Is the set of Pythagorean Triples transitive?

Question

Is the set of Pythagorean Triple's considered transitive? How would one prove its transitivity?

For the sake of formality, the set $S$ is defined as $S=\{(a,b)|\exists{c}\in{Z^+}\text{ such that }{c^2=a^2+b^2}\}$.

Any insight would be greatly appreciated! Thanks.

Answer 1

It is not. $(3,4)\in S$, $(4,3)\in S$ but $(3,3)\notin S$.

Answer 2

$3,4,5$ is a triplet. So $3*3,3*4,3*5$ and $4*3,4*4,4*5$ are triplets. So transitivity would imply $3*3,4*4,X $ will be a triplet for some integer $X = \sqrt {(3*3)^2+(4*4)^2}=\sqrt {81+256}=\sqrt {337} $.

No such $X $ so not transitive.