An E-sequence approach to the 3x + 1 problem

For any odd positive integer $x$, define $(x_n{)}_{n⩾0}$ and $(a_n ){n\geqslant 1} $bysetting$x{0}=x, ,, x_n =\cfrac{3x_{n-1} +1}{2^{a_n }}$suchthatall$x_n $areodd\mathrm{.}The3x+1problemassertsthatthereisan$x_n =1$forall$x$\mathrm{.}Usually,$(x_n ){n\geqslant 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 sequence $(a_n{)}_{n⩾1}$ of positive integers and consider all these generalized E-sequences. We then define $(a_n ){n\geqslant 1} $tobe$\Omega-$convergentto$x$ifitistheE-sequenceof$x$andtobe$\Omega-$divergent if it is not the E-sequence of any odd positive integer. We prove a remarkable fact that the $\mathrm{\Omega }-$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 $\mathrm{\Omega }-$divergence of several classes of non-periodic E-sequences. Especially, we prove that all non-periodic E-sequences $(a_n{)}_{n⩾1}$ with $\mathop {\overline {\lim } }\limits_{n\to \infty } \cfrac{b_n }{n}>\log 23$are$\Omega-$divergent by using the Wendel’s inequality and the Matthews and Watts’s formula $x_n =\cfrac{3^n x_0 }{2^{b_n }}\prod\limits{k=0}^{n-1} {(1+\cfrac{1}{3x_k })} $,where$b_n =\sum\limits_{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.