Missouri Journal of Mathematical Sciences

Generalized Pythagorean Triples and Pythagorean Triple Preserving Matrices

Mohan Tikoo and Haohao Wang

Full-text: Open access

Abstract

Traditionally, Pythagorean triples (PT) consist of three positive integers, $(x, y, z) \in \mathbb{Z}^3_+$, such that $x^2 + y^2 = z^2$, and Pythagorean triple preserving matrices (PTPM) $A$ are $3 \times 3$ matrices with entries in the real numbers $\R$, such that the product $(x, y, z)A$ is also a Pythagorean triple. In this paper, we study PT and PTPM from the view of projective geometry, and extend the results concerning PT and PTPM from integers to any commutative ring with identity. In particular, we use the method of polynomial parametrization for projective conics to obtain the general form of PT over any commutative ring with identity. In addition, we view the PTPM as projective transformations and formulate the general form of a PTPM over any commutative ring with identity.

Article information

Source
Missouri J. Math. Sci., Volume 21, Issue 1 (2009), 3-12.

Dates
First available in Project Euclid: 14 September 2011

Permanent link to this document
https://projecteuclid.org/euclid.mjms/1316032675

Digital Object Identifier
doi:10.35834/mjms/1316032675

Mathematical Reviews number (MathSciNet)
MR2503169

Zentralblatt MATH identifier
1210.11041

Subjects
Primary: 13-01: Instructional exposition (textbooks, tutorial papers, etc.)
Secondary: 14-01: Instructional exposition (textbooks, tutorial papers, etc.)

Citation

Tikoo, Mohan; Wang, Haohao. Generalized Pythagorean Triples and Pythagorean Triple Preserving Matrices. Missouri J. Math. Sci. 21 (2009), no. 1, 3--12. doi:10.35834/mjms/1316032675. https://projecteuclid.org/euclid.mjms/1316032675


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