用反证法证明：当 $n$ 是整数时，

永远不可能被 $4$ 整除。

假设存在整数 $n$，使 $n^2-2$ 能被 $4$ 整除。这个假设会先推出 $n^2$ 为偶数，从而 $n$ 为偶数；但偶数的平方必为 $4$ 的倍数，这与假设所得等式矛盾。

Suppose, for a contradiction, that there is an integer $n$ such that $n^2-2$ is divisible by $4$.

Then, for some integer $k$,

Hence

so $n^2$ is even. Therefore $n$ is even, since the square of an odd integer is odd.

Thus $n=2m$ for some integer $m$, and so

Combining this with $n^2=4k+2$ gives

and hence

This is impossible because the left-hand side is even whereas the right-hand side is odd. This contradiction shows that the original assumption is false.

Therefore, for every integer $n$,