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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:

$n^2 - 2 = 4g$, $g \in \mathbb{Z}$.
$n^2 = 4g + 2$.

Consider the case where $n$ is even.

$(2x)^2 = 4g + 2$, $x \in \mathbb{Z}$.

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

$4s = 4g + 2$, $s = x^2, s \in \mathbb{Z}$ 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.

$(2x + 1)^2 = 4g + 2$, $x \in \mathbb{Z}$

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

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

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

$4j = 4g + 1$, $j = x^2 + x, j \in \mathbb{Z}$ as integers are closed under addition

$2d = 2e + 1$, $d = 2j, e = 2g; d, e \in \mathbb{Z}$ 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 $4$.

Daniel
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    You could simplify your proof slightly. In the first case, you could go from $4x^2 = 4g + 2$ to $4(x^2-g) = 2$. This would imply that $4$ divides $2$ which is a contradiction. Similarly, in the second case, you could go from $4(x^2+x) = 4g + 1$ to $4(x^2+x-g) = 1$ so that $4$ divides $1$ which is also a contradiction. – Cameron Williams Mar 13 '17 at 16:30
  • Why would one need to consider the case where $n$ is odd? If $n^2 = 4g + 2$, then $n^2 = 2(2g + 1)$ which is by definition even. $n^2$ is even implies that $n$ is even, so only that case needs to be considered. – Aleksandr Hovhannisyan Sep 13 '22 at 15:18

4 Answers4

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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.

The proof can be done a lot quicker however (without contradiction) by looking $\mod 4$. It is quite easy to prove that squares are either $0$ or $1\mod 4$, so $n^2-2$ is either $-2$ or $-1\mod 4$, and thus, $n^2-2$ cannot be divisible by $4$.
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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 $4$.

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    This is not a proof by contradiction though –  Mar 13 '17 at 16:24
  • @vrugtehagel why do you need a proof by contradiction? – mez Mar 13 '17 at 16:26
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    @mez, that's what the OP is asking. –  Mar 13 '17 at 16:27
  • @vrugtehagel: No, it isn't (a proof by contradiction), albeit very little work would be required to convert it into one. I'm not sure how different it would be if it were, though. – Brian Tung Mar 13 '17 at 16:28
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    @BrianTung obviously, you can transform it into the standard not-really-proof-by-contradiction by assuming the statement is true, then proving that it's false directly, and then stating that the assumption was false. Don't get me wrong; I like the elegance and simplicity of this proof, I just don't think that it's what the OP was looking for. –  Mar 13 '17 at 16:32
  • I agree to vrugtehagel comments. – callculus42 Mar 13 '17 at 16:34
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    The question doesn't clearly state that proof by contradiction IS required, just that he did try this as a technique – Cato Mar 13 '17 at 16:39
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Assume $4 \mid n^2-2$, for $n \in \mathbb{z}$. 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.

Dave huff
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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.

Maged Saeed
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