1. ## a

Show that the diophantine equation x^4+4y^4=z^2 has no solutions in nonzero integers

2. Originally Posted by qpal1234
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'. And then this leads to a contradiction.

3. Originally Posted by qpal1234
aa
Do you mean this one?:

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?