Find all positive integer numbers

**harrypham** · June 1st 2012, 12:18 AM

Find all integer numbers $x,y$ such that
$(x^2+y)(x+y^2)=(x-y)^2$

**Media_Man** · June 5th 2012, 02:54 PM

Well, here's three for you for $x<y$ , none of which are positive though: $(-1,2), (-1,-1), (0,1)$ . I'm not finding any more by blind searching. By making the substitutions $y=ax, ab=1$ , you can get a quadratic on $x$ : $x^2+(a+b^2)x-(b^2-3b+1)=0$ as long as $x\ne 0$ . Solve for $x$ using quadratic formula and replace $a,b$ back in terms of $x,y$ and you can simplify to $x=-\frac{1}{2}\frac{x^3+y^3}{xy^2}\pm\frac{x-y}{2xy^2}\sqrt{x^4+2x^3y+7x^2y^2+2xy^3+y^4}$ . Not sure if that's any better, but you might be able to parametrize that bit under the radical to get a list of $x,y$ pairs that give you a perfect square. That's as far as I've gotten so far. Was there a context for this?

**Sarasij** · June 6th 2012, 08:29 AM

Originally Posted by harrypham

Find all integer numbers $x,y$ such that
$(x^2+y)(x+y^2)=(x-y)^2$

There are exactly 4 integer solutions to this eqaution as calculated by WolframAlpha search engine ... here's the result:
(x^2+y)*(x+y^2&#41 ;=(x-y)^2 - Wolfram|Alpha

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