Question 33695: The Nonzero Multiplicative Group Gives a p−1 Power Identity

Original question and givens

Use the complete prime field Z_71, with prime p=71, 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=20. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.

• G001 — Use the complete prime field Z_71, with prime p=71, 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=20. 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 = [71, [2, 3, 4, 5, 6, 7, 8]]

derived = [[1, 2, 3, 1, 5, 1, 7], Yes]

Field ORIGINAL previous power times element k=1 := mul(1, 20)

Field ORIGINAL previous power times element k=1 = 20

source roles = [power exponent1, 20, 71]

derived = [0, 20]

Field ORIGINAL previous power times element k=2 := mul(20, 20)

Field ORIGINAL previous power times element k=2 = 400

source roles = [power exponent2, 400, 71]

derived = [5, 45]

Field ORIGINAL previous power times element k=3 := mul(45, 20)

Field ORIGINAL previous power times element k=3 = 900

source roles = [power exponent3, 900, 71]

derived = [12, 48]

Field ORIGINAL previous power times element k=4 := mul(48, 20)

Field ORIGINAL previous power times element k=4 = 960

source roles = [power exponent4, 960, 71]

derived = [13, 37]

Field ORIGINAL previous power times element k=5 := mul(37, 20)

Field ORIGINAL previous power times element k=5 = 740

source roles = [power exponent5, 740, 71]

derived = [10, 30]

Field ORIGINAL previous power times element k=6 := mul(30, 20)

Field ORIGINAL previous power times element k=6 = 600

source roles = [power exponent6, 600, 71]

derived = [8, 32]

Field ORIGINAL previous power times element k=7 := mul(32, 20)

Field ORIGINAL previous power times element k=7 = 640

source roles = [power exponent7, 640, 71]

derived = [9, 1]

Field ORIGINAL previous power times element k=8 := mul(1, 20)

Field ORIGINAL previous power times element k=8 = 20

source roles = [power exponent8, 20, 71]

derived = [0, 20]

Field ORIGINAL previous power times element k=9 := mul(20, 20)

Field ORIGINAL previous power times element k=9 = 400

source roles = [power exponent9, 400, 71]

derived = [5, 45]

Field ORIGINAL previous power times element k=10 := mul(45, 20)

Field ORIGINAL previous power times element k=10 = 900

source roles = [power exponent10, 900, 71]

derived = [12, 48]

Field ORIGINAL previous power times element k=11 := mul(48, 20)

Field ORIGINAL previous power times element k=11 = 960

source roles = [power exponent11, 960, 71]

derived = [13, 37]

Field ORIGINAL previous power times element k=12 := mul(37, 20)

Field ORIGINAL previous power times element k=12 = 740

source roles = [power exponent12, 740, 71]

derived = [10, 30]

Field ORIGINAL previous power times element k=13 := mul(30, 20)

Field ORIGINAL previous power times element k=13 = 600

source roles = [power exponent13, 600, 71]

derived = [8, 32]

Field ORIGINAL previous power times element k=14 := mul(32, 20)

Field ORIGINAL previous power times element k=14 = 640

source roles = [power exponent14, 640, 71]

derived = [9, 1]

Field ORIGINAL previous power times element k=15 := mul(1, 20)

Field ORIGINAL previous power times element k=15 = 20

source roles = [power exponent15, 20, 71]

derived = [0, 20]

Field ORIGINAL previous power times element k=16 := mul(20, 20)

Field ORIGINAL previous power times element k=16 = 400

source roles = [power exponent16, 400, 71]

derived = [5, 45]

Field ORIGINAL previous power times element k=17 := mul(45, 20)

Field ORIGINAL previous power times element k=17 = 900

source roles = [power exponent17, 900, 71]

derived = [12, 48]

Field ORIGINAL previous power times element k=18 := mul(48, 20)

Field ORIGINAL previous power times element k=18 = 960

source roles = [power exponent18, 960, 71]

derived = [13, 37]

Field ORIGINAL previous power times element k=19 := mul(37, 20)

Field ORIGINAL previous power times element k=19 = 740

source roles = [power exponent19, 740, 71]

derived = [10, 30]

Field ORIGINAL previous power times element k=20 := mul(30, 20)

Field ORIGINAL previous power times element k=20 = 600

source roles = [power exponent20, 600, 71]

derived = [8, 32]

Field ORIGINAL previous power times element k=21 := mul(32, 20)

Field ORIGINAL previous power times element k=21 = 640

source roles = [power exponent21, 640, 71]

derived = [9, 1]

Field ORIGINAL previous power times element k=22 := mul(1, 20)

Field ORIGINAL previous power times element k=22 = 20

source roles = [power exponent22, 20, 71]

derived = [0, 20]

Field ORIGINAL previous power times element k=23 := mul(20, 20)

Field ORIGINAL previous power times element k=23 = 400

source roles = [power exponent23, 400, 71]

derived = [5, 45]

Field ORIGINAL previous power times element k=24 := mul(45, 20)

Field ORIGINAL previous power times element k=24 = 900

source roles = [power exponent24, 900, 71]

derived = [12, 48]

Field ORIGINAL previous power times element k=25 := mul(48, 20)

Field ORIGINAL previous power times element k=25 = 960

source roles = [power exponent25, 960, 71]

derived = [13, 37]

Field ORIGINAL previous power times element k=26 := mul(37, 20)

Field ORIGINAL previous power times element k=26 = 740

source roles = [power exponent26, 740, 71]

derived = [10, 30]

Field ORIGINAL previous power times element k=27 := mul(30, 20)

Field ORIGINAL previous power times element k=27 = 600

source roles = [power exponent27, 600, 71]

derived = [8, 32]

Field ORIGINAL previous power times element k=28 := mul(32, 20)

Field ORIGINAL previous power times element k=28 = 640

source roles = [power exponent28, 640, 71]

derived = [9, 1]

Field ORIGINAL previous power times element k=29 := mul(1, 20)

Field ORIGINAL previous power times element k=29 = 20

source roles = [power exponent29, 20, 71]

derived = [0, 20]

Field ORIGINAL previous power times element k=30 := mul(20, 20)

Field ORIGINAL previous power times element k=30 = 400

source roles = [power exponent30, 400, 71]

derived = [5, 45]

Field ORIGINAL previous power times element k=31 := mul(45, 20)

Field ORIGINAL previous power times element k=31 = 900

source roles = [power exponent31, 900, 71]

derived = [12, 48]

Field ORIGINAL previous power times element k=32 := mul(48, 20)

Field ORIGINAL previous power times element k=32 = 960

source roles = [power exponent32, 960, 71]

derived = [13, 37]

Field ORIGINAL previous power times element k=33 := mul(37, 20)

Field ORIGINAL previous power times element k=33 = 740

source roles = [power exponent33, 740, 71]

derived = [10, 30]

Field ORIGINAL previous power times element k=34 := mul(30, 20)

Field ORIGINAL previous power times element k=34 = 600

source roles = [power exponent34, 600, 71]

derived = [8, 32]

Field ORIGINAL previous power times element k=35 := mul(32, 20)

Field ORIGINAL previous power times element k=35 = 640

source roles = [power exponent35, 640, 71]

derived = [9, 1]

Field ORIGINAL previous power times element k=36 := mul(1, 20)

Field ORIGINAL previous power times element k=36 = 20

source roles = [power exponent36, 20, 71]

derived = [0, 20]

Field ORIGINAL previous power times element k=37 := mul(20, 20)

Field ORIGINAL previous power times element k=37 = 400

source roles = [power exponent37, 400, 71]

derived = [5, 45]

Field ORIGINAL previous power times element k=38 := mul(45, 20)

Field ORIGINAL previous power times element k=38 = 900

source roles = [power exponent38, 900, 71]

derived = [12, 48]

Field ORIGINAL previous power times element k=39 := mul(48, 20)

Field ORIGINAL previous power times element k=39 = 960

source roles = [power exponent39, 960, 71]

derived = [13, 37]

Field ORIGINAL previous power times element k=40 := mul(37, 20)

Field ORIGINAL previous power times element k=40 = 740

source roles = [power exponent40, 740, 71]

derived = [10, 30]

Field ORIGINAL previous power times element k=41 := mul(30, 20)

Field ORIGINAL previous power times element k=41 = 600

source roles = [power exponent41, 600, 71]

derived = [8, 32]

Field ORIGINAL previous power times element k=42 := mul(32, 20)

Field ORIGINAL previous power times element k=42 = 640

source roles = [power exponent42, 640, 71]

derived = [9, 1]

Field ORIGINAL previous power times element k=43 := mul(1, 20)

Field ORIGINAL previous power times element k=43 = 20

source roles = [power exponent43, 20, 71]

derived = [0, 20]

Field ORIGINAL previous power times element k=44 := mul(20, 20)

Field ORIGINAL previous power times element k=44 = 400

source roles = [power exponent44, 400, 71]

derived = [5, 45]

Field ORIGINAL previous power times element k=45 := mul(45, 20)

Field ORIGINAL previous power times element k=45 = 900

source roles = [power exponent45, 900, 71]

derived = [12, 48]

Field ORIGINAL previous power times element k=46 := mul(48, 20)

Field ORIGINAL previous power times element k=46 = 960

source roles = [power exponent46, 960, 71]

derived = [13, 37]

Field ORIGINAL previous power times element k=47 := mul(37, 20)

Field ORIGINAL previous power times element k=47 = 740

source roles = [power exponent47, 740, 71]

derived = [10, 30]

Field ORIGINAL previous power times element k=48 := mul(30, 20)

Field ORIGINAL previous power times element k=48 = 600

source roles = [power exponent48, 600, 71]

derived = [8, 32]

Field ORIGINAL previous power times element k=49 := mul(32, 20)

Field ORIGINAL previous power times element k=49 = 640

source roles = [power exponent49, 640, 71]

derived = [9, 1]

Field ORIGINAL previous power times element k=50 := mul(1, 20)

Field ORIGINAL previous power times element k=50 = 20

source roles = [power exponent50, 20, 71]

derived = [0, 20]

Field ORIGINAL previous power times element k=51 := mul(20, 20)

Field ORIGINAL previous power times element k=51 = 400

source roles = [power exponent51, 400, 71]

derived = [5, 45]

Field ORIGINAL previous power times element k=52 := mul(45, 20)

Field ORIGINAL previous power times element k=52 = 900

source roles = [power exponent52, 900, 71]

derived = [12, 48]

Field ORIGINAL previous power times element k=53 := mul(48, 20)

Field ORIGINAL previous power times element k=53 = 960

source roles = [power exponent53, 960, 71]

derived = [13, 37]

Field ORIGINAL previous power times element k=54 := mul(37, 20)

Field ORIGINAL previous power times element k=54 = 740

source roles = [power exponent54, 740, 71]

derived = [10, 30]

Field ORIGINAL previous power times element k=55 := mul(30, 20)

Field ORIGINAL previous power times element k=55 = 600

source roles = [power exponent55, 600, 71]

derived = [8, 32]

Field ORIGINAL previous power times element k=56 := mul(32, 20)

Field ORIGINAL previous power times element k=56 = 640

source roles = [power exponent56, 640, 71]

derived = [9, 1]

Field ORIGINAL previous power times element k=57 := mul(1, 20)

Field ORIGINAL previous power times element k=57 = 20

source roles = [power exponent57, 20, 71]

derived = [0, 20]

Field ORIGINAL previous power times element k=58 := mul(20, 20)

Field ORIGINAL previous power times element k=58 = 400

source roles = [power exponent58, 400, 71]

derived = [5, 45]

Field ORIGINAL previous power times element k=59 := mul(45, 20)

Field ORIGINAL previous power times element k=59 = 900

source roles = [power exponent59, 900, 71]

derived = [12, 48]

Field ORIGINAL previous power times element k=60 := mul(48, 20)

Field ORIGINAL previous power times element k=60 = 960

source roles = [power exponent60, 960, 71]

derived = [13, 37]

Field ORIGINAL previous power times element k=61 := mul(37, 20)

Field ORIGINAL previous power times element k=61 = 740

source roles = [power exponent61, 740, 71]

derived = [10, 30]

Field ORIGINAL previous power times element k=62 := mul(30, 20)

Field ORIGINAL previous power times element k=62 = 600

source roles = [power exponent62, 600, 71]

derived = [8, 32]

Field ORIGINAL previous power times element k=63 := mul(32, 20)

Field ORIGINAL previous power times element k=63 = 640

source roles = [power exponent63, 640, 71]

derived = [9, 1]

Field ORIGINAL previous power times element k=64 := mul(1, 20)

Field ORIGINAL previous power times element k=64 = 20

source roles = [power exponent64, 20, 71]

derived = [0, 20]

Field ORIGINAL previous power times element k=65 := mul(20, 20)

Field ORIGINAL previous power times element k=65 = 400

source roles = [power exponent65, 400, 71]

derived = [5, 45]

Field ORIGINAL previous power times element k=66 := mul(45, 20)

Field ORIGINAL previous power times element k=66 = 900

source roles = [power exponent66, 900, 71]

derived = [12, 48]

Field ORIGINAL previous power times element k=67 := mul(48, 20)

Field ORIGINAL previous power times element k=67 = 960

source roles = [power exponent67, 960, 71]

derived = [13, 37]

Field ORIGINAL previous power times element k=68 := mul(37, 20)

Field ORIGINAL previous power times element k=68 = 740

source roles = [power exponent68, 740, 71]

derived = [10, 30]

Field ORIGINAL previous power times element k=69 := mul(30, 20)

Field ORIGINAL previous power times element k=69 = 600

source roles = [power exponent69, 600, 71]

derived = [8, 32]

Field ORIGINAL previous power times element k=70 := mul(32, 20)

Field ORIGINAL previous power times element k=70 = 640

source roles = [power exponent70, 640, 71]

derived = [9, 1]

source roles = [71, 20, [20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1]]

derived = [1, Yes]

Reverse checks

• S143 · 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.
• 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

(20, [20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 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.