Question 27792: All Nonzero Prime Residues Pair with Inverses, with a Special p=2 Case

Original question and givens

Use prime modulus p=59. Give (p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence).

• G001 — Use prime modulus p=59. 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 = [59, 1]

derived = [1, 1]

source roles = [59, 2]

derived = [30, 1]

source roles = [59, 3]

derived = [20, 1]

source roles = [59, 4]

derived = [15, 1]

source roles = [59, 5]

derived = [12, 1]

source roles = [59, 6]

derived = [10, 1]

source roles = [59, 7]

derived = [17, 1]

source roles = [59, 8]

derived = [37, 1]

source roles = [59, 9]

derived = [46, 1]

source roles = [59, 10]

derived = [6, 1]

source roles = [59, 11]

derived = [43, 1]

source roles = [59, 12]

derived = [5, 1]

source roles = [59, 13]

derived = [50, 1]

source roles = [59, 14]

derived = [38, 1]

source roles = [59, 15]

derived = [4, 1]

source roles = [59, 16]

derived = [48, 1]

source roles = [59, 17]

derived = [7, 1]

source roles = [59, 18]

derived = [23, 1]

source roles = [59, 19]

derived = [28, 1]

source roles = [59, 20]

derived = [3, 1]

source roles = [59, 21]

derived = [45, 1]

source roles = [59, 22]

derived = [51, 1]

source roles = [59, 23]

derived = [18, 1]

source roles = [59, 24]

derived = [32, 1]

source roles = [59, 25]

derived = [26, 1]

source roles = [59, 26]

derived = [25, 1]

source roles = [59, 27]

derived = [35, 1]

source roles = [59, 28]

derived = [19, 1]

source roles = [59, 29]

derived = [57, 1]

source roles = [59, 30]

derived = [2, 1]

source roles = [59, 31]

derived = [40, 1]

source roles = [59, 32]

derived = [24, 1]

source roles = [59, 33]

derived = [34, 1]

source roles = [59, 34]

derived = [33, 1]

source roles = [59, 35]

derived = [27, 1]

source roles = [59, 36]

derived = [41, 1]

source roles = [59, 37]

derived = [8, 1]

source roles = [59, 38]

derived = [14, 1]

source roles = [59, 39]

derived = [56, 1]

source roles = [59, 40]

derived = [31, 1]

source roles = [59, 41]

derived = [36, 1]

source roles = [59, 42]

derived = [52, 1]

source roles = [59, 43]

derived = [11, 1]

source roles = [59, 44]

derived = [55, 1]

source roles = [59, 45]

derived = [21, 1]

source roles = [59, 46]

derived = [9, 1]

source roles = [59, 47]

derived = [54, 1]

source roles = [59, 48]

derived = [16, 1]

source roles = [59, 49]

derived = [53, 1]

source roles = [59, 50]

derived = [13, 1]

source roles = [59, 51]

derived = [22, 1]

source roles = [59, 52]

derived = [42, 1]

source roles = [59, 53]

derived = [49, 1]

source roles = [59, 54]

derived = [47, 1]

source roles = [59, 55]

derived = [44, 1]

source roles = [59, 56]

derived = [39, 1]

source roles = [59, 57]

derived = [29, 1]

source roles = [59, 58]

derived = [58, 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, 59, 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, 59, 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, 59, 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, 59, 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, 59, 5, 24, 120]

derived = [2, 2, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 := mul(2, 6)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 = 12

source roles = [Wilson ORIGINAL prime factorial, 59, 6, 2, 12]

derived = [0, 12, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 := mul(12, 7)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 = 84

source roles = [Wilson ORIGINAL prime factorial, 59, 7, 12, 84]

derived = [1, 25, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 := mul(25, 8)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 = 200

source roles = [Wilson ORIGINAL prime factorial, 59, 8, 25, 200]

derived = [3, 23, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 := mul(23, 9)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 = 207

source roles = [Wilson ORIGINAL prime factorial, 59, 9, 23, 207]

derived = [3, 30, Yes, Yes]

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

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 = 300

source roles = [Wilson ORIGINAL prime factorial, 59, 10, 30, 300]

derived = [5, 5, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 := mul(5, 11)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 = 55

source roles = [Wilson ORIGINAL prime factorial, 59, 11, 5, 55]

derived = [0, 55, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 := mul(55, 12)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 = 660

source roles = [Wilson ORIGINAL prime factorial, 59, 12, 55, 660]

derived = [11, 11, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 := mul(11, 13)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 = 143

source roles = [Wilson ORIGINAL prime factorial, 59, 13, 11, 143]

derived = [2, 25, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 := mul(25, 14)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 = 350

source roles = [Wilson ORIGINAL prime factorial, 59, 14, 25, 350]

derived = [5, 55, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 := mul(55, 15)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 = 825

source roles = [Wilson ORIGINAL prime factorial, 59, 15, 55, 825]

derived = [13, 58, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 := mul(58, 16)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 = 928

source roles = [Wilson ORIGINAL prime factorial, 59, 16, 58, 928]

derived = [15, 43, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 := mul(43, 17)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 = 731

source roles = [Wilson ORIGINAL prime factorial, 59, 17, 43, 731]

derived = [12, 23, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 := mul(23, 18)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 = 414

source roles = [Wilson ORIGINAL prime factorial, 59, 18, 23, 414]

derived = [7, 1, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 := mul(1, 19)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 = 19

source roles = [Wilson ORIGINAL prime factorial, 59, 19, 1, 19]

derived = [0, 19, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 := mul(19, 20)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 = 380

source roles = [Wilson ORIGINAL prime factorial, 59, 20, 19, 380]

derived = [6, 26, Yes, Yes]

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

Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 = 546

source roles = [Wilson ORIGINAL prime factorial, 59, 21, 26, 546]

derived = [9, 15, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 := mul(15, 22)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 = 330

source roles = [Wilson ORIGINAL prime factorial, 59, 22, 15, 330]

derived = [5, 35, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 := mul(35, 23)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 = 805

source roles = [Wilson ORIGINAL prime factorial, 59, 23, 35, 805]

derived = [13, 38, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 := mul(38, 24)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 = 912

source roles = [Wilson ORIGINAL prime factorial, 59, 24, 38, 912]

derived = [15, 27, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 := mul(27, 25)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 = 675

source roles = [Wilson ORIGINAL prime factorial, 59, 25, 27, 675]

derived = [11, 26, Yes, Yes]

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

Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 = 676

source roles = [Wilson ORIGINAL prime factorial, 59, 26, 26, 676]

derived = [11, 27, Yes, Yes]

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

Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 = 729

source roles = [Wilson ORIGINAL prime factorial, 59, 27, 27, 729]

derived = [12, 21, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 := mul(21, 28)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 = 588

source roles = [Wilson ORIGINAL prime factorial, 59, 28, 21, 588]

derived = [9, 57, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 := mul(57, 29)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 = 1653

source roles = [Wilson ORIGINAL prime factorial, 59, 29, 57, 1653]

derived = [28, 1, Yes, Yes]

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, 59, 30, 1, 30]

derived = [0, 30, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 := mul(30, 31)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 = 930

source roles = [Wilson ORIGINAL prime factorial, 59, 31, 30, 930]

derived = [15, 45, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 := mul(45, 32)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 = 1440

source roles = [Wilson ORIGINAL prime factorial, 59, 32, 45, 1440]

derived = [24, 24, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 := mul(24, 33)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 = 792

source roles = [Wilson ORIGINAL prime factorial, 59, 33, 24, 792]

derived = [13, 25, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 := mul(25, 34)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 = 850

source roles = [Wilson ORIGINAL prime factorial, 59, 34, 25, 850]

derived = [14, 24, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 := mul(24, 35)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 = 840

source roles = [Wilson ORIGINAL prime factorial, 59, 35, 24, 840]

derived = [14, 14, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 := mul(14, 36)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 = 504

source roles = [Wilson ORIGINAL prime factorial, 59, 36, 14, 504]

derived = [8, 32, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 := mul(32, 37)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 = 1184

source roles = [Wilson ORIGINAL prime factorial, 59, 37, 32, 1184]

derived = [20, 4, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 := mul(4, 38)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 = 152

source roles = [Wilson ORIGINAL prime factorial, 59, 38, 4, 152]

derived = [2, 34, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 := mul(34, 39)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 = 1326

source roles = [Wilson ORIGINAL prime factorial, 59, 39, 34, 1326]

derived = [22, 28, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 := mul(28, 40)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 = 1120

source roles = [Wilson ORIGINAL prime factorial, 59, 40, 28, 1120]

derived = [18, 58, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product41 := mul(58, 41)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product41 = 2378

source roles = [Wilson ORIGINAL prime factorial, 59, 41, 58, 2378]

derived = [40, 18, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product42 := mul(18, 42)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product42 = 756

source roles = [Wilson ORIGINAL prime factorial, 59, 42, 18, 756]

derived = [12, 48, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product43 := mul(48, 43)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product43 = 2064

source roles = [Wilson ORIGINAL prime factorial, 59, 43, 48, 2064]

derived = [34, 58, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product44 := mul(58, 44)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product44 = 2552

source roles = [Wilson ORIGINAL prime factorial, 59, 44, 58, 2552]

derived = [43, 15, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product45 := mul(15, 45)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product45 = 675

source roles = [Wilson ORIGINAL prime factorial, 59, 45, 15, 675]

derived = [11, 26, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product46 := mul(26, 46)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product46 = 1196

source roles = [Wilson ORIGINAL prime factorial, 59, 46, 26, 1196]

derived = [20, 16, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product47 := mul(16, 47)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product47 = 752

source roles = [Wilson ORIGINAL prime factorial, 59, 47, 16, 752]

derived = [12, 44, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product48 := mul(44, 48)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product48 = 2112

source roles = [Wilson ORIGINAL prime factorial, 59, 48, 44, 2112]

derived = [35, 47, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product49 := mul(47, 49)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product49 = 2303

source roles = [Wilson ORIGINAL prime factorial, 59, 49, 47, 2303]

derived = [39, 2, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product50 := mul(2, 50)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product50 = 100

source roles = [Wilson ORIGINAL prime factorial, 59, 50, 2, 100]

derived = [1, 41, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product51 := mul(41, 51)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product51 = 2091

source roles = [Wilson ORIGINAL prime factorial, 59, 51, 41, 2091]

derived = [35, 26, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product52 := mul(26, 52)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product52 = 1352

source roles = [Wilson ORIGINAL prime factorial, 59, 52, 26, 1352]

derived = [22, 54, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product53 := mul(54, 53)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product53 = 2862

source roles = [Wilson ORIGINAL prime factorial, 59, 53, 54, 2862]

derived = [48, 30, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product54 := mul(30, 54)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product54 = 1620

source roles = [Wilson ORIGINAL prime factorial, 59, 54, 30, 1620]

derived = [27, 27, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product55 := mul(27, 55)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product55 = 1485

source roles = [Wilson ORIGINAL prime factorial, 59, 55, 27, 1485]

derived = [25, 10, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product56 := mul(10, 56)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product56 = 560

source roles = [Wilson ORIGINAL prime factorial, 59, 56, 10, 560]

derived = [9, 29, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product57 := mul(29, 57)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product57 = 1653

source roles = [Wilson ORIGINAL prime factorial, 59, 57, 29, 1653]

derived = [28, 1, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product58 := mul(1, 58)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product58 = 58

source roles = [Wilson ORIGINAL prime factorial, 59, 58, 1, 58]

derived = [0, 58, Yes, Yes]

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

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

Reverse checks

• S176 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S175 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S174 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product58 with exact rational arithmetic, not rounded decimals.
• S173 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S172 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product57 with exact rational arithmetic, not rounded decimals.
• S171 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S170 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product56 with exact rational arithmetic, not rounded decimals.
• S169 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S168 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product55 with exact rational arithmetic, not rounded decimals.
• S167 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S166 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product54 with exact rational arithmetic, not rounded decimals.
• S165 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S164 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product53 with exact rational arithmetic, not rounded decimals.
• S163 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S162 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product52 with exact rational arithmetic, not rounded decimals.
• S161 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S160 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product51 with exact rational arithmetic, not rounded decimals.
• S159 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S158 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product50 with exact rational arithmetic, not rounded decimals.
• 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 product49 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 product48 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 product47 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 product46 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 product45 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 product44 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 product43 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 product42 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 product41 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 product40 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 product39 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 product38 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 product37 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 product36 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 product35 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 product34 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 product33 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 product32 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 product31 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 product30 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 product29 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 product28 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 product27 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 product26 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 product25 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 product24 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 product23 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 product22 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 product21 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 product20 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 product19 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 product18 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 product17 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 product16 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 product15 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 product14 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 product13 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 product12 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 product11 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 product10 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 product9 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 product8 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 product7 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 product6 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 product5 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 product4 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 product3 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 product2 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 product1 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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• 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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• 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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• 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

(59, [[1, 1], [2, 30], [3, 20], [4, 15], [5, 12], [6, 10], [7, 17], [8, 37], [9, 46], [10, 6], [11, 43], [12, 5], [13, 50], [14, 38], [15, 4], [16, 48], [17, 7], [18, 23], [19, 28], [20, 3], [21, 45], [22, 51], [23, 18], [24, 32], [25, 26], [26, 25], [27, 35], [28, 19], [29, 57], [30, 2], [31, 40], [32, 24], [33, 34], [34, 33], [35, 27], [36, 41], [37, 8], [38, 14], [39, 56], [40, 31], [41, 36], [42, 52], [43, 11], [44, 55], [45, 21], [46, 9], [47, 54], [48, 16], [49, 53], [50, 13], [51, 22], [52, 42], [53, 49], [54, 47], [55, 44], [56, 39], [57, 29], [58, 58]], [1, 58], [[2, 30, 1], [3, 20, 1], [4, 15, 1], [5, 12, 1], [6, 10, 1], [7, 17, 1], [8, 37, 1], [9, 46, 1], [11, 43, 1], [13, 50, 1], [14, 38, 1], [16, 48, 1], [18, 23, 1], [19, 28, 1], [21, 45, 1], [22, 51, 1], [24, 32, 1], [25, 26, 1], [27, 35, 1], [29, 57, 1], [31, 40, 1], [33, 34, 1], [36, 41, 1], [39, 56, 1], [42, 52, 1], [44, 55, 1], [47, 54, 1], [49, 53, 1]], self_inverse_residues_1_and_p_minus_1, 58, 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.