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

Whenever $1+ab$ divides $a^2+ b^2$, I need to prove that $\frac{a^2+b^2}{1+ab}$ will be a perfect square.

This is the last problem in Exercise 1.3 of number theory book by Ivan Niven. The chapter only has dealt with GCD and LCM concepts so far, along with a few division algorithms. However using the abovesaid principles I have failed to reach any solution.

So any help please?

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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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Right this is a IMO $1988$ problem. Check out this link.

  • 0
    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