# A cute discrete dynamical system

Posted by: Gary Ernest Davis on: April 17, 2013

Jim Tanton (@jamestanton) posed the problem of 6 positive integers such that the product of any two of them is a multiple of their sum (preferably 6 integers with no common factors).

Inspired by Jim’s question I asked myself what happens for 3 integers. This led to the following simple dynamical system:

Given a triple of integers $(a,b,c)$ (not necessarily positive) replace this triple by

$(ab(\textrm{mod }a+b+c),ac (\textrm{mod }a+b+c),bc (\textrm{mod }a+b+c))$

unless $(a,b,c)=(0,0,0)$, in which case leave $(0,0,0)$ as $(0,0,0)$.

This gives us a dynamical system on the lattice $\mathbf{Z^3}$ of integer triples.

The orbit of a triple $(a,b,c)$ consists of all triples obtainable from  $(a,b,c)$ by the above scheme.

## Playing around

For example, the orbit of $(2,6,19)$ is $(2,6,19)\to (12,11,6) \to (16,14,8) \to (34,14,36) \to (56,48,0) \to (88,0,0) \to (0,0,0)$

Whereas, the orbit of $(2,9,14)$ leads, in 22 steps, to the 2-cycle $(65536,10240,4096), (4096,65536,10240)$

The starting point $(2,3,14)$ leads to a 6-cycle in just 1 step.

The orbit of $(1, 2, 16)$ has length 57 , ending in a 6-cycle.

## Questions

1. For which triples $(a,b,c)$ does the orbit contain $(0,0,0)$?
2. For which triples $(a,b,c)$ does the orbit lead to a fixed point other than $(0,0,0)$? What are the possible fixed points?
3. Ditto, what are the possible period 2 points – ones that flip by replacement- and which starting triples lead to them?
4. What periods are possible for periodic points?

Have fun!