# Question 24103: Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion

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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.

R0047-24103 · q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0078



## Original question and givens

Use mode=zero_class, modulus=7, original a=21. Give (mode, modulus, normalized_class, declared_domain_qualified, boundary).


- G001 — Use mode=zero_class, modulus=7, original a=21. 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.


- [mko-euler-criterion-quadratic-residue](https://ocme.evemisslab.com/en/mko/mko-euler-criterion-quadratic-residue/)





## Step-by-step derivation

S001 · impliesBefore

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 dependenciesRule: source_ordered_fields_without_JS · Uses: G001





S002 · impliesBefore

source roles = [7]

After

derived = [[[2, 1]], 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 dependenciesRule: Euler_modulus_primality_complete_trial · Uses: G001, S001





S003 · calculationBefore

Euler ORIGINAL signed quotient times modulus := mul(3, 7)

After

Euler ORIGINAL signed quotient times modulus = 21

Conditions: none

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

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





S004 · calculationBefore

Euler ORIGINAL signed a minus modulus multiple := sub(21, 21)

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · impliesBefore

source roles = [21, 7, 3]

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 dependenciesRule: Euler_Euclidean_class_before_hypothesis · Uses: G001, S004





S006 · impliesBefore

source roles = [zero_class, 7, 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 dependenciesRule: 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, 7, 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: 88e1b5d49865452ee2d95afb082b1c0715aabece515166f069308934864d269b
- candidate record SHA-256: 18d2889e3505cdba5a30448cff58a2398fdf20b5f45f2d1fe15eb5d172945dd8
- semantic review receipt SHA-256: 9b43b9823e9bd2478d6caa5b57ffed161c2c8956af86974e901d5662ad994386
- question-first witness SHA-256: eb97596068e2d5648033bab2b8f6ca9b653bbe8cb77c0f8e1d16d0c40e46f50b
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0078.json)