Do whole number solutions exist for $3x^2 = y^2$?

Question

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?

Answer 1

$1.$ Using the FTA approach: Power of three is odd on the left side while even on that of the right side.

$2.$ $3x^2=y^2\implies \frac{y}{x}=\sqrt{3}$ LHS is rational while RHS is irrational. (This is the same approach as previous one but a bit easy to understand).

Answer 2

• Since $x$ is a whole number, $\sqrt{x^2}=|x|$ is also a whole number.
• $\sqrt{3x^2} = \sqrt 3\cdot\sqrt{x^2}$,
• $\sqrt{3}$ is not a rational number.

Answer 3

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 $x=y=0$

Answer 4

Actually, yes. Remember that zero is a whole number and identify the one solution accordingly.