Problem #WSP-000318

Problem

We call a triple of natural numbers (also known as positive integers) $(a,b,c)$ satisfying $a^2+b^2=c^2$ a Pythagorean triple. If, further, $a$, $b$ and $c$ are relatively prime, then we say that $(a,b,c)$ is a primitive Pythagorean triple.

Show that every primitive Pythagorean triple can be written in the form $(u^2-v^2,2uv,u^2+v^2)$ for some coprime positive integers $u>v$.