A Ring of Pythagorean Triples

Abstract

A one-to-one correspondence between the set of all Pythagorean triples and ${\mathbb Z} \times {\mathbb Z}$ is established, resulting in a ring of Pythagorean triples.

A triple $\langle a,b,c \rangle$ is called a Pythagorean triple if $a, b, {\text {and }} c$ are integers such that $a^2+b^2=c^2$. It seems natural to ask whether operations can be defined on the set $P$ of all Pythagorean triples in such a way as to give $P$ a ring structure. In fact, since a Pythagorean triple is determined by any two of the three integers, one might attempt to obtain a ring structure isomorphic to ${\mathbb Z} \times {\mathbb Z}$ (where ${\mathbb Z}$ represents the set of integers and the operations on ${\mathbb Z} \times {\mathbb Z}$ are defined coordinatewise) by finding a one-to-one correspondence between $P$ and ${\mathbb Z} \times {\mathbb Z}$. The establishment of such a correspondence and the resulting ring structure is the objective of this paper.