Condition on an expression to have an integer solution 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 $g$?
 A: For $\sqrt{4m-1}$ to be an integer, $m=\frac{n^2+1}{4}$
If so, then $884m-1=221n^2+220$, and this must be a square, call it $r^2$
We get $221n^2+220-r^2=0$, which by quadratic formula (after a little manipulation) requires $n=\frac{\sqrt{221(r^2-220)}}{221}$.
The easiest solution to this (I haven't looked to see if it is the only one) is seen to be $r^2-220=221$, making $r=21$ and $n=1$
This in turn gives $m=0.5$, which plugged into the original equation yields $f=10$.
A: 1)
Let $a^2=884m-1$ and $b^2=4m-1$.
Find polynomial $P(f,m)$, where $f \in \Bbb Z$ and $m \in \Bbb Q$:
? polresultant(2*f - a + b, 884*m - 1 - a^2, a)
%9 = -4*f^2 - 4*b*f + (884*m + (-b^2 - 1))
? polresultant(%, 4*m - 1 - b^2, b)
%10 = 16*f^4 + (-7104*m + 16)*f^2 + 774400*m^2
? %/16
%11 = f^4 + (-444*m + 1)*f^2 + 48400*m^2

$P(f,m)=f^4 + f^2 - 444f^2m  + 48400m^2=0$
Some numerically solutions in Wolfram.
2)
Let $f=110h$, then $h^2 + 12100 h^4 - 444 h^2 m + 4 m^2=0$ and numerically this equation have not solutions for $0<h<10^6$.
$2 \mid h$. Let $h=2t$, then $m^2 - 444 m t^2 + t^2 + 48400 t^4=t^2 - 884 t^4 + (222 t^2 - m)^2=0$.
If exist $\sqrt{884 t^4 - t^2} \in \Bbb Z$, then $m=\sqrt{884 t^4 - t^2} - 222 t^2 \in \Bbb Z$. But $m \notin \Bbb Z$. This contradiction, i.e. $110 \nmid f$.
3)
$$f=\frac{\sqrt{4gm-1}-\sqrt{4m-1}}{g-1}$$
$f,g \in \Bbb Z$ and $m \in \Bbb Q$
$P(f,g,m)=4 f^2 + f^4 - 2 f^4 g + f^4 g^2 - 8 f^2 m - 8 f^2 g m + 16 m^2=0$.
This equation have many solutions.
After simplify $4 f^2 - 4 f^4 g + (f^2 (g + 1) - 4 m)^2=0$.
Then condition on $m$ for $f$ and $g$ is $s=\sqrt{4 f^4 g-4 f^2}-f^2 (g + 1) \in \Bbb Z$ and $4 \nmid s$. $f$ is depends on $g$ from Pell equation $gf^2-1=y^2$. If for given $g$ exist solution $(f,y)$ this Pell equation, then $s=2fy-f^2(g+1)$.
