第 27750 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-composite-instance；資料={"n":38}。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-composite-instance；資料={"n":38}。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-composite-instance]

derived = [[n, prime, running_factorial_residues, factorial_residue, minus_one_residue, Wilson_congruence_fails, finite_scope]]

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

Wilson ORIGINAL factorial ORIGINAL running factorial product1 = 1

source roles = [Wilson ORIGINAL factorial, 38, 1, 1, 1]

derived = [0, 1, 是, 是]

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

Wilson ORIGINAL factorial ORIGINAL running factorial product2 = 2

source roles = [Wilson ORIGINAL factorial, 38, 2, 1, 2]

derived = [0, 2, 是, 是]

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

Wilson ORIGINAL factorial ORIGINAL running factorial product3 = 6

source roles = [Wilson ORIGINAL factorial, 38, 3, 2, 6]

derived = [0, 6, 是, 是]

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

Wilson ORIGINAL factorial ORIGINAL running factorial product4 = 24

source roles = [Wilson ORIGINAL factorial, 38, 4, 6, 24]

derived = [0, 24, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product5 := mul(24, 5)

Wilson ORIGINAL factorial ORIGINAL running factorial product5 = 120

source roles = [Wilson ORIGINAL factorial, 38, 5, 24, 120]

derived = [3, 6, 是, 是]

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

Wilson ORIGINAL factorial ORIGINAL running factorial product6 = 36

source roles = [Wilson ORIGINAL factorial, 38, 6, 6, 36]

derived = [0, 36, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product7 := mul(36, 7)

Wilson ORIGINAL factorial ORIGINAL running factorial product7 = 252

source roles = [Wilson ORIGINAL factorial, 38, 7, 36, 252]

derived = [6, 24, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product8 := mul(24, 8)

Wilson ORIGINAL factorial ORIGINAL running factorial product8 = 192

source roles = [Wilson ORIGINAL factorial, 38, 8, 24, 192]

derived = [5, 2, 是, 是]

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

Wilson ORIGINAL factorial ORIGINAL running factorial product9 = 18

source roles = [Wilson ORIGINAL factorial, 38, 9, 2, 18]

derived = [0, 18, 是, 是]

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

Wilson ORIGINAL factorial ORIGINAL running factorial product10 = 180

source roles = [Wilson ORIGINAL factorial, 38, 10, 18, 180]

derived = [4, 28, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product11 := mul(28, 11)

Wilson ORIGINAL factorial ORIGINAL running factorial product11 = 308

source roles = [Wilson ORIGINAL factorial, 38, 11, 28, 308]

derived = [8, 4, 是, 是]

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

Wilson ORIGINAL factorial ORIGINAL running factorial product12 = 48

source roles = [Wilson ORIGINAL factorial, 38, 12, 4, 48]

derived = [1, 10, 是, 是]

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

Wilson ORIGINAL factorial ORIGINAL running factorial product13 = 130

source roles = [Wilson ORIGINAL factorial, 38, 13, 10, 130]

derived = [3, 16, 是, 是]

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

Wilson ORIGINAL factorial ORIGINAL running factorial product14 = 224

source roles = [Wilson ORIGINAL factorial, 38, 14, 16, 224]

derived = [5, 34, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product15 := mul(34, 15)

Wilson ORIGINAL factorial ORIGINAL running factorial product15 = 510

source roles = [Wilson ORIGINAL factorial, 38, 15, 34, 510]

derived = [13, 16, 是, 是]

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

Wilson ORIGINAL factorial ORIGINAL running factorial product16 = 256

source roles = [Wilson ORIGINAL factorial, 38, 16, 16, 256]

derived = [6, 28, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product17 := mul(28, 17)

Wilson ORIGINAL factorial ORIGINAL running factorial product17 = 476

source roles = [Wilson ORIGINAL factorial, 38, 17, 28, 476]

derived = [12, 20, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product18 := mul(20, 18)

Wilson ORIGINAL factorial ORIGINAL running factorial product18 = 360

source roles = [Wilson ORIGINAL factorial, 38, 18, 20, 360]

derived = [9, 18, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product19 := mul(18, 19)

Wilson ORIGINAL factorial ORIGINAL running factorial product19 = 342

source roles = [Wilson ORIGINAL factorial, 38, 19, 18, 342]

derived = [9, 0, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product20 := mul(0, 20)

Wilson ORIGINAL factorial ORIGINAL running factorial product20 = 0

source roles = [Wilson ORIGINAL factorial, 38, 20, 0, 0]

derived = [0, 0, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product21 := mul(0, 21)

Wilson ORIGINAL factorial ORIGINAL running factorial product21 = 0

source roles = [Wilson ORIGINAL factorial, 38, 21, 0, 0]

derived = [0, 0, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product22 := mul(0, 22)

Wilson ORIGINAL factorial ORIGINAL running factorial product22 = 0

source roles = [Wilson ORIGINAL factorial, 38, 22, 0, 0]

derived = [0, 0, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product23 := mul(0, 23)

Wilson ORIGINAL factorial ORIGINAL running factorial product23 = 0

source roles = [Wilson ORIGINAL factorial, 38, 23, 0, 0]

derived = [0, 0, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product24 := mul(0, 24)

Wilson ORIGINAL factorial ORIGINAL running factorial product24 = 0

source roles = [Wilson ORIGINAL factorial, 38, 24, 0, 0]

derived = [0, 0, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product25 := mul(0, 25)

Wilson ORIGINAL factorial ORIGINAL running factorial product25 = 0

source roles = [Wilson ORIGINAL factorial, 38, 25, 0, 0]

derived = [0, 0, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product26 := mul(0, 26)

Wilson ORIGINAL factorial ORIGINAL running factorial product26 = 0

source roles = [Wilson ORIGINAL factorial, 38, 26, 0, 0]

derived = [0, 0, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product27 := mul(0, 27)

Wilson ORIGINAL factorial ORIGINAL running factorial product27 = 0

source roles = [Wilson ORIGINAL factorial, 38, 27, 0, 0]

derived = [0, 0, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product28 := mul(0, 28)

Wilson ORIGINAL factorial ORIGINAL running factorial product28 = 0

source roles = [Wilson ORIGINAL factorial, 38, 28, 0, 0]

derived = [0, 0, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product29 := mul(0, 29)

Wilson ORIGINAL factorial ORIGINAL running factorial product29 = 0

source roles = [Wilson ORIGINAL factorial, 38, 29, 0, 0]

derived = [0, 0, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product30 := mul(0, 30)

Wilson ORIGINAL factorial ORIGINAL running factorial product30 = 0

source roles = [Wilson ORIGINAL factorial, 38, 30, 0, 0]

derived = [0, 0, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product31 := mul(0, 31)

Wilson ORIGINAL factorial ORIGINAL running factorial product31 = 0

source roles = [Wilson ORIGINAL factorial, 38, 31, 0, 0]

derived = [0, 0, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product32 := mul(0, 32)

Wilson ORIGINAL factorial ORIGINAL running factorial product32 = 0

source roles = [Wilson ORIGINAL factorial, 38, 32, 0, 0]

derived = [0, 0, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product33 := mul(0, 33)

Wilson ORIGINAL factorial ORIGINAL running factorial product33 = 0

source roles = [Wilson ORIGINAL factorial, 38, 33, 0, 0]

derived = [0, 0, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product34 := mul(0, 34)

Wilson ORIGINAL factorial ORIGINAL running factorial product34 = 0

source roles = [Wilson ORIGINAL factorial, 38, 34, 0, 0]

derived = [0, 0, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product35 := mul(0, 35)

Wilson ORIGINAL factorial ORIGINAL running factorial product35 = 0

source roles = [Wilson ORIGINAL factorial, 38, 35, 0, 0]

derived = [0, 0, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product36 := mul(0, 36)

Wilson ORIGINAL factorial ORIGINAL running factorial product36 = 0

source roles = [Wilson ORIGINAL factorial, 38, 36, 0, 0]

derived = [0, 0, 是, 是]

Wilson ORIGINAL factorial ORIGINAL running factorial product37 := mul(0, 37)

Wilson ORIGINAL factorial ORIGINAL running factorial product37 = 0

source roles = [Wilson ORIGINAL factorial, 38, 37, 0, 0]

derived = [0, 0, 是, 是]

source roles = [38, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 6], [6, 36], [7, 24], [8, 2], [9, 18], [10, 28], [11, 4], [12, 10], [13, 16], [14, 34], [15, 16], [16, 28], [17, 20], [18, 18], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0], [27, 0], [28, 0], [29, 0], [30, 0], [31, 0], [32, 0], [33, 0], [34, 0], [35, 0], [36, 0], [37, 0]]]

derived = [否, 0, 37, 否, 是]

逐步反向核對

• S076 · 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。
• S075 · 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。
• S074 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL factorial ORIGINAL running factorial product37，不以小數近似替代。
• S073 · 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。
• S072 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL factorial ORIGINAL running factorial product36，不以小數近似替代。
• S071 · 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。
• S070 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL factorial ORIGINAL running factorial product35，不以小數近似替代。
• S069 · 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。
• S068 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL factorial ORIGINAL running factorial product34，不以小數近似替代。
• S067 · 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。
• S066 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL factorial ORIGINAL running factorial product33，不以小數近似替代。
• S065 · 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。
• S064 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL factorial ORIGINAL running factorial product32，不以小數近似替代。
• S063 · 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。
• S062 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL factorial ORIGINAL running factorial product31，不以小數近似替代。
• S061 · 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。
• S060 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL factorial ORIGINAL running factorial product30，不以小數近似替代。
• S059 · 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。
• S058 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL factorial ORIGINAL running factorial product29，不以小數近似替代。
• S057 · 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。
• S056 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL factorial ORIGINAL running factorial product28，不以小數近似替代。
• 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 factorial ORIGINAL running factorial product27，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 factorial ORIGINAL running factorial product26，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 factorial ORIGINAL running factorial product25，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 factorial ORIGINAL running factorial product24，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 factorial ORIGINAL running factorial product23，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 factorial ORIGINAL running factorial product22，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 factorial ORIGINAL running factorial product21，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 factorial ORIGINAL running factorial product20，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 factorial ORIGINAL running factorial product19，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 factorial ORIGINAL running factorial product18，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 factorial ORIGINAL running factorial product17，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 factorial ORIGINAL running factorial product16，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 factorial ORIGINAL running factorial product15，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 factorial ORIGINAL running factorial product14，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 factorial ORIGINAL running factorial product13，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 factorial ORIGINAL running factorial product12，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 factorial ORIGINAL running factorial product11，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 factorial ORIGINAL running factorial product10，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 factorial ORIGINAL running factorial product9，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 factorial ORIGINAL running factorial product8，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 factorial ORIGINAL running factorial product7，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 factorial ORIGINAL running factorial product6，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 — 以精確有理數重算 Wilson ORIGINAL factorial ORIGINAL running factorial product5，不以小數近似替代。
• 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 — 以精確有理數重算 Wilson ORIGINAL factorial ORIGINAL running factorial product4，不以小數近似替代。
• 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 — 以精確有理數重算 Wilson ORIGINAL factorial ORIGINAL running factorial product3，不以小數近似替代。
• 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 — 以精確有理數重算 Wilson ORIGINAL factorial ORIGINAL running factorial product2，不以小數近似替代。
• 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 — 以精確有理數重算 Wilson ORIGINAL factorial ORIGINAL running factorial product1，不以小數近似替代。
• 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

(38, 否, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 6], [6, 36], [7, 24], [8, 2], [9, 18], [10, 28], [11, 4], [12, 10], [13, 16], [14, 34], [15, 16], [16, 28], [17, 20], [18, 18], [19, 0], [20, 0], [21, 0], [22, 0], [23, 0], [24, 0], [25, 0], [26, 0], [27, 0], [28, 0], [29, 0], [30, 0], [31, 0], [32, 0], [33, 0], [34, 0], [35, 0], [36, 0], [37, 0]], 0, 37, 是, finite_composite_instance_not_converse_proof)

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。