Show that if $m, n$ and $m',n'$ produce the same primitive Pythagorean triple then $m=m'$ and $n=n'.$

Question

I have the definition for a primitive PT. If:

• $m,n$ are positive integers and $m >n$.
• One of $m,n$ is odd, one is even.
• $gcm(m,n)=1$

Then $(x,y,z)=((m^2-n^2),2mn,(m^2+n^2))$ is a primitive Pythagorean Triple.

I can't seem to find a way to show this, any help would be appreciated.

Answer 1

Note that $\displaystyle m^2=\frac{z+x}{2}$ and $\displaystyle n^2=\frac{z-x}{2}$.