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The $$3x+1$$ problem, generalized Pascal triangles and cellular automata. (English) Zbl 0773.11016
Let $$T:\mathbb{N}\to\mathbb{N}$$ be defined by $$T(n)=(3n+1)/2$$, $$n$$ odd, and $$T(n)=n/2$$, $$n$$ even. The sequence $$(n,T(n),T(T(n)),\dots)$$ is called the $$T$$-trajectory of $$n$$. There are several unsolved hypotheses connected with the iterations of $$T$$; for example the “$$3x+1$$” (or Collatz or Hasse or Kakutani or Syracuse)-conjecture: For every $$n\in\mathbb{N}$$ the $$T$$- trajectory starting with $$n$$ contains the value 1.
Generalized Pascal triangles are defined similarly to the classical Pascal triangle, but instead of 0 and the addition of integers operations of a finite algebra is used.
In this note two algebras (consisting of 7 or 8 elements) are constructed in such a way that some structural properties of their generalized Pascal triangles are equivalent to the “$$3x+1$$”-conjecture. In terms of cellular automata, the 8 element automaton is nilpotent if and only if the “$$3x+1$$”-conjecture holds. So one can hope that this translation to cellular automata could help to visualize the underlying number-theoretic problems.
##### MSC:
 11B83 Special sequences and polynomials 68Q80 Cellular automata (computational aspects) 68R15 Combinatorics on words
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##### References:
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