(i) 用反例证明以下命题是错误的：(2)

“如果 $n$ 是质数，那么 $3^n+2$ 也是质数。”

(ii) 使用穷尽法证明：如果 $m$ 是不能被 $3$ 整除的整数，那么 $m^2-1$ 能被 $3$ 整除。(4)

反例只需要找一个 prime $n$，使 $3^n+2$ 不是 prime。选 $n=5$ 很方便。

Take

This is prime. But

So $3^5+2$ is not prime. Therefore, the statement is false.

一个整数不被 $3$ 整除，只可能形如 $3k+1$ 或 $3k+2$。proof by exhaustion 就是把这两种情况都检查完。

If $m$ is not divisible by $3$, then either

or

where $k$ is an integer.

Case 1:

So $m^2-1$ is divisible by $3$.

Case 2:

So $m^2-1$ is divisible by $3$.

In all possible cases where $m$ is not divisible by $3$, $m^2-1$ is divisible by $3$.

先将 $m^2-1$ 因式分解。仍穷尽两个余数类别，分别找出可被 $3$ 整除的那个因子。

If $m$ is not divisible by $3$, then $m=3k+1$ or $m=3k+2$, where $k\in\mathbb Z$.

Factorise:

If $m=3k+1$, then

If $m=3k+2$, then

In each case the bracket is an integer, so $m^2-1$ is divisible by $3$. These exhaust all integers not divisible by $3$, hence the result is proved.