Vtyjkyuk

I tried to prove that $n^2 - 2$ is not divisible by 4 via proof by contradiction. Does this look right?

Suppose $n^2 - 2$ is divisible by $4$. Then:

$n^2 - 2 = 4g$, $g in mathbbZ$.

$n^2 = 4g + 2$.

Consider the case where $n$ is even.

$(2x)^2 = 4g + 2$, $x in mathbbZ$.

$4x^2 = 4g + 2$.

$4s = 4g + 2$, $s = x^2, s in mathbbZ$ as integers are closed under multiplication.

$2s = 2g + 1$

$2s$ is even, and $2g + 1$ is odd (by definition of even/odd numbers). An even number cannot equal an odd number, so we have a contradiction.

Consider the case where $n$ is odd.

$(2x + 1)^2 = 4g + 2$, $x in mathbbZ$

$4x^2 + 4x + 1 = 4g + 2$

$4x^2 + 4x = 4g + 1$

$4(x^2 + x) = 4g + 1$

$4j = 4g + 1$, $j = x^2 + x, j in mathbbZ$ as integers are closed under addition

$2d = 2e + 1$, $d = 2j, e = 2g; d, e in mathbbZ$ as integers are closed under multiplication

$2d$ is even, and $2e + 1$ is odd (by definition of even/odd numbers). An even number cannot equal an odd number, so we have a contradiction.

As both cases have a contradiction, the original supposition is false, and $n^2 - 2$ is not divisible by $4$.

Your proof is perfect.

We can shorten your proof by for example going from $4x^2=4g+2$ (in case 1) to saying "The left-hand side has remainder $0$ after division by $4$, yet the right side has remainder $2$; this is impossible" (basically, looking at the expression $mod 4$ instead of dividing by $2$ and looking $mod 2$).

Also, the second case was trivially impossible, since $n^2=4g+2$ has no solutions if $n$ is odd (since then $n^2$ is odd, but $4g+2$ is even).

Depending on the context (what you know, what you can use, etc), steps like $x^2+x=j$ with the remark that integers are closed under addition and multiplication are mostly considered so trivial that it's not worth mentioning. I repeat however that this is completely dependent on context, and if you want to make sure your audience is aware of these facts and/or steps, you should mention them. More detailed explanations with steps rarely hurt the proof.

For odd $n$, $n^2-2$ is odd.

For even $n$, $n^2$ is divisible by $4$, so that $n^2-2$ is not.

By contradiction:

Let $n$ be odd. Then $n^2-2$ is both odd and a multiple of four.

Let $n$ be even. Then $n^2-2$ and $n^2$ are both multiples of $4$, so that $2$ is a multiple of $4$.

Assume $4 mid n^2-2$, for $n in mathbbz$. This means we can rewrite $n^2-2$ as $4k$, which implies $n^2=2(2k+1)$. This is clearly impossible; this implies $4 nmid n^2$ but $2 mid n^2$. So our initial assumption is wrong.

Anohter Idea of proof goes as follows:

First of all we need to know that any square of integer is either of the form 4k or 4k+1.

Suppose, to the contrary, that 4 divides $n^2-2$, then:
$$4k = n^2-2$$
$$4k+2 = n^2$$
A contradiction. Thus $n^2-2$ cannot be divisible by 4.