Primitive Pythagorean parametrization ($(a,b,c)=(2mn,m^2-n^2,m^2+n^2)$)

Every primitive Pythagorean triple, after exchanging its legs, has this form with $m>n>0$, $\gcd (m,n)=1$ and opposite parity. Its area is $mn(m-n)(m+n)$. The factors $m,n,m-n,m+n$ are pairwise coprime, because $m\pm n$ are odd. Conversely these conditions give a primitive triple. The parametrization follows by factoring the relatively prime integers $(c+b)/2$ and $(c-b)/2$ when $a$ is the even leg.