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Math Help - nonlinear diophantine equation

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    Show that the diophantine equation x^4+4y^4=z^2 has no solutions in nonzero integers
    Last edited by qpal1234; June 6th 2008 at 05:35 PM.
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    Quote Originally Posted by qpal1234 View Post
    Show that the diophantine equation x^4+4y^4=z^2 has no solutions in nonzero integers
    I think this might work.

    Write (x^2)^2+(2y^2)^2 = z^2. Assume a non-zero solution exists. Let z be the smallest of all positive solutions. Then it must mean \gcd(x,y) = \gcd(x,z) = \gcd(y,z)=1 by minimality of z. Using Pythagorean triples construct a smaller solution (that is, use Fermat's method of infinite descent) z'<z. And then this leads to a contradiction.
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    Quote Originally Posted by qpal1234 View Post
    aa
    Do you mean this one?:

    Quote Originally Posted by qpal1234
    Show that the diophantine equation x^4+4y^4=z^2 has no solutions in nonzero integers
    The one you originally double posted?

    Why? What are you trying to hide all of a sudden?
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