A generalization of Everett's result on the Collatz \(3x+1\) problem
type: Item
instance of: scholarly article
title: A generalization of Everett's result on the Collatz \(3x+1\) problem
zbMATH Open document ID: 0648.10009
DOI: 10.1016/0196-8858(87)90018-2
author: S. H. Smith
published in: Advances in Applied Mathematics
publication date: 1987-01-01T00:00:00Z
review text: For the problem of the title \textit{C. J. Everett} [Adv. Math. 25, 42-45 (1977; Zbl 0352.10001)] has proved that almost every natural number m, \(m>1\), has an iterate less than m. The author generalizes this result as follows: Let N be a natural number, x a real number and \(\pi_ k(x)=card\{m\leq x\); m a natural number and for some natural number n, \(m_{n+p}<m\), \(0\leq p\leq k-1\}\). Then \(\pi_ k(x)/x\to 1\) as \(x\to \infty\).
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 11A63
zbMATH DE Number: 4057597
zbMATH Keywords: Collatz problem
zbMATH Keywords: \(3x+1\) conjecture
zbMATH Keywords: density
reviewed by: Ray Phillip Steiner
MaRDI profile type: Publication
full work available at URL: 0196-8858(87)90018-2
OpenAlex ID: W1976649846
cites work: Q5563135
cites work: Iteration of the number-theoretic function \(f(2n)=n\), \(f(2n+1)=3n+2\)
cites work: On the Collatz 3n + 1 Algorithm
cites work: The 3x + 1 Problem and Its Generalizations
cites work: Q3929795
cites work: A stopping time problem on the positive integers
cites work: On the existence of a density
description: scientific article; zbMATH DE number 4057597