# Prove that $\sqrt{2 + 9n}$ is never an integer

I'm trying to show the equation

$$x^2 \equiv 2 \mod 9$$

has no solutions, and I thought the best way might be to show that $\sqrt{2 + 9n}$ can never be an integer (for integer $n$). What might be a good way too go about this?

I've tried a proof by contradiction similar to the proof that $\sqrt{2}$ is irrational but can't seem to find a way to arrive upon the contradiction.

• Just try the 9 possible values of $x$... – vonbrand Feb 11 '14 at 19:25
• $2$ is not a quadratic residue mod $3$... – Watson Nov 22 '18 at 20:26

## 4 Answers

Hint any $x$ can be written as $9k$ or $9k+1$,or, ..., $9k+8$. Now square each of these numbers, and show that the remainder is never 2.

• Going mod 3 is slightly less work. – Cheerful Parsnip Feb 11 '14 at 18:10
• @GrumpyParsnip: true. :). However, it doesn't take much time to do this calculation either- and can be generalized to problems involving other numbers. – voldemort Feb 11 '14 at 18:12
• @GrumpyParsnip But $(3k+r)^2=9k^2+6kr+r^2=3k(3k+2r)+r^2$...I don't see how this can help $\pmod 9$ – chubakueno Feb 11 '14 at 18:13
• @chubakueno Then show that $\sqrt{2+3n}$ is never an integer – Hagen von Eitzen Feb 11 '14 at 18:16
• Ok, so proving mod 3 sufficient? – user127992 Feb 11 '14 at 18:26

Suppose it was an integer. Then you have that for some $k\in\mathbb{N}$,

$$\sqrt{2+9n} = k$$

Or equivalently

$$k^2=9n+2.$$

Or

$$n = \frac{1}{9}(k^2+2).$$

Can you show that this can never hold (that is, $k^2+2$ is never divisible by $9$)?

Hint: notice this cannot work if $k$ is even so $k$ must be odd. If $k$ is odd, then $k = 2l+1$ for some integer $l$. Thus, you want to inspect what happens to $k^2+2 = 4l^2+4l+3$.

• Saying that 2+9n is a square isn't the same as saying that it is n-squared. – Roger Feb 11 '14 at 18:07
• Is it mentioned anywhere that the square root is $n$ itself ? – lsp Feb 11 '14 at 18:07
• why $=n^2$? I should say $=k^2$. – drhab Feb 11 '14 at 18:07
• Oh haha. My bad. Let me fix that. – Cameron Williams Feb 11 '14 at 18:14
• I don't know how I did that. It's fixed now. – Cameron Williams Feb 11 '14 at 18:21

Hint $\,$ mod prime $\,p = 4n\!+\!3\!:\,\ \color{#0a0}{{-}1 \equiv x^2}\,\Rightarrow\, \color{#c00}{-1} \equiv (\color{#0a0}{-1})^{2n+1}\! \equiv (\color{#0a0}{x^2})^{2n+1}\! \equiv x^{p-1} \equiv \color{#c00}1,\,$ by little Fermat. Hence $\,p\mid 2 = \color{#c00}{1-(-1)},\,$ contra $\,p\,$ odd. Yours is the special case $\,n=0,\,\ p=3\,\,$ which, more simply, can be verified directly: $\,{\rm mod}\ 3\!:\ x\not\equiv 0\,\Rightarrow\,x\equiv \pm1\,\Rightarrow\, x^2\equiv 1.$

The nice thing about modulo equations, that you can just try all the possible solutions.

Mod 9 we have:

$$0^2\equiv1\\1^2\equiv1\\2^2\equiv4\\3^3\equiv0\\4^2\equiv7\\5^2\equiv7\\6^2\equiv0\\7^2=4\\8^2=1$$ $x$ nust be equal to one of them, And no square of them is 2.