Find all positive integers s.t. $\frac{1}{a^2} - \frac{1}{b^2} = \frac{1}{c^2} - \frac{1}{d^2}$ Find all positive integers $a, b, c, d$ such that :
$$\frac{1}{a^2} - \frac{1}{b^2} = \frac{1}{c^2} - \frac{1}{d^2}$$
The original problem came from atomic electron transitions :

I would like to find out non-trivial positive integer solutions; since it is trivial if $a = b$ and $c = d$, or $a = c$ and $b = d$.
I found some of the solutions, and they look like :

There is some pattern in these integers, but it seems difficult to obtain a gerneralized form of the solution.

we take, $(a,b,c,d)=(pq,rq,st,wt)$
where, $r^2=p^2+q^2$ & $w^2=s^2+t^2$
And, $(r,p,q)=((u^2+v^2),(u^2-v^2),(2uv))$
$(w,s,t)=((m^2+n^2),(m^2-n^2),(2mn))$
We have Identity,
$(1/rp)^2=(1/pq)^2-(1/rq)^2$  ---(1)
$(1/ws)^2=(1/st)^2-(1/wt)^2$  ----(2)
we want, eqn (1)=eqn (2)
Hence, $pr=sw$
or
$(u^4-v^4)=(m^4-n^4)$ ----(3)
eqn (3) has numerical solution:
$(m,n,u,v)=(59,134,133,158)$
hence,
$(p,q,r)=(7275,42028,42653)$
$(s,t,w)=(14475,15812,21437)$
$a=305753700$
$b=1792620284$
$c=228878700$
$d=338961844$
