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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 ?

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  • $\begingroup$ What does it mean to have an efficient algorithm to solve a Diophantine equation? $\endgroup$ – Qiaochu Yuan May 22 '11 at 11:29
  • $\begingroup$ 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? $\endgroup$ – Phira May 22 '11 at 11:32
  • $\begingroup$ 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. $\endgroup$ – Joel Cohen May 22 '11 at 11:47
  • $\begingroup$ It looks like a problem that cries for "vieta-jumping"... $\endgroup$ – Myself May 22 '11 at 11:58
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    $\begingroup$ Recently posted related post: When is $\frac{a^2+b}{b^2+a}$ an integer? $\endgroup$ – Martin Sleziak Nov 30 '17 at 6:09
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Follow user9325's suggestion about completing the squares, and then (look up and) apply the theory of Pell equations.

Edit: OK, I guess you didn't get anything out of user9325's suggestion, so I'll take it up for you.

$N=(x^2+y)/(x+y^2)$, $Ny^2-y=x^2-Nx$, $U^2-NV^2=1-N^3$ where $U=2Ny-1$, $V=2x-N$. This has the solution $U=-1$, $V=\pm N$. The solution $U=-1$, $V=N$ corresponds to $x=N$, $y=0$, which already shows that there's a solution for each $N$, but maybe that's too trivial. Then take any solution to $a^2-Nb^2=1$ and you get another solution, $U=-a\pm bN^2$, $V=-b\pm aN$. Now $a^2-Nb^2=1$ has lots of solutions - that's the Pellian I alluded to. For $y$ to be an integer, you need $a\equiv 1\pmod N$, so you have to study enough of the theory to see if that can be made to happen.

If $N$ is a square, say, $N=m^2$, then the Pellian doesn't apply, but you have something simpler; $(U+mV)(U-mV)=1-m^6$. Now you'll get at most finitely many solutions, since there are only finitely many ways to factor $1-m^6$. Here's one example; take $N=4=2^2$ so $m=2$ and $1-m^6=-63$; take $U+2V=63$, $U-2V=-1$ to get $U=31$, $V=16$; then $x=10$, $y=4$.

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  • $\begingroup$ Hi Gerry, That's perfect. Thank you very much for this very clear explanation. $\endgroup$ – PhilippeC May 23 '11 at 16:06
  • $\begingroup$ @Gerry Myerson: This question was asked recently. I came up with three parametric solutions for N, x, and y which did not use the Pell equation here: math.stackexchange.com/questions/2543688/… $\endgroup$ – marty cohen Nov 30 '17 at 20:17
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    $\begingroup$ @Gerry Myerson: The trick is to prove that the solution exists for square $N$. Your method guaranteees that there is some solution for $U,V$ but it’s not obvious that it translates to integer $x$ and $y$. For non square $N$ I have the whole proof here: math.stackexchange.com/questions/2802933/… $\endgroup$ – Oldboy Jun 16 '18 at 5:20
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Thanks for @Gerry Myerson's solution.

As far as I'm concernd, the solution could be stated as follows.

Solution

The equation could be rewritten as $$u^2-nv^2=1-n^3 \tag1$$where $u=2nb-1,v=2a-n$.

$\textbf{Case 1:}$ $n$ is not a square number.

Take $x,y$ such that $$x^2-ny^2=1.\tag2$$. This is always possible, because $x^2-ny^2=1$ is a Pell equation. Then, \begin{align*}(x+ yn^2)^2-n(y+xn)^2&=x^2+2xyn^2+y^2n^4-ny^2-2xyn^2-x^2n^3\\&=(x^2-ny^2)+n^3(ny^2-x^2)\\&=1-n^3.\end{align*}This shows that $u=x+yn^2$ and $v=y+xn$ is a solution for $(1)$. That's to say, all $a,b$ such that $2nb-1=x+yn^2$ and $2a-n=y+xn$ could satisfy $(1)$. Hence,$$a=\frac{(x+1)n+y}{2},b=\frac{x+yn^2+1}{2n}.\tag3$$ But, the problem comes out here. We need that $a,b \in \mathbb{N+}$, but the $a,b$ obtained from $(3)$ are not always like thus for all $x,y$.

So, how to show that we can find $x,y$ such that $(2)$ hold and $a,b \in \mathbb{N+}$?

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