  We obtain best possible results for the number of coprime positive integer
solutions, along with some effectivity, of the equation in the title when $a$
is a positive integer, $b=p^{m}$ or $b=4p^{m}$ where $m$ is a non-negative
integer, $p$ is prime and $x^{2}- \left( a^{2}+b \right) y^{2}=-4$ has an
integer solution. We prove our results by bounding the number of distinct
squares in certain binary recurrence sequences, including those associated with
such equations.
