A Reduced Collatz Dynamics Maps to a Residue Class, and Its Count of x/2 over the Count of 3∗x+1 Is Larger than ln3/ln2

Wei Ren

doi:10.1155/2020/5946759

Abstract

We propose reduced Collatz conjecture and prove that it is equivalent to Collatz conjecture but more primitive due to reduced dynamics. We study reduced dynamics (that consists of occurred computations from any starting integer to the first integer less than it) because it is the component of original dynamics (from any starting integer to 1). Reduced dynamics is denoted as a sequence of “I” that represents (3∗x+1)/2 and “O” that represents x/2. Here, 3∗x+1 and x/2 are combined together because 3∗x+1 is always even and thus followed by x/2. We discover and prove two key properties on reduced dynamics: (1) Reduced dynamics is invertible. That is, given reduced dynamics, a residue class that presents such reduced dynamics can be computed directly by our derived formula. (2) Reduced dynamics can be constructed algorithmically, instead of by computing concrete 3∗x+1 and x/2 step by step. We discover the sufficient and necessary condition that guarantees a sequence consisting of “I” and “O” to be a reduced dynamics. Counting from the beginning of a sequence, if and only if the count of x/2 over the count of 3∗x+1 is larger than ln3/ln2, reduced dynamics will be obtained (i.e., current integer will be less than starting integer).

Highlights

Preliminaries[0]2 {x | x ≡ 0 mod 2, x ∈ N∗} (4) [i]m {x | x ≡ i mod m, x ∈ N∗}, m ≥ 2, m ∈ N∗, 0 ≤
Introduction eCollatz conjecture can be stated as follows
Ren [5] proposed to use a tree-based graph to observe two key inner properties in reduced Collatz dynamics: one is ratio of the count of (x/2) over the count of 3∗ x + 1, and the other is partition

Summary

Preliminaries

[0]2 {x | x ≡ 0 mod 2, x ∈ N∗} (4) [i]m {x | x ≡ i mod m, x ∈ N∗}, m ≥ 2, m ∈ N∗, 0 ≤. (1) We call an ordered sequence fq ∈ {I, O}q in the above proof as original dynamics (referring to fq(x) 1), which consists of q occurred Collatz transformations during the computing procedure from a starting integer to 1. (4) speaking, this function can obtain the Collatz transforms from i to i + j − 1 from a given inputting transform sequence (e.g., reduced dynamics) in terms of s ∈ {I, O}|s|. Given starting integer x ∈ [3]4 (i.e., x 4t + 3, t ∈ N), the count of consecutive “I” (denoted as p) is determined by t as follows. (1) Note that, due to Corollary 1, p for Ip in RD[x] can be computed by t ((x − 3)/4) and log2((t + 1)/A) directly without conducting concrete Collatz transformations, which can shorten the computation delay for reduced dynamics.

Preparation

Conclusion