Two perfect squares in a right triangle

Question

Prove that there is no integer sided right triangle in which the lengths of two sides are simultaneously perfect squares

Answer 1

Assuming that such a triangle exist, then there is a nontrivial integer solution of $$a^4+b^2=c^4\tag{1}$$ or of: $$ a^4+b^4=c^2\tag{2} $$ that are essentially the case $n=4$ of the Fermat's last theorem. The impossibility of $(1)/(2)$ can be proved by using Fermat's descent and the fact that $\mathbb{Z}[i]$ (the ring of gaussian integers) is a unique factorization domain.