The Collatz tree as a Hilbert hotel: a proof of the 3x + 1 conjecture

Jan Kleinnijenhuis; Alissa M. Kleinnijenhuis; Mustafa G. Aydogan

The Collatz tree as a Hilbert hotel: a proof of the 3x + 1 conjecture

Jan Kleinnijenhuis
, Alissa M. Kleinnijenhuis
, Mustafa G. Aydogan

Research output: Contribution to Journal › Article › Academic

754 Downloads (Pure)

Abstract

The yet unproven Collatz conjecture maintains that repeatedly connecting even numbers n to n/2, and odd n to 3n + 1, connects all natural numbers by a unique root path to the Collatz tree with 1 as its root. The Collatz tree proves to be a Hilbert hotel. Numbers divisible by 2 or 3 depart. An infinite binary tree remains with one upward and one rightward child per number. Rightward numbers, and infinitely many generations of their upward descendants, each with a well-defined root path, depart thereafter. The Collatz tree is a Hilbert hotel because still higher upward descendants keep descending to all unoccupied nodes. The density of already departed numbers comes nevertheless arbitrarily close to 100% of the natural numbers. The latter proves the Collatz conjecture.

Original language	English
Article number	2008.13643
Pages (from-to)	1-14
Number of pages	14
Journal	arXiv
Early online date	27 Oct 2020
Publication status	Published - 17 Jan 2021

Bibliographical note

Version 3: density=1 proof simplified, Figures Collatz trees annotated with number subsets (2 theorems, 3 figures of Collatz trees, 1 table, 11 pages, 67 references). Ancillary Materials: Mathematica notebook (.nb), its PDF(.pdf), Glossary Notation and Elaboration (.pdf)

Keywords

math.GM
11B37, 11B50, 11B05, 11F06, 05C05, 05C38, 05C85
G.2.2

Access to Document

2008.13643v3Final published version, 416 KB

https://arxiv.org/abs/2008.13643

Persistent URL (handle)

Persistent link

Cite this

@article{ff30e3ac52bb4c82a6fddda4de8da060,

title = "The Collatz tree as a Hilbert hotel: a proof of the 3x + 1 conjecture",

abstract = "The yet unproven Collatz conjecture maintains that repeatedly connecting even numbers n to n/2, and odd n to 3n + 1, connects all natural numbers by a unique root path to the Collatz tree with 1 as its root. The Collatz tree proves to be a Hilbert hotel. Numbers divisible by 2 or 3 depart. An infinite binary tree remains with one upward and one rightward child per number. Rightward numbers, and infinitely many generations of their upward descendants, each with a well-defined root path, depart thereafter. The Collatz tree is a Hilbert hotel because still higher upward descendants keep descending to all unoccupied nodes. The density of already departed numbers comes nevertheless arbitrarily close to 100\% of the natural numbers. The latter proves the Collatz conjecture. ",

keywords = "math.GM, 11B37, 11B50, 11B05, 11F06, 05C05, 05C38, 05C85, G.2.2",

author = "Jan Kleinnijenhuis and Kleinnijenhuis, \{Alissa M.\} and Aydogan, \{Mustafa G.\}",

note = "Version 3: density=1 proof simplified, Figures Collatz trees annotated with number subsets (2 theorems, 3 figures of Collatz trees, 1 table, 11 pages, 67 references). Ancillary Materials: Mathematica notebook (.nb), its PDF(.pdf), Glossary Notation and Elaboration (.pdf)",

year = "2021",

month = jan,

day = "17",

language = "English",

pages = "1--14",

journal = "arXiv",

issn = "2331-8422",

publisher = "Cornell University",

}

TY - JOUR

T1 - The Collatz tree as a Hilbert hotel

T2 - a proof of the 3x + 1 conjecture

AU - Kleinnijenhuis, Jan

AU - Kleinnijenhuis, Alissa M.

AU - Aydogan, Mustafa G.

N1 - Version 3: density=1 proof simplified, Figures Collatz trees annotated with number subsets (2 theorems, 3 figures of Collatz trees, 1 table, 11 pages, 67 references). Ancillary Materials: Mathematica notebook (.nb), its PDF(.pdf), Glossary Notation and Elaboration (.pdf)

PY - 2021/1/17

Y1 - 2021/1/17

N2 - The yet unproven Collatz conjecture maintains that repeatedly connecting even numbers n to n/2, and odd n to 3n + 1, connects all natural numbers by a unique root path to the Collatz tree with 1 as its root. The Collatz tree proves to be a Hilbert hotel. Numbers divisible by 2 or 3 depart. An infinite binary tree remains with one upward and one rightward child per number. Rightward numbers, and infinitely many generations of their upward descendants, each with a well-defined root path, depart thereafter. The Collatz tree is a Hilbert hotel because still higher upward descendants keep descending to all unoccupied nodes. The density of already departed numbers comes nevertheless arbitrarily close to 100% of the natural numbers. The latter proves the Collatz conjecture.

AB - The yet unproven Collatz conjecture maintains that repeatedly connecting even numbers n to n/2, and odd n to 3n + 1, connects all natural numbers by a unique root path to the Collatz tree with 1 as its root. The Collatz tree proves to be a Hilbert hotel. Numbers divisible by 2 or 3 depart. An infinite binary tree remains with one upward and one rightward child per number. Rightward numbers, and infinitely many generations of their upward descendants, each with a well-defined root path, depart thereafter. The Collatz tree is a Hilbert hotel because still higher upward descendants keep descending to all unoccupied nodes. The density of already departed numbers comes nevertheless arbitrarily close to 100% of the natural numbers. The latter proves the Collatz conjecture.

KW - math.GM

KW - 11B37, 11B50, 11B05, 11F06, 05C05, 05C38, 05C85

KW - G.2.2

M3 - Article

SN - 2331-8422

SP - 1

EP - 14

JO - arXiv

JF - arXiv

M1 - 2008.13643

ER -

The Collatz tree as a Hilbert hotel: a proof of the 3x + 1 conjecture

Abstract

Bibliographical note

Keywords

Access to Document

Persistent URL (handle)

Fingerprint

Cite this