Question 27794: All Nonzero Prime Residues Pair with Inverses, with a Special p=2 Case

Original question and givens

Use prime modulus p=67. Give (p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence).

• G001 — Use prime modulus p=67. 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 = [67, 1]

derived = [1, 1]

source roles = [67, 2]

derived = [34, 1]

source roles = [67, 3]

derived = [45, 1]

source roles = [67, 4]

derived = [17, 1]

source roles = [67, 5]

derived = [27, 1]

source roles = [67, 6]

derived = [56, 1]

source roles = [67, 7]

derived = [48, 1]

source roles = [67, 8]

derived = [42, 1]

source roles = [67, 9]

derived = [15, 1]

source roles = [67, 10]

derived = [47, 1]

source roles = [67, 11]

derived = [61, 1]

source roles = [67, 12]

derived = [28, 1]

source roles = [67, 13]

derived = [31, 1]

source roles = [67, 14]

derived = [24, 1]

source roles = [67, 15]

derived = [9, 1]

source roles = [67, 16]

derived = [21, 1]

source roles = [67, 17]

derived = [4, 1]

source roles = [67, 18]

derived = [41, 1]

source roles = [67, 19]

derived = [60, 1]

source roles = [67, 20]

derived = [57, 1]

source roles = [67, 21]

derived = [16, 1]

source roles = [67, 22]

derived = [64, 1]

source roles = [67, 23]

derived = [35, 1]

source roles = [67, 24]

derived = [14, 1]

source roles = [67, 25]

derived = [59, 1]

source roles = [67, 26]

derived = [49, 1]

source roles = [67, 27]

derived = [5, 1]

source roles = [67, 28]

derived = [12, 1]

source roles = [67, 29]

derived = [37, 1]

source roles = [67, 30]

derived = [38, 1]

source roles = [67, 31]

derived = [13, 1]

source roles = [67, 32]

derived = [44, 1]

source roles = [67, 33]

derived = [65, 1]

source roles = [67, 34]

derived = [2, 1]

source roles = [67, 35]

derived = [23, 1]

source roles = [67, 36]

derived = [54, 1]

source roles = [67, 37]

derived = [29, 1]

source roles = [67, 38]

derived = [30, 1]

source roles = [67, 39]

derived = [55, 1]

source roles = [67, 40]

derived = [62, 1]

source roles = [67, 41]

derived = [18, 1]

source roles = [67, 42]

derived = [8, 1]

source roles = [67, 43]

derived = [53, 1]

source roles = [67, 44]

derived = [32, 1]

source roles = [67, 45]

derived = [3, 1]

source roles = [67, 46]

derived = [51, 1]

source roles = [67, 47]

derived = [10, 1]

source roles = [67, 48]

derived = [7, 1]

source roles = [67, 49]

derived = [26, 1]

source roles = [67, 50]

derived = [63, 1]

source roles = [67, 51]

derived = [46, 1]

source roles = [67, 52]

derived = [58, 1]

source roles = [67, 53]

derived = [43, 1]

source roles = [67, 54]

derived = [36, 1]

source roles = [67, 55]

derived = [39, 1]

source roles = [67, 56]

derived = [6, 1]

source roles = [67, 57]

derived = [20, 1]

source roles = [67, 58]

derived = [52, 1]

source roles = [67, 59]

derived = [25, 1]

source roles = [67, 60]

derived = [19, 1]

source roles = [67, 61]

derived = [11, 1]

source roles = [67, 62]

derived = [40, 1]

source roles = [67, 63]

derived = [50, 1]

source roles = [67, 64]

derived = [22, 1]

source roles = [67, 65]

derived = [33, 1]

source roles = [67, 66]

derived = [66, 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, 67, 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, 67, 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, 67, 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, 67, 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, 67, 5, 24, 120]

derived = [1, 53, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 := mul(53, 6)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 = 318

source roles = [Wilson ORIGINAL prime factorial, 67, 6, 53, 318]

derived = [4, 50, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 := mul(50, 7)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 = 350

source roles = [Wilson ORIGINAL prime factorial, 67, 7, 50, 350]

derived = [5, 15, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 := mul(15, 8)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 = 120

source roles = [Wilson ORIGINAL prime factorial, 67, 8, 15, 120]

derived = [1, 53, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 := mul(53, 9)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 = 477

source roles = [Wilson ORIGINAL prime factorial, 67, 9, 53, 477]

derived = [7, 8, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 := mul(8, 10)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 = 80

source roles = [Wilson ORIGINAL prime factorial, 67, 10, 8, 80]

derived = [1, 13, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 := mul(13, 11)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 = 143

source roles = [Wilson ORIGINAL prime factorial, 67, 11, 13, 143]

derived = [2, 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, 67, 12, 9, 108]

derived = [1, 41, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 := mul(41, 13)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 = 533

source roles = [Wilson ORIGINAL prime factorial, 67, 13, 41, 533]

derived = [7, 64, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 := mul(64, 14)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 = 896

source roles = [Wilson ORIGINAL prime factorial, 67, 14, 64, 896]

derived = [13, 25, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 := mul(25, 15)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 = 375

source roles = [Wilson ORIGINAL prime factorial, 67, 15, 25, 375]

derived = [5, 40, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 := mul(40, 16)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 = 640

source roles = [Wilson ORIGINAL prime factorial, 67, 16, 40, 640]

derived = [9, 37, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 := mul(37, 17)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 = 629

source roles = [Wilson ORIGINAL prime factorial, 67, 17, 37, 629]

derived = [9, 26, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 := mul(26, 18)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 = 468

source roles = [Wilson ORIGINAL prime factorial, 67, 18, 26, 468]

derived = [6, 66, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 := mul(66, 19)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 = 1254

source roles = [Wilson ORIGINAL prime factorial, 67, 19, 66, 1254]

derived = [18, 48, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 := mul(48, 20)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 = 960

source roles = [Wilson ORIGINAL prime factorial, 67, 20, 48, 960]

derived = [14, 22, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 := mul(22, 21)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 = 462

source roles = [Wilson ORIGINAL prime factorial, 67, 21, 22, 462]

derived = [6, 60, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 := mul(60, 22)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 = 1320

source roles = [Wilson ORIGINAL prime factorial, 67, 22, 60, 1320]

derived = [19, 47, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 := mul(47, 23)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 = 1081

source roles = [Wilson ORIGINAL prime factorial, 67, 23, 47, 1081]

derived = [16, 9, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 := mul(9, 24)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 = 216

source roles = [Wilson ORIGINAL prime factorial, 67, 24, 9, 216]

derived = [3, 15, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 := mul(15, 25)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 = 375

source roles = [Wilson ORIGINAL prime factorial, 67, 25, 15, 375]

derived = [5, 40, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 := mul(40, 26)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 = 1040

source roles = [Wilson ORIGINAL prime factorial, 67, 26, 40, 1040]

derived = [15, 35, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 := mul(35, 27)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 = 945

source roles = [Wilson ORIGINAL prime factorial, 67, 27, 35, 945]

derived = [14, 7, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 := mul(7, 28)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 = 196

source roles = [Wilson ORIGINAL prime factorial, 67, 28, 7, 196]

derived = [2, 62, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 := mul(62, 29)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 = 1798

source roles = [Wilson ORIGINAL prime factorial, 67, 29, 62, 1798]

derived = [26, 56, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 := mul(56, 30)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 = 1680

source roles = [Wilson ORIGINAL prime factorial, 67, 30, 56, 1680]

derived = [25, 5, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 := mul(5, 31)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 = 155

source roles = [Wilson ORIGINAL prime factorial, 67, 31, 5, 155]

derived = [2, 21, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 := mul(21, 32)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 = 672

source roles = [Wilson ORIGINAL prime factorial, 67, 32, 21, 672]

derived = [10, 2, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 := mul(2, 33)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 = 66

source roles = [Wilson ORIGINAL prime factorial, 67, 33, 2, 66]

derived = [0, 66, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 := mul(66, 34)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 = 2244

source roles = [Wilson ORIGINAL prime factorial, 67, 34, 66, 2244]

derived = [33, 33, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 := mul(33, 35)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 = 1155

source roles = [Wilson ORIGINAL prime factorial, 67, 35, 33, 1155]

derived = [17, 16, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 := mul(16, 36)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 = 576

source roles = [Wilson ORIGINAL prime factorial, 67, 36, 16, 576]

derived = [8, 40, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 := mul(40, 37)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 = 1480

source roles = [Wilson ORIGINAL prime factorial, 67, 37, 40, 1480]

derived = [22, 6, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 := mul(6, 38)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 = 228

source roles = [Wilson ORIGINAL prime factorial, 67, 38, 6, 228]

derived = [3, 27, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 := mul(27, 39)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 = 1053

source roles = [Wilson ORIGINAL prime factorial, 67, 39, 27, 1053]

derived = [15, 48, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 := mul(48, 40)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 = 1920

source roles = [Wilson ORIGINAL prime factorial, 67, 40, 48, 1920]

derived = [28, 44, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product41 := mul(44, 41)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product41 = 1804

source roles = [Wilson ORIGINAL prime factorial, 67, 41, 44, 1804]

derived = [26, 62, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product42 := mul(62, 42)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product42 = 2604

source roles = [Wilson ORIGINAL prime factorial, 67, 42, 62, 2604]

derived = [38, 58, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product43 := mul(58, 43)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product43 = 2494

source roles = [Wilson ORIGINAL prime factorial, 67, 43, 58, 2494]

derived = [37, 15, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product44 := mul(15, 44)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product44 = 660

source roles = [Wilson ORIGINAL prime factorial, 67, 44, 15, 660]

derived = [9, 57, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product45 := mul(57, 45)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product45 = 2565

source roles = [Wilson ORIGINAL prime factorial, 67, 45, 57, 2565]

derived = [38, 19, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product46 := mul(19, 46)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product46 = 874

source roles = [Wilson ORIGINAL prime factorial, 67, 46, 19, 874]

derived = [13, 3, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product47 := mul(3, 47)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product47 = 141

source roles = [Wilson ORIGINAL prime factorial, 67, 47, 3, 141]

derived = [2, 7, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product48 := mul(7, 48)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product48 = 336

source roles = [Wilson ORIGINAL prime factorial, 67, 48, 7, 336]

derived = [5, 1, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product49 := mul(1, 49)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product49 = 49

source roles = [Wilson ORIGINAL prime factorial, 67, 49, 1, 49]

derived = [0, 49, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product50 := mul(49, 50)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product50 = 2450

source roles = [Wilson ORIGINAL prime factorial, 67, 50, 49, 2450]

derived = [36, 38, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product51 := mul(38, 51)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product51 = 1938

source roles = [Wilson ORIGINAL prime factorial, 67, 51, 38, 1938]

derived = [28, 62, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product52 := mul(62, 52)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product52 = 3224

source roles = [Wilson ORIGINAL prime factorial, 67, 52, 62, 3224]

derived = [48, 8, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product53 := mul(8, 53)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product53 = 424

source roles = [Wilson ORIGINAL prime factorial, 67, 53, 8, 424]

derived = [6, 22, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product54 := mul(22, 54)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product54 = 1188

source roles = [Wilson ORIGINAL prime factorial, 67, 54, 22, 1188]

derived = [17, 49, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product55 := mul(49, 55)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product55 = 2695

source roles = [Wilson ORIGINAL prime factorial, 67, 55, 49, 2695]

derived = [40, 15, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product56 := mul(15, 56)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product56 = 840

source roles = [Wilson ORIGINAL prime factorial, 67, 56, 15, 840]

derived = [12, 36, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product57 := mul(36, 57)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product57 = 2052

source roles = [Wilson ORIGINAL prime factorial, 67, 57, 36, 2052]

derived = [30, 42, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product58 := mul(42, 58)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product58 = 2436

source roles = [Wilson ORIGINAL prime factorial, 67, 58, 42, 2436]

derived = [36, 24, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product59 := mul(24, 59)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product59 = 1416

source roles = [Wilson ORIGINAL prime factorial, 67, 59, 24, 1416]

derived = [21, 9, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product60 := mul(9, 60)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product60 = 540

source roles = [Wilson ORIGINAL prime factorial, 67, 60, 9, 540]

derived = [8, 4, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product61 := mul(4, 61)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product61 = 244

source roles = [Wilson ORIGINAL prime factorial, 67, 61, 4, 244]

derived = [3, 43, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product62 := mul(43, 62)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product62 = 2666

source roles = [Wilson ORIGINAL prime factorial, 67, 62, 43, 2666]

derived = [39, 53, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product63 := mul(53, 63)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product63 = 3339

source roles = [Wilson ORIGINAL prime factorial, 67, 63, 53, 3339]

derived = [49, 56, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product64 := mul(56, 64)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product64 = 3584

source roles = [Wilson ORIGINAL prime factorial, 67, 64, 56, 3584]

derived = [53, 33, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product65 := mul(33, 65)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product65 = 2145

source roles = [Wilson ORIGINAL prime factorial, 67, 65, 33, 2145]

derived = [32, 1, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product66 := mul(1, 66)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product66 = 66

source roles = [Wilson ORIGINAL prime factorial, 67, 66, 1, 66]

derived = [0, 66, Yes, Yes]

source roles = [67, [[1, 1], [2, 34], [3, 45], [4, 17], [5, 27], [6, 56], [7, 48], [8, 42], [9, 15], [10, 47], [11, 61], [12, 28], [13, 31], [14, 24], [15, 9], [16, 21], [17, 4], [18, 41], [19, 60], [20, 57], [21, 16], [22, 64], [23, 35], [24, 14], [25, 59], [26, 49], [27, 5], [28, 12], [29, 37], [30, 38], [31, 13], [32, 44], [33, 65], [34, 2], [35, 23], [36, 54], [37, 29], [38, 30], [39, 55], [40, 62], [41, 18], [42, 8], [43, 53], [44, 32], [45, 3], [46, 51], [47, 10], [48, 7], [49, 26], [50, 63], [51, 46], [52, 58], [53, 43], [54, 36], [55, 39], [56, 6], [57, 20], [58, 52], [59, 25], [60, 19], [61, 11], [62, 40], [63, 50], [64, 22], [65, 33], [66, 66]]]

derived = [[1, 66], [[2, 34, 1], [3, 45, 1], [4, 17, 1], [5, 27, 1], [6, 56, 1], [7, 48, 1], [8, 42, 1], [9, 15, 1], [10, 47, 1], [11, 61, 1], [12, 28, 1], [13, 31, 1], [14, 24, 1], [16, 21, 1], [18, 41, 1], [19, 60, 1], [20, 57, 1], [22, 64, 1], [23, 35, 1], [25, 59, 1], [26, 49, 1], [29, 37, 1], [30, 38, 1], [32, 44, 1], [33, 65, 1], [36, 54, 1], [39, 55, 1], [40, 62, 1], [43, 53, 1], [46, 51, 1], [50, 63, 1], [52, 58, 1]], 66, Yes]

Reverse checks

• S200 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S199 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S198 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product66 with exact rational arithmetic, not rounded decimals.
• S197 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S196 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product65 with exact rational arithmetic, not rounded decimals.
• S195 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S194 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product64 with exact rational arithmetic, not rounded decimals.
• S193 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S192 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product63 with exact rational arithmetic, not rounded decimals.
• S191 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S190 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product62 with exact rational arithmetic, not rounded decimals.
• S189 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S188 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product61 with exact rational arithmetic, not rounded decimals.
• S187 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S186 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product60 with exact rational arithmetic, not rounded decimals.
• S185 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S184 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product59 with exact rational arithmetic, not rounded decimals.
• S183 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S182 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product58 with exact rational arithmetic, not rounded decimals.
• S181 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S180 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product57 with exact rational arithmetic, not rounded decimals.
• S179 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S178 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product56 with exact rational arithmetic, not rounded decimals.
• S177 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S176 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product55 with exact rational arithmetic, not rounded decimals.
• 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 product54 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 product53 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 product52 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 product51 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 product50 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 product49 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 product48 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 product47 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 product46 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 product45 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 product44 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 product43 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 product42 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 product41 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 product40 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 product39 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 product38 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 product37 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 product36 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 product35 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 product34 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 product33 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 product32 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 product31 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 product30 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 product29 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 product28 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 product27 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 product26 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 product25 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 product24 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 product23 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 product22 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 product21 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 product20 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 product19 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 product18 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 product17 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 product16 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 product15 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 product14 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 product13 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 product12 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 product11 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 product10 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 product9 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 product8 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 product7 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 product6 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 product5 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 product4 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 product3 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 product2 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 product1 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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• 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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• 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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• 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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• 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

(67, [[1, 1], [2, 34], [3, 45], [4, 17], [5, 27], [6, 56], [7, 48], [8, 42], [9, 15], [10, 47], [11, 61], [12, 28], [13, 31], [14, 24], [15, 9], [16, 21], [17, 4], [18, 41], [19, 60], [20, 57], [21, 16], [22, 64], [23, 35], [24, 14], [25, 59], [26, 49], [27, 5], [28, 12], [29, 37], [30, 38], [31, 13], [32, 44], [33, 65], [34, 2], [35, 23], [36, 54], [37, 29], [38, 30], [39, 55], [40, 62], [41, 18], [42, 8], [43, 53], [44, 32], [45, 3], [46, 51], [47, 10], [48, 7], [49, 26], [50, 63], [51, 46], [52, 58], [53, 43], [54, 36], [55, 39], [56, 6], [57, 20], [58, 52], [59, 25], [60, 19], [61, 11], [62, 40], [63, 50], [64, 22], [65, 33], [66, 66]], [1, 66], [[2, 34, 1], [3, 45, 1], [4, 17, 1], [5, 27, 1], [6, 56, 1], [7, 48, 1], [8, 42, 1], [9, 15, 1], [10, 47, 1], [11, 61, 1], [12, 28, 1], [13, 31, 1], [14, 24, 1], [16, 21, 1], [18, 41, 1], [19, 60, 1], [20, 57, 1], [22, 64, 1], [23, 35, 1], [25, 59, 1], [26, 49, 1], [29, 37, 1], [30, 38, 1], [32, 44, 1], [33, 65, 1], [36, 54, 1], [39, 55, 1], [40, 62, 1], [43, 53, 1], [46, 51, 1], [50, 63, 1], [52, 58, 1]], self_inverse_residues_1_and_p_minus_1, 66, 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.