; every congruence has an explicit modulus. Also assume well-founded/strong induction on positive integers and the Bezout identity for coprime integers." } }, { "id": "mko_definition", "kind": "cited", "statement": { "zh-Hant": "existence以well-founded/induction與完整trial division小切片驗證；uniqueness保留Euclid lemma witness p|ab implies p|a or p|b，而非只比較少數答案。整數版分離negative sign，±1為units與empty product，0沒有此prime factorization。", "en": "Separate existence from uniqueness. Induction/well-founded reasoning gives existence, with complete bounded trial-division instances as computational companions. Uniqueness keeps the Euclid-lemma witness p|ab implies p|a or p|b, not just a few matching outputs. General factorizations may contain several distinct primes. Separate a negative sign, units±1 with the empty prime product and zero, which has no such factorization." }, "source": { "mko_id": "mko-unique-prime-factorization", "sha256": "0f01853edccbfb8cdf298587be277da27032f1f31ab87712ee5745644aa66cff", "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f" } } ], "revision_log": [], "numeric_checks": [ { "id": "product", "lhs": "2**3*3**2*5", "rhs": "360", "step": "S08" } ] }, "presentation": { "md_scope": { "zh-Hant": "以強歸納與明示Bezout前提證存在唯一性，再完整trial division核360。", "en": "Existence and uniqueness by strong induction and an explicit Bezout prerequisite, then complete trial division for 360." }, "md_symbols": [ [ "p_i,q_j", "含重數的prime因子", "prime factors with multiplicity" ], [ "r,s", "有限因數個數", "finite factor counts" ] ], "md_error_controls": [ { "title": "存在與唯一 / Existence and uniqueness", "zh": "強歸納給存在，Euclid引理加消去給唯一，兩責任分開。", "en": "Strong induction gives existence; Euclid's lemma and cancellation give uniqueness." }, { "title": "重數 / Multiplicity", "zh": "比較prime多重集合，不只distinct prime集合。", "en": "Compare prime multisets, not only distinct-prime sets." } ], "md_open_items": [] }, "revision": { "id": "ocme-new142-reader-adoption-20261021", "collection": "coverage-20261021", "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f", "adopted_source_sha256": "648c53ebc102df797ce25af2afb0b128976c45a091ff54b1a47d03de1a9d1ad6", "original_source_sha256": "648c53ebc102df797ce25af2afb0b128976c45a091ff54b1a47d03de1a9d1ad6", "owner_adopted": true, "bilingual_sync": "complete_source_bound", "formal_Lean": "NOT_RUN", "reader_review": "142_of_142_source_bound", "wording_review": "original_Lead_decisions_resolved", "glossary_reader_paraphrases": [], "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW", "issue_ids": [], "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a", "scope": "Owner-adopted reader revision. 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