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 nontrivial 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 nontrivial example is k= 6, m=8, corresponding to a rule that depends on four neighboring cells. Nontrivial 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.



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