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

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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 fromÂ Â 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 triplesÂ 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?

Have fun!

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