44325
2020
2020
eng
31
4th version
report
1
2020-01-28
2020-01-28
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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.
10.25932/publishup-44325
urn:nbn:de:kobv:517-opus4-443254
Creative Commons - Namensnennung, 4.0 International
Eldar Sultanow
Christian Koch
Sean Cox
eng
uncontrolled
Collatz
eng
uncontrolled
Cayley Graph
eng
uncontrolled
Free Group
eng
uncontrolled
Reachability
Sozialwissenschaften
NUMBER THEORY
open_access
Wirtschaftswissenschaften
Nicht referiert
Fourth Version
Universität Potsdam
https://publishup.uni-potsdam.de/files/44325/algebrabook.pdf