## Reduced Collatz Dynamics is Periodical and the Period equals 2 to the power of the count of x/2

preprint

posted on 28.01.2020, 03:36 by Wei Ren#### preprint

Preprints are manuscripts made publicly available before they have been submitted for formal peer review and publication. They might contain new research findings or data. Preprints can be a draft or final version of an author's research but must not have been accepted for publication at the time of submission.

We propose Reduced Collatz Conjecture that is equivalent to Collatz

Conjecture but is easier to explore, because reduced dynamics is

more primitive than original dynamics and presents better structures

(e.g., period and ratio). Reduced dynamics (that are occurred

computation sequence from a starting integer to the first integer

less than the starting integer) is the component of original

dynamics (from a starting integer to 1). Reduced dynamics of x is

represented by a sequence of computation that is either (3*x+1)/2 or

x/2, because 3*x+1 is always even and followed by x/2. We prove that

reduced dynamics is periodical and its period equals 2 to the power

of the count of x/2. More specifically, if there exists reduced

dynamics of x, then there exists reduced dynamics of x+P, where P

equals $2^L$ and L is the total count of x/2 computations in reduced

dynamics of x (equivalently, L is the length of the sequence).