Determining monotonic step-equal sequences of any limited length in the Collatz problem

Abstract

This paper proposes a formula expression for the well-known Collatz conjecture (or 3x+1 problem), which can pinpoint all the growth points in the orbits of the Collatz map for any natural numbers. The Collatz map $Col: \mathcal{N}+1 \rightarrow \mathcal{N}+1$ on the positive integers is defined as $x_{n+1}=Col(x_n)=(3 x_n +1)/2^{m_n}$ where $x_{n+1}$ is always odd and $m_n$ is the step size required to eliminate any possible even values. The Collatz orbit for any positive integer, $x_1$, is expressed by a sequence, $<x_1$; $x_2\doteq Col(x_1)$; $\cdots$ $x_{n+1}\doteq Col(x_n)$; $\cdots>$ and $x_n$ is defined as a growth point if $Col(x_n)>x_n$ holds, and we show that every growth point is in a format of ``$4y+3$'' where $y$ is any natural number. Moreover, we derive that, for any given positive integer $n$, there always exists a natural number, $x_1$, that starts a monotonic increasing or decreasing Collatz sequence of length $n$ with the same step size. For any given positive integer $n$, a class of orbits that share the same orbit rhythm of length $n$ can also be determined.