Question 33696: The Nonzero Multiplicative Group Gives a p−1 Power Identity

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0066-33696 · q-20261014-advance-20261031-r06-finite-field-0046

Original question and givens

Use the complete prime field Z_73, with prime p=73, 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=21. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.

G001 — Use the complete prime field Z_73, with prime p=73, 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=21. 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

S001 · implies

Before

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

After

derived = [fielddeep-power-cycle, Exact_prime_field_Zp]

Conditions: none

THIS source uses LOCAL-hyphen operations with exact keys/domain.Keep source field order and numeric/Boolean/string roles;integer rational texts omit/1.Original questions,English,MKOs,ASTs and statuses remain unchanged.

Step identifiers and dependencies

Rule: R66_EXACT_source_bank_keys_local_operation_domain · Uses: G001

S002 · implies

Before

source roles = [73, [2, 3, 4, 5, 6, 7, 8]]

After

derived = [[1, 1, 1, 3, 1, 3, 1], Yes]

Conditions: none

Trial EVERY divisor2 through floor(sqrt(p));a composite would have a factor there.The empty test for p=2 is valid.This qualifies THIS prime field Z_p,not every finite field’s order or every finite ring as a field.

Step identifiers and dependencies

Rule: Field_PRIME_complete_trial_divisor_qualification · Uses: G001, S001

S003 · calculation

Before

Field ORIGINAL previous power times element k=1 := mul(1, 21)

After

Field ORIGINAL previous power times element k=1 = 21

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

source roles = [power exponent1, 21, 73]

After

derived = [0, 21]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S003

S005 · calculation

Before

Field ORIGINAL previous power times element k=2 := mul(21, 21)

After

Field ORIGINAL previous power times element k=2 = 441

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

source roles = [power exponent2, 441, 73]

After

derived = [6, 3]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S005

S007 · calculation

Before

Field ORIGINAL previous power times element k=3 := mul(3, 21)

After

Field ORIGINAL previous power times element k=3 = 63

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

source roles = [power exponent3, 63, 73]

After

derived = [0, 63]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S007

S009 · calculation

Before

Field ORIGINAL previous power times element k=4 := mul(63, 21)

After

Field ORIGINAL previous power times element k=4 = 1323

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

source roles = [power exponent4, 1323, 73]

After

derived = [18, 9]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S009

S011 · calculation

Before

Field ORIGINAL previous power times element k=5 := mul(9, 21)

After

Field ORIGINAL previous power times element k=5 = 189

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

source roles = [power exponent5, 189, 73]

After

derived = [2, 43]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S011

S013 · calculation

Before

Field ORIGINAL previous power times element k=6 := mul(43, 21)

After

Field ORIGINAL previous power times element k=6 = 903

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

source roles = [power exponent6, 903, 73]

After

derived = [12, 27]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S013

S015 · calculation

Before

Field ORIGINAL previous power times element k=7 := mul(27, 21)

After

Field ORIGINAL previous power times element k=7 = 567

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · implies

Before

source roles = [power exponent7, 567, 73]

After

derived = [7, 56]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S015

S017 · calculation

Before

Field ORIGINAL previous power times element k=8 := mul(56, 21)

After

Field ORIGINAL previous power times element k=8 = 1176

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · implies

Before

source roles = [power exponent8, 1176, 73]

After

derived = [16, 8]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S017

S019 · calculation

Before

Field ORIGINAL previous power times element k=9 := mul(8, 21)

After

Field ORIGINAL previous power times element k=9 = 168

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · implies

Before

source roles = [power exponent9, 168, 73]

After

derived = [2, 22]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S019

S021 · calculation

Before

Field ORIGINAL previous power times element k=10 := mul(22, 21)

After

Field ORIGINAL previous power times element k=10 = 462

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · implies

Before

source roles = [power exponent10, 462, 73]

After

derived = [6, 24]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S021

S023 · calculation

Before

Field ORIGINAL previous power times element k=11 := mul(24, 21)

After

Field ORIGINAL previous power times element k=11 = 504

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · implies

Before

source roles = [power exponent11, 504, 73]

After

derived = [6, 66]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S023

S025 · calculation

Before

Field ORIGINAL previous power times element k=12 := mul(66, 21)

After

Field ORIGINAL previous power times element k=12 = 1386

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · implies

Before

source roles = [power exponent12, 1386, 73]

After

derived = [18, 72]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S025

S027 · calculation

Before

Field ORIGINAL previous power times element k=13 := mul(72, 21)

After

Field ORIGINAL previous power times element k=13 = 1512

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · implies

Before

source roles = [power exponent13, 1512, 73]

After

derived = [20, 52]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S027

S029 · calculation

Before

Field ORIGINAL previous power times element k=14 := mul(52, 21)

After

Field ORIGINAL previous power times element k=14 = 1092

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · implies

Before

source roles = [power exponent14, 1092, 73]

After

derived = [14, 70]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S029

S031 · calculation

Before

Field ORIGINAL previous power times element k=15 := mul(70, 21)

After

Field ORIGINAL previous power times element k=15 = 1470

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · implies

Before

source roles = [power exponent15, 1470, 73]

After

derived = [20, 10]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S031

S033 · calculation

Before

Field ORIGINAL previous power times element k=16 := mul(10, 21)

After

Field ORIGINAL previous power times element k=16 = 210

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · implies

Before

source roles = [power exponent16, 210, 73]

After

derived = [2, 64]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S033

S035 · calculation

Before

Field ORIGINAL previous power times element k=17 := mul(64, 21)

After

Field ORIGINAL previous power times element k=17 = 1344

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · implies

Before

source roles = [power exponent17, 1344, 73]

After

derived = [18, 30]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S035

S037 · calculation

Before

Field ORIGINAL previous power times element k=18 := mul(30, 21)

After

Field ORIGINAL previous power times element k=18 = 630

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · implies

Before

source roles = [power exponent18, 630, 73]

After

derived = [8, 46]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S037

S039 · calculation

Before

Field ORIGINAL previous power times element k=19 := mul(46, 21)

After

Field ORIGINAL previous power times element k=19 = 966

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · implies

Before

source roles = [power exponent19, 966, 73]

After

derived = [13, 17]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S039

S041 · calculation

Before

Field ORIGINAL previous power times element k=20 := mul(17, 21)

After

Field ORIGINAL previous power times element k=20 = 357

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · implies

Before

source roles = [power exponent20, 357, 73]

After

derived = [4, 65]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S041

S043 · calculation

Before

Field ORIGINAL previous power times element k=21 := mul(65, 21)

After

Field ORIGINAL previous power times element k=21 = 1365

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · implies

Before

source roles = [power exponent21, 1365, 73]

After

derived = [18, 51]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S043

S045 · calculation

Before

Field ORIGINAL previous power times element k=22 := mul(51, 21)

After

Field ORIGINAL previous power times element k=22 = 1071

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · implies

Before

source roles = [power exponent22, 1071, 73]

After

derived = [14, 49]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S045

S047 · calculation

Before

Field ORIGINAL previous power times element k=23 := mul(49, 21)

After

Field ORIGINAL previous power times element k=23 = 1029

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · implies

Before

source roles = [power exponent23, 1029, 73]

After

derived = [14, 7]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S047

S049 · calculation

Before

Field ORIGINAL previous power times element k=24 := mul(7, 21)

After

Field ORIGINAL previous power times element k=24 = 147

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · implies

Before

source roles = [power exponent24, 147, 73]

After

derived = [2, 1]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S049

S051 · calculation

Before

Field ORIGINAL previous power times element k=25 := mul(1, 21)

After

Field ORIGINAL previous power times element k=25 = 21

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · implies

Before

source roles = [power exponent25, 21, 73]

After

derived = [0, 21]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S051

S053 · calculation

Before

Field ORIGINAL previous power times element k=26 := mul(21, 21)

After

Field ORIGINAL previous power times element k=26 = 441

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · implies

Before

source roles = [power exponent26, 441, 73]

After

derived = [6, 3]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S053

S055 · calculation

Before

Field ORIGINAL previous power times element k=27 := mul(3, 21)

After

Field ORIGINAL previous power times element k=27 = 63

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S054

S056 · implies

Before

source roles = [power exponent27, 63, 73]

After

derived = [0, 63]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S055

S057 · calculation

Before

Field ORIGINAL previous power times element k=28 := mul(63, 21)

After

Field ORIGINAL previous power times element k=28 = 1323

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S056

S058 · implies

Before

source roles = [power exponent28, 1323, 73]

After

derived = [18, 9]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S057

S059 · calculation

Before

Field ORIGINAL previous power times element k=29 := mul(9, 21)

After

Field ORIGINAL previous power times element k=29 = 189

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S058

S060 · implies

Before

source roles = [power exponent29, 189, 73]

After

derived = [2, 43]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S059

S061 · calculation

Before

Field ORIGINAL previous power times element k=30 := mul(43, 21)

After

Field ORIGINAL previous power times element k=30 = 903

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S060

S062 · implies

Before

source roles = [power exponent30, 903, 73]

After

derived = [12, 27]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S061

S063 · calculation

Before

Field ORIGINAL previous power times element k=31 := mul(27, 21)

After

Field ORIGINAL previous power times element k=31 = 567

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S062

S064 · implies

Before

source roles = [power exponent31, 567, 73]

After

derived = [7, 56]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S063

S065 · calculation

Before

Field ORIGINAL previous power times element k=32 := mul(56, 21)

After

Field ORIGINAL previous power times element k=32 = 1176

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S064

S066 · implies

Before

source roles = [power exponent32, 1176, 73]

After

derived = [16, 8]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S065

S067 · calculation

Before

Field ORIGINAL previous power times element k=33 := mul(8, 21)

After

Field ORIGINAL previous power times element k=33 = 168

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S066

S068 · implies

Before

source roles = [power exponent33, 168, 73]

After

derived = [2, 22]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S067

S069 · calculation

Before

Field ORIGINAL previous power times element k=34 := mul(22, 21)

After

Field ORIGINAL previous power times element k=34 = 462

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S068

S070 · implies

Before

source roles = [power exponent34, 462, 73]

After

derived = [6, 24]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S069

S071 · calculation

Before

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

After

Field ORIGINAL previous power times element k=35 = 504

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S070

S072 · implies

Before

source roles = [power exponent35, 504, 73]

After

derived = [6, 66]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S071

S073 · calculation

Before

Field ORIGINAL previous power times element k=36 := mul(66, 21)

After

Field ORIGINAL previous power times element k=36 = 1386

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S072

S074 · implies

Before

source roles = [power exponent36, 1386, 73]

After

derived = [18, 72]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S073

S075 · calculation

Before

Field ORIGINAL previous power times element k=37 := mul(72, 21)

After

Field ORIGINAL previous power times element k=37 = 1512

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S074

S076 · implies

Before

source roles = [power exponent37, 1512, 73]

After

derived = [20, 52]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S075

S077 · calculation

Before

Field ORIGINAL previous power times element k=38 := mul(52, 21)

After

Field ORIGINAL previous power times element k=38 = 1092

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S076

S078 · implies

Before

source roles = [power exponent38, 1092, 73]

After

derived = [14, 70]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S077

S079 · calculation

Before

Field ORIGINAL previous power times element k=39 := mul(70, 21)

After

Field ORIGINAL previous power times element k=39 = 1470

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S078

S080 · implies

Before

source roles = [power exponent39, 1470, 73]

After

derived = [20, 10]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S079

S081 · calculation

Before

Field ORIGINAL previous power times element k=40 := mul(10, 21)

After

Field ORIGINAL previous power times element k=40 = 210

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S080

S082 · implies

Before

source roles = [power exponent40, 210, 73]

After

derived = [2, 64]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S081

S083 · calculation

Before

Field ORIGINAL previous power times element k=41 := mul(64, 21)

After

Field ORIGINAL previous power times element k=41 = 1344

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S082

S084 · implies

Before

source roles = [power exponent41, 1344, 73]

After

derived = [18, 30]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S083

S085 · calculation

Before

Field ORIGINAL previous power times element k=42 := mul(30, 21)

After

Field ORIGINAL previous power times element k=42 = 630

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S084

S086 · implies

Before

source roles = [power exponent42, 630, 73]

After

derived = [8, 46]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S085

S087 · calculation

Before

Field ORIGINAL previous power times element k=43 := mul(46, 21)

After

Field ORIGINAL previous power times element k=43 = 966

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S086

S088 · implies

Before

source roles = [power exponent43, 966, 73]

After

derived = [13, 17]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S087

S089 · calculation

Before

Field ORIGINAL previous power times element k=44 := mul(17, 21)

After

Field ORIGINAL previous power times element k=44 = 357

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S088

S090 · implies

Before

source roles = [power exponent44, 357, 73]

After

derived = [4, 65]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S089

S091 · calculation

Before

Field ORIGINAL previous power times element k=45 := mul(65, 21)

After

Field ORIGINAL previous power times element k=45 = 1365

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S090

S092 · implies

Before

source roles = [power exponent45, 1365, 73]

After

derived = [18, 51]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S091

S093 · calculation

Before

Field ORIGINAL previous power times element k=46 := mul(51, 21)

After

Field ORIGINAL previous power times element k=46 = 1071

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S092

S094 · implies

Before

source roles = [power exponent46, 1071, 73]

After

derived = [14, 49]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S093

S095 · calculation

Before

Field ORIGINAL previous power times element k=47 := mul(49, 21)

After

Field ORIGINAL previous power times element k=47 = 1029

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S094

S096 · implies

Before

source roles = [power exponent47, 1029, 73]

After

derived = [14, 7]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S095

S097 · calculation

Before

Field ORIGINAL previous power times element k=48 := mul(7, 21)

After

Field ORIGINAL previous power times element k=48 = 147

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S096

S098 · implies

Before

source roles = [power exponent48, 147, 73]

After

derived = [2, 1]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S097

S099 · calculation

Before

Field ORIGINAL previous power times element k=49 := mul(1, 21)

After

Field ORIGINAL previous power times element k=49 = 21

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S098

S100 · implies

Before

source roles = [power exponent49, 21, 73]

After

derived = [0, 21]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S099

S101 · calculation

Before

Field ORIGINAL previous power times element k=50 := mul(21, 21)

After

Field ORIGINAL previous power times element k=50 = 441

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S100

S102 · implies

Before

source roles = [power exponent50, 441, 73]

After

derived = [6, 3]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S101

S103 · calculation

Before

Field ORIGINAL previous power times element k=51 := mul(3, 21)

After

Field ORIGINAL previous power times element k=51 = 63

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S102

S104 · implies

Before

source roles = [power exponent51, 63, 73]

After

derived = [0, 63]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S103

S105 · calculation

Before

Field ORIGINAL previous power times element k=52 := mul(63, 21)

After

Field ORIGINAL previous power times element k=52 = 1323

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S104

S106 · implies

Before

source roles = [power exponent52, 1323, 73]

After

derived = [18, 9]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S105

S107 · calculation

Before

Field ORIGINAL previous power times element k=53 := mul(9, 21)

After

Field ORIGINAL previous power times element k=53 = 189

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S106

S108 · implies

Before

source roles = [power exponent53, 189, 73]

After

derived = [2, 43]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S107

S109 · calculation

Before

Field ORIGINAL previous power times element k=54 := mul(43, 21)

After

Field ORIGINAL previous power times element k=54 = 903

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S108

S110 · implies

Before

source roles = [power exponent54, 903, 73]

After

derived = [12, 27]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S109

S111 · calculation

Before

Field ORIGINAL previous power times element k=55 := mul(27, 21)

After

Field ORIGINAL previous power times element k=55 = 567

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S110

S112 · implies

Before

source roles = [power exponent55, 567, 73]

After

derived = [7, 56]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S111

S113 · calculation

Before

Field ORIGINAL previous power times element k=56 := mul(56, 21)

After

Field ORIGINAL previous power times element k=56 = 1176

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S112

S114 · implies

Before

source roles = [power exponent56, 1176, 73]

After

derived = [16, 8]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S113

S115 · calculation

Before

Field ORIGINAL previous power times element k=57 := mul(8, 21)

After

Field ORIGINAL previous power times element k=57 = 168

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S114

S116 · implies

Before

source roles = [power exponent57, 168, 73]

After

derived = [2, 22]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S115

S117 · calculation

Before

Field ORIGINAL previous power times element k=58 := mul(22, 21)

After

Field ORIGINAL previous power times element k=58 = 462

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S116

S118 · implies

Before

source roles = [power exponent58, 462, 73]

After

derived = [6, 24]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S117

S119 · calculation

Before

Field ORIGINAL previous power times element k=59 := mul(24, 21)

After

Field ORIGINAL previous power times element k=59 = 504

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S118

S120 · implies

Before

source roles = [power exponent59, 504, 73]

After

derived = [6, 66]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S119

S121 · calculation

Before

Field ORIGINAL previous power times element k=60 := mul(66, 21)

After

Field ORIGINAL previous power times element k=60 = 1386

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S120

S122 · implies

Before

source roles = [power exponent60, 1386, 73]

After

derived = [18, 72]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S121

S123 · calculation

Before

Field ORIGINAL previous power times element k=61 := mul(72, 21)

After

Field ORIGINAL previous power times element k=61 = 1512

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S122

S124 · implies

Before

source roles = [power exponent61, 1512, 73]

After

derived = [20, 52]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S123

S125 · calculation

Before

Field ORIGINAL previous power times element k=62 := mul(52, 21)

After

Field ORIGINAL previous power times element k=62 = 1092

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S124

S126 · implies

Before

source roles = [power exponent62, 1092, 73]

After

derived = [14, 70]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S125

S127 · calculation

Before

Field ORIGINAL previous power times element k=63 := mul(70, 21)

After

Field ORIGINAL previous power times element k=63 = 1470

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S126

S128 · implies

Before

source roles = [power exponent63, 1470, 73]

After

derived = [20, 10]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S127

S129 · calculation

Before

Field ORIGINAL previous power times element k=64 := mul(10, 21)

After

Field ORIGINAL previous power times element k=64 = 210

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S128

S130 · implies

Before

source roles = [power exponent64, 210, 73]

After

derived = [2, 64]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S129

S131 · calculation

Before

Field ORIGINAL previous power times element k=65 := mul(64, 21)

After

Field ORIGINAL previous power times element k=65 = 1344

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S130

S132 · implies

Before

source roles = [power exponent65, 1344, 73]

After

derived = [18, 30]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S131

S133 · calculation

Before

Field ORIGINAL previous power times element k=66 := mul(30, 21)

After

Field ORIGINAL previous power times element k=66 = 630

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S132

S134 · implies

Before

source roles = [power exponent66, 630, 73]

After

derived = [8, 46]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S133

S135 · calculation

Before

Field ORIGINAL previous power times element k=67 := mul(46, 21)

After

Field ORIGINAL previous power times element k=67 = 966

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S134

S136 · implies

Before

source roles = [power exponent67, 966, 73]

After

derived = [13, 17]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S135

S137 · calculation

Before

Field ORIGINAL previous power times element k=68 := mul(17, 21)

After

Field ORIGINAL previous power times element k=68 = 357

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S136

S138 · implies

Before

source roles = [power exponent68, 357, 73]

After

derived = [4, 65]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S137

S139 · calculation

Before

Field ORIGINAL previous power times element k=69 := mul(65, 21)

After

Field ORIGINAL previous power times element k=69 = 1365

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S138

S140 · implies

Before

source roles = [power exponent69, 1365, 73]

After

derived = [18, 51]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S139

S141 · calculation

Before

Field ORIGINAL previous power times element k=70 := mul(51, 21)

After

Field ORIGINAL previous power times element k=70 = 1071

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S140

S142 · implies

Before

source roles = [power exponent70, 1071, 73]

After

derived = [14, 49]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S141

S143 · calculation

Before

Field ORIGINAL previous power times element k=71 := mul(49, 21)

After

Field ORIGINAL previous power times element k=71 = 1029

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S142

S144 · implies

Before

source roles = [power exponent71, 1029, 73]

After

derived = [14, 7]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S143

S145 · calculation

Before

Field ORIGINAL previous power times element k=72 := mul(7, 21)

After

Field ORIGINAL previous power times element k=72 = 147

Conditions: none

Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S144

S146 · implies

Before

source roles = [power exponent72, 147, 73]

After

derived = [2, 1]

Conditions: none

Use integer Euclidean division: value=n*quotient+remainder with0≤remainder<n.Normalize negative representatives too;do not use floating remainders.

Step identifiers and dependencies

Rule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S145

S147 · implies

Before

source roles = [73, 21, [21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 1, 21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 1, 21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 1]]

After

derived = [1, Yes]

Conditions: none

Retain EVERY exponent1..p−1 and the final1.The nonzero residues form the multiplicative group,but this identity does NOT claim least order p−1 or certify a primitive root.

Step identifiers and dependencies

Rule: Field_NONZERO_p_minus_1_power_identity_NOT_exact_order · Uses: G001, S146

Reverse checks

S147 · 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.
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

(21, [21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 1, 21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 1, 21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 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.

Source hashes and review receipts

input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: 7944a5b22c7f05c14a7eac1ba5148770ca63bf2282062a75032a0f5eef83957f
candidate record SHA-256: 225ec960648ab9715ad5cf9a2a603bc8ca25efccd1b0491f28426da33eedb960
semantic review receipt SHA-256: 99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267
question-first witness SHA-256: d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON