Let $(u,v)$ be a pair of non-zero integers. We say that $(u,v)$ is a pair of *simultaneous squares* if for all primes $p$ dividing $u$, we have $\left(\frac{v}{p}\right) = 1$ and for all primes $q$ dividing $v$, we have $\left(\frac{u}{q} \right) = 1$. Here $\left(\frac{\cdot}{m}\right)$ denotes the Legendre symbol. 


I want to estimate the number of pairs of simultaneous squares inside a box, say $0 < \max\{|u|, |v|\} \leq X$. Are there any asymptotic results known?



Density of "simultaneous squares"
This should be asymptotic to an expression of the form
$$\frac{cX^2}{\log X}$$
as $X \to \infty$, for some $c > 0$ (this should be interpreted as $(X/\sqrt{\log X}\,)^2$). Proving an upper bound of the correct order of magnitude in this case should not be too difficult using the large sieve (I can provide more details if you would like).


Proving the precise asymptotic formula is a bit more tricky. In the special case that $u$ and $v$ are odd, square-free, and coprime, this has been achieved in the paper


Friedlander, Iwaniec - Ternary quadratic forms with rational zeros


(see in particular section 2). 


I expect that the methods from this paper could be adapted to handle your case.


