I have an expression $$f=\frac{\sqrt{884m-1}-\sqrt{4m-1}}{2}$$ What is the condition on $m$ for which $f$ is an integer. Also is it possible to prove $f/110$ can never be an integer? 
P.S: $m$ need not be an integer. From what I can see $m$ should be of the form $k+0.5$ for some $k$. I was wondering if a tighter condition on $m$ be derived.
The solutions I found by assuming that odd squares are of the form $8r+1$ for some $r$ are:
$$m=0.5,2.5,18090.5,117992.5,4034072.5,...$$ Can a pattern be derived for this?
More generally, what is the condition for an integral solution of$$f=\frac{\sqrt{4gm-1}-\sqrt{4m-1}}{g-1}$$ for some integer $g$?