Open Access
Translator Disclaimer
Winter 2000 Algebraic Structures of Some Sets of Pythagorean Triples I
Marek Wójtowicz
Missouri J. Math. Sci. 12(1): 31-35 (Winter 2000). DOI: 10.35834/2000/1201031

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

Download 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

Information

Published: Winter 2000
First available in Project Euclid: 5 October 2019

zbMATH: 1119.13300
MathSciNet: MR1741830
Digital Object Identifier: 10.35834/2000/1201031

Rights: Copyright © 2000 Central Missouri State University, Department of Mathematics and Computer Science

JOURNAL ARTICLE
5 PAGES


SHARE
Vol.12 • No. 1 • Winter 2000
Back to Top