# Question 33645: Characteristic Is the Additive Order of One

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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.

R0066-33645 · q-20261014-advance-20261031-r06-finite-field-0020



## Original question and givens

Use the complete prime field Z_71, with prime p=71, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.


- G001 — Use the complete prime field Z_71, with prime p=71, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.





## 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-characteristic, [domain, operation, p]]

After

derived = [fielddeep-characteristic, 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 = [71, [2, 3, 4, 5, 6, 7, 8]]

After

derived = [[1, 2, 3, 1, 5, 1, 7], 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 · impliesBefore

source roles = [successive sum of one k=0, 0, 71]

After

derived = [0, 0]

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, S002





S004 · impliesBefore

source roles = [successive sum of one k=1, 1, 71]

After

derived = [0, 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, S003





S005 · impliesBefore

source roles = [successive sum of one k=2, 2, 71]

After

derived = [0, 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, S004





S006 · impliesBefore

source roles = [successive sum of one k=3, 3, 71]

After

derived = [0, 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, S005





S007 · impliesBefore

source roles = [successive sum of one k=4, 4, 71]

After

derived = [0, 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, S006





S008 · impliesBefore

source roles = [successive sum of one k=5, 5, 71]

After

derived = [0, 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, S007





S009 · impliesBefore

source roles = [successive sum of one k=6, 6, 71]

After

derived = [0, 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, S008





S010 · impliesBefore

source roles = [successive sum of one k=7, 7, 71]

After

derived = [0, 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, S009





S011 · impliesBefore

source roles = [successive sum of one k=8, 8, 71]

After

derived = [0, 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, S010





S012 · impliesBefore

source roles = [successive sum of one k=9, 9, 71]

After

derived = [0, 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, S011





S013 · impliesBefore

source roles = [successive sum of one k=10, 10, 71]

After

derived = [0, 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, S012





S014 · impliesBefore

source roles = [successive sum of one k=11, 11, 71]

After

derived = [0, 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, S013





S015 · impliesBefore

source roles = [successive sum of one k=12, 12, 71]

After

derived = [0, 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, S014





S016 · impliesBefore

source roles = [successive sum of one k=13, 13, 71]

After

derived = [0, 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, S015





S017 · impliesBefore

source roles = [successive sum of one k=14, 14, 71]

After

derived = [0, 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, S016





S018 · impliesBefore

source roles = [successive sum of one k=15, 15, 71]

After

derived = [0, 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, S017





S019 · impliesBefore

source roles = [successive sum of one k=16, 16, 71]

After

derived = [0, 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, S018





S020 · impliesBefore

source roles = [successive sum of one k=17, 17, 71]

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, S019





S021 · impliesBefore

source roles = [successive sum of one k=18, 18, 71]

After

derived = [0, 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, S020





S022 · impliesBefore

source roles = [successive sum of one k=19, 19, 71]

After

derived = [0, 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, S021





S023 · impliesBefore

source roles = [successive sum of one k=20, 20, 71]

After

derived = [0, 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, S022





S024 · impliesBefore

source roles = [successive sum of one k=21, 21, 71]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S023





S025 · impliesBefore

source roles = [successive sum of one k=22, 22, 71]

After

derived = [0, 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, S024





S026 · impliesBefore

source roles = [successive sum of one k=23, 23, 71]

After

derived = [0, 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, S025





S027 · impliesBefore

source roles = [successive sum of one k=24, 24, 71]

After

derived = [0, 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, S026





S028 · impliesBefore

source roles = [successive sum of one k=25, 25, 71]

After

derived = [0, 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, S027





S029 · impliesBefore

source roles = [successive sum of one k=26, 26, 71]

After

derived = [0, 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, S028





S030 · impliesBefore

source roles = [successive sum of one k=27, 27, 71]

After

derived = [0, 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, S029





S031 · impliesBefore

source roles = [successive sum of one k=28, 28, 71]

After

derived = [0, 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, S030





S032 · impliesBefore

source roles = [successive sum of one k=29, 29, 71]

After

derived = [0, 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, S031





S033 · impliesBefore

source roles = [successive sum of one k=30, 30, 71]

After

derived = [0, 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, S032





S034 · impliesBefore

source roles = [successive sum of one k=31, 31, 71]

After

derived = [0, 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, S033





S035 · impliesBefore

source roles = [successive sum of one k=32, 32, 71]

After

derived = [0, 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, S034





S036 · impliesBefore

source roles = [successive sum of one k=33, 33, 71]

After

derived = [0, 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, S035





S037 · impliesBefore

source roles = [successive sum of one k=34, 34, 71]

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, S036





S038 · impliesBefore

source roles = [successive sum of one k=35, 35, 71]

After

derived = [0, 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, S037





S039 · impliesBefore

source roles = [successive sum of one k=36, 36, 71]

After

derived = [0, 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, S038





S040 · impliesBefore

source roles = [successive sum of one k=37, 37, 71]

After

derived = [0, 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, S039





S041 · impliesBefore

source roles = [successive sum of one k=38, 38, 71]

After

derived = [0, 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, S040





S042 · impliesBefore

source roles = [successive sum of one k=39, 39, 71]

After

derived = [0, 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, S041





S043 · impliesBefore

source roles = [successive sum of one k=40, 40, 71]

After

derived = [0, 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, S042





S044 · impliesBefore

source roles = [successive sum of one k=41, 41, 71]

After

derived = [0, 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, S043





S045 · impliesBefore

source roles = [successive sum of one k=42, 42, 71]

After

derived = [0, 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, S044





S046 · impliesBefore

source roles = [successive sum of one k=43, 43, 71]

After

derived = [0, 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, S045





S047 · impliesBefore

source roles = [successive sum of one k=44, 44, 71]

After

derived = [0, 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, S046





S048 · impliesBefore

source roles = [successive sum of one k=45, 45, 71]

After

derived = [0, 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, S047





S049 · impliesBefore

source roles = [successive sum of one k=46, 46, 71]

After

derived = [0, 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, S048





S050 · impliesBefore

source roles = [successive sum of one k=47, 47, 71]

After

derived = [0, 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, S049





S051 · impliesBefore

source roles = [successive sum of one k=48, 48, 71]

After

derived = [0, 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, S050





S052 · impliesBefore

source roles = [successive sum of one k=49, 49, 71]

After

derived = [0, 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, S051





S053 · impliesBefore

source roles = [successive sum of one k=50, 50, 71]

After

derived = [0, 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, S052





S054 · impliesBefore

source roles = [successive sum of one k=51, 51, 71]

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, S053





S055 · impliesBefore

source roles = [successive sum of one k=52, 52, 71]

After

derived = [0, 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, S054





S056 · impliesBefore

source roles = [successive sum of one k=53, 53, 71]

After

derived = [0, 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, S055





S057 · impliesBefore

source roles = [successive sum of one k=54, 54, 71]

After

derived = [0, 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, S056





S058 · impliesBefore

source roles = [successive sum of one k=55, 55, 71]

After

derived = [0, 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, S057





S059 · impliesBefore

source roles = [successive sum of one k=56, 56, 71]

After

derived = [0, 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, S058





S060 · impliesBefore

source roles = [successive sum of one k=57, 57, 71]

After

derived = [0, 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, S059





S061 · impliesBefore

source roles = [successive sum of one k=58, 58, 71]

After

derived = [0, 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, S060





S062 · impliesBefore

source roles = [successive sum of one k=59, 59, 71]

After

derived = [0, 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, S061





S063 · impliesBefore

source roles = [successive sum of one k=60, 60, 71]

After

derived = [0, 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, S062





S064 · impliesBefore

source roles = [successive sum of one k=61, 61, 71]

After

derived = [0, 61]

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 · impliesBefore

source roles = [successive sum of one k=62, 62, 71]

After

derived = [0, 62]

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, S064





S066 · impliesBefore

source roles = [successive sum of one k=63, 63, 71]

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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S065





S067 · impliesBefore

source roles = [successive sum of one k=64, 64, 71]

After

derived = [0, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S066





S068 · impliesBefore

source roles = [successive sum of one k=65, 65, 71]

After

derived = [0, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S067





S069 · impliesBefore

source roles = [successive sum of one k=66, 66, 71]

After

derived = [0, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S068





S070 · impliesBefore

source roles = [successive sum of one k=67, 67, 71]

After

derived = [0, 67]

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 · impliesBefore

source roles = [successive sum of one k=68, 68, 71]

After

derived = [0, 68]

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, S070





S072 · impliesBefore

source roles = [successive sum of one k=69, 69, 71]

After

derived = [0, 69]

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 · impliesBefore

source roles = [successive sum of one k=70, 70, 71]

After

derived = [0, 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 dependenciesRule: Finite_MOD_euclidean_quotient_remainder · Uses: G001, S072





S074 · impliesBefore

source roles = [successive sum of one k=71, 71, 71]

After

derived = [1, 0]

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 · impliesBefore

source roles = [71, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 0]]

After

derived = [71]

Conditions: none

List sums of0 through p copies of1.Residues1..p−1 are nonzero and p returns0,so the FIRST POSITIVE additive return is p.Zero is not a positive return;this is not multiplicative order.

Step identifiers and dependenciesRule: Field_CHARACTERISTIC_first_POSITIVE_additive_return · Uses: G001, S074









## Reverse checks


- S075 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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 — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof.
- 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

(71, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 0], 71, additive_order_of_one)

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: ca1aabec37de406259d58d263bef5dbc918acd6ff18ae7787b5729a4f81df887
- candidate record SHA-256: 3d23178dd8d067aaa82054b7b4f72a44ac2546ae0932534399e5dc3dcee28dd6
- 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-0020.json)