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Question 23733: Domain Extensions and Premature Methods Are Different Legendre Boundaries

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0046-23733 · q-20261014-advance-20261112-r06-legendre-symbol-0083

Original question and givens

Use mode=composite_modulus, modulus=21, original a=8. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

  • G001 — Use mode=composite_modulus, modulus=21, original a=8. Give (mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary).

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.

Step-by-step derivation

S001 · implies

Before

source operation=legendre-domain-method-boundary;original snapshot and mathematics retained

After

explicit ordered source output fields=[mode, modulus, normalized_class, unsupported_domain_or_method_certificate, boundary]

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 dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · calculation

Before

Legendre ORIGINAL boundary class quotient times modulus := mul(0, 21)

After

Legendre ORIGINAL boundary class quotient times modulus = 0

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

Legendre ORIGINAL boundary class ORIGINAL signed value minus multiple := sub(8, 0)

After

Legendre ORIGINAL boundary class ORIGINAL signed value minus multiple = 8

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

Legendre ORIGINAL boundary class:ORIGINAL a=8,positive modulus=21;actual quotient=0

After

normalized class=8;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 dependencies

Rule: legendre_ORIGINAL_signed_class_Euclidean_normalization · Uses: G001, S003

S005 · implies

Before

Legendre composite boundary modulus:ORIGINAL modulus=21;ALL possible trial divisors through square root=[2, 3, 4]

After

ALL [divisor,remainder] rows=[[2, 1], [3, 0], [4, 1]];prime=No

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 dependencies

Rule: legendre_complete_prime_trial_not_Euler_classification · Uses: G001, S004

S006 · implies

Before

ORIGINAL mode=composite_modulus;modulus=21,a=8,normalized class=8

After

unsupported_domain_or_method_certificate fixed No;modulus outside declared odd-prime Legendre domain

Conditions: none

For premature_euler,the No neither refutes Euler criterion nor puts a valid input out of domain.Modulus2/composite cases have a domain restriction.No Jacobi or other extension is silently adopted.

Step identifiers and dependencies

Rule: legendre_DOMAIN_or_METHOD_No_responsibilities_distinct · Uses: G001, S005

Reverse checks

  • 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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 boundary class ORIGINAL signed value minus multiple with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute Legendre ORIGINAL boundary class quotient times modulus with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(composite_modulus, 21, 8, No, Legendre_symbol_domain_requires_an_odd_prime_modulus)

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: 2f68578d413c867d942ec7668cb858360abb5349498285c498c6805faa42f072
  • candidate record SHA-256: f8aefc375a3aaa77a9a54097eb36998589b602540cfd1fc9706abafb330dc547
  • semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
  • question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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