Is there the exclusive list of Pythagorean triples? If there is, what is the general pattern for it?
What about Euclid's formula? Does it generate all the Pythagorean triples?
More explicitly, if $a,b,c$ is a Pythagorean triple, then does it satisfy $$ a = m^2+n^2, b=2mn, c= m^2 + n^2 $$ for some numbers $m,n$?
If you don't mind, could you give me a full rigorous statement of what I tried to describe right above (that is, the necessary condition of Pythagorean triples)?