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Some results on the Collatz problem

2000
*
Acta Informatica
*

Finally, we present

doi:10.1007/s002360000039
fatcat:3hyc5ji7cnf5zcxj5ztmqw4boy
*some*interesting*results*related to*the*number of "even" and "odd" branches in*the*Collatz's tree. ...*The*paper refers to*the*Collatz's conjecture. In*the*first part, we present*some*equivalent forms of this conjecture and a slight generalization of a former*result*from [1]. ... Matthias Jantzen and Research Assistent Olaf Kummer (Fachbereich Informatik, Universität Hamburg) for their very useful remarks, suggestions and comments which improved*the*paper. ...##
###
A Novel Theoretical Framework Formulated for Information Discovery from Number System and Collatz Conjecture Data

2015
*
Procedia Computer Science
*

*The*proposed number system, as well as its most general framework which is based

*on*

*the*modelling

*results*derived from an investigation of

*the*

*Collatz*conjecture (i.e.,

*the*3x+1

*problem*), has emerged as ... Here, we introduce and demonstrate many known applications of this prime framework and show

*the*subsequent

*results*of further analyses as new evidences to justify

*the*claimed fascinating capabilities of ... Acknowledgements

*The*author acknowledges

*the*support received from Abertay University, Dundee, United Kingdom. ...

##
###
The theory of the Collatz process and the method of dynamical balls
[article]

2022
*
arXiv
*
pre-print

We develop tools to study

arXiv:1910.13828v3
fatcat:zhga4bm3efhjrelccvsyriabqi
*problems*requiring to determine*the*convergence of certain sequences generated by iterating*on*a fixed integer. ... This theory also has a subtle connection with*the*infamous*problem*of*the*distribution of Sophie germain primes. We also provide several formulation of*the**Collatz*conjecture in this language. ... Indeed it sets out*the*idea that studying*the**problem**on**the*set of integers is in*some*way silly. ...##
###
Boolean Hypercubes, Mersenne Numbers, and the Collatz Conjecture

2020
*
Journal of Mathematical Sciences and Modelling
*

Application of a defined ad hoc

doi:10.33187/jmsm.776898
fatcat:dubl2tehx5bxtdwx7s76uyvhji
*Collatz*operator to*the*Boolean Hypercube recursive partition of N, permits to find*some*hints of*the*behavior of natural numbers under*the*(3x + 1) algorithm, and finally ... This study is based*on**the*trivial transcription of*the*vertices of a Boolean N-Dimensional Hypercube H N into a subset S N of*the*decimal natural numbers N. ... Also, Professor Jacek Karwowski's deep insight and fruitful discussion about*the*present manuscript has enlightened*the*initial contents and polished*the*nuances*on**the*proof of*Collatz*conjecture. ...##
###
A generalization of the Collatz problem and conjecture
[article]

2008
*
arXiv
*
pre-print

We introduce an infinite set of integer mappings that generalize

arXiv:0810.5169v1
fatcat:ip5f4uzaerc4zhokfuoemubgzq
*the*well-known*Collatz*-Ulam mapping and we conjecture that an infinite subset of these mappings feature*the*remarkable property of*the**Collatz*... Pistella for valuable help in*the*computer investigation. ... original*Collatz*conjecture, although*one*might hope that it provide*some*hint for solving that*problem*. ...##
###
Methods for constructing Collatz numbers

2018
*
Notes on Number Theory and Discrete Mathematics
*

*The*

*Collatz*conjecture is among

*the*unsolved

*problems*in mathematics. ... It says that if we take any natural number x; divide it by two if x is even, and multiply it by 3 and add 1 if x is odd; and repeat this rule

*on*

*the*

*resulting*numbers, eventually we obtain 1. ... Paul Erdös made

*the*following strong comment

*on*this conjecture: "Mathematics may not be ready for such

*problems*." [6] .

*The*

*problem*has also been known under other names [2, 3, 4, 5] . ...

##
###
On the existence of recurrent structures & statistical bias in the Collatz path sequences
[article]

2017
*
arXiv
*
pre-print

*The*completeness of

*the*

*problem*in these finite number of recurrent forms (here, six recurrent forms), such that

*the*elements of

*the*

*Collatz*path sequence switches in-between till

*one*of them reaches a ... This is followed by a closed form finite state machine (FSM) model of these recurrences using a set of linear congruence equations

*resulting*in a different terminating condition for

*the*

*Collatz*

*problem*... Terminating

*Collatz*path sequence is

*the*any

*one*of (a),

*The*details of

*the*enumeration

*results*are given in Table III . ...

##
###
A Topological Path to the Collatz Conjecture: A historical point of view

2020
*
Advances in dynamical systems and applications (ADSA)
*

This map κ induces

doi:10.37622/adsa/15.2.2020.143-151
fatcat:xuhvnclcknbcjk32wzn6odbjie
*the*so called primal topology τ κ*on*N. We prove that*the*space (N, τ κ ) is path-connected and compact if and only if*the**Collatz*conjecture is true. ... Also known as "*the*3x + 1*problem*",*the**Collatz*conjecture establishes that for every natural number n, there exists a natural number r such that*the*r-fold composite κ r (n) = 1, where κ : N → N is*the*... BASICS OF*THE**COLLATZ*CONJECTURE In order to understand*the**Collatz*conjecture we set*some*conventions*on*notation: Let Z + denote*the*set of nonnegative integers. ...##
###
The Periodicity to a Kind of Generalized Collatz Problem
[article]

2019
*
arXiv
*
pre-print

*The*

*Collatz*

*problem*is related to

*the*fixed point

*problem*, and is widely used in mathematics. It has attracted a wide range of math enthusiasts, but is still difficult to solve. ... So, this article aimed to study

*the*extension of

*the*

*Collatz*

*problem*, more widely, in k-adic. We define a new sequence called Z transformation sequence. ...

*The*essence of

*the*

*Collatz*

*problem*is a fixed-point

*problem*, and its research

*results*are important for solving fixed-point

*problems*. ...

##
###
THE PERIODICITY TO A KIND OF GENERALIZED COLLATZ PROBLEM

English

2020
*
Journal of Algebra Number Theory Advances and Applications
*

English

*The*

*Collatz*

*problem*is related to

*the*fixed point

*problem*, and is widely used in mathematics. It has attracted a wide range of mathematics enthusiasts, but is still difficult to solve. ... So, this article aimed to study

*the*extension of

*the*

*Collatz*

*problem*, more widely, in adic. We define a new sequence called Z transformation sequence. ... In [3] ,

*the*following mapping is given Matthews [4] studied

*some*maps

*on*

*the*rings of integers in an algebraic number field, such as In

*the*present article, we will extend

*the*

*Collatz*

*problem*in p-adic ...

##
###
A Clustering Perspective of the Collatz Conjecture

2021
*
Mathematics
*

This manuscript focuses

doi:10.3390/math9040314
fatcat:ptmhq26o3rgt7me4jpksqbeqha
*on**one*of*the*most famous open*problems*in mathematics, namely*the**Collatz*conjecture. ...*The*second part is dedicated to*the*visualization of behaviors of*the**Collatz*iteration function and*the*analysis of*the**results*. ... Conflicts of Interest:*The*authors declare no conflict of interest. ...##
###
On the Collatz Problem
[article]

2008
*
arXiv
*
pre-print

Introducing

arXiv:0804.3716v1
fatcat:cxg3r7f3qvan3cpfej7kwfuuwm
*some*new functions,*the**Collatz*-2 and*Collatz*-3 sequences, as well as deducing*results*related to*Collatz*-2 and*Collatz*-3 sequences. ... Taking a new approach towards analyzing*the**Collatz**Problem*, or, 3x+1 conjecture. ...*Collatz**problem*Possibility b.*on**the*other hand denotes*the*existence of a cycle which contradicts*the**Collatz**problem*Case 1, b If we have cycles in a particular level(n), then*Collatz*-2 sequence for ...##
###
The relationship between stopping time and number of odd terms in Collatz sequences
[article]

2019
*
arXiv
*
pre-print

*One*elusive open

*problem*in Mathematics is whether all such sequences end in 1 (

*Collatz*conjecture),

*the*alternative being

*the*possibility of cycles or of unbounded sequences. ... This

*result*is presented as a conjecture, and with

*the*hope that it could be useful for constructing a proof of

*the*

*Collatz*conjecture. ...

*The*alternative would be

*the*existence of

*some*special N for which

*the*sequence diverges or features an indefinite cycle. For a review

*on*general properties of

*Collatz*sequences, see [1] . ...

##
###
A Conjecture on the Collatz-Kakutani Path Length for the Mersenne Primes
[article]

2012
*
arXiv
*
pre-print

We present here a new conjecture for

arXiv:1104.2804v3
fatcat:h6zvd6flofdynp22gwstvs3aaq
*the*nature of*the*Mersenne prime numbers by connecting it with*the**Collatz*-Kakutani*problem*. ... By introducing a natural path length*on**the*basis of*the**Collatz*-Kakutani tree, we conjecture that this path length of a Mersenne prime from*the*root of*the**Collatz*-Kakutani tree is approximately proportional ...*The**Collatz*-Kakutani conjecture is also*one*of*the*unsolved*problems*in number theory. This conjecture is a halting*problem*that*the*following operations will stop for an arbitrary positive number. ...##
###
Two remarks on the Collatz cycle conjecture
[article]

2014
*
arXiv
*
pre-print

We also reformulate

arXiv:1010.6206v3
fatcat:gym7jggf4bhpphiiiiuv5qnkkm
*the**Collatz*cycle conjecture which is rather a algorithmic*problem*into a purely arithmetic*problem*. ... We give a short proof of Belaga's*result**on*bounds to perigees of (3x+d)-cycles of a given oddlength. ... This reformulation may give a way to reduce*the**problem*to well-known solved*problems*or conjectures such as*the*abc conjecture. ...
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