I was trying a British Math Olympiad problem, $\sqrt a + \sqrt b = \sqrt{2009}$, find all integers a and b. After solving this problem, https://youtu.be/quECgYPNCXw in a really similar fashion to this solution, i thought to try and come up with a generalisation,
if $\sqrt a + \sqrt b = \sqrt c$ , where $\sqrt c = m \sqrt n$ , where $m$ and $n$ are integers and $n$ is as small as possible, then only solutions would be of form $\sqrt a = x \sqrt n , \sqrt b = (m-x)\sqrt n $
I mean to me, this seems really intuitive and I am sure it won't be very tough to actually prove this but I haven't ever seen this 'theorem' (?) before
With this, we could do a lot of really useful things, like the above problem could be easily solved as we could write $\sqrt{2009}$ as $7\sqrt {41}$ so it would be really trivial to find $a$ and $b$,
also, we could argue that is $c \sqrt a + \sqrt b = \sqrt c$ has no solutions for $(a,b)$ in positive integers if $c$ is not of form $m^2n$ where $m$ and $n$ are positive integers too,
to me, this seems really cool with a lot of applications, could any of you tell me if this is true and actually write a mathematical statement for it. also, if you think you know a nice proof, do let me know. Any help is appreciated, thanks.
