Apologies if it's a duplicate question. I was not able to find such question though. I don't know how to proceed on this.

  • $\begingroup$ $x^2+xy-y^2=\pm 1$ can be a starting point. $\endgroup$ – Mohammad Zuhair Khan Feb 18 '19 at 5:35
  • $\begingroup$ Yes. and consecutive fibonacci numbers are the solutions to it. But how to prove it? How to solve a polynomial equation with two variables and only one equation is given? $\endgroup$ – Ratul Sarkar Feb 18 '19 at 5:38
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    $\begingroup$ Please give more context. Providing context not only assures that this is not simply copied from a homework assignment, but also allows answers to be better directed at where the problem lies and to be within the proper scope. Please avoid "I have no clue" questions. Defining keywords and trying a simpler, similar problem often helps. $\endgroup$ – robjohn Feb 18 '19 at 7:46

The hint.

Prove by induction that $$f_{n+1}^2-f_{n+1}f_n-f_n^2=(-1)^n.$$

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  • $\begingroup$ @Ratul Sarkar I added something. See now. $\endgroup$ – Michael Rozenberg Feb 18 '19 at 5:49
  • $\begingroup$ Thanks Mikchael. This is Cassini's identity. Learnt a beautiful thing today. $\endgroup$ – Ratul Sarkar Feb 18 '19 at 5:58
  • $\begingroup$ You are welcome, @Ratul! $\endgroup$ – Michael Rozenberg Feb 18 '19 at 6:28

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