## Missouri Journal of Mathematical Sciences

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

### Generalized Pythagorean Triples and Pythagorean Triple Preserving Matrices

#### 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