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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 unavailable. 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. Furthermore, we derive and prove several formulas that can be used to traverse the graph. The analysis covers the Collatz problem both in it’s original form 3x + 1 as well as in the generalized variant kx + 1. Finally, we transform the Collatz graph into a binary tree, following the approach of Kleinnijenhuis, which could form the basis for a comprehensive proof of the conjecture.

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Author details:Eldar SultanowORCiDGND, Christian Koch, Sean Cox
Publication type:Report
Date of first publication:2020/11/11
Completion year:2020
Publishing institution:Universität Potsdam
Release date:2020/11/11
Tag:Binary Tree; Cayley Graph; Collatz Conjecture; Cycle; Cyclic Group; Free Group; Multiplicative Group; Tree
Print run:Fifth version
Page number:47
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 / Fifth Version
License (German):License LogoCreative Commons - Namensnennung, 4.0 International