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

Abstract

We use the group $(\Z^2,+)$ and two associated homomorphisms, $τ_0, τ_1$, 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 $τ_{x_1}τ_{x_2}... τ_{x_n} [1,2]$ is equal to the sum of the pair $τ_{x_n}τ_{x_{n-1}} ... τ_{x_1} [1,2]$. Further, we give a conjecture as to the well-ordering of the sums of these integers.