关于∀n∈ℕ，p(n)=n²+n+41为质数表述的困惑

Hey there! I totally get why this might feel confusing at first—let me walk you through exactly why that statement is there even though it's proven false.

First off, that full statement ∀n∈ℕ, p(n) is prime is a deliberate, classic example of a "plausible but false universal claim". The point of including it in the book is to hammer home a critical logic lesson for computer science and math:

• A universal statement (one that claims something is true for all members of a set) can be completely disproven with just one single counterexample.
• You can't assume a rule holds forever just because it works for dozens, hundreds, or even thousands of test cases. In this case, p(n) gives primes for n=0 to 39, but n=40 breaks it wide open—this is a perfect way to show how easy it is to jump to the wrong conclusion without rigorous proof.

Second, this example ties into a fun bit of math history: it's Euler's famous prime-generating polynomial. Euler noticed this formula produces 40 consecutive primes, which was a huge surprise back in the day. The book uses this well-known case to make the lesson more relatable and memorable, instead of inventing some random, boring formula.

Finally, the book is probably setting you up for later topics—like inductive reasoning pitfalls, or how to formally disprove universal statements. By presenting the "too-good-to-be-true" claim first, then showing its flaw, you'll remember the logic rules way better than if you just read a dry definition.

内容的提问来源于stack exchange，提问作者Rami Chasygov