<s>[INST] Find the solutions of the diophantine equation $(x^2-y^2)(z^2-w^2)=2xyzw$ Let $x,y,z,w$ be postive integers. Find all solutions of:
$$(x^2-y^2)(z^2-w^2)=2xyzw$$
This gives: 
$$\left(\dfrac{x}{y}-\dfrac{y}{x}\right)\left(\dfrac{z}{w}-\dfrac{w}{z}\right)=2$$
$$\left(p-\dfrac{1}{p}\right)\left(q-\dfrac{1}{q}\right)=2$$
 [/INST] There are NO solutions in rationals or integers.
The equation
\begin{equation*}
\left( p-\frac{1}{p} \right) \left( q-\frac{1}{q} \right) =2
\end{equation*}
is equivalent to the quadratic
\begin{equation*}
p^2 + \frac{2q}{1-q^2} p - 1=0
\end{equation*}
For this to have rational solutions the discriminant must be a rational square, so there must exist $D \in \mathbb{Q}$ such that
\begin{equation*}
D^2=q^4-q^2+1
\end{equation*}
The quartic has a rational point when $q=0$, so is birationally equivalent to an elliptic curve. The curve is
\begin{equation*}
V^2=U^3-7U^2+12U=U(U-3)(U-4)
\end{equation*}
with
\begin{equation*}
q=\frac{V}{2U-6}
\end{equation*}
Pari-gp gives $7$ finite torsion points, $(0,0)$, $(3,0)$, $(4,0)$, $(2, \pm 2)$ and $(6,\pm 6)$. Denis Simon's ellrank package give rank $0$, so the torsion points are the only rational points.
The torsion points give $q=0$, $q=\pm 1$ or $q$ undefined, all of which are unacceptable values for a positive integer solution.
</s>