Research Article | Open Access
A New Approach on Proving Collatz Conjecture
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.
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 .
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  and R. E. Crandall , 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 . 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 .
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.
(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.
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.
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.
Proposition 8. If the induction (especially, Step 2) can be proved, Collatz Conjecture is True.
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, .
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.
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 ).
Proposition 15. ,
Proof. (1) , thus occurs. , thus .
(2) If , (by assumption).
If , , where . Thus, occurs. . , thus further transformation occurs. , thus .
(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,
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.
(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.
(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: ,