Cellular automata, the Collatz conjecture and powers of 3/2

Abstract

We discuss one-dimensional reversible cellular automata F x3 and F x3/2 that multiply numbers by 3 and 3/2, respectively, in base 6. They have the property that the orbits of all non-uniform 0-finite configurations contain as factors all finite words over the state alphabet {0,1,...,5}. Multiplication by 3/2 is conjectured to even have an orbit of 0-finite configurations that is dense in the usual product topology. An open problem by K. Mahler about Z-numbers has a natural interpretation in terms the automaton F x3/2. We also remark that the automaton F x3 that multiplies by 3 can be slightly modified to simulate the Collatz function. We state several open problems concerning pattern generation by cellular automata. © 2012 Springer-Verlag.