What are the integer solutions of the equation $(x^2 -1)(y^2 -1)=2(7xy-24)$

Question

Can you help with this one? I've been trying fruitlessly for hours

$$(x^2 -1)(y^2 -1)=2(7xy-24)$$

Answer 1

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 $(x,y)=(3,4), (4,3),(0,7),(7,0)$.