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Some results on the Collatz problem

Ştefan Andrei, Manfred Kudlek, Radu Ştefan Niculescu
2000 Acta Informatica  
Finally, we present 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.  ... 
doi:10.1007/s002360000039 fatcat:3hyc5ji7cnf5zcxj5ztmqw4boy

A Novel Theoretical Framework Formulated for Information Discovery from Number System and Collatz Conjecture Data

Michael A. Idowu
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.  ... 
doi:10.1016/j.procs.2015.09.165 fatcat:a752xprivzazjjotlmrbgiyyq4

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

Theophilus Agama
2022 arXiv   pre-print
We develop tools to study 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.  ... 
arXiv:1910.13828v3 fatcat:zhga4bm3efhjrelccvsyriabqi

Boolean Hypercubes, Mersenne Numbers, and the Collatz Conjecture

Ramon CARBÓ-DORCA
2020 Journal of Mathematical Sciences and Modelling  
Application of a defined ad hoc 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.  ... 
doi:10.33187/jmsm.776898 fatcat:dubl2tehx5bxtdwx7s76uyvhji

A generalization of the Collatz problem and conjecture [article]

M. Bruschi
2008 arXiv   pre-print
We introduce an infinite set of integer mappings that generalize 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.  ... 
arXiv:0810.5169v1 fatcat:ip5f4uzaerc4zhokfuoemubgzq

Methods for constructing Collatz numbers

Abdullah N. Arslan
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] .  ... 
doi:10.7546/nntdm.2018.24.2.47-54 fatcat:7v5c6opgsfctpe6a4eue2csgqa

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

Sawon Pratiher, Subhasis Kundu
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 .  ... 
arXiv:1608.03600v6 fatcat:6v3zfis5ancd7o7tnucztwtese

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

Fernando Mejias, Jorge Vielma, Angel Guale
2020 Advances in dynamical systems and applications (ADSA)  
This map κ induces 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.  ... 
doi:10.37622/adsa/15.2.2020.143-151 fatcat:xuhvnclcknbcjk32wzn6odbjie

The Periodicity to a Kind of Generalized Collatz Problem [article]

Sensen Chen, Qing-You Sun, Yushu Zhu
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.  ... 
arXiv:1903.03365v3 fatcat:dolax4lu2bcypda6mxuz2sivya

THE PERIODICITY TO A KIND OF GENERALIZED COLLATZ PROBLEM
English

Sensen Chen, Hangzhou Normal University, Qing-You Sun, Yushu Zhu, Hangzhou Normal University, Hangzhou Normal University
2020 Journal of Algebra Number Theory Advances and Applications  
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  ... 
doi:10.18642/jantaa_7100122089 fatcat:bqukh77otnbipbz24wmjflfnka

A Clustering Perspective of the Collatz Conjecture

José A. Tenreiro Machado, Alexandra Galhano, Daniel Cao Labora
2021 Mathematics  
This manuscript focuses 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.  ... 
doi:10.3390/math9040314 fatcat:ptmhq26o3rgt7me4jpksqbeqha

On the Collatz Problem [article]

Roupam Ghosh
2008 arXiv   pre-print
Introducing 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  ... 
arXiv:0804.3716v1 fatcat:cxg3r7f3qvan3cpfej7kwfuuwm

The relationship between stopping time and number of odd terms in Collatz sequences [article]

Rafael Ruggiero
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] .  ... 
arXiv:1911.01229v2 fatcat:gbw35osctbdnhcg55jwolbelgy

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

Toru Ohira, Hiroshi Watanabe
2012 arXiv   pre-print
We present here a new conjecture for 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.  ... 
arXiv:1104.2804v3 fatcat:h6zvd6flofdynp22gwstvs3aaq

Two remarks on the Collatz cycle conjecture [article]

Masayoshi Kaneda
2014 arXiv   pre-print
We also reformulate 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.  ... 
arXiv:1010.6206v3 fatcat:gym7jggf4bhpphiiiiuv5qnkkm
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