# Question 33668: Frobenius Fixes Every Element of This Prime Field

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0066-33668 · q-20261014-advance-20261031-r06-finite-field-0093



## Original question and givens

Use the complete prime field Z_61, with prime p=61, 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. element=17, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.


- G001 — Use the complete prime field Z_61, with prime p=61, 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. element=17, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.





## 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](https://ocme.evemisslab.com/en/mko/mko-finite-field/)





## Step-by-step derivation

S001 · impliesBefore

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

After

derived = [fielddeep-frobenius, 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 dependenciesRule: R66_EXACT_source_bank_keys_local_operation_domain · Uses: G001





S002 · impliesBefore

source roles = [61, [2, 3, 4, 5, 6, 7]]

After

derived = [[1, 1, 1, 1, 1, 5], 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 dependenciesRule: Field_PRIME_complete_trial_divisor_qualification · Uses: G001, S001





S003 · calculationBefore

Field ORIGINAL previous power times element k=1 := mul(1, 17)

After

Field ORIGINAL previous power times element k=1 = 17

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

source roles = [power exponent1, 17, 61]

After

derived = [0, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S003





S005 · calculationBefore

Field ORIGINAL previous power times element k=2 := mul(17, 17)

After

Field ORIGINAL previous power times element k=2 = 289

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · impliesBefore

source roles = [power exponent2, 289, 61]

After

derived = [4, 45]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S005





S007 · calculationBefore

Field ORIGINAL previous power times element k=3 := mul(45, 17)

After

Field ORIGINAL previous power times element k=3 = 765

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · impliesBefore

source roles = [power exponent3, 765, 61]

After

derived = [12, 33]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S007





S009 · calculationBefore

Field ORIGINAL previous power times element k=4 := mul(33, 17)

After

Field ORIGINAL previous power times element k=4 = 561

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · impliesBefore

source roles = [power exponent4, 561, 61]

After

derived = [9, 12]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S009





S011 · calculationBefore

Field ORIGINAL previous power times element k=5 := mul(12, 17)

After

Field ORIGINAL previous power times element k=5 = 204

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · impliesBefore

source roles = [power exponent5, 204, 61]

After

derived = [3, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S011





S013 · calculationBefore

Field ORIGINAL previous power times element k=6 := mul(21, 17)

After

Field ORIGINAL previous power times element k=6 = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · impliesBefore

source roles = [power exponent6, 357, 61]

After

derived = [5, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S013





S015 · calculationBefore

Field ORIGINAL previous power times element k=7 := mul(52, 17)

After

Field ORIGINAL previous power times element k=7 = 884

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S014





S016 · impliesBefore

source roles = [power exponent7, 884, 61]

After

derived = [14, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S015





S017 · calculationBefore

Field ORIGINAL previous power times element k=8 := mul(30, 17)

After

Field ORIGINAL previous power times element k=8 = 510

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S016





S018 · impliesBefore

source roles = [power exponent8, 510, 61]

After

derived = [8, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S017





S019 · calculationBefore

Field ORIGINAL previous power times element k=9 := mul(22, 17)

After

Field ORIGINAL previous power times element k=9 = 374

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S018





S020 · impliesBefore

source roles = [power exponent9, 374, 61]

After

derived = [6, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S019





S021 · calculationBefore

Field ORIGINAL previous power times element k=10 := mul(8, 17)

After

Field ORIGINAL previous power times element k=10 = 136

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S020





S022 · impliesBefore

source roles = [power exponent10, 136, 61]

After

derived = [2, 14]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S021





S023 · calculationBefore

Field ORIGINAL previous power times element k=11 := mul(14, 17)

After

Field ORIGINAL previous power times element k=11 = 238

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S022





S024 · impliesBefore

source roles = [power exponent11, 238, 61]

After

derived = [3, 55]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S023





S025 · calculationBefore

Field ORIGINAL previous power times element k=12 := mul(55, 17)

After

Field ORIGINAL previous power times element k=12 = 935

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S024





S026 · impliesBefore

source roles = [power exponent12, 935, 61]

After

derived = [15, 20]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S025





S027 · calculationBefore

Field ORIGINAL previous power times element k=13 := mul(20, 17)

After

Field ORIGINAL previous power times element k=13 = 340

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S026





S028 · impliesBefore

source roles = [power exponent13, 340, 61]

After

derived = [5, 35]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S027





S029 · calculationBefore

Field ORIGINAL previous power times element k=14 := mul(35, 17)

After

Field ORIGINAL previous power times element k=14 = 595

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S028





S030 · impliesBefore

source roles = [power exponent14, 595, 61]

After

derived = [9, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S029





S031 · calculationBefore

Field ORIGINAL previous power times element k=15 := mul(46, 17)

After

Field ORIGINAL previous power times element k=15 = 782

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S030





S032 · impliesBefore

source roles = [power exponent15, 782, 61]

After

derived = [12, 50]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S031





S033 · calculationBefore

Field ORIGINAL previous power times element k=16 := mul(50, 17)

After

Field ORIGINAL previous power times element k=16 = 850

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S032





S034 · impliesBefore

source roles = [power exponent16, 850, 61]

After

derived = [13, 57]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S033





S035 · calculationBefore

Field ORIGINAL previous power times element k=17 := mul(57, 17)

After

Field ORIGINAL previous power times element k=17 = 969

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S034





S036 · impliesBefore

source roles = [power exponent17, 969, 61]

After

derived = [15, 54]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S035





S037 · calculationBefore

Field ORIGINAL previous power times element k=18 := mul(54, 17)

After

Field ORIGINAL previous power times element k=18 = 918

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S036





S038 · impliesBefore

source roles = [power exponent18, 918, 61]

After

derived = [15, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S037





S039 · calculationBefore

Field ORIGINAL previous power times element k=19 := mul(3, 17)

After

Field ORIGINAL previous power times element k=19 = 51

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S038





S040 · impliesBefore

source roles = [power exponent19, 51, 61]

After

derived = [0, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S039





S041 · calculationBefore

Field ORIGINAL previous power times element k=20 := mul(51, 17)

After

Field ORIGINAL previous power times element k=20 = 867

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S040





S042 · impliesBefore

source roles = [power exponent20, 867, 61]

After

derived = [14, 13]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S041





S043 · calculationBefore

Field ORIGINAL previous power times element k=21 := mul(13, 17)

After

Field ORIGINAL previous power times element k=21 = 221

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S042





S044 · impliesBefore

source roles = [power exponent21, 221, 61]

After

derived = [3, 38]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S043





S045 · calculationBefore

Field ORIGINAL previous power times element k=22 := mul(38, 17)

After

Field ORIGINAL previous power times element k=22 = 646

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S044





S046 · impliesBefore

source roles = [power exponent22, 646, 61]

After

derived = [10, 36]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S045





S047 · calculationBefore

Field ORIGINAL previous power times element k=23 := mul(36, 17)

After

Field ORIGINAL previous power times element k=23 = 612

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S046





S048 · impliesBefore

source roles = [power exponent23, 612, 61]

After

derived = [10, 2]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S047





S049 · calculationBefore

Field ORIGINAL previous power times element k=24 := mul(2, 17)

After

Field ORIGINAL previous power times element k=24 = 34

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S048





S050 · impliesBefore

source roles = [power exponent24, 34, 61]

After

derived = [0, 34]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S049





S051 · calculationBefore

Field ORIGINAL previous power times element k=25 := mul(34, 17)

After

Field ORIGINAL previous power times element k=25 = 578

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S050





S052 · impliesBefore

source roles = [power exponent25, 578, 61]

After

derived = [9, 29]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S051





S053 · calculationBefore

Field ORIGINAL previous power times element k=26 := mul(29, 17)

After

Field ORIGINAL previous power times element k=26 = 493

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S052





S054 · impliesBefore

source roles = [power exponent26, 493, 61]

After

derived = [8, 5]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S053





S055 · calculationBefore

Field ORIGINAL previous power times element k=27 := mul(5, 17)

After

Field ORIGINAL previous power times element k=27 = 85

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S054





S056 · impliesBefore

source roles = [power exponent27, 85, 61]

After

derived = [1, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S055





S057 · calculationBefore

Field ORIGINAL previous power times element k=28 := mul(24, 17)

After

Field ORIGINAL previous power times element k=28 = 408

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S056





S058 · impliesBefore

source roles = [power exponent28, 408, 61]

After

derived = [6, 42]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S057





S059 · calculationBefore

Field ORIGINAL previous power times element k=29 := mul(42, 17)

After

Field ORIGINAL previous power times element k=29 = 714

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S058





S060 · impliesBefore

source roles = [power exponent29, 714, 61]

After

derived = [11, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S059





S061 · calculationBefore

Field ORIGINAL previous power times element k=30 := mul(43, 17)

After

Field ORIGINAL previous power times element k=30 = 731

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S060





S062 · impliesBefore

source roles = [power exponent30, 731, 61]

After

derived = [11, 60]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S061





S063 · calculationBefore

Field ORIGINAL previous power times element k=31 := mul(60, 17)

After

Field ORIGINAL previous power times element k=31 = 1020

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S062





S064 · impliesBefore

source roles = [power exponent31, 1020, 61]

After

derived = [16, 44]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S063





S065 · calculationBefore

Field ORIGINAL previous power times element k=32 := mul(44, 17)

After

Field ORIGINAL previous power times element k=32 = 748

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S064





S066 · impliesBefore

source roles = [power exponent32, 748, 61]

After

derived = [12, 16]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S065





S067 · calculationBefore

Field ORIGINAL previous power times element k=33 := mul(16, 17)

After

Field ORIGINAL previous power times element k=33 = 272

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S066





S068 · impliesBefore

source roles = [power exponent33, 272, 61]

After

derived = [4, 28]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S067





S069 · calculationBefore

Field ORIGINAL previous power times element k=34 := mul(28, 17)

After

Field ORIGINAL previous power times element k=34 = 476

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S068





S070 · impliesBefore

source roles = [power exponent34, 476, 61]

After

derived = [7, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S069





S071 · calculationBefore

Field ORIGINAL previous power times element k=35 := mul(49, 17)

After

Field ORIGINAL previous power times element k=35 = 833

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S070





S072 · impliesBefore

source roles = [power exponent35, 833, 61]

After

derived = [13, 40]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S071





S073 · calculationBefore

Field ORIGINAL previous power times element k=36 := mul(40, 17)

After

Field ORIGINAL previous power times element k=36 = 680

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S072





S074 · impliesBefore

source roles = [power exponent36, 680, 61]

After

derived = [11, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S073





S075 · calculationBefore

Field ORIGINAL previous power times element k=37 := mul(9, 17)

After

Field ORIGINAL previous power times element k=37 = 153

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S074





S076 · impliesBefore

source roles = [power exponent37, 153, 61]

After

derived = [2, 31]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S075





S077 · calculationBefore

Field ORIGINAL previous power times element k=38 := mul(31, 17)

After

Field ORIGINAL previous power times element k=38 = 527

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S076





S078 · impliesBefore

source roles = [power exponent38, 527, 61]

After

derived = [8, 39]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S077





S079 · calculationBefore

Field ORIGINAL previous power times element k=39 := mul(39, 17)

After

Field ORIGINAL previous power times element k=39 = 663

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S078





S080 · impliesBefore

source roles = [power exponent39, 663, 61]

After

derived = [10, 53]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S079





S081 · calculationBefore

Field ORIGINAL previous power times element k=40 := mul(53, 17)

After

Field ORIGINAL previous power times element k=40 = 901

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S080





S082 · impliesBefore

source roles = [power exponent40, 901, 61]

After

derived = [14, 47]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S081





S083 · calculationBefore

Field ORIGINAL previous power times element k=41 := mul(47, 17)

After

Field ORIGINAL previous power times element k=41 = 799

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S082





S084 · impliesBefore

source roles = [power exponent41, 799, 61]

After

derived = [13, 6]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S083





S085 · calculationBefore

Field ORIGINAL previous power times element k=42 := mul(6, 17)

After

Field ORIGINAL previous power times element k=42 = 102

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S084





S086 · impliesBefore

source roles = [power exponent42, 102, 61]

After

derived = [1, 41]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S085





S087 · calculationBefore

Field ORIGINAL previous power times element k=43 := mul(41, 17)

After

Field ORIGINAL previous power times element k=43 = 697

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S086





S088 · impliesBefore

source roles = [power exponent43, 697, 61]

After

derived = [11, 26]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S087





S089 · calculationBefore

Field ORIGINAL previous power times element k=44 := mul(26, 17)

After

Field ORIGINAL previous power times element k=44 = 442

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S088





S090 · impliesBefore

source roles = [power exponent44, 442, 61]

After

derived = [7, 15]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S089





S091 · calculationBefore

Field ORIGINAL previous power times element k=45 := mul(15, 17)

After

Field ORIGINAL previous power times element k=45 = 255

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S090





S092 · impliesBefore

source roles = [power exponent45, 255, 61]

After

derived = [4, 11]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S091





S093 · calculationBefore

Field ORIGINAL previous power times element k=46 := mul(11, 17)

After

Field ORIGINAL previous power times element k=46 = 187

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S092





S094 · impliesBefore

source roles = [power exponent46, 187, 61]

After

derived = [3, 4]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S093





S095 · calculationBefore

Field ORIGINAL previous power times element k=47 := mul(4, 17)

After

Field ORIGINAL previous power times element k=47 = 68

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S094





S096 · impliesBefore

source roles = [power exponent47, 68, 61]

After

derived = [1, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S095





S097 · calculationBefore

Field ORIGINAL previous power times element k=48 := mul(7, 17)

After

Field ORIGINAL previous power times element k=48 = 119

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S096





S098 · impliesBefore

source roles = [power exponent48, 119, 61]

After

derived = [1, 58]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S097





S099 · calculationBefore

Field ORIGINAL previous power times element k=49 := mul(58, 17)

After

Field ORIGINAL previous power times element k=49 = 986

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S098





S100 · impliesBefore

source roles = [power exponent49, 986, 61]

After

derived = [16, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S099





S101 · calculationBefore

Field ORIGINAL previous power times element k=50 := mul(10, 17)

After

Field ORIGINAL previous power times element k=50 = 170

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S100





S102 · impliesBefore

source roles = [power exponent50, 170, 61]

After

derived = [2, 48]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S101





S103 · calculationBefore

Field ORIGINAL previous power times element k=51 := mul(48, 17)

After

Field ORIGINAL previous power times element k=51 = 816

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S102





S104 · impliesBefore

source roles = [power exponent51, 816, 61]

After

derived = [13, 23]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S103





S105 · calculationBefore

Field ORIGINAL previous power times element k=52 := mul(23, 17)

After

Field ORIGINAL previous power times element k=52 = 391

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S104





S106 · impliesBefore

source roles = [power exponent52, 391, 61]

After

derived = [6, 25]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S105





S107 · calculationBefore

Field ORIGINAL previous power times element k=53 := mul(25, 17)

After

Field ORIGINAL previous power times element k=53 = 425

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S106





S108 · impliesBefore

source roles = [power exponent53, 425, 61]

After

derived = [6, 59]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S107





S109 · calculationBefore

Field ORIGINAL previous power times element k=54 := mul(59, 17)

After

Field ORIGINAL previous power times element k=54 = 1003

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S108





S110 · impliesBefore

source roles = [power exponent54, 1003, 61]

After

derived = [16, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S109





S111 · calculationBefore

Field ORIGINAL previous power times element k=55 := mul(27, 17)

After

Field ORIGINAL previous power times element k=55 = 459

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S110





S112 · impliesBefore

source roles = [power exponent55, 459, 61]

After

derived = [7, 32]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S111





S113 · calculationBefore

Field ORIGINAL previous power times element k=56 := mul(32, 17)

After

Field ORIGINAL previous power times element k=56 = 544

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S112





S114 · impliesBefore

source roles = [power exponent56, 544, 61]

After

derived = [8, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S113





S115 · calculationBefore

Field ORIGINAL previous power times element k=57 := mul(56, 17)

After

Field ORIGINAL previous power times element k=57 = 952

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S114





S116 · impliesBefore

source roles = [power exponent57, 952, 61]

After

derived = [15, 37]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S115





S117 · calculationBefore

Field ORIGINAL previous power times element k=58 := mul(37, 17)

After

Field ORIGINAL previous power times element k=58 = 629

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S116





S118 · impliesBefore

source roles = [power exponent58, 629, 61]

After

derived = [10, 19]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S117





S119 · calculationBefore

Field ORIGINAL previous power times element k=59 := mul(19, 17)

After

Field ORIGINAL previous power times element k=59 = 323

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S118





S120 · impliesBefore

source roles = [power exponent59, 323, 61]

After

derived = [5, 18]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S119





S121 · calculationBefore

Field ORIGINAL previous power times element k=60 := mul(18, 17)

After

Field ORIGINAL previous power times element k=60 = 306

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S120





S122 · impliesBefore

source roles = [power exponent60, 306, 61]

After

derived = [5, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S121





S123 · calculationBefore

Field ORIGINAL previous power times element k=61 := mul(1, 17)

After

Field ORIGINAL previous power times element k=61 = 17

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S122





S124 · impliesBefore

source roles = [power exponent61, 17, 61]

After

derived = [0, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S123





S125 · impliesBefore

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

After

derived = [17, Yes]

Conditions: none

Repeated multiplication checks element^p=element in THIS prime field,including zero.Frobenius need not fix every element of an arbitrary extension field;do not widen the conclusion.

Step identifiers and dependenciesRule: Field_FROBENIUS_prime_field_INCLUDES_zero · Uses: G001, S124









## Reverse checks


- S125 · 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.
- 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

(17, 61, 17, Yes, Frobenius_on_prime_field)

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: 750bd5be6107332869f7a1ffa03ccd2ec37e76b1f8324ce9ed2051de0f3ae8e4
- candidate record SHA-256: 35fbe5c168ce9ff2c26161042f9c08cca9a81de6c4768e97e98e55c61ee0c60c
- semantic review receipt SHA-256: 99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267
- question-first witness SHA-256: d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261031-r06-finite-field-0093.json)