Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be unbounded as n varies?
x,y are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of n (assuming n can be factored).
If yes, will large number of solutions give moderate rank EC?
If one drops $-1$ i.e. $xy(x-y)=n$ the answer is "yes" via multiples of rational point(s) and then multiplying by a cube.
EDIT: Suppose it is an open question.
EDIT: I would be interested in this computational experiment: find $n$ that gives a lot of solutions, say $100$ (I can't do it), check which points are linearly independent and this is a lower bound on the rank.
EDIT: What I find intriguing is that all integral points in this model come from factorization/divisors only.
EDIT: Current record is n=179071200 with 22 solutions with positive x,y. Due to Matthew Conroy.
Current record is n=391287046550400 with 26 solutions with positive x,y. Due to Aaron Meyerowitz
Current record is n=8659883232000 with 28 solutions with positive x,y. Found by Tapio Rajala.
Current record is n=2597882099904000 with 36 solutions with positive x,y. Found by Tapio Rajala.
EDIT: Is it possible some relation in the primes or primes of certain form to produce records? Read an article I didn't quite understand about maximizing the Selmer rank by chosing the primes carefully.