第 27787 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":37}。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":37}。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 = [37, 1]

derived = [1, 1]

source roles = [37, 2]

derived = [19, 1]

source roles = [37, 3]

derived = [25, 1]

source roles = [37, 4]

derived = [28, 1]

source roles = [37, 5]

derived = [15, 1]

source roles = [37, 6]

derived = [31, 1]

source roles = [37, 7]

derived = [16, 1]

source roles = [37, 8]

derived = [14, 1]

source roles = [37, 9]

derived = [33, 1]

source roles = [37, 10]

derived = [26, 1]

source roles = [37, 11]

derived = [27, 1]

source roles = [37, 12]

derived = [34, 1]

source roles = [37, 13]

derived = [20, 1]

source roles = [37, 14]

derived = [8, 1]

source roles = [37, 15]

derived = [5, 1]

source roles = [37, 16]

derived = [7, 1]

source roles = [37, 17]

derived = [24, 1]

source roles = [37, 18]

derived = [35, 1]

source roles = [37, 19]

derived = [2, 1]

source roles = [37, 20]

derived = [13, 1]

source roles = [37, 21]

derived = [30, 1]

source roles = [37, 22]

derived = [32, 1]

source roles = [37, 23]

derived = [29, 1]

source roles = [37, 24]

derived = [17, 1]

source roles = [37, 25]

derived = [3, 1]

source roles = [37, 26]

derived = [10, 1]

source roles = [37, 27]

derived = [11, 1]

source roles = [37, 28]

derived = [4, 1]

source roles = [37, 29]

derived = [23, 1]

source roles = [37, 30]

derived = [21, 1]

source roles = [37, 31]

derived = [6, 1]

source roles = [37, 32]

derived = [22, 1]

source roles = [37, 33]

derived = [9, 1]

source roles = [37, 34]

derived = [12, 1]

source roles = [37, 35]

derived = [18, 1]

source roles = [37, 36]

derived = [36, 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, 37, 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, 37, 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, 37, 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, 37, 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, 37, 5, 24, 120]

derived = [3, 9, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 := mul(9, 6)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 = 54

source roles = [Wilson ORIGINAL prime factorial, 37, 6, 9, 54]

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, 37, 7, 17, 119]

derived = [3, 8, 是, 是]

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

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 = 64

source roles = [Wilson ORIGINAL prime factorial, 37, 8, 8, 64]

derived = [1, 27, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 := mul(27, 9)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 = 243

source roles = [Wilson ORIGINAL prime factorial, 37, 9, 27, 243]

derived = [6, 21, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 := mul(21, 10)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 = 210

source roles = [Wilson ORIGINAL prime factorial, 37, 10, 21, 210]

derived = [5, 25, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 := mul(25, 11)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 = 275

source roles = [Wilson ORIGINAL prime factorial, 37, 11, 25, 275]

derived = [7, 16, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 := mul(16, 12)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 = 192

source roles = [Wilson ORIGINAL prime factorial, 37, 12, 16, 192]

derived = [5, 7, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 := mul(7, 13)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 = 91

source roles = [Wilson ORIGINAL prime factorial, 37, 13, 7, 91]

derived = [2, 17, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 := mul(17, 14)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 = 238

source roles = [Wilson ORIGINAL prime factorial, 37, 14, 17, 238]

derived = [6, 16, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 := mul(16, 15)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 = 240

source roles = [Wilson ORIGINAL prime factorial, 37, 15, 16, 240]

derived = [6, 18, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 := mul(18, 16)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 = 288

source roles = [Wilson ORIGINAL prime factorial, 37, 16, 18, 288]

derived = [7, 29, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 := mul(29, 17)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 = 493

source roles = [Wilson ORIGINAL prime factorial, 37, 17, 29, 493]

derived = [13, 12, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 := mul(12, 18)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 = 216

source roles = [Wilson ORIGINAL prime factorial, 37, 18, 12, 216]

derived = [5, 31, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 := mul(31, 19)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 = 589

source roles = [Wilson ORIGINAL prime factorial, 37, 19, 31, 589]

derived = [15, 34, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 := mul(34, 20)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 = 680

source roles = [Wilson ORIGINAL prime factorial, 37, 20, 34, 680]

derived = [18, 14, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 := mul(14, 21)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 = 294

source roles = [Wilson ORIGINAL prime factorial, 37, 21, 14, 294]

derived = [7, 35, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 := mul(35, 22)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 = 770

source roles = [Wilson ORIGINAL prime factorial, 37, 22, 35, 770]

derived = [20, 30, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 := mul(30, 23)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 = 690

source roles = [Wilson ORIGINAL prime factorial, 37, 23, 30, 690]

derived = [18, 24, 是, 是]

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

Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 = 576

source roles = [Wilson ORIGINAL prime factorial, 37, 24, 24, 576]

derived = [15, 21, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 := mul(21, 25)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 = 525

source roles = [Wilson ORIGINAL prime factorial, 37, 25, 21, 525]

derived = [14, 7, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 := mul(7, 26)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 = 182

source roles = [Wilson ORIGINAL prime factorial, 37, 26, 7, 182]

derived = [4, 34, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 := mul(34, 27)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 = 918

source roles = [Wilson ORIGINAL prime factorial, 37, 27, 34, 918]

derived = [24, 30, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 := mul(30, 28)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 = 840

source roles = [Wilson ORIGINAL prime factorial, 37, 28, 30, 840]

derived = [22, 26, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 := mul(26, 29)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 = 754

source roles = [Wilson ORIGINAL prime factorial, 37, 29, 26, 754]

derived = [20, 14, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 := mul(14, 30)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 = 420

source roles = [Wilson ORIGINAL prime factorial, 37, 30, 14, 420]

derived = [11, 13, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 := mul(13, 31)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 = 403

source roles = [Wilson ORIGINAL prime factorial, 37, 31, 13, 403]

derived = [10, 33, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 := mul(33, 32)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 = 1056

source roles = [Wilson ORIGINAL prime factorial, 37, 32, 33, 1056]

derived = [28, 20, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 := mul(20, 33)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 = 660

source roles = [Wilson ORIGINAL prime factorial, 37, 33, 20, 660]

derived = [17, 31, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 := mul(31, 34)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 = 1054

source roles = [Wilson ORIGINAL prime factorial, 37, 34, 31, 1054]

derived = [28, 18, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 := mul(18, 35)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 = 630

source roles = [Wilson ORIGINAL prime factorial, 37, 35, 18, 630]

derived = [17, 1, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 := mul(1, 36)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 = 36

source roles = [Wilson ORIGINAL prime factorial, 37, 36, 1, 36]

derived = [0, 36, 是, 是]

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

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

逐步反向核對

• S110 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• 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 product36，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 product35，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 product34，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 product33，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 product32，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 product31，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 product30，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 product29，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 product28，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 product27，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 product26，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 product25，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 product24，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 product23，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 product22，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 product21，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 product20，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 product19，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 product18，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 product17，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 product16，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 product15，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 product14，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 product13，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 product12，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 product11，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 product10，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 product9，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 product8，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 product7，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 product6，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 product5，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 product4，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 product3，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 product2，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 product1，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 — 回查原始資料、操作數、完整範圍與本步前提；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

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