<s>[INST] Do whole number solutions exist for $3x^2 = y^2$? I am investigating properties of square numbers and would like to find whole number solutions for this equation
$3x^2 = y^2$
or $\sqrt{3x^2} = y$
How do I prove that whole number solutions do not exist or how do I identify them?
 [/INST] We can rearrange this to get 
\begin{align}3x^2&=y^2\\
y&=\pm\sqrt{3x^2}\\
&=\pm\sqrt{3}x\end{align}
If we assume $x$ is a whole number, then $y$ can only be a whole number if $x=y=0$
Equally:
\begin{align}3x^2&=y^2\\
x^2&=\frac {y^2}{3}\\
x&=\pm\sqrt{\frac{y^2}{3}}\\
&=\pm\frac{y}{\sqrt{3}}\end{align}
Again, if we assume $y$ is a whole number, then $x$ can only be a whole number if $x=y=0$
Therefore the only solution for $x$ and $y$ whole numbers is $x=y=0$
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