integer solutions to bivariate polynomial of second degree I am trying to determine if there is a way to quickly determine if an equation of the following type $$0 = axy+x-y-A$$ has integer solutions ($a,A$ are integers).  If anyone knows how to do this or can point me in the right direction, I would greatly appreciate it!
Thank you!

We assume that $a\ne 0$. The equation can be rewritten as $a^2xy+ax-ay-aA=0$. and then as 
$$(ax-1)(ay+1)=aA-1.$$
For given $a$ and $A$, we can look for solutions to $st=aA-1$. They must satisfy additional congruential conditions modulo $a$. But even without the congruential conditions, unless $aA=1$ we only have a finite number of candidates to examine: just let $s$ run over all divisors, positive and negative, of $aA-1$.  (And if $aA=1$, the problem is very easy.)
