Q:

fundamental Pythagorean triangle

Prove that if $r \geq 3$ and r is not $\equiv 2 \mod4$, then there is a fundamental Pythagorean triangle with leg of length r.
Can someone give me a hint how to write a general element out this way?  normally you split the expression $r=m^2-n^2$ and look for r 

A:

If $r=4n$ then $(4n^2-1, 4n, 4n^2+1)=(4n^2-1,r,4n^2+1)$ is a fund.  Pyth.  triplet. 
If $r=2n+1$ then $(2n+1, 2n^2+2n, 2n^2+2n+1)=(r, 2n^2+2n, 2n^2+2n+1)$ is a fund.  Pyth.  triplet. 
The first one is obtained as $(u^2-v^2,2uv, u^2+v^2)$ with $u=2n$ and $v=1.$
The second one is obtained as $(u^2-v^2,2uv,u^2+v^2)$ with $u=n+1$ and $v=n.$ 

