Best complexity to find solutions to $x^2+y^2=z^2$ What is the algorithm with the best complexity that finds solutions in a given range to the equation $x^2+y^2=z^2$ ?

The best i could do is to iterate through all $x$ and all $y$ and store $x^2+y^2$ in a hash and then iterate through all possible $z$ and check if $z^2$ is in the hash.
I think you are asking about Pythagorean Triples (the form $x^2+y^2=z^2$ where $x$,$y$ and $z$ are integers).

Euclids Formula is generally thought to be efficient for finding primitive Pythagorean Triples.You can multiply these primitive triples with a constant to get non primitive Pythagorean Triples.

Here are some other methods of finding Pythagorean Triples