{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0001","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":2,\"a\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 1, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(2, 1, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065001,"parameters":{"p":2,"a":3,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":3,\"p\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0002","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":3,\"a\":-4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(3, 2, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065002,"parameters":{"p":3,"a":-4,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":-4,\"p\":3}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0003","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":5,\"a\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 4, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(5, 4, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065003,"parameters":{"p":5,"a":4,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":4,\"p\":5}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0004","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":7,\"a\":-5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 2, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(7, 2, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065004,"parameters":{"p":7,"a":-5,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":-5,\"p\":7}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0005","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":11,\"a\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 6, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(11, 6, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065005,"parameters":{"p":11,"a":6,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":6,\"p\":11}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0006","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":13,\"a\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 6, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(13, 6, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065006,"parameters":{"p":13,"a":-7,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":-7,\"p\":13}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0007","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":17,\"a\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 8, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(17, 8, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065007,"parameters":{"p":17,"a":8,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":8,\"p\":17}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0008","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":19,\"a\":-9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 10, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(19, 10, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065008,"parameters":{"p":19,"a":-9,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":-9,\"p\":19}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0009","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":23,\"a\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 10, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(23, 10, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065009,"parameters":{"p":23,"a":10,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":10,\"p\":23}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0010","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":29,\"a\":-11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(29, 18, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(29, 18, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065010,"parameters":{"p":29,"a":-11,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":-11,\"p\":29}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0011","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":31,\"a\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(31, 12, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(31, 12, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065011,"parameters":{"p":31,"a":12,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":12,\"p\":31}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0012","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":37,\"a\":-13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(37, 24, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(37, 24, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065012,"parameters":{"p":37,"a":-13,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":-13,\"p\":37}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0013","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":41,\"a\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(41, 14, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(41, 14, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065013,"parameters":{"p":41,"a":14,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":14,\"p\":41}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0014","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":43,\"a\":-15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(43, 28, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(43, 28, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065014,"parameters":{"p":43,"a":-15,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":-15,\"p\":43}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0015","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":47,\"a\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(47, 16, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(47, 16, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065015,"parameters":{"p":47,"a":16,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":16,\"p\":47}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0016","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":53,\"a\":-17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(53, 36, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(53, 36, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065016,"parameters":{"p":53,"a":-17,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":-17,\"p\":53}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0017","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":59,\"a\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(59, 18, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(59, 18, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065017,"parameters":{"p":59,"a":18,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":18,\"p\":59}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0018","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":61,\"a\":-19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(61, 42, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(61, 42, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065018,"parameters":{"p":61,"a":-19,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":-19,\"p\":61}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0019","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":67,\"a\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(67, 20, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(67, 20, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065019,"parameters":{"p":67,"a":20,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":20,\"p\":67}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0020","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":71,\"a\":-21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(71, 50, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(71, 50, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065020,"parameters":{"p":71,"a":-21,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":-21,\"p\":71}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0021","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":73,\"a\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(73, 22, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(73, 22, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065021,"parameters":{"p":73,"a":22,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":22,\"p\":73}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0022","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":79,\"a\":-23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(79, 56, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(79, 56, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065022,"parameters":{"p":79,"a":-23,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":-23,\"p\":79}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0023","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":83,\"a\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(83, 24, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(83, 24, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065023,"parameters":{"p":83,"a":24,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":24,\"p\":83}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0024","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":89,\"a\":-25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(89, 64, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(89, 64, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065024,"parameters":{"p":89,"a":-25,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":-25,\"p\":89}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0025","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-prime-coprime；資料={\"p\":97,\"a\":26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(97, 26, 1, 是, prime_coprime_form)","explanation_zh":"Wave5封閉層核對=(97, 26, 1, 是, prime_coprime_form)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-prime-coprime","seed":202611065025,"parameters":{"p":97,"a":26,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-prime-coprime","semantic_template":"qf-mod37-flt-prime-coprime/{\"a\":26,\"p\":97}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0026","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":2,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(2, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065026,"parameters":{"p":2,"a":2,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":2,\"p\":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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0027","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":3,\"a\":-6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(3, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065027,"parameters":{"p":3,"a":-6,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":-6,\"p\":3}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0028","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":5,\"a\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(5, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065028,"parameters":{"p":5,"a":15,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":15,\"p\":5}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0029","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":7,\"a\":-28}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(7, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065029,"parameters":{"p":7,"a":-28,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":-28,\"p\":7}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0030","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":11,\"a\":55}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(11, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065030,"parameters":{"p":11,"a":55,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":55,\"p\":11}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0031","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":13,\"a\":-78}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(13, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065031,"parameters":{"p":13,"a":-78,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":-78,\"p\":13}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0032","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":17,\"a\":119}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(17, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065032,"parameters":{"p":17,"a":119,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":119,\"p\":17}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0033","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":19,\"a\":-152}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(19, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065033,"parameters":{"p":19,"a":-152,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":-152,\"p\":19}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0034","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":23,\"a\":207}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(23, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065034,"parameters":{"p":23,"a":207,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":207,\"p\":23}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0035","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":29,\"a\":-290}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(29, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(29, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065035,"parameters":{"p":29,"a":-290,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":-290,\"p\":29}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0036","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":31,\"a\":341}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(31, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(31, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065036,"parameters":{"p":31,"a":341,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":341,\"p\":31}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0037","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":37,\"a\":-444}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(37, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(37, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065037,"parameters":{"p":37,"a":-444,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":-444,\"p\":37}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0038","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":41,\"a\":533}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(41, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(41, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065038,"parameters":{"p":41,"a":533,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":533,\"p\":41}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0039","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":43,\"a\":-602}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(43, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(43, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065039,"parameters":{"p":43,"a":-602,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":-602,\"p\":43}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0040","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":47,\"a\":705}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(47, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(47, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065040,"parameters":{"p":47,"a":705,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":705,\"p\":47}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0041","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":53,\"a\":-848}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(53, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(53, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065041,"parameters":{"p":53,"a":-848,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":-848,\"p\":53}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0042","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":59,\"a\":1003}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(59, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(59, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065042,"parameters":{"p":59,"a":1003,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":1003,\"p\":59}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0043","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":61,\"a\":-1098}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(61, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(61, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065043,"parameters":{"p":61,"a":-1098,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":-1098,\"p\":61}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0044","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":67,\"a\":1273}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(67, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(67, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065044,"parameters":{"p":67,"a":1273,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":1273,\"p\":67}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0045","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":71,\"a\":-1420}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(71, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(71, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065045,"parameters":{"p":71,"a":-1420,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":-1420,\"p\":71}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0046","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":73,\"a\":1533}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(73, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(73, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065046,"parameters":{"p":73,"a":1533,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":1533,\"p\":73}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0047","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":79,\"a\":-1738}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(79, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(79, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065047,"parameters":{"p":79,"a":-1738,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":-1738,\"p\":79}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0048","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":83,\"a\":1909}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(83, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(83, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065048,"parameters":{"p":83,"a":1909,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":1909,\"p\":83}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0049","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":89,\"a\":-2136}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(89, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(89, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065049,"parameters":{"p":89,"a":-2136,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":-2136,\"p\":89}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0050","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-all-integer；資料={\"p\":97,\"a\":2425}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(97, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)","explanation_zh":"Wave5封閉層核對=(97, 0, 0, 0, 是, all_integer_form_including_prime_divides_a)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-all-integer","seed":202611065050,"parameters":{"p":97,"a":2425,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-all-integer","semantic_template":"qf-mod37-flt-all-integer/{\"a\":2425,\"p\":97}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0051","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":4,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 2, 否, 否, 否, 0, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(4, 2, 否, 否, 否, 0, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065051,"parameters":{"modulus":4,"a":2,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":2,\"modulus\":4}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0052","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":5,\"a\":-3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, -3, 是, 是, 是, 1, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(5, -3, 是, 是, 是, 1, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065052,"parameters":{"modulus":5,"a":-3,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":-3,\"modulus\":5}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0053","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":6,\"a\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 4, 否, 否, 否, 4, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(6, 4, 否, 否, 否, 4, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065053,"parameters":{"modulus":6,"a":4,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":4,\"modulus\":6}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0054","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":7,\"a\":-5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, -5, 是, 是, 是, 1, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(7, -5, 是, 是, 是, 1, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065054,"parameters":{"modulus":7,"a":-5,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":-5,\"modulus\":7}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0055","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":8,\"a\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 6, 否, 否, 否, 0, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(8, 6, 否, 否, 否, 0, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065055,"parameters":{"modulus":8,"a":6,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":6,\"modulus\":8}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0056","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":9,\"a\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, -7, 否, 是, 否, 4, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(9, -7, 否, 是, 否, 4, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065056,"parameters":{"modulus":9,"a":-7,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":-7,\"modulus\":9}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0057","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":10,\"a\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 8, 否, 否, 否, 8, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(10, 8, 否, 否, 否, 8, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065057,"parameters":{"modulus":10,"a":8,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":8,\"modulus\":10}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0058","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":11,\"a\":-9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, -9, 是, 是, 是, 1, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(11, -9, 是, 是, 是, 1, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065058,"parameters":{"modulus":11,"a":-9,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":-9,\"modulus\":11}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0059","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":12,\"a\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 10, 否, 否, 否, 4, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(12, 10, 否, 否, 否, 4, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065059,"parameters":{"modulus":12,"a":10,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":10,\"modulus\":12}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0060","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":13,\"a\":-11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, -11, 是, 是, 是, 1, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(13, -11, 是, 是, 是, 1, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065060,"parameters":{"modulus":13,"a":-11,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":-11,\"modulus\":13}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0061","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":14,\"a\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 12, 否, 否, 否, 12, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(14, 12, 否, 否, 否, 12, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065061,"parameters":{"modulus":14,"a":12,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":12,\"modulus\":14}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0062","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":15,\"a\":-13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, -13, 否, 是, 否, 4, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(15, -13, 否, 是, 否, 4, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065062,"parameters":{"modulus":15,"a":-13,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":-13,\"modulus\":15}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0063","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":16,\"a\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 14, 否, 否, 否, 0, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(16, 14, 否, 否, 否, 0, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065063,"parameters":{"modulus":16,"a":14,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":14,\"modulus\":16}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0064","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":17,\"a\":-15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, -15, 是, 是, 是, 1, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(17, -15, 是, 是, 是, 1, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065064,"parameters":{"modulus":17,"a":-15,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":-15,\"modulus\":17}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0065","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":18,\"a\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, 16, 否, 否, 否, 4, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(18, 16, 否, 否, 否, 4, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065065,"parameters":{"modulus":18,"a":16,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":16,\"modulus\":18}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0066","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":19,\"a\":-17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, -17, 是, 是, 是, 1, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(19, -17, 是, 是, 是, 1, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065066,"parameters":{"modulus":19,"a":-17,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":-17,\"modulus\":19}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0067","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":20,\"a\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, 18, 否, 否, 否, 12, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(20, 18, 否, 否, 否, 12, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065067,"parameters":{"modulus":20,"a":18,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":18,\"modulus\":20}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0068","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":21,\"a\":-19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, -19, 否, 是, 否, 4, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(21, -19, 否, 是, 否, 4, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065068,"parameters":{"modulus":21,"a":-19,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":-19,\"modulus\":21}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0069","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":22,\"a\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, 20, 否, 否, 否, 20, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(22, 20, 否, 否, 否, 20, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065069,"parameters":{"modulus":22,"a":20,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":20,\"modulus\":22}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0070","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":23,\"a\":-21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, -21, 是, 是, 是, 1, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(23, -21, 是, 是, 是, 1, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065070,"parameters":{"modulus":23,"a":-21,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":-21,\"modulus\":23}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0071","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":24,\"a\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, 22, 否, 否, 否, 16, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(24, 22, 否, 否, 否, 16, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065071,"parameters":{"modulus":24,"a":22,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":22,\"modulus\":24}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0072","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":25,\"a\":-23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(25, -23, 否, 是, 否, 16, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(25, -23, 否, 是, 否, 16, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065072,"parameters":{"modulus":25,"a":-23,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":-23,\"modulus\":25}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0073","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":26,\"a\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(26, 24, 否, 否, 否, 24, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(26, 24, 否, 否, 否, 24, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065073,"parameters":{"modulus":26,"a":24,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":24,\"modulus\":26}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0074","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":27,\"a\":-25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(27, -25, 否, 是, 否, 13, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(27, -25, 否, 是, 否, 13, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065074,"parameters":{"modulus":27,"a":-25,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":-25,\"modulus\":27}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0075","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-hypothesis-audit；資料={\"modulus\":28,\"a\":26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(28, 26, 否, 否, 否, 20, coprime_form_not_applicable_without_both_hypotheses)","explanation_zh":"Wave5封閉層核對=(28, 26, 否, 否, 否, 20, coprime_form_not_applicable_without_both_hypotheses)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-hypothesis-audit","seed":202611065075,"parameters":{"modulus":28,"a":26,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-hypothesis-audit","semantic_template":"qf-mod37-flt-hypothesis-audit/{\"a\":26,\"modulus\":28}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0076","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 2, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 2, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065076,"parameters":{"modulus":561,"a":2,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":2,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0077","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 4, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 4, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065077,"parameters":{"modulus":561,"a":4,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":4,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0078","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 5, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 5, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065078,"parameters":{"modulus":561,"a":5,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":5,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0079","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 7, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 7, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065079,"parameters":{"modulus":561,"a":7,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":7,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0080","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 8, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 8, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065080,"parameters":{"modulus":561,"a":8,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":8,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0081","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 10, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 10, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065081,"parameters":{"modulus":561,"a":10,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":10,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0082","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 13, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 13, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065082,"parameters":{"modulus":561,"a":13,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":13,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0083","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 14, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 14, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065083,"parameters":{"modulus":561,"a":14,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":14,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0084","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 16, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 16, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065084,"parameters":{"modulus":561,"a":16,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":16,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0085","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 19, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 19, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065085,"parameters":{"modulus":561,"a":19,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":19,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0086","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 20, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 20, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065086,"parameters":{"modulus":561,"a":20,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":20,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0087","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 23, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 23, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065087,"parameters":{"modulus":561,"a":23,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":23,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0088","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 25, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 25, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065088,"parameters":{"modulus":561,"a":25,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":25,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0089","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 26, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 26, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065089,"parameters":{"modulus":561,"a":26,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":26,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0090","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":28}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 28, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 28, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065090,"parameters":{"modulus":561,"a":28,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":28,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0091","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":29}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 29, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 29, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065091,"parameters":{"modulus":561,"a":29,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":29,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0092","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":31}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 31, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 31, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065092,"parameters":{"modulus":561,"a":31,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":31,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0093","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":32}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 32, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 32, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065093,"parameters":{"modulus":561,"a":32,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":32,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0094","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":35}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 35, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 35, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065094,"parameters":{"modulus":561,"a":35,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":35,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0095","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":37}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 37, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 37, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065095,"parameters":{"modulus":561,"a":37,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":37,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0096","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":38}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 38, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 38, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065096,"parameters":{"modulus":561,"a":38,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":38,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0097","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":40}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 40, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 40, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065097,"parameters":{"modulus":561,"a":40,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":40,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0098","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":41}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 41, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 41, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065098,"parameters":{"modulus":561,"a":41,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":41,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0099","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":43}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 43, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 43, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065099,"parameters":{"modulus":561,"a":43,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":43,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r05-fermat-little-theorem-0100","batch_id":"batch-20261014-advance-20261106-r05-fermat-little-theorem-foundations","language":"zh-Hant","source_mko_ids":["mko-fermat-little-theorem"],"title_zh":"Fermat小定理","stem_zh":"exact modular exponentiation與明示prime或composite hypotheses。操作=flt-converse-counterexample；資料={\"modulus\":561,\"a\":46}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(561, 46, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)","explanation_zh":"Wave5封閉層核對=(561, 46, 否, 是, 1, converse_false_composite_can_pass_Fermat_test)；依完整明示資料獨立重算；coprime版本須p為質數且p不整除a；a的p次方版本涵蓋p整除a；不得消去不可逆a且converse為假；合數可通過特定Fermat test。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["Fermat小定理"]},"variant":{"family_id":"qf-mod37-flt-converse-counterexample","seed":202611065100,"parameters":{"modulus":561,"a":46,"domain":"Exact_modular_Fermat_little_theorem","operation":"flt-converse-counterexample","semantic_template":"qf-mod37-flt-converse-counterexample/{\"a\":46,\"modulus\":561}"}},"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、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}