On the algebraic structure of Pythagorean triples

Abstract

A Pythagorean triple is an ordered triple of integers (a,b,c) ≠ (0, 0, 0) such that a² + b² = c². It is well known that the set ℘ of all Pythagorean triples has an intrinsic structure of commutative monoid with respect to a suitable binary operation (℘,⋆). In this article, we will introduce the "commensurability" relation ℛ among Pythagorean triples, and we will see that it induces a group quotient, ℘/ℛ, which is isomorphic with the direct product of infinite (countable) copies of C_∞, the infinite cyclic group, and a cyclic group of order 4. As an application, we will see that the acute angles of Pythagorean triangles are irrational when measured in degrees.