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 p|mn , then p|m or p|n

2. ## Pythagorean Theorem

Well, for starters, you ought to specify that we're referring to the equations $\displaystyle (a,b,c)=(u^2-v^2,2uv,u^2+v^2)$. Also, this theorem is false: $\displaystyle u$ and $\displaystyle v$ must not only have opposite parities, they must be relatively prime. Example: $\displaystyle 6$ and $\displaystyle 3$ have opposite parity, yet $\displaystyle (27,36,45)$ is NOT primitive. The correct theorem should read: "If $\displaystyle (u,v)=1$, then $\displaystyle (a,b,c)$ is primitive." Moving on...

To prove by contradiction, we start with the premise "$\displaystyle u$ and $\displaystyle v$ have no common divisors AND there exists a prime $\displaystyle p$ that divides $\displaystyle a,b,c$." We will show this premise to be impossible:

Lemma: if $\displaystyle (u,v)=1$, $\displaystyle u$ and $\displaystyle v$ must have opposite parities, one even one odd. $\displaystyle p|b$ so $\displaystyle p|2uv$, so $\displaystyle p=2$ or $\displaystyle p|u$ or $\displaystyle p|v$. If $\displaystyle p=2$ then since $\displaystyle u$ and $\displaystyle v$ have opposite parities, so $\displaystyle p|u$ or $\displaystyle p|v$. WLOG, suppose $\displaystyle p|u$. Now, $\displaystyle p|a$ so $\displaystyle p|u^2-v^2$. Since $\displaystyle p|u$, $\displaystyle p|v^2$ so $\displaystyle p|v$. Thus, $\displaystyle (u,v)\neq1$, and we have our contradiction. Therefore our premise is false.