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Math Help - Find all positive integer numbers

  1. #1
    Newbie harrypham's Avatar
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    Find all positive integer numbers

    Find all integer numbers x,y such that
    (x^2+y)(x+y^2)=(x-y)^2
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  2. #2
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    Re: Find all positive integer numbers

    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?
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  3. #3
    Junior Member Sarasij's Avatar
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    Re: Find all positive integer numbers

    Quote Originally Posted by harrypham View Post
    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:
    &#40;x&#94;2&#43;y&#41;&#42;&#40;x&#43;y&#94;2&#41 ;&#61;&#40;x-y&#41;&#94;2 - Wolfram|Alpha
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