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.

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    $\begingroup$ @jerr18: Could you explain what you mean by arbitrary large? One quick observation (if I am correct) is that the number of solutions is bounded by something like $2 d(n)$, where $d(n)$ is the number of divisors of $n$ and $d(n)$ grows slowly than any power of $n$ for sufficiently large $n$. $\endgroup$ – user17762 Dec 20 '10 at 10:54
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    $\begingroup$ You should edit your first sentence. You mean more or less the opposite of what you wrote: for fixed $n$ the number of solutions is fixed (so not arbitrarily large). Presumably, you wanted to ask "is the number of solutions unbounded as $n$ varies?", which is completely different from what you wrote. $\endgroup$ – Alex B. Dec 20 '10 at 14:40
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    $\begingroup$ Dear @jerr18: I've removed the tag "soft-question," which typically applies to questions about the practice of mathematics, while your is a mathematical question. $\endgroup$ – Akhil Mathew Dec 20 '10 at 15:11
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    $\begingroup$ FYI crossposted on -overflow.net: mathoverflow.net/questions/50479/… $\endgroup$ – jerr18 Dec 27 '10 at 8:58
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    $\begingroup$ The discussion on mathoverflow is more active. $\endgroup$ – jerr18 Dec 29 '10 at 13:11

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!


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