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Question 24109: Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

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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R0047-24109 · q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0084

Original question and givens

Use mode=zero_class, modulus=29, original a=261. Give (mode, modulus, normalized_class, declared_domain_qualified, boundary).

  • G001 — Use mode=zero_class, modulus=29, original a=261. Give (mode, modulus, normalized_class, declared_domain_qualified, 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 roles = [eulercrit-domain-boundary]

After

derived = [[mode, modulus, normalized_class, declared_domain_qualified, boundary]]

Conditions: none

List the source field order without JavaScript.Gradient boundary uses its original mode to choose6,8 or7 fields;no padding or answer-derived mode.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [29]

After

derived = [[[2, 1], [3, 2], [4, 1], [5, 4]], Yes]

Conditions: none

A composite has a factor at most its square root;complete trial division qualifies the modulus,not a guessed square/Euler equality.

Step identifiers and dependencies

Rule: Euler_modulus_primality_complete_trial · Uses: G001, S001

S003 · calculation

Before

Euler ORIGINAL signed quotient times modulus := mul(9, 29)

After

Euler ORIGINAL signed quotient times modulus = 261

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Euler ORIGINAL signed a minus modulus multiple := sub(261, 261)

After

Euler ORIGINAL signed a minus modulus multiple = 0

Conditions: none

Substitute original roles and established intermediates exactly;check nonzero denominators and do not round.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

source roles = [261, 29, 9]

After

derived = [0, Yes]

Conditions: none

Use a=qp+r with0<=r<p,including negative a;retain original a where the interface requests it.

Step identifiers and dependencies

Rule: Euler_Euclidean_class_before_hypothesis · Uses: G001, S004

S006 · implies

Before

source roles = [zero_class, 29, 0, Yes]

After

derived = [No, odd prime AND nonzero modular class required]

Conditions: none

No withholds this odd-prime/nonzero domain qualification.Out-of-domain numerical equality may still happen;the criterion does not define Legendre.

Step identifiers and dependencies

Rule: Euler_boundary_failed_hypothesis_not_universal_equality_negation · 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 — Recompute Euler ORIGINAL signed a minus modulus multiple with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Euler ORIGINAL signed quotient times modulus 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

(zero_class, 29, 0, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)

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: ea110cc37b5091fa55737aa9bfbaa1ff081db998e4594c03c0ea09fd7226da1c
  • candidate record SHA-256: 248e7fbd0c9a514a36406eae1b92b5e738499af9987def3dd260d356c2b021aa
  • semantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386
  • question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50b
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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