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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.


marked as duplicate by Zev Chonoles Jun 26 '12 at 6:03

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.


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

  • $\begingroup$ Good reference. +1. $\endgroup$ – IDOK Jun 25 '12 at 10:47
  • $\begingroup$ +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. $\endgroup$ – DonAntonio Jun 25 '12 at 12:48
  • 3
    $\begingroup$ 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. $\endgroup$ – Zarrax Jun 25 '12 at 17:47
  • $\begingroup$ New and better answer without using vieta jumping here math.stackexchange.com/questions/28438/… $\endgroup$ – MathGod Jan 23 '14 at 6:53

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