Question 27797: All Nonzero Prime Residues Pair with Inverses, with a Special p=2 Case

Original question and givens

Use prime modulus p=79. Give (p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence).

• G001 — Use prime modulus p=79. 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 = [79, 1]

derived = [1, 1]

source roles = [79, 2]

derived = [40, 1]

source roles = [79, 3]

derived = [53, 1]

source roles = [79, 4]

derived = [20, 1]

source roles = [79, 5]

derived = [16, 1]

source roles = [79, 6]

derived = [66, 1]

source roles = [79, 7]

derived = [34, 1]

source roles = [79, 8]

derived = [10, 1]

source roles = [79, 9]

derived = [44, 1]

source roles = [79, 10]

derived = [8, 1]

source roles = [79, 11]

derived = [36, 1]

source roles = [79, 12]

derived = [33, 1]

source roles = [79, 13]

derived = [73, 1]

source roles = [79, 14]

derived = [17, 1]

source roles = [79, 15]

derived = [58, 1]

source roles = [79, 16]

derived = [5, 1]

source roles = [79, 17]

derived = [14, 1]

source roles = [79, 18]

derived = [22, 1]

source roles = [79, 19]

derived = [25, 1]

source roles = [79, 20]

derived = [4, 1]

source roles = [79, 21]

derived = [64, 1]

source roles = [79, 22]

derived = [18, 1]

source roles = [79, 23]

derived = [55, 1]

source roles = [79, 24]

derived = [56, 1]

source roles = [79, 25]

derived = [19, 1]

source roles = [79, 26]

derived = [76, 1]

source roles = [79, 27]

derived = [41, 1]

source roles = [79, 28]

derived = [48, 1]

source roles = [79, 29]

derived = [30, 1]

source roles = [79, 30]

derived = [29, 1]

source roles = [79, 31]

derived = [51, 1]

source roles = [79, 32]

derived = [42, 1]

source roles = [79, 33]

derived = [12, 1]

source roles = [79, 34]

derived = [7, 1]

source roles = [79, 35]

derived = [70, 1]

source roles = [79, 36]

derived = [11, 1]

source roles = [79, 37]

derived = [47, 1]

source roles = [79, 38]

derived = [52, 1]

source roles = [79, 39]

derived = [77, 1]

source roles = [79, 40]

derived = [2, 1]

source roles = [79, 41]

derived = [27, 1]

source roles = [79, 42]

derived = [32, 1]

source roles = [79, 43]

derived = [68, 1]

source roles = [79, 44]

derived = [9, 1]

source roles = [79, 45]

derived = [72, 1]

source roles = [79, 46]

derived = [67, 1]

source roles = [79, 47]

derived = [37, 1]

source roles = [79, 48]

derived = [28, 1]

source roles = [79, 49]

derived = [50, 1]

source roles = [79, 50]

derived = [49, 1]

source roles = [79, 51]

derived = [31, 1]

source roles = [79, 52]

derived = [38, 1]

source roles = [79, 53]

derived = [3, 1]

source roles = [79, 54]

derived = [60, 1]

source roles = [79, 55]

derived = [23, 1]

source roles = [79, 56]

derived = [24, 1]

source roles = [79, 57]

derived = [61, 1]

source roles = [79, 58]

derived = [15, 1]

source roles = [79, 59]

derived = [75, 1]

source roles = [79, 60]

derived = [54, 1]

source roles = [79, 61]

derived = [57, 1]

source roles = [79, 62]

derived = [65, 1]

source roles = [79, 63]

derived = [74, 1]

source roles = [79, 64]

derived = [21, 1]

source roles = [79, 65]

derived = [62, 1]

source roles = [79, 66]

derived = [6, 1]

source roles = [79, 67]

derived = [46, 1]

source roles = [79, 68]

derived = [43, 1]

source roles = [79, 69]

derived = [71, 1]

source roles = [79, 70]

derived = [35, 1]

source roles = [79, 71]

derived = [69, 1]

source roles = [79, 72]

derived = [45, 1]

source roles = [79, 73]

derived = [13, 1]

source roles = [79, 74]

derived = [63, 1]

source roles = [79, 75]

derived = [59, 1]

source roles = [79, 76]

derived = [26, 1]

source roles = [79, 77]

derived = [39, 1]

source roles = [79, 78]

derived = [78, 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, 79, 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, 79, 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, 79, 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, 79, 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, 79, 5, 24, 120]

derived = [1, 41, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 := mul(41, 6)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 = 246

source roles = [Wilson ORIGINAL prime factorial, 79, 6, 41, 246]

derived = [3, 9, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 := mul(9, 7)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 = 63

source roles = [Wilson ORIGINAL prime factorial, 79, 7, 9, 63]

derived = [0, 63, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 := mul(63, 8)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 = 504

source roles = [Wilson ORIGINAL prime factorial, 79, 8, 63, 504]

derived = [6, 30, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 := mul(30, 9)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 = 270

source roles = [Wilson ORIGINAL prime factorial, 79, 9, 30, 270]

derived = [3, 33, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 := mul(33, 10)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 = 330

source roles = [Wilson ORIGINAL prime factorial, 79, 10, 33, 330]

derived = [4, 14, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 := mul(14, 11)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 = 154

source roles = [Wilson ORIGINAL prime factorial, 79, 11, 14, 154]

derived = [1, 75, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 := mul(75, 12)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 = 900

source roles = [Wilson ORIGINAL prime factorial, 79, 12, 75, 900]

derived = [11, 31, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 := mul(31, 13)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 = 403

source roles = [Wilson ORIGINAL prime factorial, 79, 13, 31, 403]

derived = [5, 8, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 := mul(8, 14)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 = 112

source roles = [Wilson ORIGINAL prime factorial, 79, 14, 8, 112]

derived = [1, 33, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 := mul(33, 15)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 = 495

source roles = [Wilson ORIGINAL prime factorial, 79, 15, 33, 495]

derived = [6, 21, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 := mul(21, 16)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 = 336

source roles = [Wilson ORIGINAL prime factorial, 79, 16, 21, 336]

derived = [4, 20, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 := mul(20, 17)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 = 340

source roles = [Wilson ORIGINAL prime factorial, 79, 17, 20, 340]

derived = [4, 24, Yes, Yes]

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

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 = 432

source roles = [Wilson ORIGINAL prime factorial, 79, 18, 24, 432]

derived = [5, 37, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 := mul(37, 19)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 = 703

source roles = [Wilson ORIGINAL prime factorial, 79, 19, 37, 703]

derived = [8, 71, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 := mul(71, 20)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 = 1420

source roles = [Wilson ORIGINAL prime factorial, 79, 20, 71, 1420]

derived = [17, 77, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 := mul(77, 21)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 = 1617

source roles = [Wilson ORIGINAL prime factorial, 79, 21, 77, 1617]

derived = [20, 37, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 := mul(37, 22)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 = 814

source roles = [Wilson ORIGINAL prime factorial, 79, 22, 37, 814]

derived = [10, 24, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 := mul(24, 23)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 = 552

source roles = [Wilson ORIGINAL prime factorial, 79, 23, 24, 552]

derived = [6, 78, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 := mul(78, 24)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 = 1872

source roles = [Wilson ORIGINAL prime factorial, 79, 24, 78, 1872]

derived = [23, 55, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 := mul(55, 25)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 = 1375

source roles = [Wilson ORIGINAL prime factorial, 79, 25, 55, 1375]

derived = [17, 32, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 := mul(32, 26)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 = 832

source roles = [Wilson ORIGINAL prime factorial, 79, 26, 32, 832]

derived = [10, 42, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 := mul(42, 27)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 = 1134

source roles = [Wilson ORIGINAL prime factorial, 79, 27, 42, 1134]

derived = [14, 28, Yes, Yes]

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

Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 = 784

source roles = [Wilson ORIGINAL prime factorial, 79, 28, 28, 784]

derived = [9, 73, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 := mul(73, 29)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 = 2117

source roles = [Wilson ORIGINAL prime factorial, 79, 29, 73, 2117]

derived = [26, 63, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 := mul(63, 30)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 = 1890

source roles = [Wilson ORIGINAL prime factorial, 79, 30, 63, 1890]

derived = [23, 73, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 := mul(73, 31)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 = 2263

source roles = [Wilson ORIGINAL prime factorial, 79, 31, 73, 2263]

derived = [28, 51, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 := mul(51, 32)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 = 1632

source roles = [Wilson ORIGINAL prime factorial, 79, 32, 51, 1632]

derived = [20, 52, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 := mul(52, 33)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 = 1716

source roles = [Wilson ORIGINAL prime factorial, 79, 33, 52, 1716]

derived = [21, 57, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 := mul(57, 34)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 = 1938

source roles = [Wilson ORIGINAL prime factorial, 79, 34, 57, 1938]

derived = [24, 42, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 := mul(42, 35)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 = 1470

source roles = [Wilson ORIGINAL prime factorial, 79, 35, 42, 1470]

derived = [18, 48, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 := mul(48, 36)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 = 1728

source roles = [Wilson ORIGINAL prime factorial, 79, 36, 48, 1728]

derived = [21, 69, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 := mul(69, 37)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 = 2553

source roles = [Wilson ORIGINAL prime factorial, 79, 37, 69, 2553]

derived = [32, 25, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 := mul(25, 38)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 = 950

source roles = [Wilson ORIGINAL prime factorial, 79, 38, 25, 950]

derived = [12, 2, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 := mul(2, 39)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 = 78

source roles = [Wilson ORIGINAL prime factorial, 79, 39, 2, 78]

derived = [0, 78, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 := mul(78, 40)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 = 3120

source roles = [Wilson ORIGINAL prime factorial, 79, 40, 78, 3120]

derived = [39, 39, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product41 := mul(39, 41)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product41 = 1599

source roles = [Wilson ORIGINAL prime factorial, 79, 41, 39, 1599]

derived = [20, 19, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product42 := mul(19, 42)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product42 = 798

source roles = [Wilson ORIGINAL prime factorial, 79, 42, 19, 798]

derived = [10, 8, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product43 := mul(8, 43)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product43 = 344

source roles = [Wilson ORIGINAL prime factorial, 79, 43, 8, 344]

derived = [4, 28, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product44 := mul(28, 44)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product44 = 1232

source roles = [Wilson ORIGINAL prime factorial, 79, 44, 28, 1232]

derived = [15, 47, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product45 := mul(47, 45)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product45 = 2115

source roles = [Wilson ORIGINAL prime factorial, 79, 45, 47, 2115]

derived = [26, 61, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product46 := mul(61, 46)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product46 = 2806

source roles = [Wilson ORIGINAL prime factorial, 79, 46, 61, 2806]

derived = [35, 41, Yes, Yes]

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

Wilson ORIGINAL prime factorial ORIGINAL running factorial product47 = 1927

source roles = [Wilson ORIGINAL prime factorial, 79, 47, 41, 1927]

derived = [24, 31, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product48 := mul(31, 48)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product48 = 1488

source roles = [Wilson ORIGINAL prime factorial, 79, 48, 31, 1488]

derived = [18, 66, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product49 := mul(66, 49)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product49 = 3234

source roles = [Wilson ORIGINAL prime factorial, 79, 49, 66, 3234]

derived = [40, 74, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product50 := mul(74, 50)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product50 = 3700

source roles = [Wilson ORIGINAL prime factorial, 79, 50, 74, 3700]

derived = [46, 66, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product51 := mul(66, 51)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product51 = 3366

source roles = [Wilson ORIGINAL prime factorial, 79, 51, 66, 3366]

derived = [42, 48, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product52 := mul(48, 52)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product52 = 2496

source roles = [Wilson ORIGINAL prime factorial, 79, 52, 48, 2496]

derived = [31, 47, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product53 := mul(47, 53)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product53 = 2491

source roles = [Wilson ORIGINAL prime factorial, 79, 53, 47, 2491]

derived = [31, 42, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product54 := mul(42, 54)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product54 = 2268

source roles = [Wilson ORIGINAL prime factorial, 79, 54, 42, 2268]

derived = [28, 56, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product55 := mul(56, 55)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product55 = 3080

source roles = [Wilson ORIGINAL prime factorial, 79, 55, 56, 3080]

derived = [38, 78, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product56 := mul(78, 56)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product56 = 4368

source roles = [Wilson ORIGINAL prime factorial, 79, 56, 78, 4368]

derived = [55, 23, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product57 := mul(23, 57)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product57 = 1311

source roles = [Wilson ORIGINAL prime factorial, 79, 57, 23, 1311]

derived = [16, 47, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product58 := mul(47, 58)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product58 = 2726

source roles = [Wilson ORIGINAL prime factorial, 79, 58, 47, 2726]

derived = [34, 40, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product59 := mul(40, 59)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product59 = 2360

source roles = [Wilson ORIGINAL prime factorial, 79, 59, 40, 2360]

derived = [29, 69, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product60 := mul(69, 60)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product60 = 4140

source roles = [Wilson ORIGINAL prime factorial, 79, 60, 69, 4140]

derived = [52, 32, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product61 := mul(32, 61)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product61 = 1952

source roles = [Wilson ORIGINAL prime factorial, 79, 61, 32, 1952]

derived = [24, 56, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product62 := mul(56, 62)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product62 = 3472

source roles = [Wilson ORIGINAL prime factorial, 79, 62, 56, 3472]

derived = [43, 75, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product63 := mul(75, 63)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product63 = 4725

source roles = [Wilson ORIGINAL prime factorial, 79, 63, 75, 4725]

derived = [59, 64, Yes, Yes]

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

Wilson ORIGINAL prime factorial ORIGINAL running factorial product64 = 4096

source roles = [Wilson ORIGINAL prime factorial, 79, 64, 64, 4096]

derived = [51, 67, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product65 := mul(67, 65)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product65 = 4355

source roles = [Wilson ORIGINAL prime factorial, 79, 65, 67, 4355]

derived = [55, 10, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product66 := mul(10, 66)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product66 = 660

source roles = [Wilson ORIGINAL prime factorial, 79, 66, 10, 660]

derived = [8, 28, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product67 := mul(28, 67)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product67 = 1876

source roles = [Wilson ORIGINAL prime factorial, 79, 67, 28, 1876]

derived = [23, 59, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product68 := mul(59, 68)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product68 = 4012

source roles = [Wilson ORIGINAL prime factorial, 79, 68, 59, 4012]

derived = [50, 62, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product69 := mul(62, 69)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product69 = 4278

source roles = [Wilson ORIGINAL prime factorial, 79, 69, 62, 4278]

derived = [54, 12, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product70 := mul(12, 70)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product70 = 840

source roles = [Wilson ORIGINAL prime factorial, 79, 70, 12, 840]

derived = [10, 50, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product71 := mul(50, 71)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product71 = 3550

source roles = [Wilson ORIGINAL prime factorial, 79, 71, 50, 3550]

derived = [44, 74, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product72 := mul(74, 72)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product72 = 5328

source roles = [Wilson ORIGINAL prime factorial, 79, 72, 74, 5328]

derived = [67, 35, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product73 := mul(35, 73)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product73 = 2555

source roles = [Wilson ORIGINAL prime factorial, 79, 73, 35, 2555]

derived = [32, 27, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product74 := mul(27, 74)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product74 = 1998

source roles = [Wilson ORIGINAL prime factorial, 79, 74, 27, 1998]

derived = [25, 23, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product75 := mul(23, 75)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product75 = 1725

source roles = [Wilson ORIGINAL prime factorial, 79, 75, 23, 1725]

derived = [21, 66, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product76 := mul(66, 76)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product76 = 5016

source roles = [Wilson ORIGINAL prime factorial, 79, 76, 66, 5016]

derived = [63, 39, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product77 := mul(39, 77)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product77 = 3003

source roles = [Wilson ORIGINAL prime factorial, 79, 77, 39, 3003]

derived = [38, 1, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product78 := mul(1, 78)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product78 = 78

source roles = [Wilson ORIGINAL prime factorial, 79, 78, 1, 78]

derived = [0, 78, Yes, Yes]

source roles = [79, [[1, 1], [2, 40], [3, 53], [4, 20], [5, 16], [6, 66], [7, 34], [8, 10], [9, 44], [10, 8], [11, 36], [12, 33], [13, 73], [14, 17], [15, 58], [16, 5], [17, 14], [18, 22], [19, 25], [20, 4], [21, 64], [22, 18], [23, 55], [24, 56], [25, 19], [26, 76], [27, 41], [28, 48], [29, 30], [30, 29], [31, 51], [32, 42], [33, 12], [34, 7], [35, 70], [36, 11], [37, 47], [38, 52], [39, 77], [40, 2], [41, 27], [42, 32], [43, 68], [44, 9], [45, 72], [46, 67], [47, 37], [48, 28], [49, 50], [50, 49], [51, 31], [52, 38], [53, 3], [54, 60], [55, 23], [56, 24], [57, 61], [58, 15], [59, 75], [60, 54], [61, 57], [62, 65], [63, 74], [64, 21], [65, 62], [66, 6], [67, 46], [68, 43], [69, 71], [70, 35], [71, 69], [72, 45], [73, 13], [74, 63], [75, 59], [76, 26], [77, 39], [78, 78]]]

derived = [[1, 78], [[2, 40, 1], [3, 53, 1], [4, 20, 1], [5, 16, 1], [6, 66, 1], [7, 34, 1], [8, 10, 1], [9, 44, 1], [11, 36, 1], [12, 33, 1], [13, 73, 1], [14, 17, 1], [15, 58, 1], [18, 22, 1], [19, 25, 1], [21, 64, 1], [23, 55, 1], [24, 56, 1], [26, 76, 1], [27, 41, 1], [28, 48, 1], [29, 30, 1], [31, 51, 1], [32, 42, 1], [35, 70, 1], [37, 47, 1], [38, 52, 1], [39, 77, 1], [43, 68, 1], [45, 72, 1], [46, 67, 1], [49, 50, 1], [54, 60, 1], [57, 61, 1], [59, 75, 1], [62, 65, 1], [63, 74, 1], [69, 71, 1]], 78, Yes]

Reverse checks

• S236 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S235 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S234 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product78 with exact rational arithmetic, not rounded decimals.
• S233 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S232 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product77 with exact rational arithmetic, not rounded decimals.
• S231 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S230 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product76 with exact rational arithmetic, not rounded decimals.
• S229 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S228 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product75 with exact rational arithmetic, not rounded decimals.
• S227 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S226 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product74 with exact rational arithmetic, not rounded decimals.
• S225 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S224 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product73 with exact rational arithmetic, not rounded decimals.
• S223 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S222 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product72 with exact rational arithmetic, not rounded decimals.
• S221 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S220 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product71 with exact rational arithmetic, not rounded decimals.
• S219 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S218 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product70 with exact rational arithmetic, not rounded decimals.
• S217 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S216 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product69 with exact rational arithmetic, not rounded decimals.
• S215 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S214 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product68 with exact rational arithmetic, not rounded decimals.
• S213 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S212 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product67 with exact rational arithmetic, not rounded decimals.
• S211 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S210 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product66 with exact rational arithmetic, not rounded decimals.
• S209 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S208 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product65 with exact rational arithmetic, not rounded decimals.
• S207 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S206 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product64 with exact rational arithmetic, not rounded decimals.
• S205 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S204 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product63 with exact rational arithmetic, not rounded decimals.
• S203 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S202 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product62 with exact rational arithmetic, not rounded decimals.
• S201 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S200 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product61 with exact rational arithmetic, not rounded decimals.
• 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 product60 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 product59 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 product58 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 product57 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 product56 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 product55 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 product54 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 product53 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 product52 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 product51 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 product50 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 product49 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 product48 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 product47 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 product46 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 product45 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 product44 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 product43 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 product42 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 product41 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 product40 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 product39 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 product38 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 product37 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 product36 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 product35 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 product34 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 product33 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 product32 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 product31 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 product30 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 product29 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 product28 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 product27 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 product26 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 product25 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 product24 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 product23 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 product22 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 product21 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 product20 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 product19 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 product18 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 product17 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 product16 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 product15 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 product14 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 product13 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 product12 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 product11 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 product10 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 product9 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 product8 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 product7 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 product6 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 product5 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 product4 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 product3 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 product2 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 product1 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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• 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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• 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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• 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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• 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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• 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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• 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

(79, [[1, 1], [2, 40], [3, 53], [4, 20], [5, 16], [6, 66], [7, 34], [8, 10], [9, 44], [10, 8], [11, 36], [12, 33], [13, 73], [14, 17], [15, 58], [16, 5], [17, 14], [18, 22], [19, 25], [20, 4], [21, 64], [22, 18], [23, 55], [24, 56], [25, 19], [26, 76], [27, 41], [28, 48], [29, 30], [30, 29], [31, 51], [32, 42], [33, 12], [34, 7], [35, 70], [36, 11], [37, 47], [38, 52], [39, 77], [40, 2], [41, 27], [42, 32], [43, 68], [44, 9], [45, 72], [46, 67], [47, 37], [48, 28], [49, 50], [50, 49], [51, 31], [52, 38], [53, 3], [54, 60], [55, 23], [56, 24], [57, 61], [58, 15], [59, 75], [60, 54], [61, 57], [62, 65], [63, 74], [64, 21], [65, 62], [66, 6], [67, 46], [68, 43], [69, 71], [70, 35], [71, 69], [72, 45], [73, 13], [74, 63], [75, 59], [76, 26], [77, 39], [78, 78]], [1, 78], [[2, 40, 1], [3, 53, 1], [4, 20, 1], [5, 16, 1], [6, 66, 1], [7, 34, 1], [8, 10, 1], [9, 44, 1], [11, 36, 1], [12, 33, 1], [13, 73, 1], [14, 17, 1], [15, 58, 1], [18, 22, 1], [19, 25, 1], [21, 64, 1], [23, 55, 1], [24, 56, 1], [26, 76, 1], [27, 41, 1], [28, 48, 1], [29, 30, 1], [31, 51, 1], [32, 42, 1], [35, 70, 1], [37, 47, 1], [38, 52, 1], [39, 77, 1], [43, 68, 1], [45, 72, 1], [46, 67, 1], [49, 50, 1], [54, 60, 1], [57, 61, 1], [59, 75, 1], [62, 65, 1], [63, 74, 1], [69, 71, 1]], self_inverse_residues_1_and_p_minus_1, 78, 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.