第 21329 題：整數唯一質因數分解

原題與已知條件

完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={"n":42,"factors":[2,3,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 完整bounded integer factorization與exact primality。操作=upf-factorization-audit；資料={"n":42,"factors":[2,3,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-unique-prime-factorization

逐步推導

source operation=upf-factorization-audit;original snapshot and mathematics retained

explicit ordered source output fields=[n, factors, all_factors_prime, factor_product, valid_factorization, certificate]

upf candidate factor index0:ORIGINAL candidate factor=2;ALL possible trial divisors2..floor(sqrt(v))=[]

ALL [divisor,remainder] rows=[];prime=是

upf candidate complete product at original factor0 := mul(1, 2)

upf candidate complete product at original factor0 = 2

upf candidate factor index1:ORIGINAL candidate factor=3;ALL possible trial divisors2..floor(sqrt(v))=[]

ALL [divisor,remainder] rows=[];prime=是

upf candidate complete product at original factor1 := mul(2, 3)

upf candidate complete product at original factor1 = 6

upf candidate factor index2:ORIGINAL candidate factor=7;ALL possible trial divisors2..floor(sqrt(v))=[2]

ALL [divisor,remainder] rows=[[2, 1]];prime=是

upf candidate complete product at original factor2 := mul(6, 7)

upf candidate complete product at original factor2 = 42

upf candidate:ORIGINAL factors=[2, 3, 7];ALL primality flags=[是, 是, 是];original n=42

empty product convention1;actual complete product=42;all_prime=是;valid_prime_product=是

逐步反向核對

• S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S007 · premise_and_result_audit — 以精確有理數重算 upf candidate complete product at original factor2，不以小數近似替代。
• S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S005 · premise_and_result_audit — 以精確有理數重算 upf candidate complete product at original factor1，不以小數近似替代。
• S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S003 · premise_and_result_audit — 以精確有理數重算 upf candidate complete product at original factor0，不以小數近似替代。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(42, [2, 3, 7], 是, 42, 是, existence_requires_complete_prime_product)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。