第 27790 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":47}。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":47}。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 = [47, 1]

derived = [1, 1]

source roles = [47, 2]

derived = [24, 1]

source roles = [47, 3]

derived = [16, 1]

source roles = [47, 4]

derived = [12, 1]

source roles = [47, 5]

derived = [19, 1]

source roles = [47, 6]

derived = [8, 1]

source roles = [47, 7]

derived = [27, 1]

source roles = [47, 8]

derived = [6, 1]

source roles = [47, 9]

derived = [21, 1]

source roles = [47, 10]

derived = [33, 1]

source roles = [47, 11]

derived = [30, 1]

source roles = [47, 12]

derived = [4, 1]

source roles = [47, 13]

derived = [29, 1]

source roles = [47, 14]

derived = [37, 1]

source roles = [47, 15]

derived = [22, 1]

source roles = [47, 16]

derived = [3, 1]

source roles = [47, 17]

derived = [36, 1]

source roles = [47, 18]

derived = [34, 1]

source roles = [47, 19]

derived = [5, 1]

source roles = [47, 20]

derived = [40, 1]

source roles = [47, 21]

derived = [9, 1]

source roles = [47, 22]

derived = [15, 1]

source roles = [47, 23]

derived = [45, 1]

source roles = [47, 24]

derived = [2, 1]

source roles = [47, 25]

derived = [32, 1]

source roles = [47, 26]

derived = [38, 1]

source roles = [47, 27]

derived = [7, 1]

source roles = [47, 28]

derived = [42, 1]

source roles = [47, 29]

derived = [13, 1]

source roles = [47, 30]

derived = [11, 1]

source roles = [47, 31]

derived = [44, 1]

source roles = [47, 32]

derived = [25, 1]

source roles = [47, 33]

derived = [10, 1]

source roles = [47, 34]

derived = [18, 1]

source roles = [47, 35]

derived = [43, 1]

source roles = [47, 36]

derived = [17, 1]

source roles = [47, 37]

derived = [14, 1]

source roles = [47, 38]

derived = [26, 1]

source roles = [47, 39]

derived = [41, 1]

source roles = [47, 40]

derived = [20, 1]

source roles = [47, 41]

derived = [39, 1]

source roles = [47, 42]

derived = [28, 1]

source roles = [47, 43]

derived = [35, 1]

source roles = [47, 44]

derived = [31, 1]

source roles = [47, 45]

derived = [23, 1]

source roles = [47, 46]

derived = [46, 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, 47, 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, 47, 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, 47, 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, 47, 4, 6, 24]

derived = [0, 24, 是, 是]

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

Wilson ORIGINAL prime factorial ORIGINAL running factorial product5 = 120

source roles = [Wilson ORIGINAL prime factorial, 47, 5, 24, 120]

derived = [2, 26, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 := mul(26, 6)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 = 156

source roles = [Wilson ORIGINAL prime factorial, 47, 6, 26, 156]

derived = [3, 15, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 := mul(15, 7)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 = 105

source roles = [Wilson ORIGINAL prime factorial, 47, 7, 15, 105]

derived = [2, 11, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 := mul(11, 8)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 = 88

source roles = [Wilson ORIGINAL prime factorial, 47, 8, 11, 88]

derived = [1, 41, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 := mul(41, 9)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 = 369

source roles = [Wilson ORIGINAL prime factorial, 47, 9, 41, 369]

derived = [7, 40, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 := mul(40, 10)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 = 400

source roles = [Wilson ORIGINAL prime factorial, 47, 10, 40, 400]

derived = [8, 24, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 := mul(24, 11)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 = 264

source roles = [Wilson ORIGINAL prime factorial, 47, 11, 24, 264]

derived = [5, 29, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 := mul(29, 12)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 = 348

source roles = [Wilson ORIGINAL prime factorial, 47, 12, 29, 348]

derived = [7, 19, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 := mul(19, 13)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 = 247

source roles = [Wilson ORIGINAL prime factorial, 47, 13, 19, 247]

derived = [5, 12, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 := mul(12, 14)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 = 168

source roles = [Wilson ORIGINAL prime factorial, 47, 14, 12, 168]

derived = [3, 27, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 := mul(27, 15)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 = 405

source roles = [Wilson ORIGINAL prime factorial, 47, 15, 27, 405]

derived = [8, 29, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 := mul(29, 16)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 = 464

source roles = [Wilson ORIGINAL prime factorial, 47, 16, 29, 464]

derived = [9, 41, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 := mul(41, 17)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 = 697

source roles = [Wilson ORIGINAL prime factorial, 47, 17, 41, 697]

derived = [14, 39, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 := mul(39, 18)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 = 702

source roles = [Wilson ORIGINAL prime factorial, 47, 18, 39, 702]

derived = [14, 44, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 := mul(44, 19)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 = 836

source roles = [Wilson ORIGINAL prime factorial, 47, 19, 44, 836]

derived = [17, 37, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 := mul(37, 20)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 = 740

source roles = [Wilson ORIGINAL prime factorial, 47, 20, 37, 740]

derived = [15, 35, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 := mul(35, 21)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 = 735

source roles = [Wilson ORIGINAL prime factorial, 47, 21, 35, 735]

derived = [15, 30, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 := mul(30, 22)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 = 660

source roles = [Wilson ORIGINAL prime factorial, 47, 22, 30, 660]

derived = [14, 2, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 := mul(2, 23)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 = 46

source roles = [Wilson ORIGINAL prime factorial, 47, 23, 2, 46]

derived = [0, 46, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 := mul(46, 24)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 = 1104

source roles = [Wilson ORIGINAL prime factorial, 47, 24, 46, 1104]

derived = [23, 23, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 := mul(23, 25)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 = 575

source roles = [Wilson ORIGINAL prime factorial, 47, 25, 23, 575]

derived = [12, 11, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 := mul(11, 26)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 = 286

source roles = [Wilson ORIGINAL prime factorial, 47, 26, 11, 286]

derived = [6, 4, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 := mul(4, 27)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 = 108

source roles = [Wilson ORIGINAL prime factorial, 47, 27, 4, 108]

derived = [2, 14, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 := mul(14, 28)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 = 392

source roles = [Wilson ORIGINAL prime factorial, 47, 28, 14, 392]

derived = [8, 16, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 := mul(16, 29)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 = 464

source roles = [Wilson ORIGINAL prime factorial, 47, 29, 16, 464]

derived = [9, 41, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 := mul(41, 30)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 = 1230

source roles = [Wilson ORIGINAL prime factorial, 47, 30, 41, 1230]

derived = [26, 8, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 := mul(8, 31)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 = 248

source roles = [Wilson ORIGINAL prime factorial, 47, 31, 8, 248]

derived = [5, 13, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 := mul(13, 32)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 = 416

source roles = [Wilson ORIGINAL prime factorial, 47, 32, 13, 416]

derived = [8, 40, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 := mul(40, 33)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 = 1320

source roles = [Wilson ORIGINAL prime factorial, 47, 33, 40, 1320]

derived = [28, 4, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 := mul(4, 34)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 = 136

source roles = [Wilson ORIGINAL prime factorial, 47, 34, 4, 136]

derived = [2, 42, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 := mul(42, 35)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 = 1470

source roles = [Wilson ORIGINAL prime factorial, 47, 35, 42, 1470]

derived = [31, 13, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 := mul(13, 36)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 = 468

source roles = [Wilson ORIGINAL prime factorial, 47, 36, 13, 468]

derived = [9, 45, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 := mul(45, 37)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 = 1665

source roles = [Wilson ORIGINAL prime factorial, 47, 37, 45, 1665]

derived = [35, 20, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 := mul(20, 38)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 = 760

source roles = [Wilson ORIGINAL prime factorial, 47, 38, 20, 760]

derived = [16, 8, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 := mul(8, 39)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 = 312

source roles = [Wilson ORIGINAL prime factorial, 47, 39, 8, 312]

derived = [6, 30, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 := mul(30, 40)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 = 1200

source roles = [Wilson ORIGINAL prime factorial, 47, 40, 30, 1200]

derived = [25, 25, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product41 := mul(25, 41)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product41 = 1025

source roles = [Wilson ORIGINAL prime factorial, 47, 41, 25, 1025]

derived = [21, 38, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product42 := mul(38, 42)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product42 = 1596

source roles = [Wilson ORIGINAL prime factorial, 47, 42, 38, 1596]

derived = [33, 45, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product43 := mul(45, 43)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product43 = 1935

source roles = [Wilson ORIGINAL prime factorial, 47, 43, 45, 1935]

derived = [41, 8, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product44 := mul(8, 44)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product44 = 352

source roles = [Wilson ORIGINAL prime factorial, 47, 44, 8, 352]

derived = [7, 23, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product45 := mul(23, 45)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product45 = 1035

source roles = [Wilson ORIGINAL prime factorial, 47, 45, 23, 1035]

derived = [22, 1, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product46 := mul(1, 46)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product46 = 46

source roles = [Wilson ORIGINAL prime factorial, 47, 46, 1, 46]

derived = [0, 46, 是, 是]

source roles = [47, [[1, 1], [2, 24], [3, 16], [4, 12], [5, 19], [6, 8], [7, 27], [8, 6], [9, 21], [10, 33], [11, 30], [12, 4], [13, 29], [14, 37], [15, 22], [16, 3], [17, 36], [18, 34], [19, 5], [20, 40], [21, 9], [22, 15], [23, 45], [24, 2], [25, 32], [26, 38], [27, 7], [28, 42], [29, 13], [30, 11], [31, 44], [32, 25], [33, 10], [34, 18], [35, 43], [36, 17], [37, 14], [38, 26], [39, 41], [40, 20], [41, 39], [42, 28], [43, 35], [44, 31], [45, 23], [46, 46]]]

derived = [[1, 46], [[2, 24, 1], [3, 16, 1], [4, 12, 1], [5, 19, 1], [6, 8, 1], [7, 27, 1], [9, 21, 1], [10, 33, 1], [11, 30, 1], [13, 29, 1], [14, 37, 1], [15, 22, 1], [17, 36, 1], [18, 34, 1], [20, 40, 1], [23, 45, 1], [25, 32, 1], [26, 38, 1], [28, 42, 1], [31, 44, 1], [35, 43, 1], [39, 41, 1]], 46, 是]

逐步反向核對

• S140 · 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。
• S139 · 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。
• S138 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product46，不以小數近似替代。
• S137 · 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。
• S136 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product45，不以小數近似替代。
• S135 · 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。
• S134 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product44，不以小數近似替代。
• S133 · 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。
• S132 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product43，不以小數近似替代。
• S131 · 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。
• S130 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product42，不以小數近似替代。
• S129 · 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。
• S128 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product41，不以小數近似替代。
• S127 · 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。
• S126 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product40，不以小數近似替代。
• S125 · 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。
• S124 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product39，不以小數近似替代。
• S123 · 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。
• S122 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product38，不以小數近似替代。
• S121 · 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。
• S120 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product37，不以小數近似替代。
• S119 · 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。
• S118 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product36，不以小數近似替代。
• S117 · 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。
• S116 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product35，不以小數近似替代。
• S115 · 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。
• S114 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product34，不以小數近似替代。
• S113 · 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。
• S112 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product33，不以小數近似替代。
• S111 · 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。
• S110 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product32，不以小數近似替代。
• S109 · 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。
• S108 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product31，不以小數近似替代。
• S107 · 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。
• S106 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product30，不以小數近似替代。
• S105 · 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。
• S104 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product29，不以小數近似替代。
• S103 · 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。
• S102 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product28，不以小數近似替代。
• S101 · 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。
• S100 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product27，不以小數近似替代。
• S099 · 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。
• S098 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product26，不以小數近似替代。
• S097 · 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。
• S096 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product25，不以小數近似替代。
• S095 · 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。
• S094 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product24，不以小數近似替代。
• S093 · 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。
• S092 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product23，不以小數近似替代。
• S091 · 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。
• S090 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product22，不以小數近似替代。
• S089 · 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。
• S088 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product21，不以小數近似替代。
• S087 · 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。
• S086 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product20，不以小數近似替代。
• S085 · 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。
• S084 · premise_and_result_audit — 以精確有理數重算 Wilson ORIGINAL prime factorial ORIGINAL running factorial product19，不以小數近似替代。
• S083 · 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。
• S082 · 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。
• S081 · 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。
• S080 · 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。
• S079 · 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。
• S078 · 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。
• S077 · 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。
• S076 · 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。
• 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 prime factorial ORIGINAL running factorial product14，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 prime factorial ORIGINAL running factorial product13，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 prime factorial ORIGINAL running factorial product12，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 prime factorial ORIGINAL running factorial product11，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 prime factorial ORIGINAL running factorial product10，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 prime factorial ORIGINAL running factorial product9，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 prime factorial ORIGINAL running factorial product8，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 prime factorial ORIGINAL running factorial product7，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 prime factorial ORIGINAL running factorial product6，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 prime factorial ORIGINAL running factorial product5，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 prime factorial ORIGINAL running factorial product4，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 product3，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 product2，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 product1，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 — 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。
• 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 — 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。
• 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 — 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。
• 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 — 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。
• 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 — 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。
• 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 — 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。
• 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 — 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。
• 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 — 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 — 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。
• 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 — 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。
• 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 — 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。
• 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 — 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。
• 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 — 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。
• 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 — 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。
• 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

(47, [[1, 1], [2, 24], [3, 16], [4, 12], [5, 19], [6, 8], [7, 27], [8, 6], [9, 21], [10, 33], [11, 30], [12, 4], [13, 29], [14, 37], [15, 22], [16, 3], [17, 36], [18, 34], [19, 5], [20, 40], [21, 9], [22, 15], [23, 45], [24, 2], [25, 32], [26, 38], [27, 7], [28, 42], [29, 13], [30, 11], [31, 44], [32, 25], [33, 10], [34, 18], [35, 43], [36, 17], [37, 14], [38, 26], [39, 41], [40, 20], [41, 39], [42, 28], [43, 35], [44, 31], [45, 23], [46, 46]], [1, 46], [[2, 24, 1], [3, 16, 1], [4, 12, 1], [5, 19, 1], [6, 8, 1], [7, 27, 1], [9, 21, 1], [10, 33, 1], [11, 30, 1], [13, 29, 1], [14, 37, 1], [15, 22, 1], [17, 36, 1], [18, 34, 1], [20, 40, 1], [23, 45, 1], [25, 32, 1], [26, 38, 1], [28, 42, 1], [31, 44, 1], [35, 43, 1], [39, 41, 1]], self_inverse_residues_1_and_p_minus_1, 46, 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。