第 27719 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-boundary；資料={"mode":"fermat_confusion","n":67}。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-boundary；資料={"mode":"fermat_confusion","n":67}。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-boundary]

derived = [[mode, n, running_factorial_residues, factorial_residue, minus_one_residue, Wilson_congruence, Wilson_statement, Fermat_statement, theorem_identification_claim, boundary]]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product1 = 1

source roles = [Wilson boundary ORIGINAL prime, 67, 1, 1, 1]

derived = [0, 1, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product2 = 2

source roles = [Wilson boundary ORIGINAL prime, 67, 2, 1, 2]

derived = [0, 2, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product3 = 6

source roles = [Wilson boundary ORIGINAL prime, 67, 3, 2, 6]

derived = [0, 6, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product4 = 24

source roles = [Wilson boundary ORIGINAL prime, 67, 4, 6, 24]

derived = [0, 24, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product5 = 120

source roles = [Wilson boundary ORIGINAL prime, 67, 5, 24, 120]

derived = [1, 53, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product6 := mul(53, 6)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product6 = 318

source roles = [Wilson boundary ORIGINAL prime, 67, 6, 53, 318]

derived = [4, 50, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product7 := mul(50, 7)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product7 = 350

source roles = [Wilson boundary ORIGINAL prime, 67, 7, 50, 350]

derived = [5, 15, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product8 := mul(15, 8)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product8 = 120

source roles = [Wilson boundary ORIGINAL prime, 67, 8, 15, 120]

derived = [1, 53, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product9 := mul(53, 9)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product9 = 477

source roles = [Wilson boundary ORIGINAL prime, 67, 9, 53, 477]

derived = [7, 8, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product10 := mul(8, 10)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product10 = 80

source roles = [Wilson boundary ORIGINAL prime, 67, 10, 8, 80]

derived = [1, 13, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product11 = 143

source roles = [Wilson boundary ORIGINAL prime, 67, 11, 13, 143]

derived = [2, 9, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product12 := mul(9, 12)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product12 = 108

source roles = [Wilson boundary ORIGINAL prime, 67, 12, 9, 108]

derived = [1, 41, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product13 := mul(41, 13)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product13 = 533

source roles = [Wilson boundary ORIGINAL prime, 67, 13, 41, 533]

derived = [7, 64, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product14 := mul(64, 14)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product14 = 896

source roles = [Wilson boundary ORIGINAL prime, 67, 14, 64, 896]

derived = [13, 25, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product15 := mul(25, 15)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product15 = 375

source roles = [Wilson boundary ORIGINAL prime, 67, 15, 25, 375]

derived = [5, 40, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product16 := mul(40, 16)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product16 = 640

source roles = [Wilson boundary ORIGINAL prime, 67, 16, 40, 640]

derived = [9, 37, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product17 := mul(37, 17)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product17 = 629

source roles = [Wilson boundary ORIGINAL prime, 67, 17, 37, 629]

derived = [9, 26, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product18 := mul(26, 18)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product18 = 468

source roles = [Wilson boundary ORIGINAL prime, 67, 18, 26, 468]

derived = [6, 66, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product19 := mul(66, 19)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product19 = 1254

source roles = [Wilson boundary ORIGINAL prime, 67, 19, 66, 1254]

derived = [18, 48, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product20 := mul(48, 20)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product20 = 960

source roles = [Wilson boundary ORIGINAL prime, 67, 20, 48, 960]

derived = [14, 22, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product21 := mul(22, 21)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product21 = 462

source roles = [Wilson boundary ORIGINAL prime, 67, 21, 22, 462]

derived = [6, 60, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product22 := mul(60, 22)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product22 = 1320

source roles = [Wilson boundary ORIGINAL prime, 67, 22, 60, 1320]

derived = [19, 47, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product23 := mul(47, 23)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product23 = 1081

source roles = [Wilson boundary ORIGINAL prime, 67, 23, 47, 1081]

derived = [16, 9, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product24 := mul(9, 24)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product24 = 216

source roles = [Wilson boundary ORIGINAL prime, 67, 24, 9, 216]

derived = [3, 15, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product25 := mul(15, 25)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product25 = 375

source roles = [Wilson boundary ORIGINAL prime, 67, 25, 15, 375]

derived = [5, 40, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product26 := mul(40, 26)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product26 = 1040

source roles = [Wilson boundary ORIGINAL prime, 67, 26, 40, 1040]

derived = [15, 35, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product27 := mul(35, 27)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product27 = 945

source roles = [Wilson boundary ORIGINAL prime, 67, 27, 35, 945]

derived = [14, 7, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product28 := mul(7, 28)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product28 = 196

source roles = [Wilson boundary ORIGINAL prime, 67, 28, 7, 196]

derived = [2, 62, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product29 := mul(62, 29)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product29 = 1798

source roles = [Wilson boundary ORIGINAL prime, 67, 29, 62, 1798]

derived = [26, 56, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product30 := mul(56, 30)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product30 = 1680

source roles = [Wilson boundary ORIGINAL prime, 67, 30, 56, 1680]

derived = [25, 5, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product31 := mul(5, 31)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product31 = 155

source roles = [Wilson boundary ORIGINAL prime, 67, 31, 5, 155]

derived = [2, 21, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product32 := mul(21, 32)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product32 = 672

source roles = [Wilson boundary ORIGINAL prime, 67, 32, 21, 672]

derived = [10, 2, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product33 := mul(2, 33)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product33 = 66

source roles = [Wilson boundary ORIGINAL prime, 67, 33, 2, 66]

derived = [0, 66, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product34 := mul(66, 34)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product34 = 2244

source roles = [Wilson boundary ORIGINAL prime, 67, 34, 66, 2244]

derived = [33, 33, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product35 := mul(33, 35)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product35 = 1155

source roles = [Wilson boundary ORIGINAL prime, 67, 35, 33, 1155]

derived = [17, 16, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product36 := mul(16, 36)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product36 = 576

source roles = [Wilson boundary ORIGINAL prime, 67, 36, 16, 576]

derived = [8, 40, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product37 := mul(40, 37)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product37 = 1480

source roles = [Wilson boundary ORIGINAL prime, 67, 37, 40, 1480]

derived = [22, 6, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product38 := mul(6, 38)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product38 = 228

source roles = [Wilson boundary ORIGINAL prime, 67, 38, 6, 228]

derived = [3, 27, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product39 := mul(27, 39)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product39 = 1053

source roles = [Wilson boundary ORIGINAL prime, 67, 39, 27, 1053]

derived = [15, 48, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product40 := mul(48, 40)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product40 = 1920

source roles = [Wilson boundary ORIGINAL prime, 67, 40, 48, 1920]

derived = [28, 44, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product41 := mul(44, 41)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product41 = 1804

source roles = [Wilson boundary ORIGINAL prime, 67, 41, 44, 1804]

derived = [26, 62, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product42 := mul(62, 42)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product42 = 2604

source roles = [Wilson boundary ORIGINAL prime, 67, 42, 62, 2604]

derived = [38, 58, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product43 := mul(58, 43)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product43 = 2494

source roles = [Wilson boundary ORIGINAL prime, 67, 43, 58, 2494]

derived = [37, 15, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product44 := mul(15, 44)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product44 = 660

source roles = [Wilson boundary ORIGINAL prime, 67, 44, 15, 660]

derived = [9, 57, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product45 := mul(57, 45)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product45 = 2565

source roles = [Wilson boundary ORIGINAL prime, 67, 45, 57, 2565]

derived = [38, 19, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product46 := mul(19, 46)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product46 = 874

source roles = [Wilson boundary ORIGINAL prime, 67, 46, 19, 874]

derived = [13, 3, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product47 := mul(3, 47)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product47 = 141

source roles = [Wilson boundary ORIGINAL prime, 67, 47, 3, 141]

derived = [2, 7, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product48 := mul(7, 48)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product48 = 336

source roles = [Wilson boundary ORIGINAL prime, 67, 48, 7, 336]

derived = [5, 1, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product49 := mul(1, 49)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product49 = 49

source roles = [Wilson boundary ORIGINAL prime, 67, 49, 1, 49]

derived = [0, 49, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product50 := mul(49, 50)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product50 = 2450

source roles = [Wilson boundary ORIGINAL prime, 67, 50, 49, 2450]

derived = [36, 38, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product51 := mul(38, 51)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product51 = 1938

source roles = [Wilson boundary ORIGINAL prime, 67, 51, 38, 1938]

derived = [28, 62, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product52 := mul(62, 52)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product52 = 3224

source roles = [Wilson boundary ORIGINAL prime, 67, 52, 62, 3224]

derived = [48, 8, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product53 := mul(8, 53)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product53 = 424

source roles = [Wilson boundary ORIGINAL prime, 67, 53, 8, 424]

derived = [6, 22, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product54 := mul(22, 54)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product54 = 1188

source roles = [Wilson boundary ORIGINAL prime, 67, 54, 22, 1188]

derived = [17, 49, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product55 := mul(49, 55)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product55 = 2695

source roles = [Wilson boundary ORIGINAL prime, 67, 55, 49, 2695]

derived = [40, 15, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product56 := mul(15, 56)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product56 = 840

source roles = [Wilson boundary ORIGINAL prime, 67, 56, 15, 840]

derived = [12, 36, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product57 := mul(36, 57)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product57 = 2052

source roles = [Wilson boundary ORIGINAL prime, 67, 57, 36, 2052]

derived = [30, 42, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product58 := mul(42, 58)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product58 = 2436

source roles = [Wilson boundary ORIGINAL prime, 67, 58, 42, 2436]

derived = [36, 24, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product59 := mul(24, 59)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product59 = 1416

source roles = [Wilson boundary ORIGINAL prime, 67, 59, 24, 1416]

derived = [21, 9, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product60 := mul(9, 60)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product60 = 540

source roles = [Wilson boundary ORIGINAL prime, 67, 60, 9, 540]

derived = [8, 4, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product61 := mul(4, 61)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product61 = 244

source roles = [Wilson boundary ORIGINAL prime, 67, 61, 4, 244]

derived = [3, 43, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product62 := mul(43, 62)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product62 = 2666

source roles = [Wilson boundary ORIGINAL prime, 67, 62, 43, 2666]

derived = [39, 53, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product63 := mul(53, 63)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product63 = 3339

source roles = [Wilson boundary ORIGINAL prime, 67, 63, 53, 3339]

derived = [49, 56, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product64 := mul(56, 64)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product64 = 3584

source roles = [Wilson boundary ORIGINAL prime, 67, 64, 56, 3584]

derived = [53, 33, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product65 := mul(33, 65)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product65 = 2145

source roles = [Wilson boundary ORIGINAL prime, 67, 65, 33, 2145]

derived = [32, 1, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product66 := mul(1, 66)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product66 = 66

source roles = [Wilson boundary ORIGINAL prime, 67, 66, 1, 66]

derived = [0, 66, 是, 是]

source roles = [67, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 53], [6, 50], [7, 15], [8, 53], [9, 8], [10, 13], [11, 9], [12, 41], [13, 64], [14, 25], [15, 40], [16, 37], [17, 26], [18, 66], [19, 48], [20, 22], [21, 60], [22, 47], [23, 9], [24, 15], [25, 40], [26, 35], [27, 7], [28, 62], [29, 56], [30, 5], [31, 21], [32, 2], [33, 66], [34, 33], [35, 16], [36, 40], [37, 6], [38, 27], [39, 48], [40, 44], [41, 62], [42, 58], [43, 15], [44, 57], [45, 19], [46, 3], [47, 7], [48, 1], [49, 49], [50, 38], [51, 62], [52, 8], [53, 22], [54, 49], [55, 15], [56, 36], [57, 42], [58, 24], [59, 9], [60, 4], [61, 43], [62, 53], [63, 56], [64, 33], [65, 1], [66, 66]], 66]

derived = [66, 是, 否]

逐步反向核對

• S134 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• 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 boundary ORIGINAL prime ORIGINAL running factorial product66，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 boundary ORIGINAL prime ORIGINAL running factorial product65，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 boundary ORIGINAL prime ORIGINAL running factorial product64，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 boundary ORIGINAL prime ORIGINAL running factorial product63，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 boundary ORIGINAL prime ORIGINAL running factorial product62，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 boundary ORIGINAL prime ORIGINAL running factorial product61，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 boundary ORIGINAL prime ORIGINAL running factorial product60，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 boundary ORIGINAL prime ORIGINAL running factorial product59，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 boundary ORIGINAL prime ORIGINAL running factorial product58，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 boundary ORIGINAL prime ORIGINAL running factorial product57，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 boundary ORIGINAL prime ORIGINAL running factorial product56，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 boundary ORIGINAL prime ORIGINAL running factorial product55，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 boundary ORIGINAL prime ORIGINAL running factorial product54，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 boundary ORIGINAL prime ORIGINAL running factorial product53，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 boundary ORIGINAL prime ORIGINAL running factorial product52，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 boundary ORIGINAL prime ORIGINAL running factorial product51，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 boundary ORIGINAL prime ORIGINAL running factorial product50，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 boundary ORIGINAL prime ORIGINAL running factorial product49，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 boundary ORIGINAL prime ORIGINAL running factorial product48，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 boundary ORIGINAL prime ORIGINAL running factorial product47，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 boundary ORIGINAL prime ORIGINAL running factorial product46，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 boundary ORIGINAL prime ORIGINAL running factorial product45，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 boundary ORIGINAL prime ORIGINAL running factorial product44，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 boundary ORIGINAL prime ORIGINAL running factorial product43，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 boundary ORIGINAL prime ORIGINAL running factorial product42，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 boundary ORIGINAL prime ORIGINAL running factorial product41，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 boundary ORIGINAL prime ORIGINAL running factorial product40，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 boundary ORIGINAL prime ORIGINAL running factorial product39，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 boundary ORIGINAL prime ORIGINAL running factorial product38，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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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 boundary ORIGINAL prime 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

(fermat_confusion, 67, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 53], [6, 50], [7, 15], [8, 53], [9, 8], [10, 13], [11, 9], [12, 41], [13, 64], [14, 25], [15, 40], [16, 37], [17, 26], [18, 66], [19, 48], [20, 22], [21, 60], [22, 47], [23, 9], [24, 15], [25, 40], [26, 35], [27, 7], [28, 62], [29, 56], [30, 5], [31, 21], [32, 2], [33, 66], [34, 33], [35, 16], [36, 40], [37, 6], [38, 27], [39, 48], [40, 44], [41, 62], [42, 58], [43, 15], [44, 57], [45, 19], [46, 3], [47, 7], [48, 1], [49, 49], [50, 38], [51, 62], [52, 8], [53, 22], [54, 49], [55, 15], [56, 36], [57, 42], [58, 24], [59, 9], [60, 4], [61, 43], [62, 53], [63, 56], [64, 33], [65, 1], [66, 66]], 66, 66, 是, Wilson:(n-1)!==-1_mod_67, Fermat:a^(67-1)==1_mod_67_when_n_does_not_divide_a, 否, distinct_theorems_no_Fermat_power_claim_recomputed)

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。