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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.
11B83 Special sequences and polynomials
68Q80 Cellular automata (computational aspects)
68R15 Combinatorics on words
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