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Math Help - Nontrivial Solutions

  1. #1
    Member diddledabble's Avatar
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    Post Nontrivial Solutions

    Show that x^2+y^2=4xy has no notrivial solutions. I am not sure where to begin. Is it like a Fermat Pell equation?
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    If it has a nontrivial solution, it clearly has one with x,y coprime. But, for example, y^2=4xy-x^2 implies that x divides y^2... can you finish?
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  3. #3
    Member diddledabble's Avatar
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    No, you lost me. Should I rewrite as x^2+x^2-4xy=4xy? Then 2x^2=8xy
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  4. #4
    Super Member Gamma's Avatar
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    No, those two equations are not the same.

    What he is saying get the original equation in the form he has it (subtract x^2 from both sides. Then you see the RHS you can factor out an x to see that x must divide y^2. But if x|y^2, then you see that x and y must share at least 1 prime factor, or else x is \pm 1 or y is 0. y=0 yields the trivial solution. x=\pm 1 has no integer solutions, use the quadratic formula to see why (or rational root test if you want). So they must share a prime factor making them not relatively prime.

    Very nice proof bruno.
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