I am trying to understand how the Proof by Contradiction can be applied to deriving the generalized rule for Pythagorean triples. I understand the general idea behind setting:
$$\begin{align} a &= 2pq\\ b &= p^2 - q^2\\ c &= p^2 + q^2 \end{align}$$
but not sure how I could use this to imply that a and b are odd? Any suggestions on how to go about this? or should I try using a variation of the definition of an odd number?