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

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

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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 languageEnglish
Article number2008.13643
Pages (from-to)1-14
Number of pages14
Early online date27 Oct 2020
Publication statusPublished - 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)


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


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