Journal of Mathematics

Volume 2019, Article ID 6129836, 12 pages

https://doi.org/10.1155/2019/6129836

## A New Approach on Proving Collatz Conjecture

^{1}School of Computer Science, China University of Geosciences, Wuhan 430074, China^{2}Hubei Key Laboratory of Intelligent Geo-Information Processing, China University of Geosciences, Wuhan, China^{3}Guizhou Provincial Key Laboratory of Public Big Data, GuiZhou University, Guizhou, China

Correspondence should be addressed to Wei Ren; nc.ude.guc@scneriew

Received 4 January 2019; Revised 20 February 2019; Accepted 17 March 2019; Published 2 May 2019

Academic Editor: Basil K. Papadopoulos

Copyright © 2019 Wei Ren. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Collatz Conjecture (3x+1 problem) states any natural number x will return to 1 after 3x+1 computation (when x is odd) and x/2 computation (when x is even). In this paper, we propose a new approach for possibly proving Collatz Conjecture (CC). We propose Reduced Collatz Conjecture (RCC)—any natural number x will return to an integer that is less than x. We prove that RCC is equivalent to CC. For proving RCC, we propose exploring laws of Reduced Collatz Dynamics (RCD), i.e., from a starting integer to the first integer less than the starting integer. RCC can also be stated as follows: RCD of any natural number exists. We prove that RCD is the components of original Collatz dynamics (from a starting integer to 1); i.e., RCD is more primitive and presents better properties. We prove that RCD presents unified structure in terms of (3x+1)/2 and x/2, because 3x+1 is always followed by x/2. The number of forthcoming (3x+1)/2 computations can be determined directly by inputting x. We propose an induction method for proving RCC. We also discover that some starting integers present RCD with short lengths no more than 7. Hence, partial natural numbers are proved to guarantee RCC in this paper, e.g., 0 module 2; 1 module 4; 3 module 16; 11 or 23 module 32; 7, 15, or 59 module 128. The future work for proving CC can follow this direction, to prove that RCD of left portion of natural numbers exists.

#### 1. Introduction

The Collatz Conjecture is a mathematical conjecture that is first proposed by Lothar Collatz in 1937. It is also known as the conjecture, the Ulam conjecture, the Kakutani’s problem, the Thwaites conjecture, or the Syracuse problem [1–3]. “Mathematics may not be ready for such problems”, Paul Erdos once speculated about the Collatz Conjecture [4].

The conjecture can be stated simply as follows: take any positive integer number . If is even, divide it by to get . If is odd, multiply it by and add to get . Repeat the process again and again. The Collatz Conjecture is that no matter what the number (i.e., ) is taken, the process will always eventually reach . The longest progressions for initial starting numbers of less than 10 billion and 100 quadrillion are calculated by Gary T. Leavens [5] and R. E. Crandall [6], respectively. Wei Ren et al. verified can return to 1 after 481603 times of computation, and 863323 times of computation, which is the largest integer being verified in the world [7]. So far no one has tried to figure out whether all of the positive numbers eventually reach one, but we know that most of them do so. In particular, Krasikov and Lagarias proved that the number of integers finally reaching one in the interval is at least proportional to x 0.84 [8].

The paper is organized as follows: Section 2 presents our Reduced Collatz Conjecture. A mathematical induction for proving Collatz Conjecture is proposed in Section 3. Section 4 introduces for representing reduced dynamics of and explores its properties. Section 5 studies all starting numbers whose lengths of reduced dynamics are no more than 7. Finally, Section 6 concludes the paper.

*Notations*

(1): the set of integers.(2).(3)(4)(5)(6), (7): Collatz transformation. or according to the parity of current inputting ; or according to the parity of current inputting .(8): when .(9): when .(10): Collatz Conjecture.(11): Reduced Collatz Conjecture.(12): is Returnable.(13): , , , (14): reduced dynamics or code for .(15)(16): the length of ; e.g., if a code consists of or , it will be the number of in the ordered sequence .(17): concatenation of and . , or , , .(18) returns the number of elements in a set .(19), returns the maximal value in a set , and the larger one in , respectively.(20)(21): current transformed number.

#### 2. Reduced Collatz Conjecture

*Definition 1 (Collatz transformation). *

can be simply denoted as , and can be simply denoted as .

We assume Let

*Definition 2 (Collatz Conjecture ()). *, , such that .

When , is held (i.e., ). Thus, , . . More specifically, here corresponding for is an* ordered sequence *. Here “ordered sequence” implies that the parity of determines whether the intermediately forthcoming is or

In the following, we mainly are concerned with

We give the Reduced Collatz Conjecture as follows.

*Definition 3 (Reduced Collatz Conjecture ()). *, , such that and .

That is, the minimal such that is of interest, since

Proposition 4. *.*

*Proof. *, and it is obvious that , i.e.,

(1) Suppose is true. That is, , , . Thus, is true.

(2) Inversely, suppose is true. That is, , , .

If , then is true.

If , then let . As is true, ,

Let Iteratively, if , then is true. If , then , .

Thus, . is a* strictly* decreasing serial.

Besides,

Therefore, after finite times of iterations, , .

That is, ,

Thus, is true.

*Remark 5. *(1) is called starting number, and after transformation (e.g., ) is called transformed number.(2)We call an ordered sequence in above proof as* original dynamics* (as ). Simply speaking, original dynamics of a starting number is represented by (or composed of) a serial of occurred Collatz transformations during the procedure from the starting number (i.e., ) to 1. For example, the dynamics of (i.e., occurred transformations during the procedure from 5 to 1) is . That is, original dynamics of 5 is .(3)In contrast, we call in the above proof as* reduced dynamics* (as ). Simply speaking, reduced dynamics of a starting number is represented by (or composed of) a serial of occurred Collatz transformations during the procedure from the starting number (i.e., ) to the* first* transformed number that is less than the starting number (i.e., ). For example, the reduced dynamics of 5 (i.e., occurred transformations during the procedure from 5 to the first transformed number less than the starting number, namely, 4) is , in other words, .

Note that reduced dynamics is more primitive than original dynamics, because original dynamics consists of reduced dynamics. It can be easily observed from the proof of Proposition 4. Nonetheless, we formally prove it in Proposition 6 as follows.

Proposition 6. *, if such that ; then, , such that and *

*Proof. *(1) and ; thus, such that .

(2) Let . If , then such that . Thus, similarly to (1), such that .

(3) Iteratively, compute in the above way. Thus, , , ,..., is a strictly decreasing serial. Besides, , thus , Therefore, and ,

Due to above proposition, we concentrate on reduced dynamics, which is a component of original dynamics.

#### 3. Induction

This section is not preliminary for the rest of the paper, but it presents a formal mathematical induction related to Reduced Collatz Conjecture.

To simplify the statement for conjecture, we define “*Returnable*” as follows.

*Definition 7 (Returnable). * is Returnable (denoted as ), if and only if such that .

The Collatz Conjecture will be true, if the following mathematical induction can be proved.

*Induction (for Collatz Conjecture)*

(1) (recall that ).(2)If (where ) is Returnable, then will be Returnable. That is, if (where ), then can be proved.

In shorthand, the induction is as follows. where

Therefore, we* only* need to check whether current transformed number is less than designated starting number. Once current transformed number is less than the starting number, the starting number will be Returnable (i.e., ) due to the induction assumption ().

Figure 1 illustrates the rationale in our induction.