What are the integer solutions of the equation $(x^2 -1)(y^2 -1)=2(7xy-24)$ Can you help with this one? I've been trying fruitlessly for hours 
$$(x^2 -1)(y^2 -1)=2(7xy-24)$$

Let me elaborate on my hint given above in the comment section. I will focus on non-negative integer solutions because it is easy to see that if $(x,y)$ is a solution, then so is $(-x,-y)$. 
Observe  that the given equation can be written as :
\begin{align*}
(xy-7)^2 & =x^2+y^2\\
(xy-6)^2+13 & =(x+y)^2\\
(xy-6)^2-(x+y)^2 & =-13\\
(xy-6+x+y)(xy-6-x-y)&=-13.
\end{align*}
Since $13$ is a prime, so we get the following two systems
\begin{align*}
xy-6+x+y&=13 &&& xy-6+x+y=1\\
xy-6-x-y&=-1 &&& xy-6-x-y=-13
\end{align*}
These systems can be written as:
\begin{align*}
x+y&=7 &&& x+y=7\\
xy&=12 &&& xy=0
\end{align*}
So the only non-negative integer solutions are $(x,y)=(3,4), (4,3),(0,7),(7,0)$. 
