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### On heights in the Collatz 3n + 1 problem

Lynn E. Garner
1985 Discrete Mathematics
In the Collatz problem, certain runs of consecutive integers have the same height.  ...  It is found that pain of consecutive integers of the same height occur infinitely often and in infinitely many different patterns. 0012-365X/85/\$3.30 0 1985, Elsevier Science Publishers B.V.  ...  The Collatz graph is the digraph on N with edges n + T(n). A part of the Collatz graph is shown in Fig. 1 .  ...

### 3n+1 Problem and its Dynamics

Bishnu Hari Subedi, Ajaya Singh
2020 Nepal Journal of Mathematical Sciences
We briefly discuss some basic facts and results of 3n + 1 problem and Collatz conjecture. Basically, we more concentrate on the generalization of this problem and conjecture to holomorphic dynamics.  ...  The subject of this paper is the well-known 3n + 1 problem of elementary number theory.  ...  One of such problems is the 3n+1 problem. This is a wellknown problem in elementary number theory, and it can be explained to a child who has learned how to divide by 2 and multiply by 3.  ...

### The 3x+1 Problem: An Annotated Bibliography, II (2000-2009) [article]

Jeffrey C. Lagarias
2012 arXiv   pre-print
The 3x+1 problem concerns iteration of the map T(n) =(3n+1)/2 if n odd; n/2 if n even. The 3x +1 Conjecture asserts that for every positive integer n>1 the forward orbit of n includes the integer 1.  ...  This is a sequel to an annotated bibliography on the 3x+1 problem covering 1963-1999. At present the 3x+1 Conjecture remains unsolved.  ...  + 1 problem and the inverse Collatz problem.  ...

### The 3x+1 problem: An annotated bibliography (1963--1999) (sorted by author) [article]

Jeffrey C. Lagarias
2011 arXiv   pre-print
The 3x+ 1 problem concerns iteration of the map on the integers given by T(n) = (3n+1)/2 if n is odd; T(n) = n/2 if n is even.  ...  This paper is an annotated bibliography of work done on the 3x+1 problem and related problems from 1963 through 1999. At present the 3x+1 Conjecture remains unsolved.  ...  Garner (1985) , On heights in the Collatz 3n + 1 problem, Discrete Math. 55 (1985) , 57-64. (MR 86j:11005) .  ...

### On consecutive numbers of the same height in the Collatz problem

Guo-Gang Gao
1993 Discrete Mathematics
., On consecutive numbers of the same height in the Collatz problem, Discrete Mathematics 112 (1993) 261-267.  ...  The Collatz function C(n) is defined to take odd numbers n to 3n + 1 and even numbers n to n/2.  ...  The problem is also known as the 3x + 1 problem. The interested reader can find a comprehensive history and detailed discussions on this topic in [3] .  ...

### Stochastic Models for the 3x+1 and 5x+1 Problems [article]

Alex V. Kontorovich, Jeffrey C. Lagarias
2009 arXiv   pre-print
This paper discusses stochastic models for predicting the long-time behavior of the trajectories of orbits of the 3x+1 problem and, for comparison, the 5x+1 problem.  ...  The stochastic models are rigorously analyzable, and yield heuristic predictions (conjectures) for the behavior of 3x+1 orbits and 5x+1 orbits.  ...  AVK wishes to thank the hospitality of Dorian Goldfeld and Columbia University during this project.  ...

### The Collatz conjecture. A case study in mathematical problem solving

Jean Paul Van Bendegem
2005 Logic and Logical Philosophy
In addition, the problem is fairly easy to state, although the mathematics that are used in search of a proof reach formidable heights.  ...  4 → 5 → 6 ←−−−−− 7 → . . . where an arrow represents an application of the function f (in this case, the simple function f(n), defined by f(3n + 1) = 3n + 2, f(3n + 2) = 3(n + 1) and f(3(n + 1)) = 3n  ...

### The intricate labyrinth of Collatz sequences [article]

Maya Mohsin Ahmed
2016 arXiv   pre-print
In a previous article, we reduced the unsolved problem of the convergence of Collatz sequences, to convergence of Collatz sequences of odd numbers, that are divisible by 3.  ...  We also show that either the Collatz sequence of a given odd number or an equivalent Collatz sequence reverses to a multiple of 3.  ...  A comprehensive study of the Collatz conjecture can be found in [2] , [3] , and [4] . In this article, like in [1] , we focus on the subsequence of odd numbers of a Collatz sequence.  ...

### A Graph Theoretical Approach to the Collatz Problem [article]

Heinz Ebert
2021 arXiv   pre-print
Andrei et al. have shown in 2000 that the graph C of the Collatz function starting with root 8 after the initial loop is an infinite binary tree A(8).  ...  A proof that the graph C_C(1) is an infinite binary tree A_C with vertex set V(A_C(1))=ℤ^+ completes the paper.  ...  the 3n+1 problem therein 4 .  ...

### On the Distribution of the First Point of Coalescence for some Collatz Trajectories [article]

Roy Burson
2020 arXiv   pre-print
This paper is a numerical evaluation of some trajectories of the Collatz function.  ...  Afterwards, I show that the first point of coalescence of the integers n and 3n+2 appear to tend to an expected value of 4/5n.  ...  Collatz function. Fig 7 : 7 Assessment of the the first point of coalescence C(n, 3n + 2) for n ∈ [1, 100000]. Fig 8 : 8 Assessment of ς(k) for k ∈ [1, 100000].  ...

### Page 4443 of Mathematical Reviews Vol. , Issue 86j [page]

1986 Mathematical Reviews
Zném (Bratislava) Garner, Lynn E. (1-BYU) On heights in the Collatz 3n+1 problem.  ...  The Collatz problem is to show that, for every n € N, the sequence {n,T(n),T?(n), ---} eventually contains the integer 1. The height of n is the least k such that T*(n) = 1.  ...

### Page 2471 of Mathematical Reviews Vol. , Issue 2000d [page]

2000 Mathematical Reviews
(D-HANN-IM; Hannover) On the exact height of integer-detecting sequences. (English summary) J. Number Theory 73 (1998), no. 1, 1-13.  ...  If d > —1 is an odd integer not divisible by 3, the authors define the 3x +d function S,(m) as Sy(n) = odd(3n+d).  ...

### Consecutive Integers and the Collatz Conjecture [article]

Marcus Elia, Amanda Tucker
2015 arXiv   pre-print
Pairs of consecutive integers have the same height in the Collatz problem with surprising frequency. Garner gave a conjectural family of conditions for exactly when this occurs.  ...  We would also like to thank the referee for careful reading and helpful suggestions and corrections.  ...  If one naïvely searches for curves of best fit to the visible curves therein, one quickly runs into a problem.  ...

### Page 2760 of Mathematical Reviews Vol. , Issue 99d [page]

1991 Mathematical Reviews
The Collatz function is defined as the function f:No — No (where No =N {0}): f(n) =n/2 if n is even, f(n) = 3n +1 if n is odd.  ...  We are interested in the problem of selection. Selection is a critical comparison problem with numerous applications.  ...

### Convergence of Collatz Sequences: Procedure to Prove the Collatz Conjecture [article]

Ramachandra Bhat
2021 arXiv   pre-print
Conceptually, the process relies on the pre-proven sequence data and the method follows the confirmation of the convergence of the Collatz sequence for all the natural numbers in a sequential forward manner  ...  Here, the procedure for the computation is presented with example and the current status of computation (>99% completed) while the laborious computation is in progress (for the remaining < 1%).  ...  on a given positive integer, till the number reaches a value of 1(one) wherein the Collatz function is defined as: Col(n) = f (n) = (3n + 1) if n is odd ( n 2 ) if n is even (1) Collatz Conjecture:  ...
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