Question 27798: All Nonzero Prime Residues Pair with Inverses, with a Special p=2 Case

Original question and givens

Use prime modulus p=83. Give (p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence).

• G001 — Use prime modulus p=83. 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 = [83, 1]

derived = [1, 1]

source roles = [83, 2]

derived = [42, 1]

source roles = [83, 3]

derived = [28, 1]

source roles = [83, 4]

derived = [21, 1]

source roles = [83, 5]

derived = [50, 1]

source roles = [83, 6]

derived = [14, 1]

source roles = [83, 7]

derived = [12, 1]

source roles = [83, 8]

derived = [52, 1]

source roles = [83, 9]

derived = [37, 1]

source roles = [83, 10]

derived = [25, 1]

source roles = [83, 11]

derived = [68, 1]

source roles = [83, 12]

derived = [7, 1]

source roles = [83, 13]

derived = [32, 1]

source roles = [83, 14]

derived = [6, 1]

source roles = [83, 15]

derived = [72, 1]

source roles = [83, 16]

derived = [26, 1]

source roles = [83, 17]

derived = [44, 1]

source roles = [83, 18]

derived = [60, 1]

source roles = [83, 19]

derived = [35, 1]

source roles = [83, 20]

derived = [54, 1]

source roles = [83, 21]

derived = [4, 1]

source roles = [83, 22]

derived = [34, 1]

source roles = [83, 23]

derived = [65, 1]

source roles = [83, 24]

derived = [45, 1]

source roles = [83, 25]

derived = [10, 1]

source roles = [83, 26]

derived = [16, 1]

source roles = [83, 27]

derived = [40, 1]

source roles = [83, 28]

derived = [3, 1]

source roles = [83, 29]

derived = [63, 1]

source roles = [83, 30]

derived = [36, 1]

source roles = [83, 31]

derived = [75, 1]

source roles = [83, 32]

derived = [13, 1]

source roles = [83, 33]

derived = [78, 1]

source roles = [83, 34]

derived = [22, 1]

source roles = [83, 35]

derived = [19, 1]

source roles = [83, 36]

derived = [30, 1]

source roles = [83, 37]

derived = [9, 1]

source roles = [83, 38]

derived = [59, 1]

source roles = [83, 39]

derived = [66, 1]

source roles = [83, 40]

derived = [27, 1]

source roles = [83, 41]

derived = [81, 1]

source roles = [83, 42]

derived = [2, 1]

source roles = [83, 43]

derived = [56, 1]

source roles = [83, 44]

derived = [17, 1]

source roles = [83, 45]

derived = [24, 1]

source roles = [83, 46]

derived = [74, 1]

source roles = [83, 47]

derived = [53, 1]

source roles = [83, 48]

derived = [64, 1]

source roles = [83, 49]

derived = [61, 1]

source roles = [83, 50]

derived = [5, 1]

source roles = [83, 51]

derived = [70, 1]

source roles = [83, 52]

derived = [8, 1]

source roles = [83, 53]

derived = [47, 1]

source roles = [83, 54]

derived = [20, 1]

source roles = [83, 55]

derived = [80, 1]

source roles = [83, 56]

derived = [43, 1]

source roles = [83, 57]

derived = [67, 1]

source roles = [83, 58]

derived = [73, 1]

source roles = [83, 59]

derived = [38, 1]

source roles = [83, 60]

derived = [18, 1]

source roles = [83, 61]

derived = [49, 1]

source roles = [83, 62]

derived = [79, 1]

source roles = [83, 63]

derived = [29, 1]

source roles = [83, 64]

derived = [48, 1]

source roles = [83, 65]

derived = [23, 1]

source roles = [83, 66]

derived = [39, 1]

source roles = [83, 67]

derived = [57, 1]

source roles = [83, 68]

derived = [11, 1]

source roles = [83, 69]

derived = [77, 1]

source roles = [83, 70]

derived = [51, 1]

source roles = [83, 71]

derived = [76, 1]

source roles = [83, 72]

derived = [15, 1]

source roles = [83, 73]

derived = [58, 1]

source roles = [83, 74]

derived = [46, 1]

source roles = [83, 75]

derived = [31, 1]

source roles = [83, 76]

derived = [71, 1]

source roles = [83, 77]

derived = [69, 1]

source roles = [83, 78]

derived = [33, 1]

source roles = [83, 79]

derived = [62, 1]

source roles = [83, 80]

derived = [55, 1]

source roles = [83, 81]

derived = [41, 1]

source roles = [83, 82]

derived = [82, 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, 83, 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, 83, 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, 83, 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, 83, 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, 83, 5, 24, 120]

derived = [1, 37, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 := mul(37, 6)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 = 222

source roles = [Wilson ORIGINAL prime factorial, 83, 6, 37, 222]

derived = [2, 56, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 := mul(56, 7)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 = 392

source roles = [Wilson ORIGINAL prime factorial, 83, 7, 56, 392]

derived = [4, 60, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 := mul(60, 8)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 = 480

source roles = [Wilson ORIGINAL prime factorial, 83, 8, 60, 480]

derived = [5, 65, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 := mul(65, 9)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 = 585

source roles = [Wilson ORIGINAL prime factorial, 83, 9, 65, 585]

derived = [7, 4, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 := mul(4, 10)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 = 40

source roles = [Wilson ORIGINAL prime factorial, 83, 10, 4, 40]

derived = [0, 40, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 := mul(40, 11)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 = 440

source roles = [Wilson ORIGINAL prime factorial, 83, 11, 40, 440]

derived = [5, 25, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 := mul(25, 12)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 = 300

source roles = [Wilson ORIGINAL prime factorial, 83, 12, 25, 300]

derived = [3, 51, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 := mul(51, 13)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 = 663

source roles = [Wilson ORIGINAL prime factorial, 83, 13, 51, 663]

derived = [7, 82, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 := mul(82, 14)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 = 1148

source roles = [Wilson ORIGINAL prime factorial, 83, 14, 82, 1148]

derived = [13, 69, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 := mul(69, 15)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 = 1035

source roles = [Wilson ORIGINAL prime factorial, 83, 15, 69, 1035]

derived = [12, 39, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 := mul(39, 16)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 = 624

source roles = [Wilson ORIGINAL prime factorial, 83, 16, 39, 624]

derived = [7, 43, Yes, Yes]

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

Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 = 731

source roles = [Wilson ORIGINAL prime factorial, 83, 17, 43, 731]

derived = [8, 67, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 := mul(67, 18)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 = 1206

source roles = [Wilson ORIGINAL prime factorial, 83, 18, 67, 1206]

derived = [14, 44, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 := mul(44, 19)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 = 836

source roles = [Wilson ORIGINAL prime factorial, 83, 19, 44, 836]

derived = [10, 6, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 := mul(6, 20)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 = 120

source roles = [Wilson ORIGINAL prime factorial, 83, 20, 6, 120]

derived = [1, 37, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 := mul(37, 21)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 = 777

source roles = [Wilson ORIGINAL prime factorial, 83, 21, 37, 777]

derived = [9, 30, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 := mul(30, 22)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 = 660

source roles = [Wilson ORIGINAL prime factorial, 83, 22, 30, 660]

derived = [7, 79, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 := mul(79, 23)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 = 1817

source roles = [Wilson ORIGINAL prime factorial, 83, 23, 79, 1817]

derived = [21, 74, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 := mul(74, 24)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 = 1776

source roles = [Wilson ORIGINAL prime factorial, 83, 24, 74, 1776]

derived = [21, 33, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 := mul(33, 25)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product25 = 825

source roles = [Wilson ORIGINAL prime factorial, 83, 25, 33, 825]

derived = [9, 78, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 := mul(78, 26)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 = 2028

source roles = [Wilson ORIGINAL prime factorial, 83, 26, 78, 2028]

derived = [24, 36, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 := mul(36, 27)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 = 972

source roles = [Wilson ORIGINAL prime factorial, 83, 27, 36, 972]

derived = [11, 59, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 := mul(59, 28)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 = 1652

source roles = [Wilson ORIGINAL prime factorial, 83, 28, 59, 1652]

derived = [19, 75, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 := mul(75, 29)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 = 2175

source roles = [Wilson ORIGINAL prime factorial, 83, 29, 75, 2175]

derived = [26, 17, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 := mul(17, 30)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 = 510

source roles = [Wilson ORIGINAL prime factorial, 83, 30, 17, 510]

derived = [6, 12, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 := mul(12, 31)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 = 372

source roles = [Wilson ORIGINAL prime factorial, 83, 31, 12, 372]

derived = [4, 40, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 := mul(40, 32)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 = 1280

source roles = [Wilson ORIGINAL prime factorial, 83, 32, 40, 1280]

derived = [15, 35, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 := mul(35, 33)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 = 1155

source roles = [Wilson ORIGINAL prime factorial, 83, 33, 35, 1155]

derived = [13, 76, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 := mul(76, 34)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 = 2584

source roles = [Wilson ORIGINAL prime factorial, 83, 34, 76, 2584]

derived = [31, 11, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 := mul(11, 35)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 = 385

source roles = [Wilson ORIGINAL prime factorial, 83, 35, 11, 385]

derived = [4, 53, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 := mul(53, 36)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 = 1908

source roles = [Wilson ORIGINAL prime factorial, 83, 36, 53, 1908]

derived = [22, 82, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 := mul(82, 37)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 = 3034

source roles = [Wilson ORIGINAL prime factorial, 83, 37, 82, 3034]

derived = [36, 46, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 := mul(46, 38)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 = 1748

source roles = [Wilson ORIGINAL prime factorial, 83, 38, 46, 1748]

derived = [21, 5, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 := mul(5, 39)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 = 195

source roles = [Wilson ORIGINAL prime factorial, 83, 39, 5, 195]

derived = [2, 29, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 := mul(29, 40)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 = 1160

source roles = [Wilson ORIGINAL prime factorial, 83, 40, 29, 1160]

derived = [13, 81, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product41 := mul(81, 41)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product41 = 3321

source roles = [Wilson ORIGINAL prime factorial, 83, 41, 81, 3321]

derived = [40, 1, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product42 := mul(1, 42)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product42 = 42

source roles = [Wilson ORIGINAL prime factorial, 83, 42, 1, 42]

derived = [0, 42, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product43 := mul(42, 43)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product43 = 1806

source roles = [Wilson ORIGINAL prime factorial, 83, 43, 42, 1806]

derived = [21, 63, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product44 := mul(63, 44)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product44 = 2772

source roles = [Wilson ORIGINAL prime factorial, 83, 44, 63, 2772]

derived = [33, 33, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product45 := mul(33, 45)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product45 = 1485

source roles = [Wilson ORIGINAL prime factorial, 83, 45, 33, 1485]

derived = [17, 74, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product46 := mul(74, 46)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product46 = 3404

source roles = [Wilson ORIGINAL prime factorial, 83, 46, 74, 3404]

derived = [41, 1, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product47 := mul(1, 47)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product47 = 47

source roles = [Wilson ORIGINAL prime factorial, 83, 47, 1, 47]

derived = [0, 47, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product48 := mul(47, 48)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product48 = 2256

source roles = [Wilson ORIGINAL prime factorial, 83, 48, 47, 2256]

derived = [27, 15, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product49 := mul(15, 49)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product49 = 735

source roles = [Wilson ORIGINAL prime factorial, 83, 49, 15, 735]

derived = [8, 71, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product50 := mul(71, 50)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product50 = 3550

source roles = [Wilson ORIGINAL prime factorial, 83, 50, 71, 3550]

derived = [42, 64, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product51 := mul(64, 51)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product51 = 3264

source roles = [Wilson ORIGINAL prime factorial, 83, 51, 64, 3264]

derived = [39, 27, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product52 := mul(27, 52)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product52 = 1404

source roles = [Wilson ORIGINAL prime factorial, 83, 52, 27, 1404]

derived = [16, 76, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product53 := mul(76, 53)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product53 = 4028

source roles = [Wilson ORIGINAL prime factorial, 83, 53, 76, 4028]

derived = [48, 44, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product54 := mul(44, 54)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product54 = 2376

source roles = [Wilson ORIGINAL prime factorial, 83, 54, 44, 2376]

derived = [28, 52, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product55 := mul(52, 55)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product55 = 2860

source roles = [Wilson ORIGINAL prime factorial, 83, 55, 52, 2860]

derived = [34, 38, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product56 := mul(38, 56)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product56 = 2128

source roles = [Wilson ORIGINAL prime factorial, 83, 56, 38, 2128]

derived = [25, 53, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product57 := mul(53, 57)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product57 = 3021

source roles = [Wilson ORIGINAL prime factorial, 83, 57, 53, 3021]

derived = [36, 33, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product58 := mul(33, 58)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product58 = 1914

source roles = [Wilson ORIGINAL prime factorial, 83, 58, 33, 1914]

derived = [23, 5, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product59 := mul(5, 59)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product59 = 295

source roles = [Wilson ORIGINAL prime factorial, 83, 59, 5, 295]

derived = [3, 46, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product60 := mul(46, 60)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product60 = 2760

source roles = [Wilson ORIGINAL prime factorial, 83, 60, 46, 2760]

derived = [33, 21, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product61 := mul(21, 61)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product61 = 1281

source roles = [Wilson ORIGINAL prime factorial, 83, 61, 21, 1281]

derived = [15, 36, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product62 := mul(36, 62)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product62 = 2232

source roles = [Wilson ORIGINAL prime factorial, 83, 62, 36, 2232]

derived = [26, 74, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product63 := mul(74, 63)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product63 = 4662

source roles = [Wilson ORIGINAL prime factorial, 83, 63, 74, 4662]

derived = [56, 14, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product64 := mul(14, 64)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product64 = 896

source roles = [Wilson ORIGINAL prime factorial, 83, 64, 14, 896]

derived = [10, 66, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product65 := mul(66, 65)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product65 = 4290

source roles = [Wilson ORIGINAL prime factorial, 83, 65, 66, 4290]

derived = [51, 57, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product66 := mul(57, 66)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product66 = 3762

source roles = [Wilson ORIGINAL prime factorial, 83, 66, 57, 3762]

derived = [45, 27, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product67 := mul(27, 67)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product67 = 1809

source roles = [Wilson ORIGINAL prime factorial, 83, 67, 27, 1809]

derived = [21, 66, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product68 := mul(66, 68)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product68 = 4488

source roles = [Wilson ORIGINAL prime factorial, 83, 68, 66, 4488]

derived = [54, 6, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product69 := mul(6, 69)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product69 = 414

source roles = [Wilson ORIGINAL prime factorial, 83, 69, 6, 414]

derived = [4, 82, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product70 := mul(82, 70)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product70 = 5740

source roles = [Wilson ORIGINAL prime factorial, 83, 70, 82, 5740]

derived = [69, 13, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product71 := mul(13, 71)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product71 = 923

source roles = [Wilson ORIGINAL prime factorial, 83, 71, 13, 923]

derived = [11, 10, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product72 := mul(10, 72)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product72 = 720

source roles = [Wilson ORIGINAL prime factorial, 83, 72, 10, 720]

derived = [8, 56, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product73 := mul(56, 73)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product73 = 4088

source roles = [Wilson ORIGINAL prime factorial, 83, 73, 56, 4088]

derived = [49, 21, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product74 := mul(21, 74)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product74 = 1554

source roles = [Wilson ORIGINAL prime factorial, 83, 74, 21, 1554]

derived = [18, 60, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product75 := mul(60, 75)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product75 = 4500

source roles = [Wilson ORIGINAL prime factorial, 83, 75, 60, 4500]

derived = [54, 18, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product76 := mul(18, 76)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product76 = 1368

source roles = [Wilson ORIGINAL prime factorial, 83, 76, 18, 1368]

derived = [16, 40, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product77 := mul(40, 77)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product77 = 3080

source roles = [Wilson ORIGINAL prime factorial, 83, 77, 40, 3080]

derived = [37, 9, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product78 := mul(9, 78)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product78 = 702

source roles = [Wilson ORIGINAL prime factorial, 83, 78, 9, 702]

derived = [8, 38, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product79 := mul(38, 79)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product79 = 3002

source roles = [Wilson ORIGINAL prime factorial, 83, 79, 38, 3002]

derived = [36, 14, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product80 := mul(14, 80)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product80 = 1120

source roles = [Wilson ORIGINAL prime factorial, 83, 80, 14, 1120]

derived = [13, 41, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product81 := mul(41, 81)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product81 = 3321

source roles = [Wilson ORIGINAL prime factorial, 83, 81, 41, 3321]

derived = [40, 1, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product82 := mul(1, 82)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product82 = 82

source roles = [Wilson ORIGINAL prime factorial, 83, 82, 1, 82]

derived = [0, 82, Yes, Yes]

source roles = [83, [[1, 1], [2, 42], [3, 28], [4, 21], [5, 50], [6, 14], [7, 12], [8, 52], [9, 37], [10, 25], [11, 68], [12, 7], [13, 32], [14, 6], [15, 72], [16, 26], [17, 44], [18, 60], [19, 35], [20, 54], [21, 4], [22, 34], [23, 65], [24, 45], [25, 10], [26, 16], [27, 40], [28, 3], [29, 63], [30, 36], [31, 75], [32, 13], [33, 78], [34, 22], [35, 19], [36, 30], [37, 9], [38, 59], [39, 66], [40, 27], [41, 81], [42, 2], [43, 56], [44, 17], [45, 24], [46, 74], [47, 53], [48, 64], [49, 61], [50, 5], [51, 70], [52, 8], [53, 47], [54, 20], [55, 80], [56, 43], [57, 67], [58, 73], [59, 38], [60, 18], [61, 49], [62, 79], [63, 29], [64, 48], [65, 23], [66, 39], [67, 57], [68, 11], [69, 77], [70, 51], [71, 76], [72, 15], [73, 58], [74, 46], [75, 31], [76, 71], [77, 69], [78, 33], [79, 62], [80, 55], [81, 41], [82, 82]]]

derived = [[1, 82], [[2, 42, 1], [3, 28, 1], [4, 21, 1], [5, 50, 1], [6, 14, 1], [7, 12, 1], [8, 52, 1], [9, 37, 1], [10, 25, 1], [11, 68, 1], [13, 32, 1], [15, 72, 1], [16, 26, 1], [17, 44, 1], [18, 60, 1], [19, 35, 1], [20, 54, 1], [22, 34, 1], [23, 65, 1], [24, 45, 1], [27, 40, 1], [29, 63, 1], [30, 36, 1], [31, 75, 1], [33, 78, 1], [38, 59, 1], [39, 66, 1], [41, 81, 1], [43, 56, 1], [46, 74, 1], [47, 53, 1], [48, 64, 1], [49, 61, 1], [51, 70, 1], [55, 80, 1], [57, 67, 1], [58, 73, 1], [62, 79, 1], [69, 77, 1], [71, 76, 1]], 82, Yes]

Reverse checks

• S248 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S247 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S246 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product82 with exact rational arithmetic, not rounded decimals.
• S245 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S244 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product81 with exact rational arithmetic, not rounded decimals.
• S243 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S242 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product80 with exact rational arithmetic, not rounded decimals.
• S241 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S240 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product79 with exact rational arithmetic, not rounded decimals.
• S239 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S238 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product78 with exact rational arithmetic, not rounded decimals.
• S237 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S236 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product77 with exact rational arithmetic, not rounded decimals.
• 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 product76 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 product75 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 product74 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 product73 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 product72 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 product71 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 product70 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 product69 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 product68 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 product67 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 product66 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 product65 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 product64 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 product63 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 product62 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 product61 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 product60 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 product59 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 product58 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 product57 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 product56 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 product55 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 product54 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 product53 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 product52 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 product51 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 product50 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 product49 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 product48 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 product47 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 product46 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 product45 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 product44 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 product43 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 product42 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 product41 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 product40 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 product39 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 product38 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 product37 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 product36 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 product35 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 product34 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 product33 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 product32 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 product31 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 product30 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 product29 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 product28 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 product27 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 product26 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 product25 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 product24 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 product23 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 product22 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 product21 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 product20 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 product19 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 product18 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 product17 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 product16 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 product15 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 product14 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 product13 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 product12 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 product11 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 product10 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 product9 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 product8 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 product7 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 product6 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 product5 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 product4 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 product3 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 product2 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 product1 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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• 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 — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• 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

(83, [[1, 1], [2, 42], [3, 28], [4, 21], [5, 50], [6, 14], [7, 12], [8, 52], [9, 37], [10, 25], [11, 68], [12, 7], [13, 32], [14, 6], [15, 72], [16, 26], [17, 44], [18, 60], [19, 35], [20, 54], [21, 4], [22, 34], [23, 65], [24, 45], [25, 10], [26, 16], [27, 40], [28, 3], [29, 63], [30, 36], [31, 75], [32, 13], [33, 78], [34, 22], [35, 19], [36, 30], [37, 9], [38, 59], [39, 66], [40, 27], [41, 81], [42, 2], [43, 56], [44, 17], [45, 24], [46, 74], [47, 53], [48, 64], [49, 61], [50, 5], [51, 70], [52, 8], [53, 47], [54, 20], [55, 80], [56, 43], [57, 67], [58, 73], [59, 38], [60, 18], [61, 49], [62, 79], [63, 29], [64, 48], [65, 23], [66, 39], [67, 57], [68, 11], [69, 77], [70, 51], [71, 76], [72, 15], [73, 58], [74, 46], [75, 31], [76, 71], [77, 69], [78, 33], [79, 62], [80, 55], [81, 41], [82, 82]], [1, 82], [[2, 42, 1], [3, 28, 1], [4, 21, 1], [5, 50, 1], [6, 14, 1], [7, 12, 1], [8, 52, 1], [9, 37, 1], [10, 25, 1], [11, 68, 1], [13, 32, 1], [15, 72, 1], [16, 26, 1], [17, 44, 1], [18, 60, 1], [19, 35, 1], [20, 54, 1], [22, 34, 1], [23, 65, 1], [24, 45, 1], [27, 40, 1], [29, 63, 1], [30, 36, 1], [31, 75, 1], [33, 78, 1], [38, 59, 1], [39, 66, 1], [41, 81, 1], [43, 56, 1], [46, 74, 1], [47, 53, 1], [48, 64, 1], [49, 61, 1], [51, 70, 1], [55, 80, 1], [57, 67, 1], [58, 73, 1], [62, 79, 1], [69, 77, 1], [71, 76, 1]], self_inverse_residues_1_and_p_minus_1, 82, 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.