第 6778 題：階乘窗口的乘積與因子索引

原題與已知條件

N從0起；BigInt精確階乘；0≤lo=0≤hi=20≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。

• G001 — N從0起；BigInt精確階乘；0≤lo=0≤hi=20≤64。依序答 (hi!/lo!, lo+1至hi的因子索引列表, 是否空窗口)。

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-factorial

逐步推導

N includes0;F(0)=1;F(j)=j·F(j−1)

exact integer products,not floating values or a permutation proof

lo=0;hi=20;window factors=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]

hi!/lo! is the product of exactly these factors

window through factor1 := mul(1, 1)

window through factor1 = 1

window through factor2 := mul(1, 2)

window through factor2 = 2

window through factor3 := mul(2, 3)

window through factor3 = 6

window through factor4 := mul(6, 4)

window through factor4 = 24

window through factor5 := mul(24, 5)

window through factor5 = 120

window through factor6 := mul(120, 6)

window through factor6 = 720

window through factor7 := mul(720, 7)

window through factor7 = 5040

window through factor8 := mul(5040, 8)

window through factor8 = 40320

window through factor9 := mul(40320, 9)

window through factor9 = 362880

window through factor10 := mul(362880, 10)

window through factor10 = 3628800

window through factor11 := mul(3628800, 11)

window through factor11 = 39916800

window through factor12 := mul(39916800, 12)

window through factor12 = 479001600

window through factor13 := mul(479001600, 13)

window through factor13 = 6227020800

window through factor14 := mul(6227020800, 14)

window through factor14 = 87178291200

window through factor15 := mul(87178291200, 15)

window through factor15 = 1307674368000

window through factor16 := mul(1307674368000, 16)

window through factor16 = 20922789888000

window through factor17 := mul(20922789888000, 17)

window through factor17 = 355687428096000

window through factor18 := mul(355687428096000, 18)

window through factor18 = 6402373705728000

window through factor19 := mul(6402373705728000, 19)

window through factor19 = 121645100408832000

window through factor20 := mul(121645100408832000, 20)

window through factor20 = 2432902008176640000

逐步反向核對

• S022 · premise_and_result_audit — 以精確有理數重算 window through factor20，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 window through factor19，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 window through factor18，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 window through factor17，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 window through factor16，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 window through factor15，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 window through factor14，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 window through factor13，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 window through factor12，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 window through factor11，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 window through factor10，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 window through factor9，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 window through factor8，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 window through factor7，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 window through factor6，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 window through factor5，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 window through factor4，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 window through factor3，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 window through factor2，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 window through factor1，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

(2432902008176640000, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], 否)

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。