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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.

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    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#6463822014-01-23

1 Answers 1

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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

  • 0
    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#6463822014-01-23