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

Faculty of Science, Zhejiang Sci-Tech University, Hangzhou 310018, China
Symmetry 2019, 11(11), 1415; https://doi.org/10.3390/sym11111415
Received: 17 October 2019 / Revised: 6 November 2019 / Accepted: 12 November 2019 / Published: 15 November 2019
For any odd positive integer x, define $( x n ) n ⩾ 0$ and $( a n ) n ⩾ 1$ by setting 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