<s>[INST] Positive integers $(a, b, c)$ are a primitive Pythagorean triple 
Show that if $a = m^2 - n^2$ , $b = 2mn$, $c = m^2 + n^2$ , where $m$, $n$ are relatively prime, not both odd, and $m>n$, then $(a, b, c)$ is a primitive Pythagorean triple.

This is part one of a proof I am required to do.
I know that if $m$ and $n$ are not both odd, then they can be written as $2k+1$ and $2l$, respectively, or as $2k$ and $2l$. I plugged in the given values for a, b, and c into the equation $a^2 + b^2 = c^2$ and got $m^4 + 2m^2n^2 + n^4$, but this is as far as I can get, however. I know that to show $a,b,c$ are primitive I need to show their GCD is $1$, but I don't know how to do this. Can someone show me where to start?
 [/INST] Line up your ducks. And then shoot them.
Does
$(m^2 - n^2)^2 + (2mn)^2 {? \over=} (m^2+n^2)^2$
$m^4 - 2m^2n^2 + n^4 + 4m^2n^2  {? \over=} m^4 + 2m^2n^2 + n^4$
$m^4 + 2m^2n^2 {? \over=} m^4 + 2m^2n^2 + n^4$?
The answer is... yes, it does.
So $m^2-n^2, 2mn, m^2 + n^2$ are a pythogorean triple.
====
But are the a primative triplet?  That is:
And are $m^2 - n^2$ and $2mn$ relatively prime if $m,n$ are and they are not both odd?
If $p$ is a prime divisor that divides $2mn$ then either

*

*$p|2$ so $p=2$.

But $m,n$ are relatively prime so they are not both even and they are not both odd so $m^2 -n^2$ is odd and so $p\not \mid m^2 - n^2$.


*$p|m$
But $m,n$ are relatively prime $p\not \mid n$.  So $p|m^2$ but not $n^2$ so $p \not \mid m^2 -n^2$.


*$p|n$
Same argument.  $p\not \mid m$ so $p|n^2$ but not $m^2$ and therefor $p\not \mid m^2 - n^2$.
so no prime factor of $2mn$ is a facctor of $m^2 - n^2$ so $m^2-n^2$ and $2mn$ are relatively prime.
So $m^2-n^2, 2mn, m^2+n^2$ is a primitive pythagorean triplet.
</s>