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

Faculty of Science, Zhejiang Sci-Tech University, Hangzhou 310018, China

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 ${({x}_{n})}_{n\u2a7e0}$ and ${({a}_{n})}_{n\u2a7e1}$ by setting ${x}_{0}=x,{x}_{n}=\frac{3{x}_{n-1}+1}{{2}^{{a}_{n}}}$ such that all ${x}_{n}$ are odd. The $3x+1$ problem asserts that there is an ${x}_{n}=1$ for all x. Usually, ${({x}_{n})}_{n\u2a7e0}$ is called the trajectory of x. In this paper, we concentrate on ${({a}_{n})}_{n\u2a7e1}$ and call it the E-sequence of x. The idea is that we generalize E-sequences to all infinite sequences ${({a}_{n})}_{n\u2a7e1}$ of positive integers and consider all these generalized E-sequences. We then define ${({a}_{n})}_{n\u2a7e1}$ to be $\mathsf{\Omega}$ -convergent to x if it is the E-sequence of x and to be $\mathsf{\Omega}$ -divergent if it is not the E-sequence of any odd positive integer. We prove a remarkable fact that the $\mathsf{\Omega}$ -divergence of all non-periodic E-sequences implies the periodicity of ${({x}_{n})}_{n\u2a7e0}$ for all ${x}_{0}$ . The principal results of this paper are to prove the $\mathsf{\Omega}$ -divergence of several classes of non-periodic E-sequences. Especially, we prove that all non-periodic E-sequences ${({a}_{n})}_{n\u2a7e1}$ with $\underset{n\to \infty}{\overline{lim}}}\frac{{b}_{n}}{n}>{log}_{2}3$ are $\mathsf{\Omega}$ -divergent by using Wendel’s inequality and the Matthews and Watts’ formula ${x}_{n}=\frac{{3}^{n}{x}_{0}}{{2}^{{b}_{n}}}{\displaystyle \prod _{k=0}^{n-1}}(1+\frac{1}{3{x}_{k}})$ , where ${b}_{n}={\displaystyle \sum _{k=1}^{n}}{a}_{k}$ . These results present a possible way to prove the periodicity of trajectories of all positive integers in the $3x+1$ problem, and we call it the E-sequence approach.
View Full-Text

*Keywords:*3

*x*+ 1 problem; E-sequence approach; Ω-divergence of non-periodic E-sequences; Wendel’s inequality

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

**MDPI and ACS Style**

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

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.