For any odd positive integer
x, define
and
by setting
such that all
are odd. The
problem asserts that there is an
for all
x. Usually,
is called the trajectory of
x. In this paper, we concentrate on
and call it the E-sequence of
x. The idea is that we generalize E-sequences to all infinite sequences
of positive integers and consider all these generalized E-sequences. We then define
to be
-convergent to
x if it is the E-sequence of
x and to be
-divergent if it is not the E-sequence of any odd positive integer. We prove a remarkable fact that the
-divergence of all non-periodic E-sequences implies the periodicity of
for all
. The principal results of this paper are to prove the
-divergence of several classes of non-periodic E-sequences. Especially, we prove that all non-periodic E-sequences
with
are
-divergent by using Wendel’s inequality and the Matthews and Watts’ formula
, where
. These results present a possible way to prove the periodicity of trajectories of all positive integers in the
problem, and we call it the E-sequence approach.
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