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On heights in the Collatz 3n + 1 problem

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

2020
*
Nepal Journal of Mathematical Sciences
*

We briefly discuss some basic facts and results of

doi:10.3126/njmathsci.v1i0.34159
fatcat:2ogod6tmwzcttecizz3hvd4dwy
*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]

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]

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

1993
*
Discrete Mathematics
*

.,

doi:10.1016/0012-365x(93)90240-t
fatcat:fvym54nrj5edzgaavhkmxi4ifu
*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]

2009
*
arXiv
*
pre-print

This paper discusses stochastic models for predicting

arXiv:0910.1944v1
fatcat:zivkze7tqbehve6ezllocd6cbu
*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

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]

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]

2021
*
arXiv
*
pre-print

Andrei et al. have shown

arXiv:1905.07575v5
fatcat:ph7of7rmijff3kfiuxsazj737i
*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]

2020
*
arXiv
*
pre-print

This paper is a numerical evaluation of some trajectories of

arXiv:2005.09456v2
fatcat:iecwdu4cfrfkne7dbcucnlxhqe
*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]

2015
*
arXiv
*
pre-print

Pairs of consecutive integers have

arXiv:1511.09141v1
fatcat:tklsaeo7gzfadmmdeln2v3cika
*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]

2021
*
arXiv
*
pre-print

Conceptually,

arXiv:2103.03100v1
fatcat:kpeu5wcjfjdnjmcnc6baxguljq
*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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