On Using (Z^2, +) Homomorphisms to Generate Pairs of Coprime Integers

Abstract

We use the group $({\mathbb{Z}}^2,+)$ and two associated homomorphisms, ${\tau }_0,{\tau }_1$, to generate all distinct, non-zero pairs of coprime, positive integers which we describe within the context of a binary tree which we denote $T$. While this idea is related to the Stern-Brocot tree and the map of relatively prime pairs, the parents of an integer pair these trees do not necessarily correspond to the parents of the same integer pair in $T$. Our main result is a proof that for $x_i\in \{0,1\}$, the sum of the pair ${\tau }_{x_1}{\tau }_{x_2}...{\tau }_{x_n}[1,2]$ is equal to the sum of the pair ${\tau }_{x_n}{\tau }_{x_{n-1}}...{\tau }_{x_1}[1,2]$. Further, we give a conjecture as to the well-ordering of the sums of these integers.

11 pages, no figures. A serious error in terminology has been corrected. The maps τ_0 and τ_1 are homomorphisms but NOT automorphisms as they are referred to in v1