The Collatz Conjecture Challenge
Formalising the Collatz literature using proof assistants
Literature (358 entries)
Collatz high cycles do not exist
Authors: Kevin Knight
Discrete Mathematics, Vol. 349(3), pp. 114812
The Collatz function takes odd n to (3n+1)/2 and even n to n/2. Under the iterated Collatz function, every positive integer is conjectured to end up in the trivial cycle 1-2-1. Two types of rational Collatz cycles are of special interest. Consider the set S(k,x) consisting of the smallest members of k-length cycles with x odd terms. The circuit contains the smallest member of S(k,x), while the high cycle contains the largest. It is known that no circuits of positive integers exist (except 1-2-1); this paper shows that there are likewise no high cycles of positive integers.
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
$3n+1$ 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
In the Collate problem, certain runs of consecutive integers have the same height. This phenomenon is investigated using an algorithm for reconstructing the trajectory of a number from the parity vector of the number. It is found that pairs of consecutive integers of the same height occur infinitely often and in infinitely many different patterns.
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.
Almost all orbits of the Collatz map attain almost bounded values
Authors: Tao, Terence
Forum of Mathematics, Pi, Vol. 10, pp. e12
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 function T(n) is (3n+1)/2 if n is odd and n/2 if n is even. The total stopping time \sigma_\infty (n) for a positive integer n is the number of iterations of the 3x+1 function to reach 1 starting from n, and is \infty if 1 is never reached. The 3x+1 conjecture states that this function is finite. We show that infinitely many n have a finite total stopping time with \sigma_\infty(n) > 6.14316 log n. The proof uses a very large computation. It is believed that almost all positive integers have \sigma_\infty (n) > 6.95212 \log n. The method of the paper should extend to prove infinitely many integers have this property, but it would require a much larger computation.
Improved verification limit for the convergence of the Collatz conjecture
Authors: David Barina
The Journal of Supercomputing, Vol. 81(7), pp. 810
This article presents our project, which aims to verify the Collatz conjecture computationally. As a main point of the article, we introduce a new result that pushes the limit for which the conjecture is verified up to {\$}{\$}2^{\{}71{\}}{\$}{\$}. We present our baseline algorithm and then several sub-algorithms that enhance acceleration. The total acceleration from the first algorithm we used on the CPU to our best algorithm on the GPU is {\$}{\$}1{\backslash},335{\backslash}times{\$}{\$}. We further distribute individual tasks to thousands of parallel workers running on several European supercomputers. Besides the convergence verification, our program also checks for path records during the convergence test. We found four new path records.
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.
Unpredicatable Iterations
Authors: John H. Conway
Proc. 1972 Number Theory Conference, University of Colorado, pp. 49--52
On the "$3x + 1$" problem
Authors: Richard E. Crandall
Math. Comp., Vol. 32, pp. 1281--1292
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
On a Conjecture of Crandall Concerning the $qx + 1$ Problem
Authors: Zachary Franco and Carl Pomerance
Math. Comp., Vol. 64, pp. 1333--1336
There are no Collatz $m$-Cycles with $m \leq 91$
Authors: Christian Hercher
Journal of Integer Sequences, Vol. 26
Bounds for the $3x+1$ problem using difference inequalities
Authors: Ilia Krasikov and J. C. Lagarias
Acta Arithmetica, Vol. 109(3), pp. 237--258
We study difference inequality systems for the 3x+1 problem introduced by the first author in 1989. These systemes can be used to give lower bounds for the number of integers below x that contain 1 in their forward orbit under the 3x+1 map. Previous methods gave away some information in these inequalities. We give an improvement which apparantly extracts full information from the inequalities. By computer aided proof we show that at least x^{0.84} of the integers below x contain 1 in their forward orbit under the 3x+1 map.
The undecidability of the generalized Collatz problem
Authors: Stuart A. Kurtz and Janos Simon
Theory and Applications of Models of Computation, 4-th International Conference, TAMC 2007, Shanghai, China, May 22-25, 2007, pp. 542--553
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
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
On the Collatz general problem qn+1
Authors: Robert Santos
In this work the generalized Collatz problem qn+ 1 (q odd) is studied. As a natural generalization of the original 3n+ 1 problem, it consists of a discrete dynamical system of an arithmetical kind. Using standard methods of number theory and dynamical systems, general properties are established, such as the existence of finitely many periodic sequences for each q. In particular, when q is a Mersenne number, q = 2p − 1, there only exists one such cycle, known as the trivial one. Further analysis based on a probabilistic model shows that for q = 3 the asymptotic behavior of all sequences is always convergent, whereas for q ≥ 5 the asymptotic behavior of the sequences is divergent for almost all numbers (for a set of natural density one). This leads to the conclusion that the so called Collatz Conjecture is true, and that q = 3 is a very special case among the others (Crandall conjecture). Indeed, it is conjectured that the general problem qn + 1 is undecidable.
{p, q}-adic Analysis and the Collatz Conjecture
Authors: Maxwell Charles Siegel
What use can there be for a function from the p-adic numbers to the q-adic numbers, where p and q are distinct primes? The traditional answer—courtesy of the half-century old theory of non-archimedean functional analysis: not much. It turns out this judgment was premature. “(p, q)- adic analysis” of this sort appears to be naturally suited for studying the infamous Collatz map and similar arithmetical dynamical systems. Given such a map H : Z → Z, one can construct a function χH : Z_p → Z_q for an appropriate choice of distinct primes p, q with the property that x ∈ Z\ {0} is a periodic point of H if and only if there is a p-adic integer z ∈ (Q ∩ Z_p) \ {0, 1, 2, . . .} so that χH (z) = x. By generalizing Monna-Springer integration theory and establishing a (p, q)- adic analogue of the Wiener Tauberian Theorem, one can show that the question “is x ∈ Z\ {0} a periodic point of H” is essentially equivalent to “is the span of the translates of the Fourier transform of χH (z) − x dense in an appropriate non-archimedean function space?” This presents an exciting new frontier in Collatz research, and these methods can be used to study Collatz-type dynamical systems on the lattice Z^d for any d ≥ 1.
A stopping time problem on the positive integers
Authors: Riho Terras
Acta Arithmetica, Vol. 30, pp. 241--252
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
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
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Some properties of binary sequences useful for proving Collatz's conjecture
Authors: Jacek Bł ażewitz and Alberto Pettorossi
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3-Smooth Representations of Integers
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Estimates for the Syracuse problem via a probabilistic model
Authors: Konstantin Borovkov and Dietmar Pfeifer
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Which rationals are ratios of Pisot sequences?
Authors: David W. Boyd
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Sur un problème de R. Queneau
Authors: Monique Bringer
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A Note on Mignosi's Generalization of the $3x+1$ Problem
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Collatz cycles with few descents
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Acta Arithmetica, Vol. 92, pp. 181--188
A proof of the Collatz conjecture
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Functional equations associated with congruential functions
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On some Markov matrices arising from the generalized Collatz mapping
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A note on the $3x + 1$ problem
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Some observations on the $3x+1$ problem
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The $3x+1$ problem: towards a solution
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Trajectories in the $3x+1$ problem
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A Solution to the $3x+1$ Problem
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Structural and Computational Complexity of Isometries and their Shift Commutators
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A Continuous Extension of the $3x+1$ Problem to the Real Line
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Cycles in Collatz Sequences
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Selected combinatorial research problems
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Periodic solutions of arbitrary length in a simple integer iteration
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A Collatz-Type Difference Equation
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Symmetric solutions to a Collatz-like system of Difference Equations
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Small Tile Sets That Compute While Solving Mazes
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Optimal bounds for the length of rational Collatz cycles
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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
Journal of Jinan University (Science and Technology) [Jinan da xue xue bao. Zi ran ke xue ban]
The proof of the hypothesis about $"3x+1"$
Authors: Ke, Yong-Sheng
Journal of Tianjin Vocational Institute [Teacher's College] [Tianjin zhi ye ji shu shi fax xue yuan xue bao]
Finite cycles of certain periodically linear functions
Authors: Timothy P. Keller
Missouri J. Math. Sci., Vol. 11(3), pp. 152--157
Die Collatz-Ulam-Kombination CUK
Authors: Immo O. Kerner
Rostocker Informatik-Berichte No. 24
Problem $63-13^{*}$
Authors: Murray S. Klamkin
SIAM Review, Vol. 5, pp. 275--276
An algorithm to determine when certain sets have $0$ density
Authors: David A. Klarner
Journal of Algorithms, Vol. 2, pp. 31--43
A sufficient condition for certain semigroups to be free
Authors: David A. Klarner
Journal of Algebra, Vol. 74, pp. 140--148
$m$-recognizability of sets closed under certain affine functions
Authors: David A. Klarner
Discrete Applied Mathematics, Vol. 21(3), pp. 207--214
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
Authors: David A. Klarner and Richard Rado
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
J. of Symbolic Computation, Vol. 43(8), pp. 545--581
A simple group generated by involutions interchanging residue classes of the integers
Authors: Stefan Kohl
Math. Z., Vol. 264(4), pp. 927--938
We present a countable simple group which arises in a natural way from the arithmetical structure of the ring of integers.
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
Acta Arithmetica, Vol. 120, pp. 269--297
Structure Theorem for $(d, g, h)$-maps
Authors: Alex V. Kontorovich and Yakov G. Sinai
Bull. Braz. Math. Soc. (N.S.), Vol. 33, pp. 213--224
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
A Note on the $3x + 1$ Problem
Authors: Ivan Korec and Stefan Znam
Amer. Math. Monthly, Vol. 94, pp. 771--772
On conjugacies for the $3x+1$ map induced by continuous endomorphisms of the shift dynamical system
Authors: Benjamin Kraft and Kennan Monks
Discrete Math., Vol. 310(13-14), pp. 1875--1883
How many numbers satisfy the $3x + 1$ Conjecture?
Authors: Ilia Krasikov
Internatl. J. Math. & Math. Sci., Vol. 12, pp. 791--796
On the $3x+1$ Problem
Authors: James R. Kuttler
Adv. Appl. Math., Vol. 15, pp. 183--185
The $3x + 1$ problem and its generalizations
Authors: Jeffrey C. Lagarias
Amer. Math. Monthly, Vol. 92, pp. 3--23
The set of rational cycles for the $3x + 1$ problem
Authors: Jeffrey C. Lagarias
Acta Arithmetica, Vol. 56, pp. 33--53
How random are 3x + 1 function iterates?
Authors: Jeffrey C. Lagarias
The Mathemagician and Pied Puzzler: A Collection in Tribute to Martin Gardner, pp. 253--266
The $3x+1$ Problem: An Annotated Bibliography (1963--1999)
Authors: Jeffrey C. Lagarias
Wild and Wooley Numbers
Authors: Jeffrey C. Lagarias
American Mathematical Monthly, Vol. 113, pp. 97--108
The Ultimate Challenge: The $3x+1$ Problem
Authors: Jeffrey C. Lagarias
American Mathematical Society, Vol. 78, pp. 344
The 3x+1 problem: An annotated bibliography (1963--1999) (sorted by author)
Authors: Jeffrey C. Lagarias
The 3x+1 Problem: An Annotated Bibliography, II (2000-2009)
Authors: Jeffrey C. Lagarias
The $3x+1$ Problem: An Overview
Authors: Jeffrey C. Lagarias
Asymmetric Tent Map Expansions I. Eventually Periodic Points
Authors: Jeffrey C. Lagarias, Horacio A. Porta and Kenneth B. Stolarsky
J. London Math. Soc., Vol. 47, pp. 542--556
Asymmetric Tent Map Expansions II. Purely Periodic Points
Authors: Jeffey C. Lagarias, Horacio A. Porta and Kenneth B. Stolarsky
Illinois J. Math., Vol. 38, pp. 574--588
Approximate squaring
Authors: Jeffrey C. Lagarias and Neil J. A. Sloane
Experimental Math., Vol. 13, pp. 113--128
Benford's Law for the $3x+1$ Function
Authors: Jeffrey C. Lagarias and K. Soundararajan
J. London Math. Soc., Vol. 74, pp. 289--303
The $3x + 1$ Problem: Two Stochastic Models
Authors: Jeffrey C. Lagarias and Alan Weiss
Annals of Applied Probability, Vol. 2, pp. 229--261
$3x +1$ Search Programs
Authors: Gary T. Leavens and Mike Vermeulen
Computers & Mathematics, with Applications, Vol. 24(11), pp. 79--99
Two undecidable versions of Collatz's problems
Authors: Eero Lehtonen
Theor. Comp. Sci., Vol. 407, pp. 596--600
A Markov process underlying the generalized Syracuse algorithm
Authors: George M. Leigh
Acta Arithmetica, Vol. 46, pp. 125--143
On the $3X+1$ problem and holomorphic dynamics
Authors: Simon Letherman, Dierk Schleicher and Reg Wood
Experimental Math., Vol. 8(3), pp. 241--251
Injectivity and Surjectivity of Collatz Functions
Authors: Dan Levy
Discrete Math., Vol. 285, pp. 190--199
Compressive Iteration for the $3N+1$ conjecture
Authors: Li, Xiao Chun
J. Huazhong Univ. of Science and Technology (Natural Science), Vol. 30(2)
Contractible iteration for the $3n+1$ conjecture
Authors: Li, Xiao Chun
J. Huazhong Univ. Sci. Technol. Nat. Sci., Vol. 31(7), pp. 115--116
Same-flowing numbers and super contraction iteration in $3N+1$ conjecture
Authors: Li, Xiao Chun
J. Huazhong Univ. Sci. Technol. Nat. Sci., Vol. 32(10), pp. 30--32
Necessary condition for periodic numbers in the $3N+1$ conjecture
Authors: Li, Xiao Chun
J. Huazhong Univ. Sci. Technol. Nat. Sci., Vol. 33(11), pp. 102--103
Some properties of super contraction iteration in the $3N+1$ conjecture
Authors: Li, Xiao Chun
J. Huazhong Univ. Sci. Technol. Nat. Sci., Vol. 34(8), pp. 115--117
Research on the Syracuse Operator and its properties
Authors: Li, Xiao Chun and Liang, You Min
Journal of Air Force Radar Academy(3)
Equivalence of the $3N+1$ and $3N+3k$ conjecture and some related properties
Authors: Li, Xiao Chun and Liu, Jun
J. Huazhong Univ. Sci. Technol. Nat. Sci., Vol. 34(7), pp. 120-121
Study of periodic numbers in the $3N+1$ conjecture
Authors: Li, Xiao Chun and Wu, Jia Bang
J. Huazhong Univ. Sci. Technol. Nat. Sci., Vol. 32(10), pp. 100--101
The $3x+1$ conjugacy map over a Sturmian word
Authors: López, Josefine and Stoll, Peter
Integers, Vol. 9, pp. 141--162
On the nontrivial cycles in Collatz's problem
Authors: Luca, Florian
SUT Journal of Mathematics, Vol. 41(1), pp. 31--41
Eine kleine Untersuchung zu einem zahlentheoretischen Problem
Authors: Heinrich Lunkenheimer
PM (Praxis der Mathematik in der Schule), Vol. 30, pp. 4--9
An unsolved problem on the powers of $3/2$
Authors: Kurt Mahler
J. Australian Math. Soc., Vol. 8, pp. 313--321
Achilles, Turtle, and Undecidable Boundedness Problems for Small DATALOG programs
Authors: Jerzy Marcinkowski
SIAM J. Comput., Vol. 29, pp. 231--257
The powers of two and three
Authors: Dȧnuţ Marcu
Discrete Math., Vol. 89, pp. 211-212
Frontier between decidability and undecidability: a survey
Authors: Margenstern, Maurice
Theor. Comput. Sci., Vol. 231, pp. 217--251
A binomial representation of the $3X +1$ problem
Authors: Maurice Margenstern and Yuri Matiyasevich
Acta Arithmetica, Vol. 91(4), pp. 367--378
Some Borel measures associated with the generalized Collatz mapping
Authors: Keith R. Matthews
Colloq. Math., Vol. 53, pp. 191--202
A generalization of the Syracuse algorithm to $F_q [x]$
Authors: Keith R. Matthews and George M. Leigh
J. Number Theory, Vol. 25, pp. 274--278
A generalization of Hasse's generalization of the Syracuse algorithm
Authors: Keith R. Matthews and Anthony M. Watts
Acta. Arithmetica, Vol. 43, pp. 167--175
A Markov approach to the generalized Syracuse algorithm
Authors: Keith R. Matthews and Anthony M. Watts
Acta Arithmetica, Vol. 45, pp. 29--42
On the Fibonacci's Attractor and the Long Orbits in the $3n+1$ Problem
Authors: Danilo Merlini and Nicoletta Sala
International Journal of Chaos Theory and Applications, Vol. 4(2-3), pp. 75--84
Untersuchungen zum $(3n+1)$-Algorithmus. Teil I: Das Problem der Zyklen
Authors: Karl Heinz Metzger
PM (Praxis der Mathematik in der Schule), Vol. 38, pp. 255--257
Zyklenbestimmung beim $(bn+1)$-Algorithmus
Authors: Karl Heinz Metzger
PM (Praxis der Mathematik in der Schule), Vol. 41, pp. 25--27
Untersuchungen zum $(3n+1)$-Algorithmus. Teil II: Die Konstruktion des Zahlenbaums
Authors: Metzger, Karl Heinz
PM (Praxis der Mathematik in der Schule), Vol. 42, pp. 27--32
Untersuchungen zum $(3n+1)$-Algorithmus, Teil III: Gesetzmässigkeiten der Ablauffolgen
Authors: Metzger, Karl Heinz
PM (Praxis der Mathematik in der Schule), Vol. 45(1), pp. 25--32
Busy Beaver Competition and Collatz-like Problems
Authors: Pascal Michel
Archive Math. Logic, Vol. 32, pp. 351--367
Small Turing machines and generalized busy beaver competition
Authors: Michel, Pascal
Theor. Comp. Sci., Vol. 326, pp. 45--56
On a Generalization of the $3x + 1$ Problem
Authors: Filippo Mignosi
J Number Theory, Vol. 55, pp. 28--45
On certain simple cycles of the Collatz conjecture
Authors: Mimuro, Tomoaki
SUT Journal of Mathematics, Vol. 37(2), pp. 79--89
Real $3X+1$
Authors: Misiurewicz, Michal and Rodriguez, Ana
Proc. Amer. Math. Soc., Vol. 133, pp. 1109--1118
Uber Hasses Verallgemeinerung des Syracuse-Algorithmus (Kakutanis Problem)
Authors: Herbert Möller
Acta Arith., Vol. 34(3), pp. 219--226
$3X+1$ Minus the $+$
Authors: Monks, Kenneth G.
Discrete Math. Theor. Comput. Sci., Vol. 5, pp. 47--53
Endomorphisms of the shift dynamical systems, discrete derivatives, and applications
Authors: Monks, Maria
Discrete Math., Vol. 309, pp. 5196--5205
The Autoconjugacy of the $3x+1$ function
Authors: Monks, Kenneth G. and Yazinski, Jonathan
Discrete Mathematics, Vol. 275(1), pp. 219--236
Das `$3n+1$' Problem
Authors: Helmut A. Müller
Mitteilungen der Math. Ges. Hamburg, Vol. 12, pp. 231--251
Über eine Klasse 2-adischer Funktionen im Zusammenhang mit dem "$3x+1$"-Problem
Authors: Helmut A. Müller
Abh. Math. Sem. Univ. Hamburg, Vol. 64, pp. 293--302
Über Periodenlängen and die Vermutungen von Collatz und Crandall
Authors: Müller, Helmut
Mitt. Math. Ges. Hamburg, Vol. 28, pp. 121--130
A note on a paper of H. Gupta concerning powers of 2 and 3
Authors: Wł adysł aw Narkiewicz
Univ. Beograd. Publ. Elecktrotech. Fak. Ser. Mat. Fiz. No, Vol. 678-715, pp. 173-174
Computers and Mathematics Education
Authors: Jürg Nivergelt
Computers & Mathematics, with Applications, Vol. 1, pp. 121--132
Tomorrow's Math: unsolved problems for the amateur
Authors: C. Stanley Ogilvy
Oxford University Press: New York
A Generalization of the Collatz problem
Authors: Reiko Ohira and Michinori Yamashita
PC Literacy [Pasocon Literacy] (Personal Computer Users Application Technology Association], Vol. 31(4), pp. 16--21
On the $p$-Collatz problem
Authors: Reiko Ohira and Michinori Yamashita
PC Literacy [Pasocon Literacy] (Personal Computer Users Application Technology Association), Vol. 31(4), pp. 61--64
Alcune Questioni Correlate al Problema Del "3D+1"
Authors: Elio Oliveri and Giuseppe Vella
Atti della Accademia di scienze lettere e arti di Palermo, Ser. V, Vol. 28, pp. 21--52
Notes on the $3x+1$ problem
Authors: Pan, Hong-liang
Journal of Suzhou University, Natural Science [Suzhou da xue xue bao. Zi ran ke xue ban], Vol. 16(4), pp. 13--16
The $3x+1$ Fractal
Authors: Joseph L. Pe
Computers & Graphics, Vol. 28, pp. 431--435
Absolute continuity for random iterated function systems with overlaps
Authors: Yuval Peres, Károly Simon and Boris Solomak
J. London Math. Soc., Vol. 74(2), pp. 739--756
Hailstone $3n + 1$ Number graphs
Authors: Clifford A. Pickover
J. Recreational Math., Vol. 21, pp. 120--123
A transformation of iterated Collatz mappings
Authors: Margherita Pierantoni and Vladan Ćurčić
Z. Angew Math. Mech., Vol. 76(Suppl. 2), pp. 641--642
An elementary approach to some analytic asymptotics
Authors: Nicholas Pippenger
SIAM J. Math. Anal., Vol. 24(5), pp. 1361--1377
The Syracuse problem (Spanish), 5$^{\rm th}$
Authors: Susana Puddu
Latin American Colloq. on Algebra -- Santiago 1985, Vol. 5, pp. 199--200
Study on $"3x+1"$ problem
Authors: Wei Xing Qiu
J. Shanghai Univ. Nat. Sci. Ed. [Shanghai da xue xue bao. Zi ran ke xue ban], Vol. 3(4), pp. 462--464
The $3x+1$ problem and its necessary and sufficient condition
Authors: Qu, Jing-hua
Journal of Sangqiu Teacher's College(5)
Note complémentaire sur la Sextine
Authors: Raymond Queneau
Subsidia Pataphysica, Vol. 1, pp. 79--80
Sur les suites $s$-additives
Authors: Raymond Queneau
J. of Combinatorial Theory, Series A, Vol. 12, pp. 31-71
Some work on an Unsolved Palindromic Algorithm
Authors: Lee Ratzan
Pi Mu Epsilon Journal, Vol. 5, pp. 463--466
Imitation of an iteration
Authors: Daniel A. Rawsthorne
Math. Magazine, Vol. 58, pp. 172--176
Collatz Conjecture 𝟑𝒏 ± 𝟏 as a Newton Binomial Problem
Authors: Petro Kosobutskyy, Dariia Rebot
Computer Design Systems. Theory and Practice, Vol. 5(1), pp. 9
The power transformation of Newton's binomial forms two equal 3𝑛 ± 1algorithms for transformations of numbers n ℕ, each of which have one infinite cycle with a unit lower limit of oscillations. It is shown that in the reverse direction, the Kollatz sequence is formed by the lower limits of the corresponding cycles, and the last element goes to a multiple of three odd numbers. It was found that for infinite transformation cycles 3n −1 isolated from the main graph with minimum amplitudes of 5, 7, 17 lower limits of oscillations, additional conditions are fulfilled.
Approximants de Padé et mesures effectives d'irrationalité
Authors: Georges Rhin
Séminaire de Théorie des Nombres, Paris 1985--86, Vol. 71, pp. 155--164
Problem 669
Authors: H. J. J. te Riele
Nieuw Archief voor Wiskunde, Series IV, Vol. 1, pp. 80
Iteration of Number-theoretic functions
Authors: H. J. J. te Riele
Nieuw Archief voor Wiskunde, Series IV, Vol. 1, pp. 345--360
Un problème combinatoire posè par la poésie lyrique des troubadours
Authors: Jacques Roubaud
Mathématiques et Sciences Humaines [Mathematics and Social Science], Vol. 27, pp. 5--12
$N$-ine, autrement dit quenine (encore), Réflexions historiques et combinatoires sur la $n$-ine, autrement dit quenine
Authors: Jacques Roubaud
La Bibliothéque Oulipienne, numéro 66
A generalization of the Collatz problem. Building cycles and a stochastic approach
Authors: Jean-Louis Rouet and Marc R. Feix
J. Stat. Phys., Vol. 107(5/6), pp. 1283--1298
Probleme de Syracuse: "Majorations" Elementaires des Cycles
Authors: Olivier Rozier
Singularité, Vol. 2(5), pp. 8--11
Parity Sequences of the $3x+1$ Map on the 2-adic Integers and Euclidean Embedding
Authors: Olivier Rozier
Integers, Vol. 19, pp. A8
On the $(3N + 1)$-conjecture
Authors: Jürgen W. Sander
Acta Arithmetica, Vol. 55, pp. 241--248
$\omega$-rewriting the Collatz problem
Authors: Giuseppe Scollo
Fundamenta Informaticae, Vol. 64, pp. 401--412
On the arithmetic of cycles for the Collatz-Hasse (`Syracuse') conjectures
Authors: Benedict G. Seifert
Discrete Math., Vol. 68, pp. 293--298
The "$3x+1$" Problem and Finite Automata
Authors: Jeffrey O. Shallit and David W. Wilson
Bulletin of the EATCS (European Association for Theoretical Computer Science)(46), pp. 182--185
Comments on Problem $63-13^{*}$
Authors: Daniel Shanks
SIAM Review, Vol. 7, pp. 284--286
Problem $\#5$
Authors: Daniel Shanks
Western Number Theory Conference 1975, Problem List
The pure numbers generated by the Collatz sequence
Authors: Douglas J. Shaw
Fibonacci Quarterly, Vol. 44(3), pp. 194-201
Properties of cycle sets
Authors: Shi, Qian Li
J. Yangzte Univ. Natural Science. [Changjiang daxue xuebao. Zi ke ban], Vol. 2(7), pp. 191--192
Properties of the $3x+1$ problem
Authors: Shi, Qian Li
J. Yangzte Univ. Natural Science. [Changjiang daxue xuebao. Zi ke ban], Vol. 2(10), pp. 287--289
Maximum Excursion and Stopping Time Record-Holders for the $3x+1$ Problem: Computational Results
Authors: Tomás Oliveira e Silva
Math. Comp., Vol. 68(1), pp. 371-384
On the non-existence of $2$-cycles for the $3x+1$ problem
Authors: John L. Simons
Math. Comp., Vol. 74, pp. 1565--1572
A simple (inductive) proof for the non-existence of $2$-cycles for the $3x+1$ problem
Authors: John L. Simons
J. Number Theory, Vol. 123, pp. 10-17
This article generalizes a proof of Steiner for the nonexistence of 1-cycles for the 3x + 1 problem to a proof for the nonexistence of 2-cycles. A lower bound for the cycle length is derived by approximating the ratio between numbers in a cycle. An upper bound is found by applying a result of Laurent, Mignotte, and Nesterenko on linear forms in logarithms. Finally numerical calculation of convergents of log2 3 shows that 2-cycles cannot exist.
On the (non)-existence of $m$-cycles for generalized Syracuse sequences
Authors: John L. Simons
Acta Arithmetica, Vol. 131(3), pp. 217--254
Exotic Collatz Cycles
Authors: John L. Simons
Acta Arithmetica, Vol. 134, pp. 201--209
Post-transcendence conditions for the existence of $m$-cycles for the $3x+1$ problem
Authors: John L. Simons
On isomorphism between Farkas sequences and Collatz sequences
Authors: John L. Simons
Mersenne en het Syracuseprobleem [Mersenne and the Syracuse problem]
Authors: John L. Simons and Benne M. M. de Weger
Nieuw Arch. Wiskd., Vol. 5(3), pp. 218--220
Theoretical and computational bounds for $m$-cycles of the $3n+1$ problem
Authors: John L. Simons and Benne M. M. de Weger
Acta Arithmetica, Vol. 117, pp. 51--70
Statistical $(3X+1)$-Problem
Authors: Yakov G. Sinai
Comm. Pure Appl. Math., Vol. 56(7), pp. 1016--1028
Uniform distribution in the $(3x+1)$ problem
Authors: Yakov G. Sinai
Moscow Math. Journal, Vol. 3(4), pp. 1429--1440
A theorem about uniform distribution
Authors: Yakov G. Sinai
Commun. Math. Phys., Vol. 252, pp. 581--588
On the minimal cycle lengths of the Collatz sequences
Authors: Matti K. Sinisalo
On the almost convergence of Syracuse sequences
Authors: Alain Slakmon and Luc Macot
Statistics and Probability Letters, Vol. 76(15), pp. 1625--1630
The Collatz problem and Analogues
Authors: Bart Snapp and Matt Tracy
J. Integer Sequences, Vol. 11
A Theorem on the Syracuse Problem
Authors: Ray P. Steiner
Proc. 7-th Manitoba Conference on Numerical Mathematics and Computing (Univ. Manitoba-Winnipeg 1977), pp. 553--559
On the "$Qx+1$" Problem, $Q$ odd
Authors: Ray P. Steiner
Fibonacci Quarterly, Vol. 19, pp. 285--288
On the "$Qx+1$" Problem, $Q$ odd II
Authors: Ray P. Steiner
Fibonacci Quarterly, Vol. 19, pp. 293--296
The ideal Waring problem for exponents $401-200,000$
Authors: Rosemarie M. Stemmler
Math. Comp., Vol. 18, pp. 144--146
Binary expression of ancestors in the Collatz graph
Authors: Stérin, Tristan
Reachability Problems, pp. 115--130
The Collatz Process Embeds a Base Conversion Algorithm
Authors: Stérin, Tristan and Woods, Damien
RP2020: 14th International Conference on Reachability Problems, Vol. 12448, pp. 131--147
A prelude to the $3x+1$ problem
Authors: Kenneth S. Stolarsky
J. Difference Equations and Applications, Vol. 4, pp. 451--461
About recursion relation of $3x+1$ Conjecture (Chinese)
Authors: Tang, Ren Xian
J. of Hunan Univ. Science Engineering(5)
Open questions about KW-orbits and iterative roots
Authors: György Targónski
Aequationes Math., Vol. 41, pp. 277--278
On the existence of a density
Authors: Riho Terras
Acta Arithmetica, Vol. 35, pp. 101--102
My conjecture
Authors: Bryan Thwaites
Bull. Inst. Math. Appl., Vol. 21, pp. 35--41
Two Conjectures, or how to win $\pounds$ 1000
Authors: Bryan Thwaites
Math. Gazette, Vol. 80, pp. 35--36
Note on Mahler's $3/2$-problem
Authors: Robert Tijdeman
Det Kongelige Norske Videnskabers Selskab Skrifter No. 16, pp. 4
Comments on Problem 133
Authors: Charles W. Trigg and Clayton W. Dodge and Leroy F. Meyers
Eureka (now Crux Mathematicorum), Vol. 2(7), pp. 144--150
Collatz Problem IV
Authors: Toshio Urata
Epsilon (The Bulletin of the Society of Mathematics Education of Aichi Univ. of Education) [Aichi Kyoiku Daigaku Sugaku Kyoiku Gakkai shi], Vol. 41, pp. 111-116
Some holomorphic functions connected with the Collatz problem
Authors: Toshio Urata
Bulletin Aichi Univ. Education (Natural Science), Vol. 51, pp. 13--16
The Collatz problem over $2$-adic integers
Authors: Toshio Urata
Bulletin Aichi Univ. Education (Natural Science), Vol. 52, pp. 5--11
Positive values of a holomorphic function connected with the Collatz problem
Authors: Toshio Urata and Kazuhiro Hamada
Bulletin Aichi Univ. Education (Natural Science), Vol. 54, pp. 1--10
Collatz Problem III
Authors: Toshio Urata and Hisao Kajita
Epsilon (The Bulletin of the Society of Mathematics Education of Aichi Univ. of Education) [Aichi Kyoiku Daigaku Sugaku Kyoiku Gakkai shi], Vol. 40, pp. 57--65
Collatz Problem
Authors: Toshio Urata and Katsufumi Suzuki
Epsilon (The Bulletin of the Society of Mathematics Education of Aichi Univ. of Education) [Aichi Kyoiku Daigaku Sugaku Kyoiku Gakkai shi], Vol. 38, pp. 123--131
Collatz Problem II
Authors: Toshio Urata and Katsufumi Suzuki and Hisao Kajita
Epsilon (The Bulletin of the Society of Mathematics Education of Aichi Univ. of Education) [Aichi Kyoiku Daigaku Sugaku Kyoiku Gakkai shi], Vol. 39, pp. 121--129
Regularity of congruential graphs
Authors: Tanguy Urvoy
Mathematical Foundations of Computer Science 2000 (Bratislava), pp. 680--689
The Collatz Conjecture: A Case Study in Mathematical Problem Solving
Authors: Jean Paul Van Bendegem
Logic and Logical Philosophy, Vol. 14(1), pp. 7--23
Sul Comportamento delle Iterazioni di Alcune Funzioni Numeriche
Authors: Giovanni Venturini
Rend. Sci. Math. Institute Lombardo, Vol. A 116, pp. 115--130
On the $3x + 1$ Problem
Authors: Giovanni Venturini
Adv. Appl. Math, Vol. 10, pp. 344--347
Iterates of Number Theoretic Functions with Periodic Rational Coefficients (Generalization of the $3x + 1$ problem)
Authors: Giovanni Venturini
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On a Generalization of the $3x + 1$ problem
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