The Collatz problem in the light of an infinite free semigroup (Q463258)

Summary: The Collatz (or \(3m + 1\)) problem is examined in terms of a free semigroup on which suitable diophantine and rational functions are defined. The elements of the semigroup, called T-words, comprise the information about the Collatz operations which relate an odd start number to an odd end number, the group operation being the concatenation of T-words. This view puts the concept of encoding vectors, first introduced in [Acta Arith. 30, 241--252 (1976; Zbl 0348.10037)] by \textit{R. Terras}, in the proper mathematical context. A method is described which allows to determine a one-parameter family of start numbers compatible with any given T-word. The result brings to light an intimate relationship between the Collatz \((3m + 1)\) problem and the \((3m-1)\) problem. Also, criteria for the rise or fall of a Collatz sequence are derived and the important notion of anomalous T-words is established. Furthermore, the concept of T-words is used to elucidate the question what kind of cycles-trivial, nontrivial, rational-can be found in the Collatz \((3m + 1)\) problem and also in the \((3m -1)\) problem. Furthermore, the notion of the length of a Collatz sequence is discussed and applied to average sequences. Finally, a number of conjectures are proposed.