On Using (z2,+) Automorphisms to Generate Pairs of Coprime Integers

A. We use the group (Z2,+) and two associated automorphisms, τ0, τ1, 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 xi ∈ {0, 1}, the sum of the pair τx1τx2 · · · τxn [1, 2] is equal to the sum of the pair τxnτxn−1 · · · τx1 [1, 2]. Further, we give a conjecture as to the well-ordering of the sums of these integers.