43008
2019
2019
eng
15
report
1
2019-06-12
2019-06-12
--
Collatz Sequences in the Light of Graph Theory
The Collatz conjecture is a number theoretical problem, which has puzzled countless researchers using myriad approaches. Presently, there are scarcely any methodologies to describe and treat the problem from the perspective of the Algebraic Theory of Automata. Such an approach is promising with respect to facilitating the comprehension of the Collatz sequences "mechanics". The systematic technique of a state machine is both simpler and can fully be described by the use of algebraic means.
The current gap in research forms the motivation behind the present contribution. The present authors are convinced that exploring the Collatz conjecture in an algebraic manner, relying on findings and fundamentals of Graph Theory and Automata Theory, will simplify the problem as a whole.
10.25932/publishup-43008
urn:nbn:de:kobv:517-opus4-430089
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
First Version
Universität Potsdam
https://publishup.uni-potsdam.de/files/43008/collatz_sequences.pdf
44185
2020
2020
eng
29
3rd version
report
1
2020-01-06
2020-01-06
--
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 1, as conjectured by Lothar Collatz.
10.25932/publishup-44185
urn:nbn:de:kobv:517-opus4-441859
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
Third Version
Universität Potsdam
https://publishup.uni-potsdam.de/files/44185/algebrabook.pdf
43741
2019
2019
eng
21
2nd version
report
1
2019-11-05
2019-11-05
--
Collatz Sequences in the Light of Graph Theory
The Collatz conjecture is a number theoretical problem, which has puzzled countless researchers using myriad approaches. Presently, there are scarcely any methodologies to describe and treat the problem from the perspective of the Algebraic Theory of Automata. Such an approach is promising with respect to facilitating the comprehension of the Collatz sequence’s "mechanics". The systematic technique of a state machine is both simpler and can fully be described by the use of algebraic means.
The current gap in research forms the motivation behind the present contribution. The present authors are convinced that exploring the Collatz conjecture in an algebraic manner, relying on findings and fundamentals of Graph Theory and Automata Theory, will simplify the problem as a whole.
10.25932/publishup-43741
urn:nbn:de:kobv:517-opus4-437416
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
Second Version
Universität Potsdam
https://publishup.uni-potsdam.de/files/43741/collatz_sequences.pdf
44325
2020
2020
eng
31
4th version
report
1
2020-01-28
2020-01-28
--
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
48214
2020
2020
eng
47
Fifth version
report
1
2020-11-11
2020-11-11
--
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.
10.25932/publishup-48214
urn:nbn:de:kobv:517-opus4-482140
Creative Commons - Namensnennung, 4.0 International
Eldar Sultanow
Christian Koch
Sean Cox
eng
uncontrolled
Collatz Conjecture
eng
uncontrolled
Free Group
eng
uncontrolled
Multiplicative Group
eng
uncontrolled
Cyclic Group
eng
uncontrolled
Cayley Graph
eng
uncontrolled
Cycle
eng
uncontrolled
Tree
eng
uncontrolled
Binary Tree
Sozialwissenschaften
NUMBER THEORY
open_access
Wirtschaftswissenschaften
Nicht referiert
Fifth Version
Universität Potsdam
https://publishup.uni-potsdam.de/files/48214/algebrabook.pdf