<s>[INST] Can a finite sum of square roots be an integer? Can a sum of a finite number of square roots of integers be an integer? If yes can a sum of two square roots of integers be an integer?
The square roots need to be irrational.
 [/INST] At least there's an elementary way to see that if $\sqrt{a} + \sqrt{b}$ is an integer, then $a$ and $b$ are perfect squares.
Suppose $\sqrt{a} + \sqrt{b} = c\in\mathbb{Z}.$ If $c=0$ the result is trivial. Otherwise, squaring both sides we get that
$$a + b + 2\sqrt{ab} = c^2$$
and therefore $ab$ must be a perfect square. Let's say $ab = d^2$. Then $a=\frac{d^2}{b}$ and
\begin{align*}\frac{d}{\sqrt{b}} + \sqrt{b} &= c\\
d + b &= c\sqrt{b},
\end{align*}
so $b$ is a perfect square, and $a$ must be as well.
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