Reduced Collatz Dynamics is Periodical and the Period Equals 2 to the Power of the Count of x/2

TL;DR

This paper proves that the reduced Collatz dynamics are periodic with a period of 2 raised to the number of x/2 steps, linking the existence of reduced dynamics of x to that of x+P, and proposing a new approach to verify the Collatz conjecture.

Contribution

The paper introduces a periodical property of reduced Collatz dynamics, establishing a direct relation between the dynamics of x and x+P, and proposes a new verification method for the Collatz conjecture.

Findings

Reduced dynamics are periodic with period 2^L, where L is the count of x/2 steps.

Existence of reduced dynamics for x implies existence for x+P, with P=2^L.

Verifying reduced dynamics for all x could prove the Collatz conjecture.

Abstract

In this paper, we prove that reduced dynamics on Collatz conjecture is periodical, and its period equals 2 to the power of the count of x/2 computation in the reduced dynamics. More specifically, if there exists reduced dynamics of x (that is, start from an integer x and the computation will go to an integer less than x), then there must also exist reduced dynamics of x+P (that is, if starting from an integer x+P, then computation will go to an integer less than x+P), where P equals 2 to the power of L, and L is the total count of x/2 computations (i.e., computational times) in the reduced dynamics of x (note that, equivalently, L is also the length of the reduced dynamics of x). Therefore, the power (or output) of this period property, which is discovered and proved in this paper, is - the study of the existence of reduced dynamics of x will result in the existence of reduced dynamics of x+P (and iteratively x+n*P, n is a positive integer). Hence, only partition of integers needs to be verified for the existence of their reduced dynamics. Finally, if any starting integer x can be verified for the existence of its reduced dynamics, then Collatz Conjecture will be True (due to our proposed Reduced Collatz Conjecture).