On the combinatorial structure of \(3N+1\) predecessor sets
type: Item
instance of: scholarly article
title: On the combinatorial structure of \(3N+1\) predecessor sets
zbMATH Open document ID: 0847.11009
DOI: 10.1016/0012-365X(94)00243-C
author: Günther J. Wirsching
published in: Discrete Mathematics
publication date: 1996-03-24T00:00:00Z
review text: The ``Collatz''-problem (or ``\(3N+ 1\)''- or ``Hasse''- or ``Syracuse''- or ``Kakutani''-problem) is to prove that for every \(n\in \mathbb{N}\) there exists a \(k\) with \(T^{(k)}(n)= 1\) where the function \(T(n)\) takes odd numbers \(n\) to \((3n+ 1)/2\) and even numbers \(n\) to \(n/2\). In the note under review, the author studies not as usual the trajectory \((T^{(k)}(a))_{k\in \mathbb{N}_0}\) for \(a\in \mathbb{N}\), but its predecessor set which consists of all integers \(n\) whose iterates \(T^{(k)}(n)\) eventually hit \(a\). He gives a representation of these sets and studies their combinatorial structure. It turns out that the well-known coefficient stopping time conjecture in the original ``\(3N+ 1\)''-problem could be formulated in terms of the predecessor sets. In the last section, the author shows how density considerations are related to the asymptotic behaviour at the origin of a certain smooth function on the unit interval which would also convey new information on the original ``\(3N+ 1\)''-problem.
reviewed by: Helmut Müller
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 11B37
zbMATH DE Number: 858089
zbMATH Keywords: Collatz-problem
zbMATH Keywords: predecessor set
zbMATH Keywords: combinatorial structure
zbMATH Keywords: coefficient stopping time conjecture
zbMATH Keywords: \(3N+ 1\)-problem
zbMATH Keywords: density
zbMATH Keywords: asymptotic behaviour
MaRDI profile type: Publication
cites work: Q3207035
cites work: On the "3x + 1" Problem
cites work: Q3856819
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description: scientific article; zbMATH DE number 858089