第 27780 題：Wilson定理

原題與已知條件

n>1、factorial congruence與prime-residue inverse pairing責任。操作=wilson-prime-inverse-pairing；資料={"p":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — n>1、factorial congruence與prime-residue inverse pairing責任。操作=wilson-prime-inverse-pairing；資料={"p":11}。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-wilson-theorem

逐步推導

source roles = [wilson-prime-inverse-pairing]

derived = [[p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence]]

source roles = [11, 1]

derived = [1, 1]

source roles = [11, 2]

derived = [6, 1]

source roles = [11, 3]

derived = [4, 1]

source roles = [11, 4]

derived = [3, 1]

source roles = [11, 5]

derived = [9, 1]

source roles = [11, 6]

derived = [2, 1]

source roles = [11, 7]

derived = [8, 1]

source roles = [11, 8]

derived = [7, 1]

source roles = [11, 9]

derived = [5, 1]

source roles = [11, 10]

derived = [10, 1]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product1 := mul(1, 1)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product1 = 1

source roles = [Wilson ORIGINAL prime factorial, 11, 1, 1, 1]

derived = [0, 1, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product2 := mul(1, 2)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product2 = 2

source roles = [Wilson ORIGINAL prime factorial, 11, 2, 1, 2]

derived = [0, 2, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product3 := mul(2, 3)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product3 = 6

source roles = [Wilson ORIGINAL prime factorial, 11, 3, 2, 6]

derived = [0, 6, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product4 := mul(6, 4)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product4 = 24

source roles = [Wilson ORIGINAL prime factorial, 11, 4, 6, 24]

derived = [2, 2, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product5 := mul(2, 5)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product5 = 10

source roles = [Wilson ORIGINAL prime factorial, 11, 5, 2, 10]

derived = [0, 10, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 := mul(10, 6)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 = 60

source roles = [Wilson ORIGINAL prime factorial, 11, 6, 10, 60]

derived = [5, 5, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 := mul(5, 7)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 = 35

source roles = [Wilson ORIGINAL prime factorial, 11, 7, 5, 35]

derived = [3, 2, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 := mul(2, 8)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 = 16

source roles = [Wilson ORIGINAL prime factorial, 11, 8, 2, 16]

derived = [1, 5, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 := mul(5, 9)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 = 45

source roles = [Wilson ORIGINAL prime factorial, 11, 9, 5, 45]

derived = [4, 1, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 := mul(1, 10)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 = 10

source roles = [Wilson ORIGINAL prime factorial, 11, 10, 1, 10]

derived = [0, 10, 是, 是]

source roles = [11, [[1, 1], [2, 6], [3, 4], [4, 3], [5, 9], [6, 2], [7, 8], [8, 7], [9, 5], [10, 10]]]

derived = [[1, 10], [[2, 6, 1], [3, 4, 1], [5, 9, 1], [7, 8, 1]], 10, 是]

逐步反向核對

• S032 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S031 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S030 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product10，不以小數近似替代。
• S029 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S028 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product9，不以小數近似替代。
• S027 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S026 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product8，不以小數近似替代。
• S025 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S024 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product7，不以小數近似替代。
• S023 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S022 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product6，不以小數近似替代。
• S021 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S020 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product5，不以小數近似替代。
• S019 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S018 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product4，不以小數近似替代。
• S017 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S016 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product3，不以小數近似替代。
• S015 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S014 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product2，不以小數近似替代。
• S013 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S012 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product1，不以小數近似替代。
• S011 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S010 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S009 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S008 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S007 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S006 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S005 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S004 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S003 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S002 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• S001 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。

結論

(11, [[1, 1], [2, 6], [3, 4], [4, 3], [5, 9], [6, 2], [7, 8], [8, 7], [9, 5], [10, 10]], [1, 10], [[2, 6, 1], [3, 4, 1], [5, 9, 1], [7, 8, 1]], self_inverse_residues_1_and_p_minus_1, 10, 是)

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。