TITLE: To solve $ \dfrac1m+\dfrac1n-\dfrac1{mn^2}=\dfrac34$
QUESTION [5 upvotes]: How do we find all positive integers $(m,n)$ such that $ \dfrac1m+\dfrac1n-\dfrac1{mn^2}=\dfrac34$ ?

REPLY [6 votes]: Since this is equivalent to the integer equation $$4n^2+4mn-4=3mn^2$$ then $n$ must divide $4$. Similar to @MarkBennet's answer, there are only a small number of cases to check. 
If $n=1$, then $4m=3m$, so this doesn't give any solutions.
If $n=2$, then $12+8m=12m$, and $m=3$.
If $n=4$, then $60+16m=48m$, which gives no solutions.

You can check for negative solutions too this way, but it turns out there are none:
If $n=-1$, then $-4m=3m$, so this doesn't give any solutions.
If $n=-2$, then $12-8m=12m$, which gives no solutions.
If $n=-4$, then $60-16m=48m$, which gives no solutions.