第 27710 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":29}。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":29}。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, 29, 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, 29, 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, 29, 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, 29, 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, 29, 5, 24, 120]

derived = [4, 4, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product6 := mul(4, 6)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product6 = 24

source roles = [Wilson boundary ORIGINAL prime, 29, 6, 4, 24]

derived = [0, 24, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product7 := mul(24, 7)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product7 = 168

source roles = [Wilson boundary ORIGINAL prime, 29, 7, 24, 168]

derived = [5, 23, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product8 := mul(23, 8)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product8 = 184

source roles = [Wilson boundary ORIGINAL prime, 29, 8, 23, 184]

derived = [6, 10, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product9 := mul(10, 9)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product9 = 90

source roles = [Wilson boundary ORIGINAL prime, 29, 9, 10, 90]

derived = [3, 3, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product10 := mul(3, 10)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product10 = 30

source roles = [Wilson boundary ORIGINAL prime, 29, 10, 3, 30]

derived = [1, 1, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product11 := mul(1, 11)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product11 = 11

source roles = [Wilson boundary ORIGINAL prime, 29, 11, 1, 11]

derived = [0, 11, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product12 := mul(11, 12)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product12 = 132

source roles = [Wilson boundary ORIGINAL prime, 29, 12, 11, 132]

derived = [4, 16, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product13 := mul(16, 13)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product13 = 208

source roles = [Wilson boundary ORIGINAL prime, 29, 13, 16, 208]

derived = [7, 5, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product14 := mul(5, 14)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product14 = 70

source roles = [Wilson boundary ORIGINAL prime, 29, 14, 5, 70]

derived = [2, 12, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product15 := mul(12, 15)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product15 = 180

source roles = [Wilson boundary ORIGINAL prime, 29, 15, 12, 180]

derived = [6, 6, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product16 := mul(6, 16)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product16 = 96

source roles = [Wilson boundary ORIGINAL prime, 29, 16, 6, 96]

derived = [3, 9, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product17 := mul(9, 17)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product17 = 153

source roles = [Wilson boundary ORIGINAL prime, 29, 17, 9, 153]

derived = [5, 8, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product18 := mul(8, 18)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product18 = 144

source roles = [Wilson boundary ORIGINAL prime, 29, 18, 8, 144]

derived = [4, 28, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product19 := mul(28, 19)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product19 = 532

source roles = [Wilson boundary ORIGINAL prime, 29, 19, 28, 532]

derived = [18, 10, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product20 := mul(10, 20)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product20 = 200

source roles = [Wilson boundary ORIGINAL prime, 29, 20, 10, 200]

derived = [6, 26, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product21 := mul(26, 21)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product21 = 546

source roles = [Wilson boundary ORIGINAL prime, 29, 21, 26, 546]

derived = [18, 24, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product22 := mul(24, 22)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product22 = 528

source roles = [Wilson boundary ORIGINAL prime, 29, 22, 24, 528]

derived = [18, 6, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product23 := mul(6, 23)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product23 = 138

source roles = [Wilson boundary ORIGINAL prime, 29, 23, 6, 138]

derived = [4, 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, 29, 24, 22, 528]

derived = [18, 6, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product25 := mul(6, 25)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product25 = 150

source roles = [Wilson boundary ORIGINAL prime, 29, 25, 6, 150]

derived = [5, 5, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product26 := mul(5, 26)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product26 = 130

source roles = [Wilson boundary ORIGINAL prime, 29, 26, 5, 130]

derived = [4, 14, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product27 := mul(14, 27)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product27 = 378

source roles = [Wilson boundary ORIGINAL prime, 29, 27, 14, 378]

derived = [13, 1, 是, 是]

Wilson boundary ORIGINAL prime ORIGINAL running factorial product28 := mul(1, 28)

Wilson boundary ORIGINAL prime ORIGINAL running factorial product28 = 28

source roles = [Wilson boundary ORIGINAL prime, 29, 28, 1, 28]

derived = [0, 28, 是, 是]

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

derived = [28, 是, 否]

逐步反向核對

• S058 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• 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, 29, [[1, 1], [2, 2], [3, 6], [4, 24], [5, 4], [6, 24], [7, 23], [8, 10], [9, 3], [10, 1], [11, 11], [12, 16], [13, 5], [14, 12], [15, 6], [16, 9], [17, 8], [18, 28], [19, 10], [20, 26], [21, 24], [22, 6], [23, 22], [24, 6], [25, 5], [26, 14], [27, 1], [28, 28]], 28, 28, 是, Wilson:(n-1)!==-1_mod_29, Fermat:a^(29-1)==1_mod_29_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。