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

arXiv1909.13218MaRDI QIDQ6326178

Author name not available (Why is that?)

Publication date: 29 September 2019

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

C o l : m a t h c a l N + 1 i g h t a r r o w m a t h c a l N + 1

on the positive integers is defined as

x_{n + 1} = C o l (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} d o t e q C o l (x_{1})

;

c d o t s

x_{n + 1} d o t e q C o l (x_{n})

;

c d o t s >

and

x_{n}

is defined as a growth point if

C o l (x_{n}) > x_{n}

holds, and we show that every growth point is in a format of ``

4 y + 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.

Has companion code repository: https://github.com/lilj999/MonotonicCollatz

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