dc.contributor.author Batang, Zenon B. dc.date.accessioned 2019-12-24T12:37:49Z dc.date.available 2019-12-24T12:37:49Z dc.date.issued 2019-07-14 dc.identifier.uri http://hdl.handle.net/10754/660813 dc.description.abstract The Collatz conjecture asserts that repeatedly iterating $f(x) = (3x + 1)/2^{a(x)}$, where $a(x)$ is the highest exponent for which $2^{a(x)}$ exactly divides $3x+1$, always lead to $1$ for any odd positive integer $x$. Here, we present an arborescence graph constructed from iterations of $g(x) = (2^{e(x)}x - 1)/3$, which is the inverse of $f(x)$ and where $x \not \equiv [0]_3$ and $e(x)$ is any positive integer satisfying $2^{e(x)}x - 1 \equiv [0]_3$, with $[0]_3$ denoting $0\pmod{3}$. The integer patterns inferred from the resulting arborescence provide new insights into proving the validity of the conjecture. dc.description.sponsorship Critical remarks from an anonymous reviewer of an earlier version of the manuscript substantially improved the approach presented in this paper. The support from KAUST Core Labs is deeply appreciated. dc.publisher arXiv dc.relation.url https://arxiv.org/pdf/1907.07088 dc.rights Archived with thanks to arXiv dc.title Integer patterns in Collatz sequences dc.type Preprint dc.contributor.department Research Support dc.eprint.version Pre-print dc.identifier.arxivid 1907.07088 kaust.person Batang, Zenon B. refterms.dateFOA 2019-12-24T12:38:04Z kaust.acknowledged.supportUnit Core Labs kaust.acknowledged.supportUnit KAUST Core Lab
﻿

### Files in this item

Name:
Preprintfile1.pdf
Size:
387.4Kb
Format:
PDF
Description:
Pre-print