第 27788 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":41}。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":41}。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 = [41, 1]

derived = [1, 1]

source roles = [41, 2]

derived = [21, 1]

source roles = [41, 3]

derived = [14, 1]

source roles = [41, 4]

derived = [31, 1]

source roles = [41, 5]

derived = [33, 1]

source roles = [41, 6]

derived = [7, 1]

source roles = [41, 7]

derived = [6, 1]

source roles = [41, 8]

derived = [36, 1]

source roles = [41, 9]

derived = [32, 1]

source roles = [41, 10]

derived = [37, 1]

source roles = [41, 11]

derived = [15, 1]

source roles = [41, 12]

derived = [24, 1]

source roles = [41, 13]

derived = [19, 1]

source roles = [41, 14]

derived = [3, 1]

source roles = [41, 15]

derived = [11, 1]

source roles = [41, 16]

derived = [18, 1]

source roles = [41, 17]

derived = [29, 1]

source roles = [41, 18]

derived = [16, 1]

source roles = [41, 19]

derived = [13, 1]

source roles = [41, 20]

derived = [39, 1]

source roles = [41, 21]

derived = [2, 1]

source roles = [41, 22]

derived = [28, 1]

source roles = [41, 23]

derived = [25, 1]

source roles = [41, 24]

derived = [12, 1]

source roles = [41, 25]

derived = [23, 1]

source roles = [41, 26]

derived = [30, 1]

source roles = [41, 27]

derived = [38, 1]

source roles = [41, 28]

derived = [22, 1]

source roles = [41, 29]

derived = [17, 1]

source roles = [41, 30]

derived = [26, 1]

source roles = [41, 31]

derived = [4, 1]

source roles = [41, 32]

derived = [9, 1]

source roles = [41, 33]

derived = [5, 1]

source roles = [41, 34]

derived = [35, 1]

source roles = [41, 35]

derived = [34, 1]

source roles = [41, 36]

derived = [8, 1]

source roles = [41, 37]

derived = [10, 1]

source roles = [41, 38]

derived = [27, 1]

source roles = [41, 39]

derived = [20, 1]

source roles = [41, 40]

derived = [40, 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, 41, 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, 41, 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, 41, 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, 41, 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, 41, 5, 24, 120]

derived = [2, 38, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 := mul(38, 6)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 = 228

source roles = [Wilson ORIGINAL prime factorial, 41, 6, 38, 228]

derived = [5, 23, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 := mul(23, 7)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 = 161

source roles = [Wilson ORIGINAL prime factorial, 41, 7, 23, 161]

derived = [3, 38, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 := mul(38, 8)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 = 304

source roles = [Wilson ORIGINAL prime factorial, 41, 8, 38, 304]

derived = [7, 17, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 := mul(17, 9)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 = 153

source roles = [Wilson ORIGINAL prime factorial, 41, 9, 17, 153]

derived = [3, 30, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 := mul(30, 10)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 = 300

source roles = [Wilson ORIGINAL prime factorial, 41, 10, 30, 300]

derived = [7, 13, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 := mul(13, 11)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 = 143

source roles = [Wilson ORIGINAL prime factorial, 41, 11, 13, 143]

derived = [3, 20, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 := mul(20, 12)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 = 240

source roles = [Wilson ORIGINAL prime factorial, 41, 12, 20, 240]

derived = [5, 35, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 := mul(35, 13)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 = 455

source roles = [Wilson ORIGINAL prime factorial, 41, 13, 35, 455]

derived = [11, 4, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 := mul(4, 14)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 = 56

source roles = [Wilson ORIGINAL prime factorial, 41, 14, 4, 56]

derived = [1, 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, 41, 15, 15, 225]

derived = [5, 20, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 := mul(20, 16)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 = 320

source roles = [Wilson ORIGINAL prime factorial, 41, 16, 20, 320]

derived = [7, 33, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 := mul(33, 17)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 = 561

source roles = [Wilson ORIGINAL prime factorial, 41, 17, 33, 561]

derived = [13, 28, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 := mul(28, 18)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 = 504

source roles = [Wilson ORIGINAL prime factorial, 41, 18, 28, 504]

derived = [12, 12, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 := mul(12, 19)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 = 228

source roles = [Wilson ORIGINAL prime factorial, 41, 19, 12, 228]

derived = [5, 23, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 := mul(23, 20)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 = 460

source roles = [Wilson ORIGINAL prime factorial, 41, 20, 23, 460]

derived = [11, 9, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 := mul(9, 21)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 = 189

source roles = [Wilson ORIGINAL prime factorial, 41, 21, 9, 189]

derived = [4, 25, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 := mul(25, 22)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 = 550

source roles = [Wilson ORIGINAL prime factorial, 41, 22, 25, 550]

derived = [13, 17, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 := mul(17, 23)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 = 391

source roles = [Wilson ORIGINAL prime factorial, 41, 23, 17, 391]

derived = [9, 22, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 := mul(22, 24)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 = 528

source roles = [Wilson ORIGINAL prime factorial, 41, 24, 22, 528]

derived = [12, 36, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 := mul(36, 25)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 = 900

source roles = [Wilson ORIGINAL prime factorial, 41, 25, 36, 900]

derived = [21, 39, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 := mul(39, 26)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 = 1014

source roles = [Wilson ORIGINAL prime factorial, 41, 26, 39, 1014]

derived = [24, 30, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 := mul(30, 27)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 = 810

source roles = [Wilson ORIGINAL prime factorial, 41, 27, 30, 810]

derived = [19, 31, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 := mul(31, 28)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 = 868

source roles = [Wilson ORIGINAL prime factorial, 41, 28, 31, 868]

derived = [21, 7, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 := mul(7, 29)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 = 203

source roles = [Wilson ORIGINAL prime factorial, 41, 29, 7, 203]

derived = [4, 39, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 := mul(39, 30)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 = 1170

source roles = [Wilson ORIGINAL prime factorial, 41, 30, 39, 1170]

derived = [28, 22, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 := mul(22, 31)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 = 682

source roles = [Wilson ORIGINAL prime factorial, 41, 31, 22, 682]

derived = [16, 26, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 := mul(26, 32)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 = 832

source roles = [Wilson ORIGINAL prime factorial, 41, 32, 26, 832]

derived = [20, 12, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 := mul(12, 33)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 = 396

source roles = [Wilson ORIGINAL prime factorial, 41, 33, 12, 396]

derived = [9, 27, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 := mul(27, 34)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 = 918

source roles = [Wilson ORIGINAL prime factorial, 41, 34, 27, 918]

derived = [22, 16, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 := mul(16, 35)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 = 560

source roles = [Wilson ORIGINAL prime factorial, 41, 35, 16, 560]

derived = [13, 27, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 := mul(27, 36)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 = 972

source roles = [Wilson ORIGINAL prime factorial, 41, 36, 27, 972]

derived = [23, 29, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 := mul(29, 37)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 = 1073

source roles = [Wilson ORIGINAL prime factorial, 41, 37, 29, 1073]

derived = [26, 7, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 := mul(7, 38)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 = 266

source roles = [Wilson ORIGINAL prime factorial, 41, 38, 7, 266]

derived = [6, 20, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 := mul(20, 39)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 = 780

source roles = [Wilson ORIGINAL prime factorial, 41, 39, 20, 780]

derived = [19, 1, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 := mul(1, 40)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 = 40

source roles = [Wilson ORIGINAL prime factorial, 41, 40, 1, 40]

derived = [0, 40, 是, 是]

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

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

逐步反向核對

• S122 · 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。
• 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 product40，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 product39，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 product38，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 product37，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 product36，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 product35，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 product34，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 product33，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 product32，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 product31，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 product30，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 product29，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 product28，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 product27，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 product26，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 product25，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 product24，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 product23，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 product22，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 product21，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 product20，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 product19，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 product18，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 product17，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 product16，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 product15，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 product14，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 product13，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 product12，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 product11，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 product10，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 product9，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 product8，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 product7，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 product6，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 product5，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 product4，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 product3，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 product2，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 product1，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 — 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

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