# Question 23777: Divisibility by p Gives Symbol Zero, Not Plus 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.

R0046-23777 · q-20261014-advance-20261112-r06-legendre-symbol-0052



## Original question and givens

Use odd prime p=5, original a=10. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).


- G001 — Use odd prime p=5, original a=10. Give (prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate).





## 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-legendre-symbol](https://ocme.evemisslab.com/en/mko/mko-legendre-symbol/)





## Step-by-step derivation

S001 · impliesBefore

source operation=legendre-zero-class;original snapshot and mathematics retained

After

explicit ordered source output fields=[prime, a, normalized_class, Legendre_symbol, nonzero_roots, certificate]

Conditions: none

Keep the source snapshot while listing ordered fields without JavaScript.Radius uses its original mode to select6 or11 fields without padding;distinguish nonclaims from actual negations.

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

Legendre ORIGINAL odd prime:ORIGINAL modulus=5;ALL possible trial divisors through square root=[2]

After

ALL [divisor,remainder] rows=[[2, 1]];prime=Yes

Conditions: none

Qualify primality over the full trial range,not by guessing from Legendre/Euler classification.This is a modulus premise,not the square status of a.

Step identifiers and dependenciesRule: legendre_complete_prime_trial_not_Euler_classification · Uses: G001, S001





S003 · calculationBefore

Legendre ORIGINAL nonzero root1 exact square := mul(1, 1)

After

Legendre ORIGINAL nonzero root1 exact square = 1

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

ORIGINAL root x=1 in1..4;exact square=1

After

square=0*5+1;canonical square class=1;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependenciesRule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S003





S005 · calculationBefore

Legendre ORIGINAL nonzero root2 exact square := mul(2, 2)

After

Legendre ORIGINAL nonzero root2 exact square = 4

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · impliesBefore

ORIGINAL root x=2 in1..4;exact square=4

After

square=0*5+4;canonical square class=4;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependenciesRule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S005





S007 · calculationBefore

Legendre ORIGINAL nonzero root3 exact square := mul(3, 3)

After

Legendre ORIGINAL nonzero root3 exact square = 9

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · impliesBefore

ORIGINAL root x=3 in1..4;exact square=9

After

square=1*5+4;canonical square class=4;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependenciesRule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S007





S009 · calculationBefore

Legendre ORIGINAL nonzero root4 exact square := mul(4, 4)

After

Legendre ORIGINAL nonzero root4 exact square = 16

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · impliesBefore

ORIGINAL root x=4 in1..4;exact square=16

After

square=3*5+1;canonical square class=1;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependenciesRule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S009





S011 · impliesBefore

ORIGINAL prime=5;COMPLETE nonzero square table=[[1, 1], [2, 4], [3, 4], [4, 1]];nonzero square classes=[1, 4]

After

for EACH normalized class r,first r0 ->symbol0;else square-table member ->symbol1;otherwise ->symbol-1;these cases partition ALL0..p-1

Conditions: none

Separate zero first;the fact that0 is a square does not give symbol+1.Nonzero classes are classified by the complete square table.This is definition plus exhaustive evidence,not a later multiplicativity or Euler formula.

Step identifiers and dependenciesRule: legendre_ZERO_then_residue_nonresidue_three_way_definition · Uses: G001, S010





S012 · calculationBefore

Legendre ORIGINAL candidate class quotient times modulus := mul(2, 5)

After

Legendre ORIGINAL candidate class quotient times modulus = 10

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · calculationBefore

Legendre ORIGINAL candidate class ORIGINAL signed value minus multiple := sub(10, 10)

After

Legendre ORIGINAL candidate class ORIGINAL signed value minus multiple = 0

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · impliesBefore

Legendre ORIGINAL candidate class:ORIGINAL a=10,positive modulus=5;actual quotient=2

After

normalized class=0;a=quotient*modulus+class;ORIGINAL a retained in output where declared

Conditions: none

Retain the same class for negative a and normalize into0..p−1,rather than using a negative language-specific remainder.

Step identifiers and dependenciesRule: legendre_ORIGINAL_signed_class_Euclidean_normalization · Uses: G001, S013









## Reverse checks


- S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S013 · premise_and_result_audit — Recompute Legendre ORIGINAL candidate class ORIGINAL signed value minus multiple with exact rational arithmetic, not rounded decimals.
- S012 · premise_and_result_audit — Recompute Legendre ORIGINAL candidate class quotient times modulus with exact rational arithmetic, not rounded decimals.
- S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S009 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root4 exact square with exact rational arithmetic, not rounded decimals.
- S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S007 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root3 exact square with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S005 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root2 exact square with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Recompute Legendre ORIGINAL nonzero root1 exact square with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(5, 10, 0, 0, [], p_divides_a_so_Legendre_symbol_is_zero_before_residue_classification)

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: 23e395b71f4d7357b436aea1fbbe79018aee96e83fe3d2f8270420d5f2e087bf
- candidate record SHA-256: 41c489597af72c50fa0a99b90b9130e08e3c5a6b9642b3bad825c2f94f9266be
- semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
- question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261112-r06-legendre-symbol-0052.json)