The Collatz Conjecture Challenge
Formalising the Collatz literature using proof assistants
Literature (352 entries)
On total stopping times under $3X+1$ iteration
Authors: Paul Andaloro
Fibonacci Quarterly, Vol. 38, pp. 73--78
The $3x+1$ Conjugacy Map
Authors: Daniel J. Bernstein and Jeffrey C. Lagarias
Canadian J. Math., Vol. 48, pp. 1154--1169
On the existence of cycles of given length in integer sequences like $x_{n+1}= x_n/2$ if $x_n$ even, and $x_{n+1}= 3x_n + 1$ otherwise
Authors: Corrado Böhm and Giovanna Sontacchi
Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., Vol. 64, pp. 260--264
Odd Collatz Sequence and Binary Representations
Authors: Jose Capco
Here we investigate the odd numbers in Collatz sequences (sequences arising from the $3n+1$ problem). We are especially interested in methods in binary number representations of the numbers in the sequence. In the first section, we show some results for odd Collatz sequences using mostly binary arithmetics. We see how some results become more obvious in binary arithmetic than in usual method of computing the Collatz sequence. In the second section of this paper we deal with some known results and show how we can use binary representation and OCS from the first section to prove some known results. We give a generalization of a result by Andaloro \cite{And2} and show a generalized sufficient condition for the Collatz conjecture to be true: If for a fixed natural number $n$ the Collatz conjecture holds for numbers congruent to $1$ modulo $2^n$ then the Collatz conjecture is true.
$3n+1$ problem: an heuristic lower bound for the number of integers connected to 1 and less than $x$
Authors: Jean-Jacques Daudin
The $3x + 1$ problem: New Lower Bounds on Nontrivial Cycle Lengths
Authors: Shalom Eliahou
Discrete Math., Vol. 118, pp. 45--56
On heights in the Collatz $3n + 1$ problem
Authors: Lynn E. Garner
Discrete Math., Vol. 55, pp. 57--64
The 2-Adic, Binary and Decimal Periods of 1/3k Approach Full Complexity for Increasing k
Authors: Josefina López and Peter Stoll
Integers, Vol. 12(5), pp. 907--928
The sufficiency of arithmetic progressions for the $3x+1$ conjecture
Authors: Monks, Kenneth M.
Proc. Amer. Math. Soc., Vol. 134(10), pp. 2861--2872
Define $T : \mathbb{Z}^+ \to \mathbb{Z}^+$ by $T(x) = (3x+1)/2$ if $x$ is odd and $T(x) = x/2$ if $x$ is even. The $3x+1$ Conjecture states that the $T$-orbit of every positive integer contains $1$. A set of positive integers is said to be sufficient if the $T$-orbit of every positive integer intersects the $T$-orbit of an element of that set. Thus to prove the $3x+1$ Conjecture it suffices to prove it on some sufficient set. Andaloro proved that the sets $1 + 2^n\mathbb{N}$ are sufficient for $n \le 4$ and asked if $1 + 2^n\mathbb{N}$ is also sufficient for larger values of $n$. We answer this question in the affirmative by proving the stronger result that $A + B\mathbb{N}$ is sufficient for any nonnegative integers $A$ and $B$ with $B \neq 0$, i.e. every nonconstant arithmetic sequence forms a sufficient set. We then prove analogous results for the Divergent Orbits Conjecture and Nontrivial Cycles Conjecture.
Démonstraton de l'absence de cycles d'une certain forme pour le problème de Syracuse
Authors: Olivier Rozier
Singularité, Vol. 1(3), pp. 9--12
Almost all orbits of the Collatz map attain almost bounded values
Authors: Tao, Terence
Forum of Mathematics, Pi, Vol. 10, pp. e12
Why is the $3x+1$ Problem Hard?
Authors: Ethan Akin
Chapel Hill Ergodic Theory Workshops, Vol. 356, pp. 1--20
Una breve introduzione a diffusioni su insiemi frattali e ad alcuni essempi di sistemi dinamici semplici
Authors: Sergio Albeverio and Danilo Merlini and Remiglio Tartini
Note di matematica e fisica, Vol. 3, pp. 1--39
Sur la conjecture de "Syracuse-Kakutani-Collatz"
Authors: Jean-Paul Allouche
Séminaire de Théorie des Nombres 1978--1979, Expose No. 9
A linear algebra approach to the conjecture of Collatz
Authors: João F. Alves and Mário M. Graca and M. E. Sousa Dias and José Sousa Ramos
Lin. Alg. Appl., Vol. 394, pp. 277--289
The $2$-adic valuation of a sequence arising from a rational integral
Authors: Tewodrus Amdeberhan and Dante Manna and Victor H. Moll
J. Combinatorial Theory, Series A, Vol. 115(8), pp. 1474--1486
On some difference equations with eventually periodic solutions
Authors: Amal S. Amleh and Edward A. Grove and Candace M. Kent and Gerasimos Ladas
J. Math. Anal. Appl., Vol. 223, pp. 196--215
The $3X+1$ problem and directed graphs
Authors: Paul Andaloro
Fibonacci Quarterly, Vol. 40, pp. 43--54
Struggling with the $3x + 1$ problem
Authors: S. Anderson
Math. Gazette, Vol. 71, pp. 271--274
A Functional View over the Collatz's Problem
Authors: Stefan Andrei and Wei-Ngan Chin and Huu Hai Nguyen
Some results on the Collatz problem
Authors: Stefan Andrei and Manfred Kudlek and Radu Stefan Niculescu
Acta Informatica, Vol. 37, pp. 145--160
About the Collatz Conjecture
Authors: Stefan Andrei and Cristian Masalagiu
Acta Informatica, Vol. 35, pp. 167--179
Density Bounds for the $3x+1$ Problem I. Tree-Search Method
Authors: David Applegate and Jeffrey C. Lagarias
Math. Comp., Vol. 64, pp. 411--426
Density Bounds for the $3x+1$ Problem II. Krasikov Inequalities
Authors: David Applegate and Jeffrey C. Lagarias
Math. Comp., Vol. 64, pp. 427--438
On the distribution of $3x+1$ trees
Authors: David Applegate and Jeffrey C. Lagarias
Experimental Mathematics, Vol. 4, pp. 101--117
Lower bounds for the total stopping time of $3x+1$ iterates
Authors: David Applegate and Jeffrey C. Lagarias
Math. Comp., Vol. 72, pp. 1035--1049
The $3x+1$ semigroup
Authors: David Applegate and Jeffrey C. Lagarias
J. Number Theory, Vol. 177, pp. 146--159
Algorithmes pour vérifier la conjecture de Syracuse
Authors: Jacques Arsac
C. R. Acad. Sci. Paris, Vol. 303, pp. 155--159
Further Investigations of the Wondrous Numbers
Authors: Charles Ashbacher
J. Recreational Math., Vol. 24, pp. 1--15
Comment on Problem $63-13^{*}$
Authors: Arthur Oliver Lonsdale Atkin
SIAM Review, Vol. 8, pp. 234--236
On primitive $3$-smooth partitions of $n$
Authors: Michael R. Avidon
Electronic J. Combinatorics, Vol. 4
3N+1, UTM e Tag-Systems
Authors: Claudio Baiocchi
Dipartimento di Matematica dell'Università "La Sapienza" di Roma
Linear forms in the logarithms of algebraic numbers
Authors: Baker, A.
Mathematika, Vol. 13(2), pp. 204-216
Some Properties of the $3n+1$ Function
Authors: Ranan B. Banerji
Cybernetics and Systems, Vol. 27, pp. 473--486
A heuristic probabilistic argument for the Collatz sequence
Authors: Enzo Barone
Ital. J. Pure Appl. Math., Vol. 4, pp. 151--153
HAKMEM
Authors: Michael Beeler and William Gosper and Richard Schroeppel
Memo 239, Artificial Intelligence Laboratory, MIT
Reflecting on the $3x+1$ Mystery: Outline of a Scenario- Improbable or Realistic?
Authors: Edward Belaga
U. Strasbourg report
Effective polynomial upper bounds to perigees and numbers of $(3x+d)$-cycles of a given oddlength
Authors: Edward Belaga
Acta Arithmetica, Vol. 106(2), pp. 197--206
Embedding the $3x+1$ Conjecture in a $3x+d$ Context
Authors: Edward Belaga and Maurice Mignotte
Experimental Math., Vol. 7, pp. 145--151
Cyclic Structure of Dynamical Systems Associated with $3x+d$ Extensions of Collatz Problem
Authors: Edward Belaga and Maurice Mignotte
Walking Cautiously into the Collatz Wilderness: Algorithmically, Number Theoretically, and Randomly
Authors: Edward Belaga and Maurice Mignotte
Fourth Colloquium on Mathematics and Computer Science, Vol. AG, pp. 249--260
The Collatz problem and its generalizations: Experimental Data. Table 1. Primitive cycles of $3x+d$ mappings
Authors: Edward Belaga and Maurice Mignotte
The Collatz problem and its generalizations: Experimental Data. Table 2. Factorization of Collatz numbers $2^l-3^k$
Authors: Edward Belaga and Maurice Mignotte
Orbite inverse nel problema del $3n+1$
Authors: Stefano Beltraminelli and Danilo Merlini and Luca Rusconi
Note di matematica e fisica, Edizioni Cerfim Locarno, Vol. 7, pp. 325--357
Functional equations connected with the Collatz problem
Authors: Lothar Berg and Günter Meinardus
Results in Math., Vol. 25, pp. 1--12
The 3n+1 Collatz Problem and Functional Equations
Authors: Lothar Berg and Günter Meinardus
Rostock Math. Kolloq., Vol. 48, pp. 11--18
Affine Actions of a Free Semigroup on the Real Line
Authors: Vitaly Bergelson and Michal Misiurewicz and Samuel Senti
Ergodic Theory and Dynamical Systems, Vol. 26, pp. 1285--1305
A Non-Iterative 2-adic Statement of the $3x + 1$ Conjecture
Authors: Daniel J. Bernstein
Proc. Amer. Math. Soc., Vol. 121, pp. 405--408
Some properties of binary sequences useful for proving Collatz's conjecture
Authors: Jacek Bł ażewitz and Alberto Pettorossi
J. Found. Control Engr., Vol. 8, pp. 53--63
3-Smooth Representations of Integers
Authors: Richard Blecksmith, Michael McCallum, and John L. Selfridge
American Math. Monthly, Vol. 105, pp. 529--543
Estimates for the Syracuse problem via a probabilistic model
Authors: Konstantin Borovkov and Dietmar Pfeifer
Theory of Probability and its Applications, Vol. 45(2), pp. 300--310
Which rationals are ratios of Pisot sequences?
Authors: David W. Boyd
Canad. Math. Bull., Vol. 28, pp. 343--349
Sur un problème de R. Queneau
Authors: Monique Bringer
Mathématiques et Sciences Humaines [Mathematics and Social Science], Vol. 27, pp. 13--20
A Note on Mignosi's Generalization of the $3x+1$ Problem
Authors: Stefano Brocco
J. Number Theory, Vol. 52, pp. 173--178
Collatz cycles with few descents
Authors: Thomas Brox
Acta Arithmetica, Vol. 92, pp. 181--188
A proof of the Collatz conjecture
Authors: Paul S. Bruckman
International Journal of Mathematical Education in Science and Technology, Vol. 39(3), pp. 403--407
Functional equations associated with congruential functions
Authors: Serge Burckel
Theoretical Computer Science, Vol. 123, pp. 397--406
On some Markov matrices arising from the generalized Collatz mapping
Authors: Robert N. Buttsworth and Keith R. Matthews
Acta Arithmetica, Vol. 55, pp. 43--57
A note on the $3x + 1$ problem
Authors: Charles C. Cadogan
Caribbean J. Math., Vol. 3(2), pp. 69--72
Some observations on the $3x+1$ problem
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Proc. Sixth Caribbean Conference on Combinatorics & Computing, pp. 84--91
Exploring the $3x+1$ problem I
Authors: Charles C. Cadogan
Caribbean J. Math. Comput. Sci., Vol. 6, pp. 10--18
The $3x+1$ problem: towards a solution
Authors: Charles C. Cadogan
Caribbean J. Math. Comput. Sci., Vol. 10, pp. 2
Trajectories in the $3x+1$ problem
Authors: Charles C. Cadogan
J. of Combinatorial Mathematics and Combinatorial Computing, Vol. 44, pp. 177--187
A Solution to the $3x+1$ Problem
Authors: Charles C. Cadogan
Caribbean J. Math. Comp. Sci., Vol. 13, pp. 1--11
Structural and Computational Complexity of Isometries and their Shift Commutators
Authors: Mónica del Pilar Canales Chacón and Michael Vielhaber
Electronic Colloquium on Computational Complexity, Report No. 57, pp. 24
A Continuous Extension of the $3x+1$ Problem to the Real Line
Authors: Marc Chamberland
Dynamics of Continuous, Discrete and Impulsive Dynamical Systems, Vol. 2, pp. 495--509
Una actualizachio del problema $3x+1$
Authors: Mark Chamberland
Butletti de la Societat Catalana, Vol. 22, pp. 19--45
Cycles in Collatz Sequences
Authors: Busiso P. Chisala
Publ. Math. Debrecen, Vol. 45, pp. 35--39
Selected combinatorial research problems
Authors: Vasik Chvatal, David A. Klarner and Donald E. Knuth
(STAN-CS-72-292)
Second-Order Difference Equations Related to the Collatz $3n+1$ Conjecture
Authors: Dean Clark
J. Difference Equations & Appl., Vol. 1, pp. 73--85
Periodic solutions of arbitrary length in a simple integer iteration
Authors: Dean Clark
Advances in Difference Equations, Vol. 2006, pp. 1--9
A Collatz-Type Difference Equation
Authors: Dean Clark and James T. Lewis
Proc. Twenty-sixth Internationsal Conference on Combinatorics, Graph Theory and Computing (Boca Raton 1995), Vol. 111, pp. 129--135
Symmetric solutions to a Collatz-like system of Difference Equations
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Proc. Twenty-ninth Internationsal Conference on Combinatorics, Graph Theory and Computing (Baton Rouge 1998), Vol. 131, pp. 101--114
The $3x + 1$ Problem: a Quasi-Cellular Automaton
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On the Motivation and Origin of the $(3n + 1)$-Problem
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J. of Qufu Normal University, Natural Science Edition [Qufu shi fan da xue xue bao. Zi ran ke xue ban], Vol. 12(3), pp. 9--11
Unpredicatable Iterations
Authors: John H. Conway
Proc. 1972 Number Theory Conference, University of Colorado, pp. 49--52
FRACTRAN- A Simple Universal Computing Language for Arithmetic
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Open Problems in Communication and Computation, pp. 3--27
Small Tile Sets That Compute While Solving Mazes
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The complement of certain recursively defined sets
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Cyclic Sequences and Frieze Patterns, (The Fourth Felix Behrend Memorial Lecture)
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Vinculum, Vol. 8, pp. 4--7
On the "$3x + 1$" problem
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Some Comments on an Iteration Problem
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Proc. 6-th Manitoba Conf. On Numerical Mathematics, and Computing (Univ. of Manitoba-Winnipeg 1976), Congressus Numerantium XVIII, pp. 55--59
Tag Systems and Collatz-like functions
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Theoretical Computer Science, Vol. 390, pp. 92--101
Halting Problem of One Binary Horn Clause is Undecidable
Authors: Philippe Devienne, Patrick Lebègue, Jean-Chrisophe Routier
Proceedings of STACS 1993, pp. 48--57
A generalization of Everett's result on the Collatz $3x + 1$ problem
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Adv. Appl. Math., Vol. 8, pp. 405--409
A few observations on the Collatz problem
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Inter. J. Appl. Math. Stat., Vol. 14, pp. 97--107
Caractérisation des quenines et leur représentation spirale
Authors: Jean-Guillaume Dumas
Mathématiques et Sciences Humaines [Mathematics and Social Science], Vol. 184, pp. 9--23
Visualizing Generalized 3x+1 Function Dynamics
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Computers and Graphics, Vol. 25, pp. 883--898
Real dynamics of a $3$-power extension of the $3x+1$ function
Authors: Jeffrey P. Dumont and Clifford A. Reiter
Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis, Vol. 10, pp. 875--893
On Ulam's Problem
Authors: Richard Dunn
Department of Computer Science, University of Colorado, Boulder, Technical Report CU-CS-011-73, pp. 15
Game of Cards, Dynamical Systems, and a Characterization of the Floor and Ceiling Functions
Authors: Peter Eisele and Karl-Peter Hadeler
Amer. Math. Monthly, Vol. 97, pp. 466--477
Is the Syracuse Falling Time Bounded by 12?
Authors: Eliahou, Shalom and Fromentin, Jean and Simonetto, Rénald
{Combinatorial and Additive Number Theory V}, pp. 139--152
Arithmetic Functions and Integer Products
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Springer-Verlag
On a Theorem of S. Sivasankaranarayana Pillai
Authors: W. J. Ellison
Séminaire de théorie des nombres de Bordeaux, Vol. 1970-1971(10), pp. 1--10
Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics
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Old and new problems and results in combinatorial number theory
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Monographie No. 28 de L'Enseignement Mathématique, Kundig
Iteration of the number theoretic function $f(2n)=n, f(2n+1)=3n+2$
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Advances in Math., Vol. 25, pp. 42--45
Variants of the $3N+1$ problem and multiplicative semigroups
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The elimination of a family of periodic parity vectors in the $3x+1$ problem
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The $(3x + 1)/2$ Problem: A Statistical Approach
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Statistical Properties of an Iterated Arithmetic Mapping
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On the $3n+1$ Problem: Something Old, Something New
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$Z$-numbers and $\beta$-transformations
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On a Conjecture of Crandall Concerning the $qx + 1$ Problem
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Math. Comp., Vol. 64, pp. 1333--1336
Growth properties of a class of recursively defined functions
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Recurrence relations based on minimization
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J. Math. Anal. Appl., Vol. 48, pp. 534--559
Mathematical Entertainments: More Mysteries
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Mathematical Intelligencer, Vol. 13(3), pp. 54--55
On consecutive numbers of the same height in the Collatz problem
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Discrete Math., Vol. 112, pp. 261--267
A note on the generalized $3n + 1$ problem
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Mathematical Games
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Scientific American, Vol. 226(6), pp. 114--118
On the Collatz $3n+1$ algorithm
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Proc. Amer. Math. Soc., Vol. 82, pp. 19--22
Linear abgeschlossene Zahlenmengen I. [Linearly closed number sets I.]
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Das-ULAM Problem-Computergestütze Entdeckungen
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DdM (Didaktik der Mathematik), Vol. 17(2), pp. 114--134
A new statistic for the $3x+1$ problem
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Proc. Amer. Math. Soc., Vol. 130, pp. 1293--1301
Computations on the $3n+1$ Conjecture
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MAPLE Technical Newsletter, Vol. 0(6)
Results on the Collatz conjecture
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Annalele Stiintifice ale Universitatii ``Al. I. Cuza'' din lasi serie noua. Informatica (Romanian), Vol. XIII, pp. 1--16
Unsolved Problems in Number Theory
Authors: Richard K. Guy
Springer-Verlag, New York
Don't try to solve these problems!
Authors: Richard K. Guy
Amer. Math. Monthly, Vol. 90, pp. 35--41
Conway's prime producing machine
Authors: Richard K. Guy
Math. Magazine, Vol. 56, pp. 26--33
John Isbell's Game of Beanstalk and John Conway's Game of Beans Don't Talk
Authors: Richard K. Guy
Math. Magazine, Vol. 59, pp. 259--269
Optimal bounds for the length of rational Collatz cycles
Authors: Lorenz Halbeisen and Norbert Hungerbühler
Acta Arithmetica, Vol. 78, pp. 227--239
Hasse's Syracuse-Problem und die Rolle der Basen
Authors: Gisbert Hasenjager
Mathesis rationis. Festschrift für Heinrich Schepers, pp. 329--336
Computer recreations: The ups and downs of hailstone numbers
Authors: Brian Hayes
Scientific American, Vol. 250(1), pp. 10--16
$3n+1$ problem's simplifying and structure property of $\{T(n)\}$$(n \in \mathbb{N})$
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Acta Scieniarum Naturalium Universitatis Neimonggol [Nei Menggu da xue xue bao. Zi ran ke xue]
Eine Bemerkung zum Hasse-Syracuse Algorithmus
Authors: Ernst Heppner
Archiv. Math., Vol. 31, pp. 317--320
There are no Collatz $m$-Cycles with $m \leq 91$
Authors: Christian Hercher
Journal of Integer Sequences, Vol. 26
A Polynomial Analogue of the $3N+1$ Problem?
Authors: Kenneth Hicks and Gary L. Mullen and Joseph L. Yucas and Ryan Zavislak
American Math. Monthly, Vol. 115, pp. 615--622
Sets of integers closed under affine operators- the closure of finite sets
Authors: Dean G. Hoffman and David A. Klarner
Pacific Journal of Mathematics, Vol. 78(2), pp. 337--344
Sets of integers closed under affine operators- the finite basis theorems
Authors: Dean G. Hoffman and David A. Klarner
Pacific Journal of Mathematics, Vol. 83(1), pp. 135--144
About $3X+1$ problem
Authors: Bo Yang Hong
J. of Hubei Normal University, Natural Science Edition(1), pp. 1--5
Beitäge zum $3n+1$-Problem
Authors: Wernt Hotzel
Dissertation: Univsität Hamburg
One-to-one correspondence between the natural numbers and the parity vectors in the Collatz problem
Authors: Huang, Guo Lin and Wu, Jia Bang
J. of South-Central University for the Nationalities, Natural Science Ed. [Zhong nan min zu da xue xue bao. Zi ran ke xue ban], Vol. 19, pp. 59--61
Three open questions in the theory of one-symbol Smullyan systems
Authors: Stephen D. Isard and Harold M. Zwicky
SIGACT News, pp. 11-19
A Hardware-Software Cooperative Approach to the Exhaustive verification of the Collatz conjecture
Authors: Yasuaki Ito and Koji Nakano
International Symp. on Parallel and Distributed Processing with Applications, pp. 63--70
13, 31 and the $3x + 1$ problem
Authors: Frazer Jarvis
Eureka, Vol. 49, pp. 22--25
A chaotic extension of the Collatz function to $\mathbb{Z}_2 [i]$
Authors: John A. Joseph
Fibonacci Quarterly, Vol. 36, pp. 309--317
Cellular Automata, the Collatz Conjecture and Powers of 3/2
Authors: Kari, Jarkko
Developments in Language Theory, pp. 40--49
We discuss one-dimensional reversible cellular automata F{\texttimes}3 and F{\texttimes}3/2 that multiply numbers by 3 and 3/2, respectively, in base 6. They have the property that the orbits of all non-uniform 0-finite configurations contain as factors all finite words over the state alphabet {{}0,1,{łdots},5{}}. Multiplication by 3/2 is conjectured to even have an orbit of 0-finite configurations that is dense in the usual product topology. An open problem by K. Mahler about Z-numbers has a natural interpretation in terms the automaton F{\texttimes}3/2. We also remark that the automaton F{\texttimes}3 that multiplies by 3 can be slightly modified to simulate the Collatz function. We state several open problems concerning pattern generation by cellular automata.
Cellular automata and powers of p/q
Authors: Jarkko Kari and Johan Kopra
RAIRO Theor. Informatics Appl., Vol. 51(4), pp. 191--204
Small universal one-state linear operator algorithms
Authors: Frantisek Kascak
Proc. MFCS '92, Lecture Notes in Computer Science No. 629, pp. 327--335
Arithmetic in the form
Authors: Louis H. Kauffman
Cybernetics and Systems, Vol. 26, pp. 1--57
An Algorithm for Reducing the Size of an Integer
Authors: David Kay
Pi Mu Epsilon Journal, Vol. 4, pp. 338
Generalization Concerning the $3x+1$ Problem
Authors: Ke, Wei
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The proof of the hypothesis about $"3x+1"$
Authors: Ke, Yong-Sheng
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Finite cycles of certain periodically linear functions
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Die Collatz-Ulam-Kombination CUK
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Problem $63-13^{*}$
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An algorithm to determine when certain sets have $0$ density
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Journal of Algorithms, Vol. 2, pp. 31--43
A sufficient condition for certain semigroups to be free
Authors: David A. Klarner
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$m$-recognizability of sets closed under certain affine functions
Authors: David A. Klarner
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Some fascinating integer sequences
Authors: David A. Klarner and Karel Post
Discrete Mathematics, Vol. 106/107, pp. 303--309
Linear combinations of sets of consecutive integers
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American Math. Monthly, Vol. 80(9), pp. 985--989
Arithmetic properties of certain recursively defined sets
Authors: David A. Klarner and Richard Rado
Pacific J. Math., Vol. 53, pp. 445--463
Wildness of iteration of certain residue class-wise affine mappings
Authors: Stefan Kohl
Advances in Applied Math., Vol. 39, pp. 322--328
On conjugates of Collatz-type mappings
Authors: Stefan Kohl
Int. J. Number Theory, Vol. 4(1), pp. 117--120
Algorithms for a class of infinite permutation groups
Authors: Stefan Kohl
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A simple group generated by involutions interchanging residue classes of the integers
Authors: Stefan Kohl
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The real $3x+1$ problem
Authors: Pavlos B. Konstadinidis
Acta Arithmetica, Vol. 122, pp. 35--44
Benford's law, values of $L$-functions, and the $3x+1$ problem
Authors: Alex V. Kontorovich and Steven J. Miller
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Structure Theorem for $(d, g, h)$-maps
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The $3x + 1$ Problem, Generalized Pascal Triangles, and Cellular Automata
Authors: Ivan Korec
Math. Slovaca, Vol. 42, pp. 547--563
A Density Estimate for the $3x+1$ Problem
Authors: Ivan Korec
Math. Slovaca, Vol. 44, pp. 85--89
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Fibonacci Quarterly, Vol. 46/47(2), pp. 115--125
A generalization of Kakutani's conjecture
Authors: Zhang, Cheng Yu
Nature Magazine [Zi ran za zhi (Shanghai)], Vol. 13, pp. 267--269
Problems on mapping sequences
Authors: Zhang, Zhongfu and Yang, Shiming
Mathmedia [Shu xue chuan bo ji kan], Vol. 22, pp. 76--88
Some discussion on the $3x+1$ problem
Authors: Zhou, Chuan Zhong
J. South China Normal Univ. Natur. Sci. Ed. [ Hua nan shi fan da xue xue bao. Zi ran ke xue ban], pp. 103--105
Some recurrence relations connected with the $3x+1$ problem
Authors: Zhou, Chuan Zhong
J. South China Normal Univ. Natur. Sci. Ed. [Hua nan shi fan da xue xue bao. Zi ran ke xue ban], pp. 7--8
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