The following property has been stated without proof in a problem solving book (not as a problem, hence no solution). I also looked at the number theory text I have, and I cannot find it.
All pairwise prime triples of integers satisfying $x^2+y^2=z^2$ are given by $x=|u^2-v^2|\;,\;y=2uv\;,\; z=u^2+v^2\;,\;\text{gcd}(u,v)=1\;,\; u\neq v \;\text{mod} \;2$
Particularly, why $u,v$ should be coprime and $u\neq v\; \bmod \;2$ as otherwise this is trivial algebra. A reference or an intuitive explanation of this result would be appreciated.