<s>[INST] Number theory - rational number Are there any $x, y$ that fit in below 
$\sqrt{4y^2-3x^2}$ such that an rational number is yielded.
Appreciate if explanation is given.
 [/INST] We can find infinitely many solutions by considering the Pell equation $y^2-3s^2=1$. 
We look for perfect squares $4y^2-3x^2$, where $x$ is even, say $2s$. Then we want $4y^2-12s^2$ to be a perfect square, say $(2t)^2$. So we want to solve the Diophantine equation $y^2-3s^2=t^2$.
There are already infinitely many solutions with $t=1$. For $(2+\sqrt{3})(2-\sqrt{3})=1$.  Thus $(2+\sqrt{3})^n(2-\sqrt{3})^n=1$. 
But $(2+\sqrt{3})^n=y_n+s_n\sqrt{3}$ for integers $y_n$, $s_n$. It follows that $y_n^2-3s_n^2=1$. 
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