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 (not necessarily positive) replace this triple by

unless , in which case leave as .

This gives us a dynamical system on the lattice of integer triples.

The orbit of a triple consists of all triples obtainable from   by the above scheme.

Playing around

For example, the orbit of is

Whereas, the orbit of leads, in 22 steps, to the 2-cycle

The starting point leads to a 6-cycle in just 1 step.

The orbit of has length 57 , ending in a 6-cycle.

Questions

1. For which triples  does the orbit contain ?
2. For which triples does the orbit lead to a fixed point other than ? 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!