A Reduced Collatz Dynamics Maps to a Residue Class, and Its Count of over the Count of Is Larger than ln3/ln2
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 ()/2 and “O” that represents . Here, and are combined together because is always even and thus followed by . 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 and 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 over the count of is larger than ln3/ln2, reduced dynamics will be obtained (i.e., current integer will be less than starting integer).
The Collatz conjecture can be stated simply as follows. Take any positive integer number . If is even, divide it by 2 to get . If is odd, multiply it by 3 and add 1 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 1.
Ren et al.  verified that can return to 1 after 481603 times of computation, and 863323 times of computation, which is the largest integer being verified in the world. Ren  also pointed out a new approach for the possible proof of Collatz conjecture. Ren  proposed to use a tree-based graph to observe two key inner properties in reduced Collatz dynamics: one is ratio of the count of over the count of , and the other is partition (all positive integers are partitioned regularly corresponding to ongoing dynamics). Ren and Xiao  also proposed an automata method for fast computing Collatz dynamics. All source code and output data by computer programs in those related papers can be accessed in public repository .
Notation 1. (1): positive integers(2)(3); (4)
Proposition 1. always follows after .
Proof. When , then next computation is . Obviously, ; thus, the next computation must be consequently.
We thus can represent required computation as and , which are denoted by and , respectively.
Notation 2. , .
Note that and can be simply denoted as and , or and , respectively. Obviously, , , and . That is, the reason of notation, , represents “Increase” and, , represents “dOwn.”
Definition 1. Collatz transformation, denoted as , where if , and if .
Remark 1. (1)We assume .(2)Obviously, , where , , and “” is concatenation of Collatz transformations. For simplicity, we just denote as .(3), . Note that whether is or in , is determined by or .
Definition 2. (Collatz conjecture). and , such that , where .
Obviously, Collatz conjecture is held when . In the following, we mainly concern .
Definition 3. (reduced Collatz conjecture). , and , such that and , , .
Obviously, must be the minimal positive integer such that .
Theorem 1. Collatz conjecture is equivalent to reduced Collatz conjecture.
Proof. , it is obvious that , i.e., .(1)Suppose Collatz conjecture is true. That is, , , . Thus, . Hence, reduced Collatz conjecture is true.(2)Inversely, suppose reduced Collatz conjecture is true. That is, , , .If , then Collatz conjecture is true.
If , then let . As reduced Collatz conjecture is true, , .
For better notation, let . Iteratively, if , then Collatz conjecture is true. If , then , .
Thus, . is a strictly decreasing sequence.
Therefore, after finite times of iterations, , .
That is, , , .
Thus, Collatz conjecture is true.
Remark 2. (1)We call an ordered sequence in the above proof as original dynamics (referring to ), which consists of occurred Collatz transformations during the computing procedure from a starting integer to 1. For example, the original dynamics of 5 is due to .(2)In contrast, we call in the above proof as reduced dynamics (referring to ), which is represented by a sequence of occurred Collatz transformations during the computing procedure from a starting integer (i.e., ) to the first transformed integer that is less than the starting integer (i.e., ). For example, the reduced dynamics of 5 is due to .(3)Obviously, reduced dynamics is more primitive than original dynamics because original dynamics consists of reduced dynamics. Simply speaking, reduced dynamics are building blocks of original dynamics.Due to above theorem, we concentrate on reduced dynamics.
Notation 3. (). It denotes reduced dynamics of that are represented by . Formally, , , if such that and , where ; then, let and is called as reduced dynamics of , denoted as .
Remark 3. (1)Simply speaking (or recall that), represents occurred Collatz transformations in terms of and during the computing process from starting integer to the first transformed integer that is less than .(2)Recall that, is an ordered sequence consisting of and . Besides, , , and . Furthermore, this sequence implicitly matches the parity of all occurred intermediate transformed integers that are taken as input of .(3)Roughly speaking, in , is called the starting integer. are called transformed integers. is the first transformed integer that is less than the starting integer . In other words, , and . (). Besides, the parity of determines the selection of the intermediately next after .(4)Obviously, .(5)For example, , , , , and . Indeed, we design computer programs that output all . From the data we discover the property—period and its relation to the number of computing in reduced dynamics—which will be proved in the remainder of this paper.(6)In fact, we proved some results on for specific , e.g., , , and .(7) can be denoted in short as . can be denoted in short as . In other words, we denote as , and we denote as , where . We also assume and . means no transformation occurs.(8)We will formally prove that the ratio exists in any reduced Collatz dynamics. That is, the count of over the count of is larger than in this paper. The ratio can also be observed and verified in my proposed tree-based graph .
Example 1. , if and only if(1)“” is due to (2), and thus it continues(3)“” is due to (4), and thus it endsTo better present above the implicity in reduced dynamics, we introduce two functions as follows.
Definition 4. . It takes input and , and outputs . If and , or if and , then output . Otherwise, output .
Remark 4. Simply speaking, this function checks whether the forthcoming Collatz transformation (i.e., ) matches with the current transformed integer .
Definition 5. . It takes input , where , , and , and outputs , where , , , and , and “” returns length in terms of the total count of or .
Remark 5. (1)For example, .(2)Especially, . returns the last transformation in . returns the first transformation in . returns the th transformation in .(3)In other words, is a selected segment in that starts from the location and has the length of . Indeed, that is, the reason we call this function as “Get Substring.”(4)Simply speaking, this function can obtain the Collatz transforms from to from a given inputting transform sequence (e.g., reduced dynamics) in terms of .(5)Note that itself is a function. In other words, it can be looked as .For example, , , , and .(6)It is worth to stress that, although in the above definition , it can be extended to by assuming .
Proposition 2. Suppose . If exists, then(1), where (2), where (3), where
Proof. Straightforward by the definition of .
Remark 6. (1) is the last transformed integer, or the first transformed integer that is less than the starting integer.(2) () are all intermediate transformed integers.(3)When , . is the first transformation.(4)If () is the current transformed integer; then, is the next intermediate Collatz transformation.
Proposition 3. , , if exists; then, is unique.
Proof. Straightforward. Given , either or is deterministic and unique. Similarly, given , is deterministic and unique, where . Thus, is unique for any given .
Remark 7. We assume , although . In other words, we assume the reduced dynamics of is . In the following, we always concern .
Proposition 4. Given , if exists, then ends by .
Proof. Straightforward due to . Suppose , , , and . Then, ; thus, . Contradiction occurs.
Proposition 5. and .
Proof. (1), thus occurs. , thus .(2)If , (by assumption).If , , where . Thus, occurs. . , thus further transformation occurs. , thus .
Proposition 6. Given , if exists, then , .
Proof. Let , . Obviously, . . .(1)If , then . Thus, the next transformation is “.” Thus, current occurred transformations are “.” Besides, . Further transformation thus occurs. Hence, if exists, then .(2)If , then . Thus, current occurred transformations are “.” 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, there exists further transformation after .
In summary, if exists, then .
Theorem 2 (format theorem). Given , if exists, then , where and . Besides, when ; when .
Proof. The range of is straightforward due to Propositions 5 and 6; thus, we mainly concern .
By Proposition 5, 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 the current transformed number is less than the 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.(2.1)If , (); thus, will occur consequently.(2.2)If , ; thus, will occur consequently. , 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 head (i.e., ) for . Observing the following equation for after consecutive times of “”,Note that the 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 the current transformed number be in . Thus, we need to explore the requirement on for given such thatRepresent as . That is, . Obviously, this representation is unique. We thus need to prove that the requirement in equation (3) is satisfied if and only if . Note that we will see that here is indeed determined by .
For , we have :.
When , we have exactlyIt is easy to see that is the one and only one for the requirement in equation (3), as desired.
Corollary 1(t determines p corollary). Given starting integer (i.e., ), the count of consecutive “” (denoted as ) is determined by as follows.
If , then ; if , then , where , and .
Proof. It is straightforward by Theorem 2.
Corollary 2. Given the starting integer , the first count of Collatz transformations must be and can be determined by , and the transformed integer after transformations is
Remark 8. (1)Note that, due to Corollary 1, for in can be computed by and directly without conducting concrete Collatz transformations, which can shorten the computation delay for reduced dynamics.(2)Besides, by equation (2) or Corollary 2, if , we then have because , which matches with the result by manually computing.(3)Indeed, equation (2) can be extended to include all cases (i.e., for ). If , by assuming , . If , . Therefore, for .
Corollary 3. Suppose . If exists, then .
Remark 9. Simply speaking, above corollary states that if exists, then the first Collatz transformations for must be . Indeed, the resumption on the existence of the reduced dynamics of can be omitted. That is, the first Collatz transformations in original dynamics of must also be .
Example 2. , as .
Next, corollary states that reduced dynamics consists of segment or segments with a unified form as .
Corollary 4. Given , if exists, then
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 is in , . Otherwise, ).
3. Derive x from
Notation 4. .
That is, , if exists, then will be included in , which is a set of existing reduced dynamics.
In this section, we will study two problems as follows:(1)Inverse problem: given , is it possible to derive starting integer such that ?(2)What is the sufficient and necessary conditions for any existing reduced dynamics, that is, given , how to decide whether ?Before exploring general situations, we give two trivial cases.
Proposition 7. and .
Theorem 3 (subset theorem). Suppose , , , . We have (1.1) (1.2) (2.1) (2.2)