第 27783 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":19}。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":19}。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 = [19, 1]

derived = [1, 1]

source roles = [19, 2]

derived = [10, 1]

source roles = [19, 3]

derived = [13, 1]

source roles = [19, 4]

derived = [5, 1]

source roles = [19, 5]

derived = [4, 1]

source roles = [19, 6]

derived = [16, 1]

source roles = [19, 7]

derived = [11, 1]

source roles = [19, 8]

derived = [12, 1]

source roles = [19, 9]

derived = [17, 1]

source roles = [19, 10]

derived = [2, 1]

source roles = [19, 11]

derived = [7, 1]

source roles = [19, 12]

derived = [8, 1]

source roles = [19, 13]

derived = [3, 1]

source roles = [19, 14]

derived = [15, 1]

source roles = [19, 15]

derived = [14, 1]

source roles = [19, 16]

derived = [6, 1]

source roles = [19, 17]

derived = [9, 1]

source roles = [19, 18]

derived = [18, 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, 19, 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, 19, 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, 19, 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, 19, 4, 6, 24]

derived = [1, 5, 是, 是]

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

Wilson ORIGINAL prime factorial ORIGINAL running factorial product5 = 25

source roles = [Wilson ORIGINAL prime factorial, 19, 5, 5, 25]

derived = [1, 6, 是, 是]

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

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 = 36

source roles = [Wilson ORIGINAL prime factorial, 19, 6, 6, 36]

derived = [1, 17, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 := mul(17, 7)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 = 119

source roles = [Wilson ORIGINAL prime factorial, 19, 7, 17, 119]

derived = [6, 5, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 := mul(5, 8)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 = 40

source roles = [Wilson ORIGINAL prime factorial, 19, 8, 5, 40]

derived = [2, 2, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 := mul(2, 9)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 = 18

source roles = [Wilson ORIGINAL prime factorial, 19, 9, 2, 18]

derived = [0, 18, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 := mul(18, 10)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 = 180

source roles = [Wilson ORIGINAL prime factorial, 19, 10, 18, 180]

derived = [9, 9, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 := mul(9, 11)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 = 99

source roles = [Wilson ORIGINAL prime factorial, 19, 11, 9, 99]

derived = [5, 4, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 := mul(4, 12)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 = 48

source roles = [Wilson ORIGINAL prime factorial, 19, 12, 4, 48]

derived = [2, 10, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 := mul(10, 13)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 = 130

source roles = [Wilson ORIGINAL prime factorial, 19, 13, 10, 130]

derived = [6, 16, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 := mul(16, 14)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 = 224

source roles = [Wilson ORIGINAL prime factorial, 19, 14, 16, 224]

derived = [11, 15, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 := mul(15, 15)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 = 225

source roles = [Wilson ORIGINAL prime factorial, 19, 15, 15, 225]

derived = [11, 16, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 := mul(16, 16)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 = 256

source roles = [Wilson ORIGINAL prime factorial, 19, 16, 16, 256]

derived = [13, 9, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 := mul(9, 17)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 = 153

source roles = [Wilson ORIGINAL prime factorial, 19, 17, 9, 153]

derived = [8, 1, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 := mul(1, 18)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 = 18

source roles = [Wilson ORIGINAL prime factorial, 19, 18, 1, 18]

derived = [0, 18, 是, 是]

source roles = [19, [[1, 1], [2, 10], [3, 13], [4, 5], [5, 4], [6, 16], [7, 11], [8, 12], [9, 17], [10, 2], [11, 7], [12, 8], [13, 3], [14, 15], [15, 14], [16, 6], [17, 9], [18, 18]]]

derived = [[1, 18], [[2, 10, 1], [3, 13, 1], [4, 5, 1], [6, 16, 1], [7, 11, 1], [8, 12, 1], [9, 17, 1], [14, 15, 1]], 18, 是]

逐步反向核對

• S056 · 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。
• S055 · 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。
• S054 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product18，不以小數近似替代。
• S053 · 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。
• S052 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product17，不以小數近似替代。
• S051 · 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。
• S050 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product16，不以小數近似替代。
• S049 · 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。
• S048 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product15，不以小數近似替代。
• S047 · 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。
• S046 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product14，不以小數近似替代。
• S045 · 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。
• S044 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product13，不以小數近似替代。
• S043 · 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。
• S042 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product12，不以小數近似替代。
• S041 · 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。
• S040 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product11，不以小數近似替代。
• S039 · 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。
• S038 · 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。
• S037 · 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。
• S036 · 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。
• S035 · 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。
• S034 · 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。
• S033 · 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。
• S032 · 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。
• 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 product6，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 product5，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 product4，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 product3，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 product2，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 product1，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 — 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。
• 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 — 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。
• 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 — 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。
• 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 — 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。
• 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

(19, [[1, 1], [2, 10], [3, 13], [4, 5], [5, 4], [6, 16], [7, 11], [8, 12], [9, 17], [10, 2], [11, 7], [12, 8], [13, 3], [14, 15], [15, 14], [16, 6], [17, 9], [18, 18]], [1, 18], [[2, 10, 1], [3, 13, 1], [4, 5, 1], [6, 16, 1], [7, 11, 1], [8, 12, 1], [9, 17, 1], [14, 15, 1]], self_inverse_residues_1_and_p_minus_1, 18, 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。