Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be unbounded as n varies?
This answer is primarily intended to remove this question from the Unanswered queue.
While no conclusive theoretical answers were given, there was a more fruitful discussion at the crosspost on MathOverflow, together with some interesting numerical data.
Further, the crosspost are also contains a number of nice references; check it out if you haven't!