An E-Sequence Approach to the 3x1 Problem+

Abstract

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.