Theorem

Any primitive Pythagorean triple $(a,b,c)$ can be written as $$\tag{1}a=u^{2}-v^{2}, \ b=2uv, \ c=u^{2}+v^{2}, \ (u,v)=1$$with $u,v\in\mathbb{Z}$ of opposite parity and $u>v$. Conversely, if $u$ and $v$ are any two coprime integers of opposite parity, then $(1)$ gives a primitive Pythagorean triple.