第 27786 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":31}。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":31}。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 = [31, 1]

derived = [1, 1]

source roles = [31, 2]

derived = [16, 1]

source roles = [31, 3]

derived = [21, 1]

source roles = [31, 4]

derived = [8, 1]

source roles = [31, 5]

derived = [25, 1]

source roles = [31, 6]

derived = [26, 1]

source roles = [31, 7]

derived = [9, 1]

source roles = [31, 8]

derived = [4, 1]

source roles = [31, 9]

derived = [7, 1]

source roles = [31, 10]

derived = [28, 1]

source roles = [31, 11]

derived = [17, 1]

source roles = [31, 12]

derived = [13, 1]

source roles = [31, 13]

derived = [12, 1]

source roles = [31, 14]

derived = [20, 1]

source roles = [31, 15]

derived = [29, 1]

source roles = [31, 16]

derived = [2, 1]

source roles = [31, 17]

derived = [11, 1]

source roles = [31, 18]

derived = [19, 1]

source roles = [31, 19]

derived = [18, 1]

source roles = [31, 20]

derived = [14, 1]

source roles = [31, 21]

derived = [3, 1]

source roles = [31, 22]

derived = [24, 1]

source roles = [31, 23]

derived = [27, 1]

source roles = [31, 24]

derived = [22, 1]

source roles = [31, 25]

derived = [5, 1]

source roles = [31, 26]

derived = [6, 1]

source roles = [31, 27]

derived = [23, 1]

source roles = [31, 28]

derived = [10, 1]

source roles = [31, 29]

derived = [15, 1]

source roles = [31, 30]

derived = [30, 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, 31, 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, 31, 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, 31, 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, 31, 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, 31, 5, 24, 120]

derived = [3, 27, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 := mul(27, 6)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 = 162

source roles = [Wilson ORIGINAL prime factorial, 31, 6, 27, 162]

derived = [5, 7, 是, 是]

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

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 = 49

source roles = [Wilson ORIGINAL prime factorial, 31, 7, 7, 49]

derived = [1, 18, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 := mul(18, 8)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 = 144

source roles = [Wilson ORIGINAL prime factorial, 31, 8, 18, 144]

derived = [4, 20, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 := mul(20, 9)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 = 180

source roles = [Wilson ORIGINAL prime factorial, 31, 9, 20, 180]

derived = [5, 25, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 := mul(25, 10)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 = 250

source roles = [Wilson ORIGINAL prime factorial, 31, 10, 25, 250]

derived = [8, 2, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 := mul(2, 11)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 = 22

source roles = [Wilson ORIGINAL prime factorial, 31, 11, 2, 22]

derived = [0, 22, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 := mul(22, 12)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 = 264

source roles = [Wilson ORIGINAL prime factorial, 31, 12, 22, 264]

derived = [8, 16, 是, 是]

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

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 = 208

source roles = [Wilson ORIGINAL prime factorial, 31, 13, 16, 208]

derived = [6, 22, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 := mul(22, 14)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 = 308

source roles = [Wilson ORIGINAL prime factorial, 31, 14, 22, 308]

derived = [9, 29, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 := mul(29, 15)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 = 435

source roles = [Wilson ORIGINAL prime factorial, 31, 15, 29, 435]

derived = [14, 1, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 := mul(1, 16)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 = 16

source roles = [Wilson ORIGINAL prime factorial, 31, 16, 1, 16]

derived = [0, 16, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 := mul(16, 17)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 = 272

source roles = [Wilson ORIGINAL prime factorial, 31, 17, 16, 272]

derived = [8, 24, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 := mul(24, 18)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 = 432

source roles = [Wilson ORIGINAL prime factorial, 31, 18, 24, 432]

derived = [13, 29, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 := mul(29, 19)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 = 551

source roles = [Wilson ORIGINAL prime factorial, 31, 19, 29, 551]

derived = [17, 24, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 := mul(24, 20)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 = 480

source roles = [Wilson ORIGINAL prime factorial, 31, 20, 24, 480]

derived = [15, 15, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 := mul(15, 21)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 = 315

source roles = [Wilson ORIGINAL prime factorial, 31, 21, 15, 315]

derived = [10, 5, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 := mul(5, 22)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 = 110

source roles = [Wilson ORIGINAL prime factorial, 31, 22, 5, 110]

derived = [3, 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, 31, 23, 17, 391]

derived = [12, 19, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 := mul(19, 24)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 = 456

source roles = [Wilson ORIGINAL prime factorial, 31, 24, 19, 456]

derived = [14, 22, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 := mul(22, 25)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 = 550

source roles = [Wilson ORIGINAL prime factorial, 31, 25, 22, 550]

derived = [17, 23, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 := mul(23, 26)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 = 598

source roles = [Wilson ORIGINAL prime factorial, 31, 26, 23, 598]

derived = [19, 9, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 := mul(9, 27)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 = 243

source roles = [Wilson ORIGINAL prime factorial, 31, 27, 9, 243]

derived = [7, 26, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 := mul(26, 28)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 = 728

source roles = [Wilson ORIGINAL prime factorial, 31, 28, 26, 728]

derived = [23, 15, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 := mul(15, 29)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 = 435

source roles = [Wilson ORIGINAL prime factorial, 31, 29, 15, 435]

derived = [14, 1, 是, 是]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 := mul(1, 30)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 = 30

source roles = [Wilson ORIGINAL prime factorial, 31, 30, 1, 30]

derived = [0, 30, 是, 是]

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

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

逐步反向核對

• S092 · premise_and_result_audit — 回查原始資料、操作數、完整範圍與本步前提；有限見證、條件式定理引用與形式證明分開。
• 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 product30，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 product29，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 product28，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 product27，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 product26，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 product25，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 product24，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 product23，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 product22，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 product21，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 product20，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 product19，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 product18，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 product17，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 product16，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 product15，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 product14，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 product13，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 product12，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 product11，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 product10，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 product9，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 product8，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 product7，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 product6，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 product5，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 product4，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 ORIGINAL prime factorial ORIGINAL running factorial product3，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 ORIGINAL prime factorial ORIGINAL running factorial product2，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 ORIGINAL prime factorial ORIGINAL running factorial product1，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 — 回查原始資料、操作數、完整範圍與本步前提；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

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