Pythagorean Triples and Inner Products

Abstract

Linear algebra over rings particularly over Z—is a fascinating topic that straddles course boundaries and can help bridge the present unnecessary gap between introductions to algebra and number theory. Moreover, a brief introduction to integral linear algebra can be helpful in further studies. For example, in both coding theory and solid state physics one is led to study lattices over the integers—roughly speaking, periodic discrete arrays of points in space (a more precise definition will be given below) for which the standard linear algebra curriculum, dealing only with spaces, is inadequate. We refer the reader to the books by Conway–Sloane [3] or Ebeling [6] for lattices in coding theory, and Kittel [7] or Senechal [14] for lattices in solid state physics. Following the introduction, we review inner product spaces and the representation of inner products by symmetric matrices. The inner products we study will not necessarily be positive definite. (Indeed, our coefficients will not always be real or complex numbers.) In a background section, we introduce unimodular matrices over Z and then Z-lattices on rational inner product spaces; and then we explore the fundamental role of unimodular matrices in connection with basis changes for lattices. The following section is the heart of the article. There we give a new perspective on Pythagorean triples through the medium of lattices on inner product spaces, using the machinery of the earlier sections. A Pythagorean triple is a triple (a, b, c) of integers such that a2 b2 = c2; and the triple (a, b, c) is primitive if gcd(a , b, c) = 1. The construction of primitive Pythagorean triples, and in particular the demonstration that there are infinitely many of them, is a truly ancient subject and a standard topic in elementary number theory courses.