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Math Help - Infinite Solutions

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    Member diddledabble's Avatar
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    Infinite Solutions

    Show how to determine that there are infinite solutions to x^2 + y^2 =z^4 where x, y, and z > 0 and (x,y,z)=1
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    Quote Originally Posted by diddledabble View Post
    Show how to determine that there are infinite solutions to x^2 + y^2 =z^4 where x, y, and z > 0 and (x,y,z)=1
    x=a^4-6a^2b^2+b^4, \ y=4ab(a^2-b^2), \ z=a^2+b^2. you don't need to worry about positivity of x,y,z because if (x,y,z) is a solution, then (\pm x, \pm y, \pm z) will be a solution too.

    in order to make sure that \gcd(x,y,z)=1, you should choose a,b so that \gcd(a,b)=1 and not both a,b are odd.
    Last edited by NonCommAlg; July 11th 2009 at 06:42 PM.
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