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Collatz Sequences in the Light of Graph Theory

  • It is well known that the inverted Collatz sequence can be represented as a graph or a tree. Similarly, it is acknowledged that in order to prove the Collatz conjecture, one must demonstrate that this tree covers all odd natural numbers. A structured reachability analysis is hitherto not available. This paper investigates the problem from a graph theory perspective. We define a tree that consists of nodes labeled with Collatz sequence numbers. This tree will be transformed into a sub-tree that only contains odd labeled nodes. The analysis of this tree will provide new insights into the structure of Collatz sequences. The findings are of special interest to possible cycles within a sequence. Next, we describe the conditions which must be fulfilled by a cycle. Finally, we demonstrate how these conditions could be used to prove that the only possible cycle within a Collatz sequence is the trivial cycle, starting with the number one, as conjectured by Lothar Collatz.

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Author details:Eldar SultanowORCiDGND, Christian Koch, Sean Cox
Publication type:Report
Date of first publication:2020/01/28
Completion year:2020
Publishing institution:Universität Potsdam
Release date:2020/01/28
Tag:Cayley Graph; Collatz; Free Group; Reachability
Print run:4th version
Page number:31
Organizational units:Wirtschafts- und Sozialwissenschaftliche Fakultät / Wirtschaftswissenschaften
DDC classification:3 Sozialwissenschaften / 30 Sozialwissenschaften, Soziologie / 300 Sozialwissenschaften
MSC classification:11-XX NUMBER THEORY
Peer review:Nicht referiert
Collection(s):Universität Potsdam / Sondersammlungen / Collatz Sequences in the Light of Graph Theory / Fourth Version
License (German):License LogoCreative Commons - Namensnennung, 4.0 International