We combine the deep methods of Frey, Ribet, Serre and Wiles with some
results of Darmon, Merel and Poonen to solve certain explicit
diophantine equations. In particular, we prove that the area of a
primitive Pythagorean triangle is never a perfect power, and that each
of the equations $X^4 - 4Y^4 = Z^p$, $X^4 + 4Y^p = Z^2$ has no
non-trivial solution. Proofs are short and rest heavily on results
whose proofs required Wiles' deep machinery.