Document Type

Thesis - Open Access

Award Date


Degree Name

Master of Science (MS)


Mathematics and Statistics


For positive integers a, b, c, we call the ordered triple (a, b, c) a Pythagorean triple provided a2 + b2 + c2. A Pythagorean triple is said to be primitive if a and b are relatively prime. It is a well-known fact that a primitive Pythagorean triple (a, b, c) in which a is even can be represented by a = 2xy, b = x2 – y2, c = x2 + y2, where x and y are positive integers. Teigen and Hadwin note that it is not possible to represent all Pythagorean triples in this way. In particular, they observe that (12, 9, 15) does not have such a representation. Furthermore, Teigen and Hadwin present a new way of representing and generating all Pythagorean triples. The purpose of this paper is to generalize the well-known results about Pythagorean triples as well as the results of Teigen and Hadwin and Arpaia to the Euclidean domain of the Gaussian integers, which we denote by Z(i), where Z(i)={a + bi׀a, b ε Z} with Z the set of rational integers. Throughout the remainder of this paper, unless stated otherwise, the letters of the Greek alphabet will be used to represent integers in the integral domain Z(i). Latin letters with the exception of i, which is the imaginary unit for the complex number system, will represent rational integers.

Library of Congress Subject Headings

Number theory



Number of Pages



South Dakota State University