Power cellular automata

Multiplication by m in base k corresponds to a local cellular automaton operation on digit sequences when every prime that divides m also divides k. The first non-trivial cases for which this is so are k=6, m=2^^{i} 3^^{j} and k=10, m=2^^{i} 5^^{j}. When m itself divides k, the cellular automaton rule is {_, b_, c_} -> m Mod[b, k/m] + Quotient[c, k/m]; in other cases the rule can be obtained by composition. A similar result holds for rational m, obtained for example by allowing i and j above to be negative. In all cases the cellular automaton rule, like the original operation on numbers, is invertible. The inverse rule, corresponding to multiplication by 1/m, can be obtained by applying the rule for multiplication by the integer k^^{q}/m, then shifting right by q positions. (See page 903.)

The condition for locality in negative bases (see page 902) is more stringent. The first non-trivial example is k= -6, m=8, corresponding to a rule that depends on four neighboring cells.

Non-trivial examples of multiplication by m in base k all appear to be class 3 systems (see page 250), with small changes in initial conditions growing at a roughly fixed rate.