第 27713 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":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-boundary；資料={"mode":"fermat_confusion","n":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-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, 41, 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, 41, 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, 41, 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, 41, 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, 41, 5, 24, 120]

derived = [2, 38, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product6 = 228

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

derived = [5, 23, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product7 = 161

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

derived = [3, 38, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product8 = 304

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

derived = [7, 17, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product9 = 153

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

derived = [3, 30, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product10 = 300

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

derived = [7, 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, 41, 11, 13, 143]

derived = [3, 20, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product12 = 240

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

derived = [5, 35, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product13 = 455

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

derived = [11, 4, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product14 = 56

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

derived = [1, 15, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product15 = 225

source roles = [Wilson boundary ORIGINAL prime, 41, 15, 15, 225]

derived = [5, 20, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product16 = 320

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

derived = [7, 33, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product17 = 561

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

derived = [13, 28, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product18 = 504

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

derived = [12, 12, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product19 = 228

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

derived = [5, 23, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product20 = 460

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

derived = [11, 9, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product21 = 189

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

derived = [4, 25, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product22 = 550

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

derived = [13, 17, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product23 = 391

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

derived = [9, 22, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product24 = 528

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

derived = [12, 36, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product25 = 900

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

derived = [21, 39, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product26 = 1014

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

derived = [24, 30, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product27 = 810

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

derived = [19, 31, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product28 = 868

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

derived = [21, 7, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product29 = 203

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

derived = [4, 39, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product30 = 1170

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

derived = [28, 22, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product31 = 682

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

derived = [16, 26, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product32 = 832

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

derived = [20, 12, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product33 = 396

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

derived = [9, 27, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product34 = 918

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

derived = [22, 16, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product35 = 560

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

derived = [13, 27, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product36 = 972

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

derived = [23, 29, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product37 = 1073

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

derived = [26, 7, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product38 = 266

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

derived = [6, 20, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product39 = 780

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

derived = [19, 1, 是, 是]

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

Wilson boundary ORIGINAL prime ORIGINAL running factorial product40 = 40

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

derived = [0, 40, 是, 是]

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

derived = [40, 是, 否]

逐步反向核對

• S082 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• 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, 41, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 38], [6, 23], [7, 38], [8, 17], [9, 30], [10, 13], [11, 20], [12, 35], [13, 4], [14, 15], [15, 20], [16, 33], [17, 28], [18, 12], [19, 23], [20, 9], [21, 25], [22, 17], [23, 22], [24, 36], [25, 39], [26, 30], [27, 31], [28, 7], [29, 39], [30, 22], [31, 26], [32, 12], [33, 27], [34, 16], [35, 27], [36, 29], [37, 7], [38, 20], [39, 1], [40, 40]], 40, 40, 是, Wilson:(n-1)!==-1_mod_41, Fermat:a^(41-1)==1_mod_41_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。