{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-proof"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-proof-01","title":{"zh-Hant":"數學證明：證明 n²+n 恆為偶數，並以反例否定「n²+n+41 恆為質數」","en":"Mathematical proof: proving n²+n is always even, and refuting 'n²+n+41 is always prime' by a counterexample"},"summary":{"zh-Hant":"(1) 在整數的指定語境中，由明示的前提與規則證明定理 T：對每個整數 n，2∣(n²+n)；再由 T 以反證法推出沒有整數 n 使 n²+n 為奇數。(2) 檢驗主張 C：「對每個自然數 n，n²+n+41 是質數」；說明 n=0 到 39 的 40 個正例為何不構成證明，並以 n=40 的反例否定 C。","en":"(1) In a stated context for the integers, prove from explicit premises and rules the theorem T: for every integer n, 2∣(n²+n); then derive from T by contradiction that no integer n makes n²+n odd. (2) Test the claim C: 'for every natural number n, n²+n+41 is prime'; explain why the 40 positive cases n=0 to 39 do not constitute a proof, and refute C by the counterexample n=40."},"mko_ids":["mko-proof"]}],"title":"概念推導導讀"}