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

Prove that if $c=\frac{a^2+b^2}{ab+1}\in \mathbb{N}$ then $c$ is a square given $a, b\in \mathbb{N}$.

I know that this's a very hard question, I welcome any ideas, guesses and hints, and I accept all kinds of method to solve this . Thank you.

P.S. Please don't be restricted by the tag "number theory".

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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

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