TY - RPRT
A1 - Sultanow, Eldar
A1 - Koch, Christian
A1 - Cox, Sean
T1 - Collatz Sequences in the Light of Graph Theory
N2 - 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.
KW - Collatz
KW - Cayley Graph
KW - Free Group
KW - Reachability
Y1 - 2019
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-430089
ER -
TY - RPRT
A1 - Sultanow, Eldar
A1 - Koch, Christian
A1 - Cox, Sean
T1 - Collatz Sequences in the Light of Graph Theory
N2 - 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.
KW - Collatz
KW - Cayley Graph
KW - Free Group
KW - Reachability
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-441859
ET - 3rd version
ER -
TY - RPRT
A1 - Sultanow, Eldar
A1 - Koch, Christian
A1 - Cox, Sean
T1 - Collatz Sequences in the Light of Graph Theory
N2 - 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.
KW - Collatz
KW - Cayley Graph
KW - Free Group
KW - Reachability
Y1 - 2019
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-437416
ET - 2nd version
ER -
TY - RPRT
A1 - Sultanow, Eldar
A1 - Koch, Christian
A1 - Cox, Sean
T1 - Collatz Sequences in the Light of Graph Theory
N2 - 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.
KW - Collatz
KW - Cayley Graph
KW - Free Group
KW - Reachability
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-443254
ET - 4th version
ER -
TY - RPRT
A1 - Sultanow, Eldar
A1 - Koch, Christian
A1 - Cox, Sean
T1 - Collatz Sequences in the Light of Graph Theory
N2 - 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.
KW - Collatz Conjecture
KW - Free Group
KW - Multiplicative Group
KW - Cyclic Group
KW - Cayley Graph
KW - Cycle
KW - Tree
KW - Binary Tree
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-482140
ET - Fifth version
ER -