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On heights in the Collatz 3n + 1 problem
1985
Discrete Mathematics
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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 . ...
doi:10.1016/s0012-365x(85)80020-0
fatcat:6poosstm3baxjpgc6nvp5x3rcy
3n+1 Problem and its Dynamics
2020
Nepal Journal of Mathematical Sciences
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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. ...
doi:10.3126/njmathsci.v1i0.34159
fatcat:2ogod6tmwzcttecizz3hvd4dwy
The 3x+1 Problem: An Annotated Bibliography, II (2000-2009)
[article]
2012
arXiv
pre-print
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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. ...
arXiv:math/0608208v6
fatcat:a3wdbqlvz5hqfimoxzpjmwbdyi
The 3x+1 problem: An annotated bibliography (1963--1999) (sorted by author)
[article]
2011
arXiv
pre-print
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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) . ...
arXiv:math/0309224v13
fatcat:ayplvta5vbamriea77ps5i4zhy
On consecutive numbers of the same height in the Collatz problem
1993
Discrete Mathematics
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., 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] . ...
doi:10.1016/0012-365x(93)90240-t
fatcat:fvym54nrj5edzgaavhkmxi4ifu
Stochastic Models for the 3x+1 and 5x+1 Problems
[article]
2009
arXiv
pre-print
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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. ...
arXiv:0910.1944v1
fatcat:zivkze7tqbehve6ezllocd6cbu
The Collatz conjecture. A case study in mathematical problem solving
2005
Logic and Logical Philosophy
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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 ...
doi:10.12775/llp.2005.002
fatcat:jndw4v3movevrifbla6fz3mfdm
The intricate labyrinth of Collatz sequences
[article]
2016
arXiv
pre-print
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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. ...
arXiv:1602.01617v1
fatcat:gyzg2zk6m5c2vmncg7hihblkym
A Graph Theoretical Approach to the Collatz Problem
[article]
2021
arXiv
pre-print
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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 . ...
arXiv:1905.07575v5
fatcat:ph7of7rmijff3kfiuxsazj737i
On the Distribution of the First Point of Coalescence for some Collatz Trajectories
[article]
2020
arXiv
pre-print
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2020 pre-print arXiv:2005.09456v1 fatcat:3yaudmtiundhzftak36vunkrci
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]. ...
arXiv:2005.09456v2
fatcat:iecwdu4cfrfkne7dbcucnlxhqe
Page 4443 of Mathematical Reviews Vol. , Issue 86j
[page]
1986
Mathematical Reviews
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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
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(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]
2015
arXiv
pre-print
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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. ...
arXiv:1511.09141v1
fatcat:tklsaeo7gzfadmmdeln2v3cika
Page 2760 of Mathematical Reviews Vol. , Issue 99d
[page]
1991
Mathematical Reviews
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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]
2021
arXiv
pre-print
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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: ...
arXiv:2103.03100v1
fatcat:kpeu5wcjfjdnjmcnc6baxguljq
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