第 31896 題：後繼階乘遞迴

原題與已知條件

已知 20!=2432902008176640000。用 (n+1)!=(n+1)×n! 求 21!。

• G001 — 已知 20!=2432902008176640000。用 (n+1)!=(n+1)×n! 求 21!。

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

source roles = [qf-factorial-successor-recurrence, no-domain-field, successor_recurrence, no-form-field]

derived = [[exact_integer_STRING]]

Factorial requested n ORIGINAL factorial factor2 := mul(1, 2)

Factorial requested n ORIGINAL factorial factor2 = 2

Factorial requested n ORIGINAL factorial factor3 := mul(2, 3)

Factorial requested n ORIGINAL factorial factor3 = 6

Factorial requested n ORIGINAL factorial factor4 := mul(6, 4)

Factorial requested n ORIGINAL factorial factor4 = 24

Factorial requested n ORIGINAL factorial factor5 := mul(24, 5)

Factorial requested n ORIGINAL factorial factor5 = 120

Factorial requested n ORIGINAL factorial factor6 := mul(120, 6)

Factorial requested n ORIGINAL factorial factor6 = 720

Factorial requested n ORIGINAL factorial factor7 := mul(720, 7)

Factorial requested n ORIGINAL factorial factor7 = 5040

Factorial requested n ORIGINAL factorial factor8 := mul(5040, 8)

Factorial requested n ORIGINAL factorial factor8 = 40320

Factorial requested n ORIGINAL factorial factor9 := mul(40320, 9)

Factorial requested n ORIGINAL factorial factor9 = 362880

Factorial requested n ORIGINAL factorial factor10 := mul(362880, 10)

Factorial requested n ORIGINAL factorial factor10 = 3628800

Factorial requested n ORIGINAL factorial factor11 := mul(3628800, 11)

Factorial requested n ORIGINAL factorial factor11 = 39916800

Factorial requested n ORIGINAL factorial factor12 := mul(39916800, 12)

Factorial requested n ORIGINAL factorial factor12 = 479001600

Factorial requested n ORIGINAL factorial factor13 := mul(479001600, 13)

Factorial requested n ORIGINAL factorial factor13 = 6227020800

Factorial requested n ORIGINAL factorial factor14 := mul(6227020800, 14)

Factorial requested n ORIGINAL factorial factor14 = 87178291200

Factorial requested n ORIGINAL factorial factor15 := mul(87178291200, 15)

Factorial requested n ORIGINAL factorial factor15 = 1307674368000

Factorial requested n ORIGINAL factorial factor16 := mul(1307674368000, 16)

Factorial requested n ORIGINAL factorial factor16 = 20922789888000

Factorial requested n ORIGINAL factorial factor17 := mul(20922789888000, 17)

Factorial requested n ORIGINAL factorial factor17 = 355687428096000

Factorial requested n ORIGINAL factorial factor18 := mul(355687428096000, 18)

Factorial requested n ORIGINAL factorial factor18 = 6402373705728000

Factorial requested n ORIGINAL factorial factor19 := mul(6402373705728000, 19)

Factorial requested n ORIGINAL factorial factor19 = 121645100408832000

Factorial requested n ORIGINAL factorial factor20 := mul(121645100408832000, 20)

Factorial requested n ORIGINAL factorial factor20 = 2432902008176640000

source roles = [Factorial requested n, 20]

derived = [2432902008176640000]

source roles = [20, 2432902008176640000]

derived = [2432902008176640000]

Factorial ORIGINAL successor index n plus1 := sum(20, 1)

Factorial ORIGINAL successor index n plus1 = 21

Factorial ORIGINAL successor factor times verified n factorial := mul(21, 2432902008176640000)

Factorial ORIGINAL successor factor times verified n factorial = 51090942171709440000

source roles = [20, 2432902008176640000, 21]

derived = [51090942171709440000]

逐步反向核對

• S025 · premise_and_result_audit — 反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。
• S024 · premise_and_result_audit — 以精確有理數重算 Factorial ORIGINAL successor factor times verified n factorial，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Factorial ORIGINAL successor index n plus1，不以小數近似替代。
• S022 · premise_and_result_audit — 反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。
• S021 · premise_and_result_audit — 反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。
• S020 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor20，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor19，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor18，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor17，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor16，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor15，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor14，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor13，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor12，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor11，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor10，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor9，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor8，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor7，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor6，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor5，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor4，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor3，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Factorial requested n ORIGINAL factorial factor2，不以小數近似替代。
• S001 · premise_and_result_audit — 反向代回原輸入、欄位角色、論域及本步條件；一致性不表示結論能唯一恢復前提。有限核對、獨立審查、形式證明與發布分開標記。

結論

51090942171709440000

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。