{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0082", "pilot_number": "R0047-24107", "number": 24107, "family_id": "qf-mod44-eulercrit-domain-boundary", "title": { "zh-Hant": "第 24107 題：Euler二次剩餘判準", "en": "Question 24107: Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0082", "batch_id": "batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-euler-criterion-quadratic-residue" ], "title_zh": "Euler二次剩餘判準", "stem_zh": "odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(even_prime, 2, 1, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)", "explanation_zh": "Wave6第七層核對=(even_prime, 2, 1, 否, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)；依完整明示資料獨立重算；Legendre值由square table先獨立取得，p-1代表-1；finite table只核指定instance而非一般定理證明；p=2、composite與zero class不在domain；不得用Euler criterion反定義Legendre。" }, "difficulty": { "band": "undergraduate_core", "level": 1, "skills": [ "Euler二次剩餘判準" ] }, "variant": { "family_id": "qf-mod44-eulercrit-domain-boundary", "seed": 202611134082, "parameters": { "mode": "even_prime", "modulus": 2, "a": 7, "domain": "Odd_prime_Euler_criterion", "operation": "eulercrit-domain-boundary", "semantic_template": "qf-mod44-eulercrit-domain-boundary/{\"a\":7,\"mode\":\"even_prime\",\"modulus\":2}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-15", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-0082", "batch_id": "batch-20261015-advance-20261113-r04-euler-criterion-quadratic-residue-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "a2baa5a0ffc34d739ef56e54102f668f89ab929f66e28a264f3389514b5d679a", "family_id": "qf-mod44-eulercrit-domain-boundary", "seed": 202611134082, "semantic_template": "qf-mod44-eulercrit-domain-boundary/{\"a\":7,\"mode\":\"even_prime\",\"modulus\":2}" }, "fields": { "title": "Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion", "stem": "Use mode=even_prime, modulus=2, original a=7. Give (mode, modulus, normalized_class, declared_domain_qualified, boundary).", "answer": { "kind": "expression", "value": "(even_prime, 2, 1, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)", "explanation": "Validate the named failed hypothesis and normalize the class. No means this odd-prime/nonzero domain is not qualified; it does not assert that every out-of-domain numerical equality must fail or redefine the Legendre symbol. The exact result, retaining the requested order, is (even_prime, 2, 1, No, Euler_criterion_requires_an_odd_prime_and_a_nonzero_class_and_does_not_define_the_Legendre_symbol)." }, "skills": [ "Zero, Even Prime and Composite Inputs Are Outside This Declared Criterion" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "odd prime、nonzero class、complete square table與independent repeated-squaring。操作=eulercrit-domain-boundary；資料={\"mode\":\"even_prime\",\"modulus\":2,\"a\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use mode=even_prime, modulus=2, original a=7. Give (mode, modulus, normalized_class, declared_domain_qualified, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [eulercrit-domain-boundary]", "after": "derived = [[mode, modulus, normalized_class, declared_domain_qualified, boundary]]", "relation": "implies", "rule_id": "source_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "本頁列完整來源欄序，不需JavaScript；gradient boundary依原mode選6／8／7欄，不補欄也不由答案猜mode。", "en": "List the source field order without JavaScript.Gradient boundary uses its original mode to choose6,8 or7 fields;no padding or answer-derived mode." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [2]", "after": "derived = [[], 是]", "relation": "implies", "rule_id": "Euler_modulus_primality_complete_trial", "conditions": [], "explanation": { "zh-Hant": "合數必有不大於平方根的因數，故完整試除驗模數資格；這不由平方性或Euler等式猜測。", "en": "A composite has a factor at most its square root;complete trial division qualifies the modulus,not a guessed square/Euler equality." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "Euler ORIGINAL signed quotient times modulus := mul(3, 2)", "after": "Euler ORIGINAL signed quotient times modulus = 6", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Euler ORIGINAL signed quotient times modulus，不以小數近似替代。", "en": "Recompute Euler ORIGINAL signed quotient times modulus with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "3", "2" ], "output": "6", "label": "Euler ORIGINAL signed quotient times modulus" } }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "Euler ORIGINAL signed a minus modulus multiple := sub(7, 6)", "after": "Euler ORIGINAL signed a minus modulus multiple = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原題角色與已證中間量精確代值；分母先核非零，不以近似數替代。", "en": "Substitute original roles and established intermediates exactly;check nonzero denominators and do not round." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Euler ORIGINAL signed a minus modulus multiple，不以小數近似替代。", "en": "Recompute Euler ORIGINAL signed a minus modulus multiple with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "7", "6" ], "output": "1", "label": "Euler ORIGINAL signed a minus modulus multiple" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "source roles = [7, 2, 3]", "after": "derived = [1, 是]", "relation": "implies", "rule_id": "Euler_Euclidean_class_before_hypothesis", "conditions": [], "explanation": { "zh-Hant": "a=qp+r、0<=r