On Using (Z^2, +) Homomorphisms to Generate Pairs of Coprime Integers
| dc.creator | Benson, Brian A. | |
| dc.date | 2008-02-05 | |
| dc.date | 2008-02-17 | |
| dc.date.accessioned | 2026-07-07T09:21:01Z | |
| dc.date.available | 2026-07-07T09:21:01Z | |
| dc.description | 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. | |
| dc.description | 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 | |
| dc.identifier | https://arxiv.org/abs/0802.0547 | |
| dc.identifier | http://arxiv.org/abs/0802.0547 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/154877 | |
| dc.subject | Number Theory | |
| dc.title | On Using (Z^2, +) Homomorphisms to Generate Pairs of Coprime Integers | |
| dc.type | text |