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I am looking for information about the following diophantine equation :

$N = \displaystyle\frac{x^2+y}{x+y^2}$

Has it been studied ? Is there any efficient algorithm to solve it?
Any links?

I have tried to solve it by myself this week-end, but haven't made any progress ...

Thanks in advance
Philippe

P.S:
My first post. Sorry for being unclear.
Does this equation have solutions in integers x,y for all integer N > 0 ?

  • 0
    What does it mean to have an efficient algorithm to solve a Diophantine equation?2011-05-22
  • 0
    What are the methods in your course? Do you expect to find a general solution? Have you tried to simplify the expression and complete the squares?2011-05-22
  • 0
    Is $N$ a fixed number or an unknown in your equation ? Because taking $N = 1$ and $x = y$ for example gives you a bunch of (trivial) solutions.2011-05-22
  • 0
    It looks like a problem that cries for "vieta-jumping"...2011-05-22
  • 1
    Recently posted related post: [When is $\frac{a^2+b}{b^2+a}$ an integer?](https://math.stackexchange.com/q/2543688)2017-11-30
  • 0
    A recent post about the same equation: [Prove or disprove that, for any $n \in \mathbb{N_+}$, there exist $a,b \in \mathbb{N_+} $ such that $\frac{a^2+b}{a+b^2}=n.$](https://math.stackexchange.com/q/2802933). And the MO cross-post: [Prove or disprove that, for any $n \in \mathbb{N_+}$, there exist $a,b \in \mathbb{N_+} $ such that $\dfrac{a^2+b}{a+b^2}=n.$](https://mathoverflow.net/q/302416).2018-06-09

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