Next Article in Journal
Study of Indoor Ventilation Based on Large-Scale DNS by a Domain Decomposition Method
Next Article in Special Issue
Nonlinear Dynamic Modelling of Two-Point and Symmetrically Supported Pipeline Brackets with Elastic-Porous Metal Rubber Damper
Previous Article in Journal
Dynamic Soft Sensor Development for Time-Varying and Multirate Data Processes Based on Discount and Weighted ARMA Models
Previous Article in Special Issue
About the Orbit Structure of Sequences of Maps of Integers
Open AccessArticle

An E-Sequence Approach to the 3x + 1 Problem

Faculty of Science, Zhejiang Sci-Tech University, Hangzhou 310018, China
Symmetry 2019, 11(11), 1415;
Received: 17 October 2019 / Revised: 6 November 2019 / Accepted: 12 November 2019 / Published: 15 November 2019
(This article belongs to the Special Issue Symmetry and Dynamical Systems)
For any odd positive integer x, define ( x n ) n 0 and ( a n ) n 1 by setting x 0 = x ,   x n = 3 x n 1 + 1 2 a n such that all x n are odd. The 3 x + 1 problem asserts that there is an x n = 1 for all x. Usually, ( x n ) n 0 is called the trajectory of x. In this paper, we concentrate on ( a n ) n 1 and call it the E-sequence of x. The idea is that we generalize E-sequences to all infinite sequences ( a n ) n 1 of positive integers and consider all these generalized E-sequences. We then define ( a n ) n 1 to be Ω -convergent to x if it is the E-sequence of x and to be Ω -divergent if it is not the E-sequence of any odd positive integer. We prove a remarkable fact that the Ω -divergence of all non-periodic E-sequences implies the periodicity of ( x n ) n 0 for all x 0 . The principal results of this paper are to prove the Ω -divergence of several classes of non-periodic E-sequences. Especially, we prove that all non-periodic E-sequences ( a n ) n 1 with lim ¯ n b n n > log 2 3 are Ω -divergent by using Wendel’s inequality and the Matthews and Watts’ formula x n = 3 n x 0 2 b n k = 0 n 1 ( 1 + 1 3 x k ) , where b n = k = 1 n a k . These results present a possible way to prove the periodicity of trajectories of all positive integers in the 3 x + 1 problem, and we call it the E-sequence approach. View Full-Text
Keywords: 3x + 1 problem; E-sequence approach; Ω-divergence of non-periodic E-sequences; Wendel’s inequality 3x + 1 problem; E-sequence approach; Ω-divergence of non-periodic E-sequences; Wendel’s inequality
MDPI and ACS Style

Wang, S. An E-Sequence Approach to the 3x + 1 Problem. Symmetry 2019, 11, 1415.

Show more citation formats Show less citations formats
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map by Country/Region

Back to TopTop