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Alternative proof that $(a^2+b^2)/(ab+1)$ is a square when it's an integer

I came across this problem, but couldn't solve it.

Let $a,b>0$ be two integers such that $(1+ab)\mid (a^2+b^2)$. Show that the integer $\frac{(a^2+b^2)}{(1+ab)}$ must be a perfect square.

It's a double star problem in Number theory (by Niven). Thanks in advance.

Community
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Aang
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    Let the double star come here too !! ;). I gave +1 and a star. Let me wait for another one.. – IDOK Jun 25 '12 at 10:35
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    Done. It's now a double star problem in math.SE ;-). – JBC Jun 25 '12 at 10:43
  • @JBC : Ha ha, Yes.. – IDOK Jun 25 '12 at 10:43
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    related: http://math.stackexchange.com/questions/28438/alternative-proof-that-a2b2-ab1-is-a-square-when-its-an-integer http://math.stackexchange.com/questions/141383/seemingly-invalid-step-in-the-proof-of-fraca2b2ab1-is-a-perfect-squar/ – anon Jun 25 '12 at 10:51
  • New and better solution without using vieta jumping method here http://math.stackexchange.com/questions/28438/alternative-proof-that-a2b2-ab1-is-a-square-when-its-an-integer/646382#646382 – MathGod Jan 23 '14 at 06:54

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It was an IMO(International Mathematical Olympiad)problem, Terence Tao among few others solved it. There is a technique that solves similar problems, here is a link http://www.georgmohr.dk/tr/tr09taltvieta.pdf

clark
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  • Good reference. +1. – IDOK Jun 25 '12 at 10:47
  • +1 Excellent reference. This reinforces what one teacher once told me about the IMO's: the team/student that wins is not always the best mathematician but the one who had the best team to get the best tricky-solving resources. – DonAntonio Jun 25 '12 at 12:48
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    Not that it matters, but Terence Tao didn't figure it out. Another Fields medalist, Ngo Bao Chau did... he got a perfect score on the IMO that year. – Zarrax Jun 25 '12 at 17:47
  • New and better answer without using vieta jumping here http://math.stackexchange.com/questions/28438/alternative-proof-that-a2b2-ab1-is-a-square-when-its-an-integer/646382#646382 – MathGod Jan 23 '14 at 06:53