TY  - RPRT
A1  - Sultanow, Eldar
A1  - Volkov, Denis
A1  - Cox, Sean
T1  - Introducing a Finite State Machine for processing Collatz Sequences
N2  - The present work will introduce a Finite State Machine (FSM) that processes any Collatz Sequence; further, we will endeavor to investigate its behavior in relationship to transformations of a special infinite input. Moreover, we will prove that the machine’s word transformation is equivalent to the standard Collatz number transformation and subsequently discuss the possibilities for use of this approach at solving similar problems. The benefit of this approach is that the investigation of the word transformation performed by the Finite State Machine is less complicated than the traditional number-theoretical transformation.
KW  - Collatz Conjecture
KW  - State Machine
KW  - Graph
KW  - Double Colored Edges
Y1  - 2017
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-399223
ET  - 1st version
ER  - 
TY  - RPRT
A1  - Sultanow, Eldar
A1  - Volkov, Denis
A1  - Cox, Sean
T1  - Introducing a Finite State Machine for processing Collatz Sequences
N2  - The present work will introduce a Finite State Machine (FSM) that processes any Collatz Sequence; further, we will endeavor to investigate its behavior in relationship to transformations of a special infinite input. Moreover, we will prove that the machine’s word transformation is equivalent to the standard Collatz number transformation and subsequently discuss the possibilities for use of this approach at solving similar problems. The benefit of this approach is that the investigation of the word transformation performed by the Finite State Machine is less complicated than the traditional number-theoretical transformation.
KW  - Collatz Conjecture
KW  - State Machine
KW  - Graph
KW  - Double Colored Edges
Y1  - 2017
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-404738
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 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  -