A cute discrete dynamical system

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.


  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!

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