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.
Citation
Marek Wójtowicz. "Algebraic Structures of Some Sets of Pythagorean Triples I." Missouri J. Math. Sci. 12 (1) 31 - 35, Winter 2000. https://doi.org/10.35834/2000/1201031
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