关于素数的题目（高手进）

不是。
当n = 41 的时候
n^2-n+41 = 41*41 - 41 + 41 = 41*41

是合数。
Is n^2-n+41 a prime number for all positive integeral values of n
answer: No.

证明就是举个反例。41就是一个反例。否定一个命题只要举一个反例即可。
Is n^2-n+41 a prime number for all positive integeral values of n 从逻辑上讲是全称肯定。对于这样的命题只要找一个反例就可以否定。

Illustrative Examples
Example 1:

01. If P(n) is the statement n2-n+41 is prime, prove that P(1), P(2) are true but P(41) is not true.

Suggested answer:

Let P(n) = n2- n + 41 is a prime number

P(1) = (1)2- 1 + 41 = 41 is a prime number

P(1) is true.

P(2) = (2)2 - 2 + 41 = 43 is a prime number

P(2) is also true.

P(41) = (41)2 - 41 + 41 = (41)2 is a prime number

But (41)2 = 41 x 41 = 1681 which is not true.

P(41) is false .

Remark: Principle of mathematical induction does not hold good in this case. We cannot make a general assertion f