Question 27793: All Nonzero Prime Residues Pair with Inverses, with a Special p=2 Case

Original question and givens

Use prime modulus p=61. Give (p, all_inverse_pairs, self_inverse_residues, distinct_cancelling_pairs, self_inverse_scope, factorial_residue, Wilson_congruence).

• G001 — Use prime modulus p=61. 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 = [61, 1]

derived = [1, 1]

source roles = [61, 2]

derived = [31, 1]

source roles = [61, 3]

derived = [41, 1]

source roles = [61, 4]

derived = [46, 1]

source roles = [61, 5]

derived = [49, 1]

source roles = [61, 6]

derived = [51, 1]

source roles = [61, 7]

derived = [35, 1]

source roles = [61, 8]

derived = [23, 1]

source roles = [61, 9]

derived = [34, 1]

source roles = [61, 10]

derived = [55, 1]

source roles = [61, 11]

derived = [50, 1]

source roles = [61, 12]

derived = [56, 1]

source roles = [61, 13]

derived = [47, 1]

source roles = [61, 14]

derived = [48, 1]

source roles = [61, 15]

derived = [57, 1]

source roles = [61, 16]

derived = [42, 1]

source roles = [61, 17]

derived = [18, 1]

source roles = [61, 18]

derived = [17, 1]

source roles = [61, 19]

derived = [45, 1]

source roles = [61, 20]

derived = [58, 1]

source roles = [61, 21]

derived = [32, 1]

source roles = [61, 22]

derived = [25, 1]

source roles = [61, 23]

derived = [8, 1]

source roles = [61, 24]

derived = [28, 1]

source roles = [61, 25]

derived = [22, 1]

source roles = [61, 26]

derived = [54, 1]

source roles = [61, 27]

derived = [52, 1]

source roles = [61, 28]

derived = [24, 1]

source roles = [61, 29]

derived = [40, 1]

source roles = [61, 30]

derived = [59, 1]

source roles = [61, 31]

derived = [2, 1]

source roles = [61, 32]

derived = [21, 1]

source roles = [61, 33]

derived = [37, 1]

source roles = [61, 34]

derived = [9, 1]

source roles = [61, 35]

derived = [7, 1]

source roles = [61, 36]

derived = [39, 1]

source roles = [61, 37]

derived = [33, 1]

source roles = [61, 38]

derived = [53, 1]

source roles = [61, 39]

derived = [36, 1]

source roles = [61, 40]

derived = [29, 1]

source roles = [61, 41]

derived = [3, 1]

source roles = [61, 42]

derived = [16, 1]

source roles = [61, 43]

derived = [44, 1]

source roles = [61, 44]

derived = [43, 1]

source roles = [61, 45]

derived = [19, 1]

source roles = [61, 46]

derived = [4, 1]

source roles = [61, 47]

derived = [13, 1]

source roles = [61, 48]

derived = [14, 1]

source roles = [61, 49]

derived = [5, 1]

source roles = [61, 50]

derived = [11, 1]

source roles = [61, 51]

derived = [6, 1]

source roles = [61, 52]

derived = [27, 1]

source roles = [61, 53]

derived = [38, 1]

source roles = [61, 54]

derived = [26, 1]

source roles = [61, 55]

derived = [10, 1]

source roles = [61, 56]

derived = [12, 1]

source roles = [61, 57]

derived = [15, 1]

source roles = [61, 58]

derived = [20, 1]

source roles = [61, 59]

derived = [30, 1]

source roles = [61, 60]

derived = [60, 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, 61, 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, 61, 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, 61, 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, 61, 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, 61, 5, 24, 120]

derived = [1, 59, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 := mul(59, 6)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 = 354

source roles = [Wilson ORIGINAL prime factorial, 61, 6, 59, 354]

derived = [5, 49, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 := mul(49, 7)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 = 343

source roles = [Wilson ORIGINAL prime factorial, 61, 7, 49, 343]

derived = [5, 38, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 := mul(38, 8)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 = 304

source roles = [Wilson ORIGINAL prime factorial, 61, 8, 38, 304]

derived = [4, 60, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 := mul(60, 9)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 = 540

source roles = [Wilson ORIGINAL prime factorial, 61, 9, 60, 540]

derived = [8, 52, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 := mul(52, 10)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product10 = 520

source roles = [Wilson ORIGINAL prime factorial, 61, 10, 52, 520]

derived = [8, 32, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 := mul(32, 11)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product11 = 352

source roles = [Wilson ORIGINAL prime factorial, 61, 11, 32, 352]

derived = [5, 47, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 := mul(47, 12)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product12 = 564

source roles = [Wilson ORIGINAL prime factorial, 61, 12, 47, 564]

derived = [9, 15, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 := mul(15, 13)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product13 = 195

source roles = [Wilson ORIGINAL prime factorial, 61, 13, 15, 195]

derived = [3, 12, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 := mul(12, 14)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product14 = 168

source roles = [Wilson ORIGINAL prime factorial, 61, 14, 12, 168]

derived = [2, 46, Yes, Yes]

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

Wilson ORIGINAL prime factorial ORIGINAL running factorial product15 = 690

source roles = [Wilson ORIGINAL prime factorial, 61, 15, 46, 690]

derived = [11, 19, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 := mul(19, 16)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product16 = 304

source roles = [Wilson ORIGINAL prime factorial, 61, 16, 19, 304]

derived = [4, 60, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 := mul(60, 17)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product17 = 1020

source roles = [Wilson ORIGINAL prime factorial, 61, 17, 60, 1020]

derived = [16, 44, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 := mul(44, 18)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product18 = 792

source roles = [Wilson ORIGINAL prime factorial, 61, 18, 44, 792]

derived = [12, 60, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 := mul(60, 19)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product19 = 1140

source roles = [Wilson ORIGINAL prime factorial, 61, 19, 60, 1140]

derived = [18, 42, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 := mul(42, 20)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product20 = 840

source roles = [Wilson ORIGINAL prime factorial, 61, 20, 42, 840]

derived = [13, 47, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 := mul(47, 21)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product21 = 987

source roles = [Wilson ORIGINAL prime factorial, 61, 21, 47, 987]

derived = [16, 11, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 := mul(11, 22)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product22 = 242

source roles = [Wilson ORIGINAL prime factorial, 61, 22, 11, 242]

derived = [3, 59, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 := mul(59, 23)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product23 = 1357

source roles = [Wilson ORIGINAL prime factorial, 61, 23, 59, 1357]

derived = [22, 15, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 := mul(15, 24)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product24 = 360

source roles = [Wilson ORIGINAL prime factorial, 61, 24, 15, 360]

derived = [5, 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, 61, 25, 55, 1375]

derived = [22, 33, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 := mul(33, 26)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product26 = 858

source roles = [Wilson ORIGINAL prime factorial, 61, 26, 33, 858]

derived = [14, 4, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 := mul(4, 27)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product27 = 108

source roles = [Wilson ORIGINAL prime factorial, 61, 27, 4, 108]

derived = [1, 47, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 := mul(47, 28)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product28 = 1316

source roles = [Wilson ORIGINAL prime factorial, 61, 28, 47, 1316]

derived = [21, 35, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 := mul(35, 29)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product29 = 1015

source roles = [Wilson ORIGINAL prime factorial, 61, 29, 35, 1015]

derived = [16, 39, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 := mul(39, 30)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product30 = 1170

source roles = [Wilson ORIGINAL prime factorial, 61, 30, 39, 1170]

derived = [19, 11, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 := mul(11, 31)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product31 = 341

source roles = [Wilson ORIGINAL prime factorial, 61, 31, 11, 341]

derived = [5, 36, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 := mul(36, 32)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product32 = 1152

source roles = [Wilson ORIGINAL prime factorial, 61, 32, 36, 1152]

derived = [18, 54, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 := mul(54, 33)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product33 = 1782

source roles = [Wilson ORIGINAL prime factorial, 61, 33, 54, 1782]

derived = [29, 13, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 := mul(13, 34)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product34 = 442

source roles = [Wilson ORIGINAL prime factorial, 61, 34, 13, 442]

derived = [7, 15, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 := mul(15, 35)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product35 = 525

source roles = [Wilson ORIGINAL prime factorial, 61, 35, 15, 525]

derived = [8, 37, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 := mul(37, 36)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product36 = 1332

source roles = [Wilson ORIGINAL prime factorial, 61, 36, 37, 1332]

derived = [21, 51, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 := mul(51, 37)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product37 = 1887

source roles = [Wilson ORIGINAL prime factorial, 61, 37, 51, 1887]

derived = [30, 57, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 := mul(57, 38)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product38 = 2166

source roles = [Wilson ORIGINAL prime factorial, 61, 38, 57, 2166]

derived = [35, 31, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 := mul(31, 39)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product39 = 1209

source roles = [Wilson ORIGINAL prime factorial, 61, 39, 31, 1209]

derived = [19, 50, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 := mul(50, 40)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product40 = 2000

source roles = [Wilson ORIGINAL prime factorial, 61, 40, 50, 2000]

derived = [32, 48, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product41 := mul(48, 41)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product41 = 1968

source roles = [Wilson ORIGINAL prime factorial, 61, 41, 48, 1968]

derived = [32, 16, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product42 := mul(16, 42)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product42 = 672

source roles = [Wilson ORIGINAL prime factorial, 61, 42, 16, 672]

derived = [11, 1, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product43 := mul(1, 43)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product43 = 43

source roles = [Wilson ORIGINAL prime factorial, 61, 43, 1, 43]

derived = [0, 43, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product44 := mul(43, 44)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product44 = 1892

source roles = [Wilson ORIGINAL prime factorial, 61, 44, 43, 1892]

derived = [31, 1, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product45 := mul(1, 45)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product45 = 45

source roles = [Wilson ORIGINAL prime factorial, 61, 45, 1, 45]

derived = [0, 45, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product46 := mul(45, 46)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product46 = 2070

source roles = [Wilson ORIGINAL prime factorial, 61, 46, 45, 2070]

derived = [33, 57, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product47 := mul(57, 47)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product47 = 2679

source roles = [Wilson ORIGINAL prime factorial, 61, 47, 57, 2679]

derived = [43, 56, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product48 := mul(56, 48)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product48 = 2688

source roles = [Wilson ORIGINAL prime factorial, 61, 48, 56, 2688]

derived = [44, 4, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product49 := mul(4, 49)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product49 = 196

source roles = [Wilson ORIGINAL prime factorial, 61, 49, 4, 196]

derived = [3, 13, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product50 := mul(13, 50)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product50 = 650

source roles = [Wilson ORIGINAL prime factorial, 61, 50, 13, 650]

derived = [10, 40, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product51 := mul(40, 51)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product51 = 2040

source roles = [Wilson ORIGINAL prime factorial, 61, 51, 40, 2040]

derived = [33, 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, 61, 52, 27, 1404]

derived = [23, 1, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product53 := mul(1, 53)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product53 = 53

source roles = [Wilson ORIGINAL prime factorial, 61, 53, 1, 53]

derived = [0, 53, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product54 := mul(53, 54)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product54 = 2862

source roles = [Wilson ORIGINAL prime factorial, 61, 54, 53, 2862]

derived = [46, 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, 61, 55, 56, 3080]

derived = [50, 30, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product56 := mul(30, 56)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product56 = 1680

source roles = [Wilson ORIGINAL prime factorial, 61, 56, 30, 1680]

derived = [27, 33, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product57 := mul(33, 57)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product57 = 1881

source roles = [Wilson ORIGINAL prime factorial, 61, 57, 33, 1881]

derived = [30, 51, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product58 := mul(51, 58)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product58 = 2958

source roles = [Wilson ORIGINAL prime factorial, 61, 58, 51, 2958]

derived = [48, 30, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product59 := mul(30, 59)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product59 = 1770

source roles = [Wilson ORIGINAL prime factorial, 61, 59, 30, 1770]

derived = [29, 1, Yes, Yes]

Wilson ORIGINAL prime factorial ORIGINAL running factorial product60 := mul(1, 60)

Wilson ORIGINAL prime factorial ORIGINAL running factorial product60 = 60

source roles = [Wilson ORIGINAL prime factorial, 61, 60, 1, 60]

derived = [0, 60, Yes, Yes]

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

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

Reverse checks

• S182 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• 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 product60 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 product59 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 product58 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 product57 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 product56 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 product55 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 product54 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 product53 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 product52 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 product51 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 product50 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 product49 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 product48 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 product47 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 product46 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 product45 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 product44 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 product43 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 product42 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 product41 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 product40 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 product39 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 product38 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 product37 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 product36 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 product35 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 product34 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 product33 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 product32 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 product31 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 product30 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 product29 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 product28 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 product27 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 product26 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 product25 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 product24 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 product23 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 product22 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 product21 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 product20 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 product19 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 product18 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 product17 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 product16 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 product15 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 product14 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 product13 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 product12 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 product11 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 product10 with exact rational arithmetic, not rounded decimals.
• S079 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S078 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product9 with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S076 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product8 with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S074 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product7 with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S072 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product6 with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S070 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product5 with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S068 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product4 with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S066 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product3 with exact rational arithmetic, not rounded decimals.
• S065 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S064 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product2 with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S062 · premise_and_result_audit — Recompute Wilson ORIGINAL prime factorial ORIGINAL running factorial product1 with exact rational arithmetic, not rounded decimals.
• S061 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
• S060 · premise_and_result_audit — 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

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