TITLE: Equation $\sqrt{x}+\sqrt{y}=\sqrt{2013}$ in rationals
QUESTION [8 upvotes]: Can we find all rational numbers $x,y$ such that $\sqrt{x}+\sqrt{y}=\sqrt{2013}$?
Certainly possible answers are $(2013,0)$ and $(0,2013)$.
If we square the equation, we get $x+y+2\sqrt{xy}=2013$, so $\sqrt{xy}$ must be rational.

REPLY [3 votes]: $$\sqrt x+\sqrt y=\sqrt a\iff\sqrt\frac xa+\sqrt\frac ya=1\iff\sqrt\frac xa=1-\sqrt\frac ya\iff\frac xa=1+\frac ya-2\sqrt\frac ya$$ $$\sqrt\frac ya=\frac{1+\frac ya-\frac xa}2\in\mathbb{Q}\iff y=a\cdot\left(\frac mn\right)^2$$ Can you take it from here ? :-)