Journal of Mathematics
Volume 2019 (2019), Article ID 6129836, 12 pages
https://doi.org/10.1155/2019/6129836

Research Article

A New Approach on Proving Collatz Conjecture

Wei Ren1,2,3

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

Correspondence should be addressed to Wei Ren; weirencs@cug.edu.cn

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.

Figure 1: Induction rationale. Once current transformed number is less than the starting number, the starting number will be called “Returnable”. That is, once , then

Proposition 8. If the induction (especially, Step 2) can be proved, Collatz Conjecture is True.

Proof. Straightforward.

Besides, it is trivial to check that .

If is odd in the induction, the induction is trivial to be proved. We state it as a proposition as follows.

Proposition 9.

where

Proof. , thus is even. That is, when , . , so . Thus, .

Therefore, for the proof of induction we only need to prove the case that is even.

If in the induction is even with , the induction is straightforward to be proved. We state it as a proposition as follows.

Proposition 10.

where

Proof. Thus, . Next, let us check whether . , , . As is Returnable, . Thus, .

Therefore, we only need to prove the case that is even with in the induction due to the above Propositions 9 and 10. We give reduced version of induction as follows.

Induction (Reduced Version of Induction for Collatz Conjecture)

(1) (Straightforward).(2)If () is Returnable, then will be Returnable. That is, if (), then can be proved.(3)If () is Returnable, then will be Returnable. That is, if (), then can be proved.(4)If () is Returnable, then will be Returnable. That is, if (), then can be proved.

As Steps (2) and (3) can be proved by Propositions 9 and 10, respectively. In shorthand, the reduced version of induction for Collatz Conjecture that needs to be proved is only Step (4) as follows:

where .

4. CODE(x) and Its Properties

Theorem 11. always follows after in .

Proof. In the definition of , when , , which is even obviously. Thus, next must be consequently. Therefore, always follows after

Therefore, we introduce new notations (i.e., and ) for simplicity.

(1) always occurs after ; thus, can be written together and denoted as . That is, .(2) is used to denote (for better vision contrastively).(3) and may also be called Collatz transformations.

For example, reduced dynamics of are . The transformation procedures are , , and . It can also be simplified as and . Thus, reduced dynamics of can be written as or “” in short.

Besides, can be simply written as . That is, , where is a composite function, e.g., Formally, , where

Definition 12. , , if such that and , where , then let and is called
code (or reduced dynamics) for , denoted as .

Note that is an ordered sequence consisting of and Besides, Recall that Furthermore, this sequence implicitly matches the parity of all intermediate transformed numbers that are taken as input of .

For example, implies the following results:

(1);(2);(3)“” is due to ;(4)“” is due to .

Theorem 13. , , if , such that and , where , and letting , then is unique.
That is,

Proof (straightforward). , , such that , where ; let Given , either or is deterministic and unique. Similarly, given , is deterministic and unique. (Recall that, the parity of determines the intermediately forthcoming transformation). Thus, is unique for any given .

Remark 14.

(1)We assume , although In other words, the code for is . In the following, we are always concerned with .(2)If is finite for ( returns the length of , or the number of and in the ordered sequence ), then exists; if exists, then is finite.(3)If is true, then , exists; if , exists, then is true.(4)In , is called starting number. are called transformed numbers. is the first transformed number that is less than the starting number . That is, , and Besides, the parity of determines the selection of the intermediately next after .(5)Each includes one computation since . We denote the count of in as . It equals the count of or in the reduced dynamics of . As , roughly speaking, indeed equals the times of “going Up (becoming larger)” in the reduced dynamics of .(6)Each includes one computation, and itself is one computation. We denote the count of in as . It equals the count of or in the reduced dynamics of . As , roughly speaking, indeed equals the times of “going Down (becoming smaller)” in the reduced dynamics of . Note that the count of and in also equals . In other words, equals the length of . That is, .(7)We do not assume the existence of for , which is exactly what needs to be proved in Reduced Collatz Conjecture. We introduce notation for the following reasons:

(i)The presentation will be more convenient.(ii) are outputted by our computer program. We may discover some properties in them by observation, and they will be proved formally.(iii)We can explore inner laws for without the detail of (independent to ).

The following propositions again confirm Propositions 9 and 10.

Proposition 15. ,

Proof. (1) , thus occurs. , thus .
(2) If , (by assumption).
If , , where . Thus, occurs. . , thus further transformation occurs. , thus .
In summary,

(In the following, is shortened as , .)

If exists, they can be looked as a whole - , and presents certain properties.

Proposition 16. If exists, then .

Proof. Letting , . Obviously, .
.
.
(1) If , then . Thus, the next transformation is “”. Thus, the first five Collatz transformations are “” (i.e., “”). Besides, Further transformation thus occurs. Hence, if exists, then
(2) If , then . Thus, the first six Collatz transformations are “” (i.e., “”). Besides, . Further transformation occurs. Hence, if exists, then .
If , then more “” occurs. Obviously, . Further transformation occurs. Hence, if exists, then . If , then Further transformation occurs consequently. Hence, if exists, then .
Suppose . There exists at least one “” in transformations; otherwise, , which contradicts with . Besides, , ; thus, after further transformation occurs.
In summary, if exists, then .

Next corollary states that (or ) presents unified format.

Proposition 17. , where More specifically,

Proof. We here assume and . According to Propositions 15 and 16, we have

It can be written as follows.
(1) as ; as .
(2) when as .
(3) when , as
Thus, (9) can be rewritten as (8).

More specifically, we have the following theorem that give more details on

Theorem 18 (format theorem). , where Besides,

when ; when

Proof. is trivial due to Proposition 17, so we are mainly concerned with .
By Proposition 15 or (7), if , then and . If , then and .
Next, we concentrate on .
, which can be manually and easily verified.
Let .
(1) Case I: . As , is conducted consequently. As and , the checking on whether current transformed number is less than starting number may be omitted in some straightforward cases.
; thus, transformation is conducted consequently.
Thus, is conducted consequently.
Thus, further transformation is conducted consequently.
Therefore, .
(2) Case II: . As , is conducted consequently.
; thus, is conducted consequently.
Thus, is conducted consequently.
It depends on the partition of (more specifically, or ) whether is even or odd.
(It comes from following observations: , )
(2.1) If , (); thus, will occur consequently.
(2.2) If , ; thus, will occur consequently.
Besides, suppose current transformed number is denoted as .
, whose parity depends on the parity of .
, whose parity depends on the parity of .
In other words, the judgement on the parity of is undecidable, unless the domain ( or ) is partitioned further.
For exploring more general results, we put it in another way as follows.
Suppose there exist at most times of “” at code head (i.e., ) for . Observing following equation for after consecutive times of “”:

Note that above computation implicitly includes two requirements due to times of consecutive as follows.
(i) All intermediate transformed numbers during processes (i.e., computing times of consecutive “”) satisfy

where
(ii) Besides,

where , as only (or at most) consecutive occur.
In other words, can also be looked as the minimal value to let current transformed number be in . Thus, we need to explore the requirement on for given such that

We call this requirement (i.e., (14)) as REQ.
Represent as That is, . Obviously, this representation is unique. We thus need to prove that REQ is satisfied if and only if . Note that we will see that here is indeed determined by
For , we have

When , we have exactly

It is easy to see that is the one and only one for REQ, as desired.

Corollary 19 ( determine corollary). Given starting number (i.e., ), the number of consecutive “” (denoted as ) is determined by according to (17) as follows:

Corollary 20. Given starting number , the first times of Collatz transformations must be “” and can be determined by by (17), and the transformed number after transformations is

Proof. It is straightforward due to (11), Theorem 18, and Corollary 19.
Alternatively, only to compute the transformed number , we can prove it by induction as follows.
(1) .

(2) Suppose ; we have

Remark 21. (1) Note that, due to Corollary 19, for in can be computed by and directly without conducting concrete Collatz transformations, which can accelerate the computation of dynamics.
(2) Besides, by (11) or Corollary 20, if , we then have , because , which matches with the result by manually computing.
(3) Indeed, (11) can be extended to include all cases (i.e., for ). If , by assuming , ; if , Therefore, for

Corollary 22. Given starting number , the first times of Collatz transformations must be “” and can be determined by , and the transformed number after transformations is

Proof. It is straightforward due to (11), Theorem 18, and Corollaries 19 and 20.
If , then ,
If , then , due to Corollary 20.

Next corollary states that the head of code for is

Corollary 23.

Proof. Suppose , due to Theorem 18.
(1) , ; thus, due to Theorem 18. Thus, .
Indeed, can be manually computed (recall that )
(2) , .
(2.1) , , or , thus due to Theorem 18. Thus,
Indeed, can be manually computed (recall that )
(2.2) , .   Thus, . Thus, Hence, owing to Theorem 18.
Summarizing (1) and (2),

Example 24. (1) . Thus, .
(2) . Thus, .

Proposition 25. . (That is, must end with “”, or the last transformation in is “”.)

Proof (straightforward). It can be easily understood intuitively. That is, reduced dynamics should be ended by “going Down”, not “going Up” (recall items and in Remark 14).
If ends by “”, suppose , , , .
; thus, . Contradiction occurs.
Put it in another way, if , then Together with , we have . Thus, reduced dynamics of ends after and

Next corollary gives more details on that has a unified form as

That is, each code consists of one or more segments, and each segment has a unified form as .

Corollary 26.

where

Proof (straightforward). , thus occurs. After times of transformations, and thus follows. After times of transformations, , thus occurs. Indeed, can be determined by by ,
Iteratively, each segment has a unified form , where .
The first segment is listed solely, because the distinction between the first segment and the other segments is that but . (In other words, when and only when an intermediate transformed number occurs, . Otherwise, .)

Put it in another way, if intermediate transformed numbers in during reduced dynamics are tackled explicitly, we have the following corollary.

Corollary 27.

where

5. Short Codes -

In this section, we explore and their codes whose length is less than 7, called short codes. The exploration of this section helps build empirical understanding for Sections 3 and 4 (e.g., induction and ). Before the exploration, a lemma is given as preliminaries as follows.

Lemma 28. .
(1) ;
(2) ;
(3) ,

Proof. (1)
=
≡
.
(2)
,
.
(Besides, ⇒)
(3) The proof is the combination of above (1) and (2).
Specifically,

,

Proposition 29. , if

Proof. .
, and Thus, “” occurs consequently.
. Thus, “” occurs consequently.
As , the code ends hereby with “”. That is, .

Example 30. Thus, It can be verified that

Proposition 31. , if

Proof. .
.
due to Lemma 28 (1). “” occurs consequently.
Thus, next transformation is “”. Besides, .
.  .   Thus, “” occurs consequently.
= = = <; the code ends with “”. That is, .

Example 32. ; , .

Proposition 33. , if

Proof. , .
, thus next transformation is “”.
, thus next transformation is “”.
. . , thus next transformations are double “”.
Check whether current transformed number is less than the starting number as follows:
.
, as

Example 34. ; , .

Proposition 35. , if
.

Proof. .
.
.
.
.
Thus, double “” follow immediately. In summary, above occurred dynamics to current transformed number is thus .
Next, we check whether it is the final code (or reduced dynamics is terminated at this transformed number) by checking whether current transformed number is less than the staring number (i.e., ).

Therefore, , which can be also written as .

Remark 36. Indeed, (11) can be used for computing current transformed number (denoted as ) after to simplify above process in the proof. After ,
,
,
,
,
, as due to

Example 37. ; ,

The following two propositions originally stem from our observations on codes outputted by our computer programs.

Proposition 38. , if

Proof. .
, thus “” will follow consequently.
, thus “” will follow.
, thus “” will follow.
, thus “” will follow.
, thus double “” will follow.
Next, current transformed number will be compared with starting number .
.
.
.
.
.
.
Therefore, the reduced dynamics ends with “”. That is, .

Proposition 39. , if

Proof. .
.
.
.
, thus triple “” will follow.
.
.
.
.
.
.
Therefore, .

In summary, aforementioned codes that have short lengths are listed in (31) as follows:

Note that here short length means is short, where is the length of measured by the count of “” and “” in . For example, ,

In summary, (7) and (31) are presented together in (32) (). It justifies that is unique (recall Theorem 13), as all intersection sets for in (32) are empty. The Format Theorem (Theorem 18) is confirmed as well.

We discover that (32) enumerates all possible codes for

Proposition 40.  ∪ .

Proof. If , then because Format Theorem (Theorem 18) and end by “” (Proposition 25). However, Thus,
If , then because Format Theorem and end by “” (Proposition 25). Thus, or . As (recall Proposition 35), is impossible. Hence, there exists only one type of code for (i.e., ).
Similarly, we can prove that there exists exactly two types of codes for , since . , but and . Thus, only “” and “” are possible. Similarly, we can prove that there exists three types of codes for .
Besides, is impossible, as , , , , and (and by Format Theorem and Proposition 25).

In the following, we use to denote

Corollary 41. Let .

Proof. Let .

.

Indeed, have already been discussed in Propositions 9 and 10. Recall that is the major concern in reduced version of induction for Collatz Conjecture; that is, , where . We can prove that portion of in induction is Returnable as follows.

Corollary 42. Let .
.

In other words, we can prove that portion of is Returnable as follows:

Corollary 43. Let .
.

Thus, we have already proved that portion of induction cases is Returnable. For the proof of Collatz Conjecture, we need to prove that all cases are Returnable. That is, all cases for are Returnable to make , or all cases for are Returnable to make

Note that, indeed, without relying on the induction in Section 3, we only need to prove As is equivalent to (recall Proposition 4), we thus solely need to prove is true. In other words, (recall Remark 14 (3)).

6. Conclusion

In this paper, we propose a new direction for proving Collatz Conjecture, by proving Reduced Collatz Conjecture. Reduced Collatz Conjecture is equivalent to Collatz Conjecture but easier to explore for inherent properties. It is because all dynamics going to transformed number 1 will consist of multiple reduced dynamics, in which transformed number is less than the corresponding starting number instead of 1.

We also present an induction method for Collatz Conjecture for better understanding of Reduced Collatz Conjecture and reduced version of the induction that is easier to tackle.

We denoted reduced dynamics as (called code) and explore some fundamental properties of it, especially the structure of reduced dynamics (i.e., unified format concatenated by regular segments).

The starting numbers and their codes whose lengths are no more than 7 (i.e., ) are also given and proved; thus, portion of has already guaranteed Reduced Collatz Conjecture. That is, ). (Recall that codes for and are trivial - and .)

The future work for the proof of Collatz Conjecture can follow this direction, to just prove that exists for the left portion of . Indeed, we also discovered the bound between the counts of and the counts of for any valid code is related to , the period in terms of is related to the length of , and how to compute residue class of directly when is given, which will be presented in our other papers.

Data Availability

The data (Source Code in C, Computer Program Outputs) used to support the findings of this study have been deposited in the repository. (1) Wei Ren, Exploring properties in Reduced Collatz Dynamics, IEEE Dataport, 2018. https://doi.org/10.21227/ge9h-8a77 (2018). (2) Wei Ren, Verifying whether extremely large integer guarantees Collatz Conjecture (can return to 1 finally), IEEE Dataport, https://doi.org/10.21227/fs3z-vc10 (2018). (3) Wei Ren, Exploring the ratio between the count of x/2 and the count of (3x+1)/2 in original dynamics for extremely large starting integers asymptotically, IEEE Dataport. https://doi.org/10.21227/rxx6-8322 (2018). (4) Wei Ren, Exploring the inverse mapping from a dynamics to a residue class - inputting a reduced dynamics or partial dynamics and outputting a residue class, IEEE Dataport. https://doi.org/10.21227/qmzw-gn71 (2018). (5) Wei Ren, Reduced Collatz Dynamics for Integers from 3 to 999999, IEEE Dataport. https://doi.org/10.21227/hq8c-x340 (2018). (6) Wei Ren, Collatz Automata and Compute Residue Class from Reduced Dynamics by Formula, IEEE Dataport. https://doi.org/10.21227/7z84-ms87 (2018).

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The research was financially supported by Major Scientific and Technological Special Project of Guizhou Province (20183001), the Open Funding of Guizhou Provincial Key Laboratory of Public Big Data (2018BDKFJJ009, 2017BDKFJJ006), and Open Funding of Hubei Provincial Key Laboratory of Intelligent Geo-Information Processing (KLIGIP2016A05).

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