  We investigate the solvability of the Diophantine equation in the title,
where $d>1$ is a square-free integer, $p, q$ are distinct odd primes and
$x,y,a,b$ are unknown positive integers with $\gcd(x,y)=1$. We describe all the
integer solutions of this equation, and then use the main finding to deduce
some results concerning the integers solutions of some of its variants. The
methods adopted here are elementary in nature and are primarily based on the
existence of the primitive divisors of certain Lehmer numbers.
