Form of Pythagorean Triples
All the solutions of the primitive Pythagorean equation:
x**2 + y**2 = z**2
are given by:
x=2st, y=s**2 - t**2, z=s**2 + t**2
Proof:
WLOG x even, y,z odd, so
z-y=2u, z+y=2v
x**2=z**2 - y**2
=(z-y)(z+y)
(x/2)**2=((z-y)/2)((z+y)/2)=uv
hcf (y,z)=1 => hcf (u,v)=1
=> u=s**2, v=t**2, hcf (s,t)=1
z=u+v=s**2 + t**2
y=u-v=s**2 - t**2
x**2=4uv=4s**2t**2
Conversely:
x**2 + y**2 = (2st)**2 + (s**2 - t**2)**2 = (s**2 + t**2)**2 = z**2