Pythagorean triangles: partitions into series and recalculation algorithm

Abstract

Pythagorean triangles, or Pythagorean triples, have been the subject of such extensive popular science literature, including articles from Kvant and Kvantik, that it makes no sense to list all the sources in a bibliography; simply searching for the relevant terms in an online search
engine is sufficient. You can find websites that explicitly list all Pythagorean triangles in which the hypotenuse does not exceed a certain large number, such as 1,000,000. Each publication, along with well-known facts for centuries, highlights new developments and intriguing numerical details.
Below, we propose one possible algorithm for enumerating all irreducible Pythagorean triangles, which boils down to dividing them into series, each with its own invariant.