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