On generalized circuit of the Collatz conjecture
type: Item
instance of: scholarly article
title: On generalized circuit of the Collatz conjecture
zbMATH Open document ID: 1136.11018
author: Tomoaki Mimuro
published in: Tokyo Journal of Mathematics
publication date: 2006-03-29T00:00:00Z
review text: The ``Collatz-Conjecture'' (or ``\(3x+1\)''- or ``Hasse''- or ``Syracuse''- or ``Kakutani''-problem) is to prove that for every \(n \in \mathbb N\) (or \(n\in\mathbb Q)\) there exists an iteration \(k\) of the Collatz-function \(C(n)\) with \(C^{(k)}(n)=1\), where \(C(n)\) takes odd numbers \(n\) to \((3n+1)/2\) and even numbers \(n\) to \(n/2\). A number \(m\) is called a cycle-number if some iteration of \(C\) will give \(C^{(k)}(m)=m\). It is well-known that the sequence of a cycle cannot be a circuit (i.e. the sequence consists of one strictly rising and one strictly descending part) until \(m=1\). In the note under review the author shows that a somewhat generalized cycle cannot exist among the natural numbers. His result depends essentially on a theorem from Diophantine approximation theory proved by G.~J. Wirsching.
reviewed by: Helmut Müller
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 11B75
zbMATH DE Number: 5015948
zbMATH Keywords: Collatz Conjecture
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1244208209
OpenAlex ID: W2066040106
Wikidata QID: Q122984452
cites work: Q3207035
cites work: Q3929795
cites work: Q4534406
cites work: The \(3x+1\) problem: New lower bounds on nontrivial cycle lengths
cites work: Linear forms in two logarithms and interpolation determinants
cites work: The dynamical system generated by the \(3n+1\) function
cites work: The Collatz problem
DOI: 10.3836/TJM/1244208209
description: scientific article; zbMATH DE number 5015948