Show that the diophantine equation x^4+4y^4=z^2 has no solutions in nonzero integers
I think this might work.
Write . Assume a non-zero solution exists. Let be the smallest of all positive solutions. Then it must mean by minimality of . Using Pythagorean triples construct a smaller solution (that is, use Fermat's method of infinite descent) . And then this leads to a contradiction.