Algebraic Structures of Some Sets of Pythagorean Triples I

Abstract

Let ${\mathcal P}$ denote the set of all Pythagorean triples $\{ (a,b,c) \in {\mathbb Z} ^3 : a^2 + b^2 = c^2 \}$, and let ${\mathcal P}_n = \{ (a,b,c) \in {\mathcal P} : c-b = n \}$, for $n \ne 0$, and ${\mathcal P} _0 = \{ (0,j,j) : j \in {\mathbb Z} \}$. It is shown that the ring operations defined by A. Grytczuk on ${\mathcal P} _n$'s are determined by shifts and an injection acting from suitable subsets of ${\mathbb Z} + i {\mathbb Z}$ into ${\mathbb Z} ^3$ (Section 2), and that all ${\mathcal P} _n$'s are distributive lattices (Theorem 2). The ring and the lattice structures of $\Pi = \{ (a,b,c) \in {\mathcal P} : a = 2xy,\ b = x^2 - y^2,\ c = x^2 + y^2 \}$ and some of its subsets are discussed in Theorems 3, 4, and 5.