Question 27791: All Nonzero Prime Residues Pair with Inverses, with a Special p=2 Case

Original question and givens

Use prime modulus p=53. Give (p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence).

• G001 — Use prime modulus p=53. Give (p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-wilson-theorem

Step-by-step derivation

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 = [53, 1]

derived = [1, 1]

source roles = [53, 2]

derived = [27, 1]

source roles = [53, 3]

derived = [18, 1]

source roles = [53, 4]

derived = [40, 1]

source roles = [53, 5]

derived = [32, 1]

source roles = [53, 6]

derived = [9, 1]

source roles = [53, 7]

derived = [38, 1]

source roles = [53, 8]

derived = [20, 1]

source roles = [53, 9]

derived = [6, 1]

source roles = [53, 10]

derived = [16, 1]

source roles = [53, 11]

derived = [29, 1]

source roles = [53, 12]

derived = [31, 1]

source roles = [53, 13]

derived = [49, 1]

source roles = [53, 14]

derived = [19, 1]

source roles = [53, 15]

derived = [46, 1]

source roles = [53, 16]

derived = [10, 1]

source roles = [53, 17]

derived = [25, 1]

source roles = [53, 18]

derived = [3, 1]

source roles = [53, 19]

derived = [14, 1]

source roles = [53, 20]

derived = [8, 1]

source roles = [53, 21]

derived = [48, 1]

source roles = [53, 22]

derived = [41, 1]

source roles = [53, 23]

derived = [30, 1]

source roles = [53, 24]

derived = [42, 1]

source roles = [53, 25]

derived = [17, 1]

source roles = [53, 26]

derived = [51, 1]

source roles = [53, 27]

derived = [2, 1]

source roles = [53, 28]

derived = [36, 1]

source roles = [53, 29]

derived = [11, 1]

source roles = [53, 30]

derived = [23, 1]

source roles = [53, 31]

derived = [12, 1]

source roles = [53, 32]

derived = [5, 1]

source roles = [53, 33]

derived = [45, 1]

source roles = [53, 34]

derived = [39, 1]

source roles = [53, 35]

derived = [50, 1]

source roles = [53, 36]

derived = [28, 1]

source roles = [53, 37]

derived = [43, 1]

source roles = [53, 38]

derived = [7, 1]

source roles = [53, 39]

derived = [34, 1]

source roles = [53, 40]

derived = [4, 1]

source roles = [53, 41]

derived = [22, 1]

source roles = [53, 42]

derived = [24, 1]

source roles = [53, 43]

derived = [37, 1]

source roles = [53, 44]

derived = [47, 1]

source roles = [53, 45]

derived = [33, 1]

source roles = [53, 46]

derived = [15, 1]

source roles = [53, 47]

derived = [44, 1]

source roles = [53, 48]

derived = [21, 1]

source roles = [53, 49]

derived = [13, 1]

source roles = [53, 50]

derived = [35, 1]

source roles = [53, 51]

derived = [26, 1]

source roles = [53, 52]

derived = [52, 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, 53, 1, 1, 1]

derived = [0, 1, Yes, Yes]

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, 53, 2, 1, 2]

derived = [0, 2, Yes, Yes]

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, 53, 3, 2, 6]

derived = [0, 6, Yes, Yes]

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, 53, 4, 6, 24]

derived = [0, 24, Yes, Yes]

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, 53, 5, 24, 120]

derived = [2, 14, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 := mul(14, 6)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 = 84

source roles = [Wilson ORIGINAL prime factorial, 53, 6, 14, 84]

derived = [1, 31, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 := mul(31, 7)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 = 217

source roles = [Wilson ORIGINAL prime factorial, 53, 7, 31, 217]

derived = [4, 5, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 := mul(5, 8)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 = 40

source roles = [Wilson ORIGINAL prime factorial, 53, 8, 5, 40]

derived = [0, 40, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 := mul(40, 9)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 = 360

source roles = [Wilson ORIGINAL prime factorial, 53, 9, 40, 360]

derived = [6, 42, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 := mul(42, 10)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 = 420

source roles = [Wilson ORIGINAL prime factorial, 53, 10, 42, 420]

derived = [7, 49, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 := mul(49, 11)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 = 539

source roles = [Wilson ORIGINAL prime factorial, 53, 11, 49, 539]

derived = [10, 9, Yes, Yes]

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

Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 = 108

source roles = [Wilson ORIGINAL prime factorial, 53, 12, 9, 108]

derived = [2, 2, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 := mul(2, 13)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 = 26

source roles = [Wilson ORIGINAL prime factorial, 53, 13, 2, 26]

derived = [0, 26, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 := mul(26, 14)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 = 364

source roles = [Wilson ORIGINAL prime factorial, 53, 14, 26, 364]

derived = [6, 46, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 := mul(46, 15)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 = 690

source roles = [Wilson ORIGINAL prime factorial, 53, 15, 46, 690]

derived = [13, 1, Yes, Yes]

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, 53, 16, 1, 16]

derived = [0, 16, Yes, Yes]

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, 53, 17, 16, 272]

derived = [5, 7, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 := mul(7, 18)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 = 126

source roles = [Wilson ORIGINAL prime factorial, 53, 18, 7, 126]

derived = [2, 20, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 := mul(20, 19)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 = 380

source roles = [Wilson ORIGINAL prime factorial, 53, 19, 20, 380]

derived = [7, 9, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 := mul(9, 20)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 = 180

source roles = [Wilson ORIGINAL prime factorial, 53, 20, 9, 180]

derived = [3, 21, Yes, Yes]

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

Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 = 441

source roles = [Wilson ORIGINAL prime factorial, 53, 21, 21, 441]

derived = [8, 17, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 := mul(17, 22)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 = 374

source roles = [Wilson ORIGINAL prime factorial, 53, 22, 17, 374]

derived = [7, 3, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 := mul(3, 23)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 = 69

source roles = [Wilson ORIGINAL prime factorial, 53, 23, 3, 69]

derived = [1, 16, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 := mul(16, 24)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 = 384

source roles = [Wilson ORIGINAL prime factorial, 53, 24, 16, 384]

derived = [7, 13, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 := mul(13, 25)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 = 325

source roles = [Wilson ORIGINAL prime factorial, 53, 25, 13, 325]

derived = [6, 7, Yes, Yes]

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, 53, 26, 7, 182]

derived = [3, 23, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 := mul(23, 27)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 = 621

source roles = [Wilson ORIGINAL prime factorial, 53, 27, 23, 621]

derived = [11, 38, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 := mul(38, 28)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 = 1064

source roles = [Wilson ORIGINAL prime factorial, 53, 28, 38, 1064]

derived = [20, 4, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 := mul(4, 29)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 = 116

source roles = [Wilson ORIGINAL prime factorial, 53, 29, 4, 116]

derived = [2, 10, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 := mul(10, 30)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 = 300

source roles = [Wilson ORIGINAL prime factorial, 53, 30, 10, 300]

derived = [5, 35, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 := mul(35, 31)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 = 1085

source roles = [Wilson ORIGINAL prime factorial, 53, 31, 35, 1085]

derived = [20, 25, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 := mul(25, 32)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 = 800

source roles = [Wilson ORIGINAL prime factorial, 53, 32, 25, 800]

derived = [15, 5, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 := mul(5, 33)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 = 165

source roles = [Wilson ORIGINAL prime factorial, 53, 33, 5, 165]

derived = [3, 6, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 := mul(6, 34)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 = 204

source roles = [Wilson ORIGINAL prime factorial, 53, 34, 6, 204]

derived = [3, 45, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 := mul(45, 35)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 = 1575

source roles = [Wilson ORIGINAL prime factorial, 53, 35, 45, 1575]

derived = [29, 38, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 := mul(38, 36)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 = 1368

source roles = [Wilson ORIGINAL prime factorial, 53, 36, 38, 1368]

derived = [25, 43, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 := mul(43, 37)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 = 1591

source roles = [Wilson ORIGINAL prime factorial, 53, 37, 43, 1591]

derived = [30, 1, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 := mul(1, 38)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 = 38

source roles = [Wilson ORIGINAL prime factorial, 53, 38, 1, 38]

derived = [0, 38, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 := mul(38, 39)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 = 1482

source roles = [Wilson ORIGINAL prime factorial, 53, 39, 38, 1482]

derived = [27, 51, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 := mul(51, 40)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 = 2040

source roles = [Wilson ORIGINAL prime factorial, 53, 40, 51, 2040]

derived = [38, 26, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product41 := mul(26, 41)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product41 = 1066

source roles = [Wilson ORIGINAL prime factorial, 53, 41, 26, 1066]

derived = [20, 6, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product42 := mul(6, 42)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product42 = 252

source roles = [Wilson ORIGINAL prime factorial, 53, 42, 6, 252]

derived = [4, 40, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product43 := mul(40, 43)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product43 = 1720

source roles = [Wilson ORIGINAL prime factorial, 53, 43, 40, 1720]

derived = [32, 24, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product44 := mul(24, 44)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product44 = 1056

source roles = [Wilson ORIGINAL prime factorial, 53, 44, 24, 1056]

derived = [19, 49, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product45 := mul(49, 45)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product45 = 2205

source roles = [Wilson ORIGINAL prime factorial, 53, 45, 49, 2205]

derived = [41, 32, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product46 := mul(32, 46)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product46 = 1472

source roles = [Wilson ORIGINAL prime factorial, 53, 46, 32, 1472]

derived = [27, 41, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product47 := mul(41, 47)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product47 = 1927

source roles = [Wilson ORIGINAL prime factorial, 53, 47, 41, 1927]

derived = [36, 19, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product48 := mul(19, 48)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product48 = 912

source roles = [Wilson ORIGINAL prime factorial, 53, 48, 19, 912]

derived = [17, 11, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product49 := mul(11, 49)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product49 = 539

source roles = [Wilson ORIGINAL prime factorial, 53, 49, 11, 539]

derived = [10, 9, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product50 := mul(9, 50)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product50 = 450

source roles = [Wilson ORIGINAL prime factorial, 53, 50, 9, 450]

derived = [8, 26, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product51 := mul(26, 51)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product51 = 1326

source roles = [Wilson ORIGINAL prime factorial, 53, 51, 26, 1326]

derived = [25, 1, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product52 := mul(1, 52)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product52 = 52

source roles = [Wilson ORIGINAL prime factorial, 53, 52, 1, 52]

derived = [0, 52, Yes, Yes]

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

derived = [[1, 52], [[2, 27, 1], [3, 18, 1], [4, 40, 1], [5, 32, 1], [6, 9, 1], [7, 38, 1], [8, 20, 1], [10, 16, 1], [11, 29, 1], [12, 31, 1], [13, 49, 1], [14, 19, 1], [15, 46, 1], [17, 25, 1], [21, 48, 1], [22, 41, 1], [23, 30, 1], [24, 42, 1], [26, 51, 1], [28, 36, 1], [33, 45, 1], [34, 39, 1], [35, 50, 1], [37, 43, 1], [44, 47, 1]], 52, Yes]

Reverse checks

• S158 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S157 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S156 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product52 with exact rational arithmetic, not rounded decimals.
• S155 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S154 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product51 with exact rational arithmetic, not rounded decimals.
• S153 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S152 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product50 with exact rational arithmetic, not rounded decimals.
• S151 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S150 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product49 with exact rational arithmetic, not rounded decimals.
• S149 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S148 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product48 with exact rational arithmetic, not rounded decimals.
• S147 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S146 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product47 with exact rational arithmetic, not rounded decimals.
• S145 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S144 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product46 with exact rational arithmetic, not rounded decimals.
• S143 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S142 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product45 with exact rational arithmetic, not rounded decimals.
• S141 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S140 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product44 with exact rational arithmetic, not rounded decimals.
• S139 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S138 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product43 with exact rational arithmetic, not rounded decimals.
• S137 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S136 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product42 with exact rational arithmetic, not rounded decimals.
• S135 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S134 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product41 with exact rational arithmetic, not rounded decimals.
• S133 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S132 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 with exact rational arithmetic, not rounded decimals.
• S131 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S130 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 with exact rational arithmetic, not rounded decimals.
• S129 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S128 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 with exact rational arithmetic, not rounded decimals.
• S127 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S126 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 with exact rational arithmetic, not rounded decimals.
• S125 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S124 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 with exact rational arithmetic, not rounded decimals.
• S123 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S122 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 with exact rational arithmetic, not rounded decimals.
• S121 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S120 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S118 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S116 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 with exact rational arithmetic, not rounded decimals.
• S115 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S114 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S112 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 with exact rational arithmetic, not rounded decimals.
• S111 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S110 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 with exact rational arithmetic, not rounded decimals.
• S109 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S108 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 with exact rational arithmetic, not rounded decimals.
• S107 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S106 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 with exact rational arithmetic, not rounded decimals.
• S105 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S104 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 with exact rational arithmetic, not rounded decimals.
• S103 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S102 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 with exact rational arithmetic, not rounded decimals.
• S101 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S100 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 with exact rational arithmetic, not rounded decimals.
• S099 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S098 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S096 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 with exact rational arithmetic, not rounded decimals.
• S095 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S094 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 with exact rational arithmetic, not rounded decimals.
• S093 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S092 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 with exact rational arithmetic, not rounded decimals.
• S091 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S090 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 with exact rational arithmetic, not rounded decimals.
• S089 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S088 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 with exact rational arithmetic, not rounded decimals.
• S087 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S086 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 with exact rational arithmetic, not rounded decimals.
• S085 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S084 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S082 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S080 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 with exact rational arithmetic, not rounded decimals.
• S079 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S078 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S076 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S074 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S072 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S070 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S068 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S066 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 with exact rational arithmetic, not rounded decimals.
• S065 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S064 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S062 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product5 with exact rational arithmetic, not rounded decimals.
• S061 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S060 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product4 with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S058 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product3 with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S056 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product2 with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S054 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product1 with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S052 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S051 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S050 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S049 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S048 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S047 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S046 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S045 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S044 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S043 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S042 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S041 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S040 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S039 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S038 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S037 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S036 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S035 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S034 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S033 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S032 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S031 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S030 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S029 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S028 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S027 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S026 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S025 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S024 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S023 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S022 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S021 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S020 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S019 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S018 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S017 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S016 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S015 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S014 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S013 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S012 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S011 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S010 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S009 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S008 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S007 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S006 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S005 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S004 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S003 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S002 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S001 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.

Result

(53, [[1, 1], [2, 27], [3, 18], [4, 40], [5, 32], [6, 9], [7, 38], [8, 20], [9, 6], [10, 16], [11, 29], [12, 31], [13, 49], [14, 19], [15, 46], [16, 10], [17, 25], [18, 3], [19, 14], [20, 8], [21, 48], [22, 41], [23, 30], [24, 42], [25, 17], [26, 51], [27, 2], [28, 36], [29, 11], [30, 23], [31, 12], [32, 5], [33, 45], [34, 39], [35, 50], [36, 28], [37, 43], [38, 7], [39, 34], [40, 4], [41, 22], [42, 24], [43, 37], [44, 47], [45, 33], [46, 15], [47, 44], [48, 21], [49, 13], [50, 35], [51, 26], [52, 52]], [1, 52], [[2, 27, 1], [3, 18, 1], [4, 40, 1], [5, 32, 1], [6, 9, 1], [7, 38, 1], [8, 20, 1], [10, 16, 1], [11, 29, 1], [12, 31, 1], [13, 49, 1], [14, 19, 1], [15, 46, 1], [17, 25, 1], [21, 48, 1], [22, 41, 1], [23, 30, 1], [24, 42, 1], [26, 51, 1], [28, 36, 1], [33, 45, 1], [34, 39, 1], [35, 50, 1], [37, 43, 1], [44, 47, 1]], self_inverse_residues_1_and_p_minus_1, 52, Yes)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.