Algebraic Structures of Some Sets of Pythagorean Triples II

Abstract

A natural bijection from $\mathbb{Z} ^2$ onto the set of all Pythagorean triples ${\mathcal P} = \{ (a,b,c) \in \mathbb{Z} ^3 : a^2 + b^2 = c^2 \}$ is given (Theorem 6). Consequently, all algebraic structures of $\mathbb{Z} ^2$ are carried in a natural way onto ${\mathcal P}$ (Theorems 7, 8, and 9). This solves the open problem of defining ring operations under which ${\mathcal P}$ is essentially a different ring than the one constructed by B. Dawson (Example, Section 5). This article and the enumeration of its sections and theorems is a continuation of the author's paper [3].