Question 33699: The Nonzero Multiplicative Group Gives a p−1 Power Identity

Original question and givens

Use the complete prime field Z_89, with prime p=89, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=24. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.

• G001 — Use the complete prime field Z_89, with prime p=89, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=24. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.

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-finite-field

Step-by-step derivation

source roles = [qf-mod31-fielddeep-power-cycle, [domain, element, operation, p]]

derived = [fielddeep-power-cycle, Exact_prime_field_Zp]

source roles = [89, [2, 3, 4, 5, 6, 7, 8, 9]]

derived = [[1, 2, 1, 4, 5, 5, 1, 8], Yes]

Field ORIGINAL previous power times element k=1 := mul(1, 24)

Field ORIGINAL previous power times element k=1 = 24

source roles = [power exponent1, 24, 89]

derived = [0, 24]

Field ORIGINAL previous power times element k=2 := mul(24, 24)

Field ORIGINAL previous power times element k=2 = 576

source roles = [power exponent2, 576, 89]

derived = [6, 42]

Field ORIGINAL previous power times element k=3 := mul(42, 24)

Field ORIGINAL previous power times element k=3 = 1008

source roles = [power exponent3, 1008, 89]

derived = [11, 29]

Field ORIGINAL previous power times element k=4 := mul(29, 24)

Field ORIGINAL previous power times element k=4 = 696

source roles = [power exponent4, 696, 89]

derived = [7, 73]

Field ORIGINAL previous power times element k=5 := mul(73, 24)

Field ORIGINAL previous power times element k=5 = 1752

source roles = [power exponent5, 1752, 89]

derived = [19, 61]

Field ORIGINAL previous power times element k=6 := mul(61, 24)

Field ORIGINAL previous power times element k=6 = 1464

source roles = [power exponent6, 1464, 89]

derived = [16, 40]

Field ORIGINAL previous power times element k=7 := mul(40, 24)

Field ORIGINAL previous power times element k=7 = 960

source roles = [power exponent7, 960, 89]

derived = [10, 70]

Field ORIGINAL previous power times element k=8 := mul(70, 24)

Field ORIGINAL previous power times element k=8 = 1680

source roles = [power exponent8, 1680, 89]

derived = [18, 78]

Field ORIGINAL previous power times element k=9 := mul(78, 24)

Field ORIGINAL previous power times element k=9 = 1872

source roles = [power exponent9, 1872, 89]

derived = [21, 3]

Field ORIGINAL previous power times element k=10 := mul(3, 24)

Field ORIGINAL previous power times element k=10 = 72

source roles = [power exponent10, 72, 89]

derived = [0, 72]

Field ORIGINAL previous power times element k=11 := mul(72, 24)

Field ORIGINAL previous power times element k=11 = 1728

source roles = [power exponent11, 1728, 89]

derived = [19, 37]

Field ORIGINAL previous power times element k=12 := mul(37, 24)

Field ORIGINAL previous power times element k=12 = 888

source roles = [power exponent12, 888, 89]

derived = [9, 87]

Field ORIGINAL previous power times element k=13 := mul(87, 24)

Field ORIGINAL previous power times element k=13 = 2088

source roles = [power exponent13, 2088, 89]

derived = [23, 41]

Field ORIGINAL previous power times element k=14 := mul(41, 24)

Field ORIGINAL previous power times element k=14 = 984

source roles = [power exponent14, 984, 89]

derived = [11, 5]

Field ORIGINAL previous power times element k=15 := mul(5, 24)

Field ORIGINAL previous power times element k=15 = 120

source roles = [power exponent15, 120, 89]

derived = [1, 31]

Field ORIGINAL previous power times element k=16 := mul(31, 24)

Field ORIGINAL previous power times element k=16 = 744

source roles = [power exponent16, 744, 89]

derived = [8, 32]

Field ORIGINAL previous power times element k=17 := mul(32, 24)

Field ORIGINAL previous power times element k=17 = 768

source roles = [power exponent17, 768, 89]

derived = [8, 56]

Field ORIGINAL previous power times element k=18 := mul(56, 24)

Field ORIGINAL previous power times element k=18 = 1344

source roles = [power exponent18, 1344, 89]

derived = [15, 9]

Field ORIGINAL previous power times element k=19 := mul(9, 24)

Field ORIGINAL previous power times element k=19 = 216

source roles = [power exponent19, 216, 89]

derived = [2, 38]

Field ORIGINAL previous power times element k=20 := mul(38, 24)

Field ORIGINAL previous power times element k=20 = 912

source roles = [power exponent20, 912, 89]

derived = [10, 22]

Field ORIGINAL previous power times element k=21 := mul(22, 24)

Field ORIGINAL previous power times element k=21 = 528

source roles = [power exponent21, 528, 89]

derived = [5, 83]

Field ORIGINAL previous power times element k=22 := mul(83, 24)

Field ORIGINAL previous power times element k=22 = 1992

source roles = [power exponent22, 1992, 89]

derived = [22, 34]

Field ORIGINAL previous power times element k=23 := mul(34, 24)

Field ORIGINAL previous power times element k=23 = 816

source roles = [power exponent23, 816, 89]

derived = [9, 15]

Field ORIGINAL previous power times element k=24 := mul(15, 24)

Field ORIGINAL previous power times element k=24 = 360

source roles = [power exponent24, 360, 89]

derived = [4, 4]

Field ORIGINAL previous power times element k=25 := mul(4, 24)

Field ORIGINAL previous power times element k=25 = 96

source roles = [power exponent25, 96, 89]

derived = [1, 7]

Field ORIGINAL previous power times element k=26 := mul(7, 24)

Field ORIGINAL previous power times element k=26 = 168

source roles = [power exponent26, 168, 89]

derived = [1, 79]

Field ORIGINAL previous power times element k=27 := mul(79, 24)

Field ORIGINAL previous power times element k=27 = 1896

source roles = [power exponent27, 1896, 89]

derived = [21, 27]

Field ORIGINAL previous power times element k=28 := mul(27, 24)

Field ORIGINAL previous power times element k=28 = 648

source roles = [power exponent28, 648, 89]

derived = [7, 25]

Field ORIGINAL previous power times element k=29 := mul(25, 24)

Field ORIGINAL previous power times element k=29 = 600

source roles = [power exponent29, 600, 89]

derived = [6, 66]

Field ORIGINAL previous power times element k=30 := mul(66, 24)

Field ORIGINAL previous power times element k=30 = 1584

source roles = [power exponent30, 1584, 89]

derived = [17, 71]

Field ORIGINAL previous power times element k=31 := mul(71, 24)

Field ORIGINAL previous power times element k=31 = 1704

source roles = [power exponent31, 1704, 89]

derived = [19, 13]

Field ORIGINAL previous power times element k=32 := mul(13, 24)

Field ORIGINAL previous power times element k=32 = 312

source roles = [power exponent32, 312, 89]

derived = [3, 45]

Field ORIGINAL previous power times element k=33 := mul(45, 24)

Field ORIGINAL previous power times element k=33 = 1080

source roles = [power exponent33, 1080, 89]

derived = [12, 12]

Field ORIGINAL previous power times element k=34 := mul(12, 24)

Field ORIGINAL previous power times element k=34 = 288

source roles = [power exponent34, 288, 89]

derived = [3, 21]

Field ORIGINAL previous power times element k=35 := mul(21, 24)

Field ORIGINAL previous power times element k=35 = 504

source roles = [power exponent35, 504, 89]

derived = [5, 59]

Field ORIGINAL previous power times element k=36 := mul(59, 24)

Field ORIGINAL previous power times element k=36 = 1416

source roles = [power exponent36, 1416, 89]

derived = [15, 81]

Field ORIGINAL previous power times element k=37 := mul(81, 24)

Field ORIGINAL previous power times element k=37 = 1944

source roles = [power exponent37, 1944, 89]

derived = [21, 75]

Field ORIGINAL previous power times element k=38 := mul(75, 24)

Field ORIGINAL previous power times element k=38 = 1800

source roles = [power exponent38, 1800, 89]

derived = [20, 20]

Field ORIGINAL previous power times element k=39 := mul(20, 24)

Field ORIGINAL previous power times element k=39 = 480

source roles = [power exponent39, 480, 89]

derived = [5, 35]

Field ORIGINAL previous power times element k=40 := mul(35, 24)

Field ORIGINAL previous power times element k=40 = 840

source roles = [power exponent40, 840, 89]

derived = [9, 39]

Field ORIGINAL previous power times element k=41 := mul(39, 24)

Field ORIGINAL previous power times element k=41 = 936

source roles = [power exponent41, 936, 89]

derived = [10, 46]

Field ORIGINAL previous power times element k=42 := mul(46, 24)

Field ORIGINAL previous power times element k=42 = 1104

source roles = [power exponent42, 1104, 89]

derived = [12, 36]

Field ORIGINAL previous power times element k=43 := mul(36, 24)

Field ORIGINAL previous power times element k=43 = 864

source roles = [power exponent43, 864, 89]

derived = [9, 63]

Field ORIGINAL previous power times element k=44 := mul(63, 24)

Field ORIGINAL previous power times element k=44 = 1512

source roles = [power exponent44, 1512, 89]

derived = [16, 88]

Field ORIGINAL previous power times element k=45 := mul(88, 24)

Field ORIGINAL previous power times element k=45 = 2112

source roles = [power exponent45, 2112, 89]

derived = [23, 65]

Field ORIGINAL previous power times element k=46 := mul(65, 24)

Field ORIGINAL previous power times element k=46 = 1560

source roles = [power exponent46, 1560, 89]

derived = [17, 47]

Field ORIGINAL previous power times element k=47 := mul(47, 24)

Field ORIGINAL previous power times element k=47 = 1128

source roles = [power exponent47, 1128, 89]

derived = [12, 60]

Field ORIGINAL previous power times element k=48 := mul(60, 24)

Field ORIGINAL previous power times element k=48 = 1440

source roles = [power exponent48, 1440, 89]

derived = [16, 16]

Field ORIGINAL previous power times element k=49 := mul(16, 24)

Field ORIGINAL previous power times element k=49 = 384

source roles = [power exponent49, 384, 89]

derived = [4, 28]

Field ORIGINAL previous power times element k=50 := mul(28, 24)

Field ORIGINAL previous power times element k=50 = 672

source roles = [power exponent50, 672, 89]

derived = [7, 49]

Field ORIGINAL previous power times element k=51 := mul(49, 24)

Field ORIGINAL previous power times element k=51 = 1176

source roles = [power exponent51, 1176, 89]

derived = [13, 19]

Field ORIGINAL previous power times element k=52 := mul(19, 24)

Field ORIGINAL previous power times element k=52 = 456

source roles = [power exponent52, 456, 89]

derived = [5, 11]

Field ORIGINAL previous power times element k=53 := mul(11, 24)

Field ORIGINAL previous power times element k=53 = 264

source roles = [power exponent53, 264, 89]

derived = [2, 86]

Field ORIGINAL previous power times element k=54 := mul(86, 24)

Field ORIGINAL previous power times element k=54 = 2064

source roles = [power exponent54, 2064, 89]

derived = [23, 17]

Field ORIGINAL previous power times element k=55 := mul(17, 24)

Field ORIGINAL previous power times element k=55 = 408

source roles = [power exponent55, 408, 89]

derived = [4, 52]

Field ORIGINAL previous power times element k=56 := mul(52, 24)

Field ORIGINAL previous power times element k=56 = 1248

source roles = [power exponent56, 1248, 89]

derived = [14, 2]

Field ORIGINAL previous power times element k=57 := mul(2, 24)

Field ORIGINAL previous power times element k=57 = 48

source roles = [power exponent57, 48, 89]

derived = [0, 48]

Field ORIGINAL previous power times element k=58 := mul(48, 24)

Field ORIGINAL previous power times element k=58 = 1152

source roles = [power exponent58, 1152, 89]

derived = [12, 84]

Field ORIGINAL previous power times element k=59 := mul(84, 24)

Field ORIGINAL previous power times element k=59 = 2016

source roles = [power exponent59, 2016, 89]

derived = [22, 58]

Field ORIGINAL previous power times element k=60 := mul(58, 24)

Field ORIGINAL previous power times element k=60 = 1392

source roles = [power exponent60, 1392, 89]

derived = [15, 57]

Field ORIGINAL previous power times element k=61 := mul(57, 24)

Field ORIGINAL previous power times element k=61 = 1368

source roles = [power exponent61, 1368, 89]

derived = [15, 33]

Field ORIGINAL previous power times element k=62 := mul(33, 24)

Field ORIGINAL previous power times element k=62 = 792

source roles = [power exponent62, 792, 89]

derived = [8, 80]

Field ORIGINAL previous power times element k=63 := mul(80, 24)

Field ORIGINAL previous power times element k=63 = 1920

source roles = [power exponent63, 1920, 89]

derived = [21, 51]

Field ORIGINAL previous power times element k=64 := mul(51, 24)

Field ORIGINAL previous power times element k=64 = 1224

source roles = [power exponent64, 1224, 89]

derived = [13, 67]

Field ORIGINAL previous power times element k=65 := mul(67, 24)

Field ORIGINAL previous power times element k=65 = 1608

source roles = [power exponent65, 1608, 89]

derived = [18, 6]

Field ORIGINAL previous power times element k=66 := mul(6, 24)

Field ORIGINAL previous power times element k=66 = 144

source roles = [power exponent66, 144, 89]

derived = [1, 55]

Field ORIGINAL previous power times element k=67 := mul(55, 24)

Field ORIGINAL previous power times element k=67 = 1320

source roles = [power exponent67, 1320, 89]

derived = [14, 74]

Field ORIGINAL previous power times element k=68 := mul(74, 24)

Field ORIGINAL previous power times element k=68 = 1776

source roles = [power exponent68, 1776, 89]

derived = [19, 85]

Field ORIGINAL previous power times element k=69 := mul(85, 24)

Field ORIGINAL previous power times element k=69 = 2040

source roles = [power exponent69, 2040, 89]

derived = [22, 82]

Field ORIGINAL previous power times element k=70 := mul(82, 24)

Field ORIGINAL previous power times element k=70 = 1968

source roles = [power exponent70, 1968, 89]

derived = [22, 10]

Field ORIGINAL previous power times element k=71 := mul(10, 24)

Field ORIGINAL previous power times element k=71 = 240

source roles = [power exponent71, 240, 89]

derived = [2, 62]

Field ORIGINAL previous power times element k=72 := mul(62, 24)

Field ORIGINAL previous power times element k=72 = 1488

source roles = [power exponent72, 1488, 89]

derived = [16, 64]

Field ORIGINAL previous power times element k=73 := mul(64, 24)

Field ORIGINAL previous power times element k=73 = 1536

source roles = [power exponent73, 1536, 89]

derived = [17, 23]

Field ORIGINAL previous power times element k=74 := mul(23, 24)

Field ORIGINAL previous power times element k=74 = 552

source roles = [power exponent74, 552, 89]

derived = [6, 18]

Field ORIGINAL previous power times element k=75 := mul(18, 24)

Field ORIGINAL previous power times element k=75 = 432

source roles = [power exponent75, 432, 89]

derived = [4, 76]

Field ORIGINAL previous power times element k=76 := mul(76, 24)

Field ORIGINAL previous power times element k=76 = 1824

source roles = [power exponent76, 1824, 89]

derived = [20, 44]

Field ORIGINAL previous power times element k=77 := mul(44, 24)

Field ORIGINAL previous power times element k=77 = 1056

source roles = [power exponent77, 1056, 89]

derived = [11, 77]

Field ORIGINAL previous power times element k=78 := mul(77, 24)

Field ORIGINAL previous power times element k=78 = 1848

source roles = [power exponent78, 1848, 89]

derived = [20, 68]

Field ORIGINAL previous power times element k=79 := mul(68, 24)

Field ORIGINAL previous power times element k=79 = 1632

source roles = [power exponent79, 1632, 89]

derived = [18, 30]

Field ORIGINAL previous power times element k=80 := mul(30, 24)

Field ORIGINAL previous power times element k=80 = 720

source roles = [power exponent80, 720, 89]

derived = [8, 8]

Field ORIGINAL previous power times element k=81 := mul(8, 24)

Field ORIGINAL previous power times element k=81 = 192

source roles = [power exponent81, 192, 89]

derived = [2, 14]

Field ORIGINAL previous power times element k=82 := mul(14, 24)

Field ORIGINAL previous power times element k=82 = 336

source roles = [power exponent82, 336, 89]

derived = [3, 69]

Field ORIGINAL previous power times element k=83 := mul(69, 24)

Field ORIGINAL previous power times element k=83 = 1656

source roles = [power exponent83, 1656, 89]

derived = [18, 54]

Field ORIGINAL previous power times element k=84 := mul(54, 24)

Field ORIGINAL previous power times element k=84 = 1296

source roles = [power exponent84, 1296, 89]

derived = [14, 50]

Field ORIGINAL previous power times element k=85 := mul(50, 24)

Field ORIGINAL previous power times element k=85 = 1200

source roles = [power exponent85, 1200, 89]

derived = [13, 43]

Field ORIGINAL previous power times element k=86 := mul(43, 24)

Field ORIGINAL previous power times element k=86 = 1032

source roles = [power exponent86, 1032, 89]

derived = [11, 53]

Field ORIGINAL previous power times element k=87 := mul(53, 24)

Field ORIGINAL previous power times element k=87 = 1272

source roles = [power exponent87, 1272, 89]

derived = [14, 26]

Field ORIGINAL previous power times element k=88 := mul(26, 24)

Field ORIGINAL previous power times element k=88 = 624

source roles = [power exponent88, 624, 89]

derived = [7, 1]

source roles = [89, 24, [24, 42, 29, 73, 61, 40, 70, 78, 3, 72, 37, 87, 41, 5, 31, 32, 56, 9, 38, 22, 83, 34, 15, 4, 7, 79, 27, 25, 66, 71, 13, 45, 12, 21, 59, 81, 75, 20, 35, 39, 46, 36, 63, 88, 65, 47, 60, 16, 28, 49, 19, 11, 86, 17, 52, 2, 48, 84, 58, 57, 33, 80, 51, 67, 6, 55, 74, 85, 82, 10, 62, 64, 23, 18, 76, 44, 77, 68, 30, 8, 14, 69, 54, 50, 43, 53, 26, 1]]

derived = [1, Yes]

Reverse checks

• S179 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S178 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S177 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=88 with exact rational arithmetic, not rounded decimals.
• S176 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S175 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=87 with exact rational arithmetic, not rounded decimals.
• S174 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S173 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=86 with exact rational arithmetic, not rounded decimals.
• S172 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S171 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=85 with exact rational arithmetic, not rounded decimals.
• S170 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S169 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=84 with exact rational arithmetic, not rounded decimals.
• S168 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S167 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=83 with exact rational arithmetic, not rounded decimals.
• S166 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S165 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=82 with exact rational arithmetic, not rounded decimals.
• S164 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S163 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=81 with exact rational arithmetic, not rounded decimals.
• S162 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S161 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=80 with exact rational arithmetic, not rounded decimals.
• S160 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S159 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=79 with exact rational arithmetic, not rounded decimals.
• S158 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S157 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=78 with exact rational arithmetic, not rounded decimals.
• S156 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S155 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=77 with exact rational arithmetic, not rounded decimals.
• S154 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S153 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=76 with exact rational arithmetic, not rounded decimals.
• S152 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S151 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=75 with exact rational arithmetic, not rounded decimals.
• S150 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S149 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=74 with exact rational arithmetic, not rounded decimals.
• S148 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S147 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=73 with exact rational arithmetic, not rounded decimals.
• S146 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S145 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=72 with exact rational arithmetic, not rounded decimals.
• S144 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S143 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=71 with exact rational arithmetic, not rounded decimals.
• S142 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S141 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=70 with exact rational arithmetic, not rounded decimals.
• S140 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S139 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=69 with exact rational arithmetic, not rounded decimals.
• S138 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S137 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=68 with exact rational arithmetic, not rounded decimals.
• S136 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S135 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=67 with exact rational arithmetic, not rounded decimals.
• S134 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S133 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=66 with exact rational arithmetic, not rounded decimals.
• S132 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S131 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=65 with exact rational arithmetic, not rounded decimals.
• S130 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S129 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=64 with exact rational arithmetic, not rounded decimals.
• S128 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S127 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=63 with exact rational arithmetic, not rounded decimals.
• S126 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S125 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=62 with exact rational arithmetic, not rounded decimals.
• S124 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S123 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=61 with exact rational arithmetic, not rounded decimals.
• S122 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S121 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=60 with exact rational arithmetic, not rounded decimals.
• S120 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S119 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=59 with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S117 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=58 with exact rational arithmetic, not rounded decimals.
• S116 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S115 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=57 with exact rational arithmetic, not rounded decimals.
• S114 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S113 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=56 with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S111 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=55 with exact rational arithmetic, not rounded decimals.
• S110 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S109 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=54 with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S107 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=53 with exact rational arithmetic, not rounded decimals.
• S106 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S105 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=52 with exact rational arithmetic, not rounded decimals.
• S104 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S103 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=51 with exact rational arithmetic, not rounded decimals.
• S102 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S101 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=50 with exact rational arithmetic, not rounded decimals.
• S100 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S099 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=49 with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S097 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=48 with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S095 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=47 with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S093 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=46 with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S091 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=45 with exact rational arithmetic, not rounded decimals.
• S090 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S089 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=44 with exact rational arithmetic, not rounded decimals.
• S088 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S087 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=43 with exact rational arithmetic, not rounded decimals.
• S086 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S085 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=42 with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S083 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=41 with exact rational arithmetic, not rounded decimals.
• S082 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S081 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=40 with exact rational arithmetic, not rounded decimals.
• S080 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S079 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=39 with exact rational arithmetic, not rounded decimals.
• S078 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S077 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=38 with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S075 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=37 with exact rational arithmetic, not rounded decimals.
• S074 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S073 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=36 with exact rational arithmetic, not rounded decimals.
• S072 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S071 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=35 with exact rational arithmetic, not rounded decimals.
• S070 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S069 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=34 with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S067 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=33 with exact rational arithmetic, not rounded decimals.
• S066 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S065 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=32 with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S063 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=31 with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S061 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=30 with exact rational arithmetic, not rounded decimals.
• S060 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S059 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=29 with exact rational arithmetic, not rounded decimals.
• S058 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S057 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=28 with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S055 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=27 with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S053 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=26 with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S051 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=25 with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S049 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=24 with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S047 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=23 with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S045 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=22 with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S043 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=21 with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S041 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=20 with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S039 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=19 with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S037 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=18 with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S035 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=17 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S033 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=16 with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S031 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=15 with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S029 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=14 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S027 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=13 with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S025 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=12 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S023 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=11 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S021 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=10 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S019 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=9 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S017 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=8 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S015 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=7 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S013 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=6 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S011 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=5 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S009 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=4 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S007 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=3 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S005 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=2 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S003 · premise_and_result_audit — Recompute Field ORIGINAL previous power times element k=1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
• S001 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.

Result

(24, [24, 42, 29, 73, 61, 40, 70, 78, 3, 72, 37, 87, 41, 5, 31, 32, 56, 9, 38, 22, 83, 34, 15, 4, 7, 79, 27, 25, 66, 71, 13, 45, 12, 21, 59, 81, 75, 20, 35, 39, 46, 36, 63, 88, 65, 47, 60, 16, 28, 49, 19, 11, 86, 17, 52, 2, 48, 84, 58, 57, 33, 80, 51, 67, 6, 55, 74, 85, 82, 10, 62, 64, 23, 18, 76, 44, 77, 68, 30, 8, 14, 69, 54, 50, 43, 53, 26, 1], 1, Yes, nonzero_multiplicative_group_cycle)

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.