Consider the equation \begin{equation} x^2+y^2+z^2=a^2,~~~~~~~~~~(*) \end{equation} where $x,y,z \text{ and } a $ are positive integers. It is easy to prove that exist infinite triples $x,y,z$ with $GCD(x,y,z)=1$ that satisfy $(*)$. Are there triples $x,y,z$ pairwise coprime satisfying $(*)$?
Note: By pairwise coprime we mean $GCD(x,y)=GCD(x,z)=GCD(y,z)=1$.
mod $4$ there are only two possibilities for the left side:
$1+1+1$ and $1+1+0$, neither are quadratic residues $\mod 4$.