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The $$3x+1$$ problem: A quasi cellular automaton. (English) Zbl 0662.10010
The “Collatz-Ulam-Syracuse-Kakutani” problem, at its simplest, is the study of the iterates of $$f(x)$$ defined equal to $$(3x+1)/2$$ if x is an odd natural number, and equal to $$x/2$$ if $$x$$ is an even natural number. Whether $$f$$ iterates to 1 is an unresolved issue. The authors show empirical results of the first 717 iterates of $$f$$ for a variety of initial values for $$x$$, of the order $$10^{300}$$. The graphical representations of the iterates expressed in base 2 reveal regular patterns and chaotic behavior, some of which seems partly dependent on the process for generating the random seeds. Their figures also show a resemblance to one-dimensional cellular automata patterns, not unsurprisingly given the effect in base 2 of the definition of the f function. Heuristic arguments, illustrated by the figures, are given for the steady behavior of large iterates, and, for the decay in the size of successive iterates.
Reviewer: G.Lord

##### MSC:
 11B37 Recurrences 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 68Q80 Cellular automata (computational aspects) 68Q45 Formal languages and automata