
Proof by contradiction
If u and v have different parities( one is even and one is odd), then ( a,b,c) is primitive.
Fact: If (a, b, c) is not primitive, then there exists a prime number p and natural numbers a`, b`, c` , such that a=pa`, b=pb` , c=pc`
Fact 2: if p is a prime number, and m,n are elements of N such tha pmn , then pm or pn

Pythagorean Theorem
Well, for starters, you ought to specify that we're referring to the equations . Also, this theorem is false: and must not only have opposite parities, they must be relatively prime. Example: and have opposite parity, yet is NOT primitive. The correct theorem should read: "If , then is primitive." Moving on...
To prove by contradiction, we start with the premise " and have no common divisors AND there exists a prime that divides ." We will show this premise to be impossible:
Lemma: if , and must have opposite parities, one even one odd. so , so or or . If then since and have opposite parities, so or . WLOG, suppose . Now, so . Since , so . Thus, , and we have our contradiction. Therefore our premise is false.