第 21724 題：Möbius函數

原題與已知條件

positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={"n":600}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — positive integer domain與完整UPF multiplicities。操作=mobius-canonical-value；資料={"n":600}。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-mobius-function

逐步推導

source operation=mobius-canonical-value;original snapshot and mathematics retained

explicit ordered source output fields=[n, prime_multiset, distinct_primes, mu, certificate]

mobius original n:actual positive input for mu=600

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

ORIGINAL positive integer absolute magnitude=600>1

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

upf extract divisor2 occurrence0 := div(600, 2)

upf extract divisor2 occurrence0 = 300

upf extract divisor2 occurrence1 := div(300, 2)

upf extract divisor2 occurrence1 = 150

upf extract divisor2 occurrence2 := div(150, 2)

upf extract divisor2 occurrence2 = 75

next divisor=2;previous remaining=600;all smaller divisors exhausted

extracted multiplicity=3;next remaining=75;remaining modulo2=1;extracted multiset=[2, 2, 2]

upf extract divisor3 occurrence3 := div(75, 3)

upf extract divisor3 occurrence3 = 25

next divisor=3;previous remaining=75;all smaller divisors exhausted

extracted multiplicity=1;next remaining=25;remaining modulo3=1;extracted multiset=[2, 2, 2, 3]

next divisor=4;previous remaining=25;all smaller divisors exhausted

extracted multiplicity=0;next remaining=25;remaining modulo4=1;extracted multiset=[2, 2, 2, 3]

upf extract divisor5 occurrence4 := div(25, 5)

upf extract divisor5 occurrence4 = 5

upf extract divisor5 occurrence5 := div(5, 5)

upf extract divisor5 occurrence5 = 1

next divisor=5;previous remaining=25;all smaller divisors exhausted

extracted multiplicity=2;next remaining=1;remaining modulo5=1;extracted multiset=[2, 2, 2, 3, 5, 5]

upf canonical complete product factor0 := mul(1, 2)

upf canonical complete product factor0 = 2

upf canonical complete product factor1 := mul(2, 2)

upf canonical complete product factor1 = 4

upf canonical complete product factor2 := mul(4, 2)

upf canonical complete product factor2 = 8

upf canonical complete product factor3 := mul(8, 3)

upf canonical complete product factor3 = 24

upf canonical complete product factor4 := mul(24, 5)

upf canonical complete product factor4 = 120

upf canonical complete product factor5 := mul(120, 5)

upf canonical complete product factor5 = 600

exhausted trial divisors below6;terminal remaining=1;d*d>remaining;ORIGINAL n=600

complete increasing prime multiset=[2, 2, 2, 3, 5, 5];full product=600;a composite terminal remainder would have an already tested smaller divisor

mobius original n:input=600;COMPLETE prime multiset=[2, 2, 2, 3, 5, 5];distinct primes=[2, 3, 5]

mu=0;empty product case=否;repeated prime=是

逐步反向核對

• S021 · 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。
• S020 · 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。
• S019 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor5，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor4，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor3，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor2，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor1，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 upf canonical complete product factor0，不以小數近似替代。
• S013 · 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。
• S012 · premise_and_result_audit — 以精確有理數重算 upf extract divisor5 occurrence5，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 upf extract divisor5 occurrence4，不以小數近似替代。
• S010 · 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。
• S009 · 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。
• S008 · premise_and_result_audit — 以精確有理數重算 upf extract divisor3 occurrence3，不以小數近似替代。
• S007 · 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。
• S006 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence2，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence1，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 upf extract divisor2 occurrence0，不以小數近似替代。
• S003 · 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。
• 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

(600, [2, 2, 2, 3, 5, 5], [2, 3, 5], 0, square_prime_factor_gives_zero)

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。