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

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

2008-02-05

arxiv.org/abs/0802.0547v2

Mathematics

Number Theory

11 pages, no figures. A serious error in terminology has been corrected. The maps \tau_0 and \tau_1 are homomorphisms but NOT

Scientific paper

We use the group $(\Z^2,+)$ 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.

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