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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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    @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 $n$.2010-12-20
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    Tried to explain with edit. EDIT: I know there is an upper bound of the # of solutions for a single n because the number of divisors is finite. Intuitively, large number of divisors doesn't guarantee large number of solutions. Is there a constant K s.t. numberofsolutions≤K∀n2010-12-20
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    If someone gets the question, suggestions on how to improve it will be appreciated.2010-12-20
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    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.2010-12-20
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    Thank you, tried to fix it.2010-12-20
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    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.2010-12-20
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    Thank you for helping me. What find intruguing is that integral points come from factorization only.2010-12-20
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    What is the maximum number of solutions you've found? $n=48232800$, giving 10 solutions, is the best I've been able to come up with. The record-setting $n$ appear to have monotonically decreasing prime factorizations, but, unlike highly composite numbers, there are gaps in the higher primes in the factorization (e.g. $48232800=2^4 \cdot 3^3 \cdot 5^2 \cdot 7 \cdot 13 \cdot 23$, so 11, 17 and 19 are skipped).2010-12-21
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    Update: 179071200 has 11 solutions. This has decreasing exponents in its prime factorization, though not strictly, and again gaps in the higher primes.2010-12-21
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    Thank you for your interest. Should I rephrase the question insisting on **positive** solutions?2010-12-21
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    @Matthew Conroy for your examples my program gives 20 and 22 points for positive x,y. Am I missing something (I can give the points here). I don't have better than 22 points records yet. Multiples of a record doesn't produce a records.2010-12-21
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    @Matthew Conroy have you checked the bound of the rank resulting from your 179071200 ?2010-12-21
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    The solutions generally come in pairs, so I was counting pairs of solutions. I should have said so.2010-12-21
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    If someone knows better forum he/she or I may ask there. I am surprised the way I earn repuatation points here.2010-12-22
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    FYI crossposted on -overflow.net: http://mathoverflow.net/questions/50479/can-the-number-of-solutions-xyx-y-1n-for-x-y-n-in-z-be-unbounded-as-n-var2010-12-27
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    Current record is n=391287046550400 with 26 solutions with positive x,y. Due to Aaron Meyerowitz2010-12-28
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    According to Magma the minimal Weierstrass model of the 26 pts curve is: [ 1, 0, 0, 195643523275200, 38276388199533724134935040000 ] (a_i). Positive x,y give 6 for lower bound of the rank. "Mwrank" abort()s even with the -p switch2010-12-28
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    The discussion on mathoverflow is more active.2010-12-29

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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!