arXiv:1311.5690v1 [cs.CC] 22 Nov 2013 · 2013-11-25 · arxiv:1311.5690v1 [cs.cc] 22 nov 2013 journal of the american mathematical society volume 00, number 0, pages 000–000 s 0894-0347(xx)0000-0

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JOURNAL OF THEAMERICAN MATHEMATICAL SOCIETYVolume 00, Number 0, Pages 000–000S 0894-0347(XX)0000-0

THE ERGODICITY OF THE COLLATZ PROCESS IN POSITIVE

INTEGER FIELD

BOJIN ZHENG, YANGQIAN SU, HONGRUN WU, AND LI KUANG

1. INTRODUCTION

Since 1930s, researchers have deeply investigated the 3x+1 problem. Until now,the 3x+1 problem has obtained many names, such as the Kakutani’s problem, theSyracuse problem and Ulam’s problem and so on[11, 9].

The 3x+ 1 problem can be stated from the viewpoint of computer algorithm asfollows:

For any given integer x, if x is odd, then let x := 3x + 1; if x is even, then letx := x/2; if we repeat this process, x will certainly be 1 at some time.

Mathematically, this problem can be presented as the iterations of a functionf(x), called the Collatz function shown as equation (1.1), i.e., ∀x, ∃k, fk(x) = 1,x ∈ N

+. Here, N+ is the set of positive integers.

(1.1) f(x) =

3x+ 1, if x is oddx

2, if x is even

x > 0

This problem is very hard to solve because the iteration process is very “random”,although the Collatz function is deterministic.

In spite of the difficulty of this problem, the researchers still attained manyfruitful achievements [4]. From the view of probability theory [5], the researchersexplored the existence of divergent trajectories; from the view of number theoryand diophantine approximations and other mathematical tools, the researchers dis-cussed the existence of the cycles other than 4 → 2 → 1 [10, 7, 1, 14, 15, 3, 8];from the perspective of mathematical logic and theory of algorithms, the researchersstudied the solvability of this problem [6, 16]. Moreover, this problem was also triedfrom the view of the fractal [18, 17], graph theory [2] and computation experiments[13, 11, 4] and so on. Owing to the efforts of Prof. J. C. Lagarias, the related workswere collected and commented [12].

In this paper, we treated the Collatz function as a deterministic program (pro-cess), and generalized it to the non-deterministic program and set up three models;furthermore, we mapped the programs to the Collatz graphs. By the proposedmodels and the graph theory, we proved that the Collatz conjecture holds, i.e., allthe positive integers can reach 1.

2010 Mathematics Subject Classification. Primary 11B85, 68R10; Secondary 37A99.

c©XXXX American Mathematical Society

1

http://arxiv.org/abs/1311.5690v1

2 BOJIN ZHENG, YANGQIAN SU, HONGRUN WU, AND LI KUANG

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Figure 1. The original Collatz graph of model M0

2. MODELS

The Collatz problem, which we called model M0, can be mapped to the Collatzgraph [9] as shown in Fig. 1.

In Fig. 1, every positive integer is a node. Every node has two directed edges.Every edge responds to an item of Collatz function, which represents a transforma-tion of the value of the variable x.

From the viewpoint of Collatz graph, the Collatz problem can be stated asfollows: for any given node, i.e., positive integer A, there exists an A ⇒ 1 path inCollatz graph.

If there is an A⇒ 1 path in Collatz graph, then we say A is reachable. All thereachable positive integers will form a set Ω0, thus, the Collatz conjecture can alsobe stated as Ω0 = N

+.To solve this problem, we generalized the problem to non-deterministic process

and set up three models. The first model is named MS, the second is named M1and the third is named M2. Recall that the original problem is called M0.

In the model MS, we extended the Collatz function to three items, i.e., addinga non-deterministic item that is inverse to the 3x + 1 item without the constraintof parity, shown in equation (2.1).

(2.1) fs(x) =

3x+ 1, if x mod 2 ≡ 1x

2, if x mod 2 ≡ 0

x−13

, if x mod 3 ≡ 1x > 0

THE ERGODICITY OF THE COLLATZ PROCESS IN POSITIVE INTEGER FIELD 3

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Figure 2. The Collatz graph of model MS

By using a similar method to that of the Collatz graph, we can draw a graph toreflect equation (2.1) as Fig. 2. In honor of L. Collatz, we called the generalizedgraphs as the Collatz graphs.

From Fig. 2, we can see that every positive integer, i.e., every node, has a fewoptions to connect to the other nodes. For examples, node 7 can connect to 22by 3x + 1 or connect to 2 by x−1

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Therefore, the nodes in the Collatz graphs can be categorized into a few classes.Or say, the Collatz graphs have fruitful patterns.

In Fig. 2, the red edges are different edges in contrast to Fig. 1 of model M0.Like M0, if there exists an A ⇒ 1 path, we say A is reachable. We denote the

reachable set of MS as Ωs.In the second model M1, we removed more constraints. The function of M1 is

presented as equation (2.2).

(2.2) f1(x) =

3x+ 1x

2, if x mod 2 ≡ 0

2xx−13

, if x mod 3 ≡ 1

x > 0

Compared with the mode MS, model M1 adds a new item, i.e., 2x, which isinverse to the item x/2.

With respect to equation (2.2), the Collatz graph can be shown as in Fig. 3.The structure of Fig. 3 of model M1 is the same as Fig. 2 of model MS except

that: some edges in Fig. 2 are one-way, and all the edges in Fig. 3 are two-way.


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Figure 3. The Collatz graph of model M1

Similarly, we denote the reachable set of M1 as Ω1.According to equation (2.2), f1(x) has four items, which we call “actions”.

Definition 2.1 (Action). An action is an optional transformation of functions, i.e.,an item of functions.

We use T for 3x+ 1; B for x/2; F for (x− 1)/3 and D for 2x, respectively.According to the functions, action T is inverse to F , and B is inverse to D.We further generalize M1 to the third model M2, which can be formulated as

equation (2.3).

(2.3) f2(x) =

3x+ 1x

2

2xx−13

x > 0

Obviously, in model M2, the functional values can be the rational numbers, i.e.,M2 has many nodes with the rational values.

Similarly, We denoted the reachable set of M2 as Ω2.After the introduction of the proposed models, we further explored the properties

of the proposed models. From model M2, we got some valuable clues for M1, andfurther we got the structural features of M1 and MS. Finally, we proved thatΩ0 = N

+.


3. THE MODEL M2

Theorem 3.1 (The succession theorem). The action sequence ’TDDFFBBT’ isthe succession function for any given positive integer, i.e., equation (3.1) alwaysholds for any given positive integer.

(3.1) ′TDDFFBBT ′(x) = TBBFFDDT (x) = x+ 1

Proof. The calculations are listed as follows,

′TDDFFBBT ′(x) = TBBFFDDT (x)

=(x∗3+1)∗2∗2−1

3 −1

3∗ 1

2∗ 1

2∗ 3 + 1

= 4x3∗ 1

4∗ 3 + 1

= x+ 1

Lemma 3.2. Every positive integer is reachable in M2, i.e., Ω2 = N+.

According to the succession theorem, M2 demonstrates a spiral structure. Thepositive integers are the central pillars.

Theorem 3.3 (The 2 successions theorem). The action sequence ’DFFBTT’ isthe succession of succession function for any positive integer, i.e., equation (3.2)always holds for any given positive integer.

(3.2) ′DFFBTT ′(x) = TTBFFD(x) = x+ 2


′DFFBTT ′(x) = TTBFFD(x)

=(

x∗2−13 −1

3∗ 1

2∗ 3 + 1

)

∗ 3 + 1

=(2x−4

9∗ 3

2+ 1)∗ 3 + 1

= x+ 2

According to the 2 successions theorem, there often exist shorter action sequencesto transform a number to another number in M2.

Theorem 3.4 (The 3 successions theorem). The action sequence ’DDFFBBTT’is the succession of succession of succession function for any given positive integer,i.e., equation (3.3) always holds for any given positive integer.

(3.3) ′DDFFBBTT ′(x) = TTBBFFDD(x) = x+ 3


′DDFFBBTT ′(x) = TTBBFFDD(x)

=(

x∗2∗2−13 −1

3∗ 1

2∗ 1

2∗ 3 + 1

)

∗ 3 + 1

=(4x−4

9∗ 3

4+ 1)∗ 3 + 1

= x+ 3


Theorem 3.5 (The 4 successions theorem). The action sequence ’TDDFDDFF-BBBBTT’ is the succession of succession of succession of succession function forany positive integer, i.e., equation (3.4) always holds for any given positive integer.

(3.4) ′TDDFDDFFBBBBTT ′(x) = TTBBBBFFDDFDDT (x) = x+ 4


′TDDFDDFFBBBBTT ′(x) = TTBBBBFFDDFDDT (x)

=

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3∗2∗2−1

3 −1

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2∗ 1

2∗ 1

2∗ 1

2∗ 3 + 1

)

∗ 3 + 1

=

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3 −1

16+ 1

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∗ 3 + 1

= x+ 4

Because all the edges are two-way, there exists the precursor theorems corre-sponding to the succession theorems.

Of course, there exists more than one action sequence to perform the arbitrarysuccessions and precursors.

Besides, these theorems above will be used in the model M1 to demonstrate thestructures of M1 and MS.

4. THE MODEL M1

Compared with M2, model M1 only eliminates all non-integers from M2. Here,we also need to prove that all positive integers are reachable in M1, i.e., Ω1 = N

+.From now, we use a new notational method to represent a positive integer to

facilitate the calculation. Basically, we use the 3-based numeral system.For any given positive integer A represented in the 3-based numeral system, if

A accepts a ′T ′ action, the value would be A1. Therefore, we use A1 to representthe number 3A+1. Formally, we use A11 to represent a number like (9(A)3+4)10.

To represent the carry in the 3-based numeral system, we use (A + 1)11 torepresent a number like ((A)3 ∗ 9 + 9 + 4)10.

We also use AD to represent the double value of A, i.e., the value after action“D”, and AD11 to represent a number like (2 ∗ (A)3 ∗ 9 + 4)10.

We also use A1 to represent ⌊A/2⌋, A2 to ⌊⌊A/2⌋/2⌋, and A3 to ⌊⌊⌊A/2⌋/2⌋/2⌋.Obviously, A1 is the value of A after a ′B′ action with consideration of the carry.

As to the ′F ′ action, we only need to erase the last ′1′ symbol of ′A1′.Moreover, for positive integers A and C, if there is an action sequence that can

transform A to C, we denote it as A⇒ C.

Definition 4.1 (9-cluster). For any given positive integer A, the set A00, A01,A02, A10, A11, A12, A20, A21, A22, i.e., in decimal, 9k+0, 9k+1, 9k+2, 9k+3,9k + 4, 9k + 5, 9k + 6, 9k + 7, 9k + 8 is called a 9-cluster. Here, (A)3 = (k)10.

Definition 4.2 (5-cluster). For any given positive integer A, the set A00, A01,A02, A10, A11, i.e., in decimal, 9k+0, 9k+1, 9k+2, 9k+3, 9k+4 is called a5-cluster. Here, (A)3 = (k)10.

Definition 4.3 (3-cluster). For any given positive integer A, the set A12, A20,A21, i.e.,9k + 5, 9k + 6, 9k + 7 is called a 3-cluster. Here, (A)3 = (k)10.


The proof procedure can be organized as follows:

(1) first we prove that every 5-cluster can form an internally connected sub-graph from Lemma 4.4 to Theorem 4.12.

(2) next we prove that every 3-cluster can form an internally connected sub-graph from Lemma 4.14 to 4.24 and Theorem 4.41 .

(3) then we prove that every 3-cluster can connect to its corresponding 5-cluster.

(4) also we prove that every 9-cluster can form an internally connected sub-graph.

(5) at last we show that all 9-clusters can connect to 1.

Actually, there exist simpler proofs on Ω1 = N+. However, to illustrate the

structure of the Collatz graphs, we used the proof method stated above.Here, we firstly prove that every 5-cluster can form an internally connected

subgraph.

Lemma 4.4. The action sequence ’TDDFFBBT’ can transform A10 to A11.

The proof of A10⇒ A11. The calculations are listed as follows,T (A10) = A101D(A101) = AD202D(AD202) = (ADD + 1)111F ((ADD + 1)111) = (ADD + 1)11F ((ADD + 1)11) = (ADD + 1)1B((ADD + 1)1) = AD2B(AD2) = A1T (A1) = A11

Lemma 4.5. The action sequence (’TDDFFBBT’)−1= ’FDDTTBBF’ can trans-form A11 to A10.

Because the procedure is inverse to that in Lemma 4.4 and all edges in M1 istwo-way, we omit the detailed proof procedure. Moreover, we will omit the proofprocedure on all inverse lemmas.

Lemma 4.6. The action sequence ’DFFBTT’ can transform A02 to A11.

The proof of A02⇒ A11. The calculations are listed as follows,D(A02) = AD11F (AD11) = AD1F (AD1) = AD

B(AD) = AT (A) = A1T (A1) = A11

Lemma 4.7. The action sequence (’DFFBTT’)−1= ’FFDTTB’ can transform A11to A02.

Lemma 4.8. The action sequence ’DDFFBBTT’ can transform A01 to A11.

The proof of A01⇒ A11. The calculations are listed as follows,D(A01) = AD02


D(AD02) = (ADD)11F (ADD11) = ADD1F (ADD1) = ADD

B(ADD) = AD

B(AD) = AT (A) = A1T (A1) = A11

Lemma 4.9. The action sequence (’DDFFBBTT’)−1= ’FFDDTTBB’ can trans-form A11 to A01.

Lemma 4.10. The action sequence ’TDDFDDFFBBBBTT’ can transform A00 toA11.

The proof of A00⇒ A11. The calculations are listed as follows,T (A00) = A001D(A001) = AD002D(AD002) = ADD011F (ADD011) = ADD01D(ADD01) = ADDD02D(ADDD02) = ADDDD11F (ADDDD11) = ADDDD1F (ADDDD1) = ADDDD

B(ADDD) = ADD

B(ADD) = AD

B(AD) = AT (A) = A1T (A1) = A11

Lemma 4.11. The action sequence (’TDDFDDFFBBBBTT’)−1 = ’FFDDDDT-TBBTBBF’ can transform A11 to A00.

Theorem 4.12 (The 5-cluster connection theorem). For any given positive integerA, nodes A00, A01, A02, A10 and A11 are internally connected.

This theorem follows from Lemmas 4.4, 4.5, 4.6, 4.7, 4.8, 4.9, 4.10 and 4.11.

Theorem 4.13 (The 5-cluster attaching theorem). For any given positive integerA, there is at least a path from A to a 5-cluster.

Proof. T (A) = A1T (A1) = A11

From the 5-cluster attaching theorem, the action sequence ‘TT ′ can assure thatarbitrary positive integer is connected to at least one 5-cluster.

Here, we prove that the 3-clusters are internally connected.

Lemma 4.14. The action sequence ’TDDFFBBT’ can transform A20 to A21.

The proof of A20⇒ A21. The calculations are listed as follows,T (A20) = A201D(A201) = (AD + 1)102


D((AD + 1)102) = ((AD + 1)D)211F (((AD + 1)D)211) = (AD + 1)D21F ((AD + 1)21) = (AD + 1)D2B((AD + 1)D2) = (AD + 1)1B((AD + 1)1) = A2T (A2) = A21

Lemma 4.15. The action sequence (’TDDFFBBT’)−1=’FDDTTBBF’ can trans-form A21 to A20.

Lemma 4.16. The action sequence ’DDDFFBBTBT’ can transform A12 to A21.

The proof of A12⇒ A21. The calculations are listed as follows,D(A12) = (AD + 1)01D((AD + 1)01) = (AD + 1)D02D((AD + 1)D02) = (AD + 1)DD11F ((AD + 1)DD11) = (AD + 1)DD1F ((AD + 1)DD1) = (AD + 1)DD

B((AD + 1)DD) = (AD + 1)D

B((AD + 1)D) = (AD + 1)T ((AD + 1)) = (AD + 1)1B((AD + 1)1) = A2T (A2) = A21

Lemma 4.17. The action sequence (’DDDFFBBTBT’)−1 = ’FDFDDTTBBB’ cantransform A21 to A12.

Theorem 4.18 (The 3-cluster connection theorem). For any given positive integerA, nodes A12, A20 and A21 are internally connected.

According to Lemma 4.14, 4.15, 4.16 and 4.17.From now on, we will prove that every 3-cluster can connect to its corresponding

5-cluster.

Lemma 4.19. When A is even, there exists at least one action sequence to trans-form A21 to A11, i.e., A21⇒ A11.

The proof of A21⇒ A11. The calculations are listed as follows,T (A21) = A211B(A211) = A1102′DFFBTT ′(A1102) = A1111 (Lemma 4.6)FFF (A1111) = A1′DTT ′(A1) = A11

Lemma 4.20. When A is even, there exists at least one sequence to transform A11to A21, i.e., A11⇒ A21.

Lemma 4.21. When A is odd, and A = R0, i.e., the last symbol of A is ′0′, thereexists at least one action sequence to transform A21 to A11, i.e., R021 ⇒ R011.


The proof of A21⇒ A11. The calculations are listed as follows,R021⇒ RD112 (D)⇒ RD11211 (TT)⇒ R02102 (B)⇒ R02111 (Lemma 4.6)⇒ R02 (FFF)⇒ R11 (Lemma 4.6)⇒ R01 (Lemma 4.9)⇒ R011 (T)

Lemma 4.22. When A is odd, and A = R0, there exists at least one actionsequence to transform A11 to A21, i.e., R011⇒ R021.


The proof of A21⇒ A11. The calculations are listed as follows,R121⇒ R112 (Lemma 4.17)⇒ R11211 (TT)⇒ R102102 (B, here R is even because A is odd and A = R1)⇒ R102111 (FFF)⇒ R102 (D)⇒ R11 (T)⇒ R111 (T)


Here, we firstly discuss the relationship between A22 and A11 and then comeback to discuss the circumstance when A is an odd and A = R2.

Lemma 4.25. When A is even, there exists at least one action sequence to trans-form A22 to A11, i.e., A22⇒ A11.

The proof of A22⇒ A11. The calculations are listed as follows,B(A22) = A111′FFDTT ′(A111) = A11

Lemma 4.26. When A is even, there exists at least one sequence to transform A11to A22, i.e., A11⇒ A22.

Lemma 4.27. When A is odd, and A = R0, there exists at least one actionsequence to transform A22 to A11, i.e., R022 ⇒ R011.

The proof of R022⇒ R011. The calculations are listed as follows,R022⇒ RD121 (D)⇒ RD112 (Lemma 4.17)⇒ R021 (B)⇒ R02 (F)⇒ R11 (Lemma 4.6)⇒ R01 (Lemma 4.9)⇒ R011 (T)




The proof of R122⇒ R111. The calculations are listed as follows,R122⇒ R1011⇒ R101⇒ R02⇒ R11⇒ R111


Lemma 4.31. There exists at least one action sequence to transform R0

n︷︸︸︷

2 · · ·2 to

R0

n+1︷︸︸︷

2 · · ·2, i.e., R0

n︷︸︸︷

2 · · ·2⇒ R0

n+1︷︸︸︷

2 · · ·2.


R0

n︷︸︸︷

2 · · ·2⇒ R0

n︷︸︸︷

2 · · ·21

⇒ RD1

n−1︷︸︸︷

2 · · ·212

⇒ RD1

n−1︷︸︸︷

2 · · ·221

⇒ R0

n−1︷︸︸︷

2 · · ·222⇒ R0

n+1︷︸︸︷

2 · · · 2


n+1︷︸︸︷

2 · · ·2 to

R0

n︷︸︸︷

2 · · ·2, i.e., R0

n+1︷︸︸︷

2 · · ·2⇒ R0

n︷︸︸︷

2 · · ·2.


n

︷︸︸︷

2 · · ·2 to

R1

n+1︷︸︸︷

2 · · ·2, i.e., R1

n

︷︸︸︷

2 · · ·2 ⇒ R1

n+1︷︸︸︷

2 · · · 2.

Proof. The calculations are listed as follows,When R is even,

R1

n

︷︸︸︷

2 · · ·2⇒ R1

n

︷︸︸︷

2 · · ·21

⇒ R10

n︷︸︸︷

2 · · ·22

⇒ R10

n︷︸︸︷

2 · · ·221

⇒ R10

n︷︸︸︷

2 · · ·212


⇒ R1

n

︷︸︸︷

2 · · ·221

⇒ R1

n+1︷︸︸︷

2 · · ·2When R is odd, let R = (P + 1),

(P + 1)1

n

︷︸︸︷

2 · · ·2

⇒ P 12

n︷︸︸︷

1 · · · 1

⇒ P 12

n+1︷︸︸︷

1 · · · 1

⇒ (P + 1)1

n+1︷︸︸︷

2 · · ·2

Therefore, for any given R, R1

n︷︸︸︷

2 · · ·2 ⇒ R1

n+1︷︸︸︷

2 · · · 2


n+1︷︸︸︷

2 · · ·2 to

R1

n

︷︸︸︷

2 · · ·2, i.e., R1

n+1︷︸︸︷

2 · · ·2⇒ R1

n

︷︸︸︷

2 · · ·2.

Theorem 4.35 (The 2 appending theorem). There exists at least one action se-

quence to transform R

n︷︸︸︷

2 · · ·2 to R

n+1︷︸︸︷

2 · · · 2, i.e., R

n︷︸︸︷

2 · · · 2 ⇒ R

n+1︷︸︸︷

2 · · · 2.

According to Lemma 4.31 to 4.33, this theorem is obvious.

Theorem 4.36 (The 2 backspace theorem). There exists at least one action se-

quence to transform R

n+1︷︸︸︷

2 · · ·2 to R

n

︷︸︸︷

2 · · · 2, i.e., R

n+1︷︸︸︷

2 · · · 2 ⇒ R

n

︷︸︸︷

2 · · · 2.

According to Theorem 4.35, this theorem is obvious.

Lemma 4.37. When A is odd, and A = R2, there exists at least one actionsequence to transform A22 to A11, i.e., R222 ⇒ R211.

R222 ⇒ R211. ′0R′ should include at least one ′1′ or ′0′ before a series of ′2′.Therefore, R222⇒ R2⇒ R211, or say, A22⇒ A⇒ A11.

From Lemma 4.25 to 4.37, we can obtain a conclusion as Theorem 4.38.

Theorem 4.38. For any given positive integer A, there exists at least an actionsequence to transform A22 to A11, i.e., A22⇒ A11.

This proposition follows from Lemma 4.25 to 4.37.

Lemma 4.39. For any given positive integer A, there exists at least an actionsequence to transform A11 to A22.

Now, we can come back to discuss A21 when A is odd.



Proof of R221⇒ R211. The calculations are listed as follows,R221⇒ R22⇒ R2⇒ R211

Theorem 4.41. For any given positive integer A, there exists at least one actionsequence to transform A21 to A11.

Lemma 4.42. For any given positive integer A, there exists at least one actionsequence to transform A11 to A21.

According to Theorems 4.38, 4.41 and 4.12, we can obtain Theorem 4.43.

Theorem 4.43. For any given positive integer A, the corresponding 9-cluster isinternally connected.

So we can prove that every positive integer can reach (11)3, i.e., (4)10; Of course,can also connect to 1.

Theorem 4.44. For any given positive integer A, there exists at least one actionsequence to transform A to 11.

Proof. If the numbers of symbols A is odd, then let A = 0A, i.e., add an additional0 to the head of A, to make the numbers of symbols is even.

A ∗ ∗ ⇒ A11⇒ A, here * is arbitrary one of ′0′, ′1′ and ′2′.By repeating this process, we obtain A⇒ 11.

According to the proofs above, we can obtain more conclusions.

Lemma 4.45. For any given positive integer A, there exists at least one actionsequence H in M1 such that H(A) < A.

Lemma 4.46. For any given positive integer A, there exists at least one actionsequence to transform A to 1, i.e., Ω1 = N

+.

Because M1 can be mapped into Fig. 3, Theorem 4.44 and Lemma 4.46 indicatethat there is at least one path from any given positive integer to 4 or 1.

Theorem 4.47 (The node loop existence theorem). For any given positive integerA, A is in a loop.

Proof. 1) If A is odd, A⇒ A111⇒ A1202⇒ A1211⇒ A12⇒ A1. Because A1 < Aand A1 ⇒ 1, therefore, A is in a loop.

2) If A is even, A⇒ A11 ⇒ A102 ⇒ A111 ⇒ A1. Because A1 < A and A1 ⇒ 1,therefore, A is in a loop.

5. THE MODEL MS

Compared with model M1, all nodes in M1 are still in MS. However, someedges of model MS become directed. That is, the Collatz graph in Fig. 2 is aweakly connected graph and all the positive integers are weakly connected in it.

If we can prove that, for any given positive integer A, there exists an actionsequence H such that H(A) < A, then there should exist an path from A to 1, i.e.,Ωs = N

+, because 1 is the smallest value.


Lemma 5.1. For any given A, there exists an action sequence H, such thatH(A2) < A2.

Proof. If A is even, then B(A2) = A11, F (A11) = A1. Because A1 < A2, thisproposition holds.

If A is odd, then T (A2) = A21, B(A21) = A122. For any given k >= 1, if Ak is

even, then B

k

︷︸︸︷

BT · · ·BT (A2) = Ak+1

k+1︷︸︸︷

1 · · · 1. Further,

k+1︷︸︸︷

F · · ·F (Ak+1

k+1︷︸︸︷

1 · · · 1) = Ak+1.Because Ak+1 < A2, this proposition holds.

Because there exists a k such that Ak = 0, this proposition holds.


Proof. F (A1) = A < A1


Proof. If A is even, B(A0)=A10 < A0, this proposition holds. If A is odd, T (A0) =A01, B(A01) = A112. For A112, if A1 is odd, BF (A112) = A22 < A0, thisproposition holds; if A1 is even, TB(A112) = A2022. According to Lemma 5.1, thisproposition also holds.

Therefore, this proposition always holds.

Theorem 5.4 (The descending theorem). For any given positive integer A, thereexists an action sequence H in MS such that H(A) < A.

Proof. According to Lemmas 5.3, 5.2 and 5.1, this theorem holds.

Theorem 5.5 (The edge loop existence theorem). For any given positive eveninteger A, the edge A← A1 is in a loop.

Proof. Because A is even, A1⇒ A11⇒ A102.

(1) If A1 is even, A102 ⇒ A201 ⇒ A20 . Because A20 < A and A1 ⇒ 1, theedge A← A1 is in a loop.

(2) If A1 is odd, A102⇒ A1021 ⇒ A2122.(3) If A2 is odd, A2122 ⇒ A3211 ⇒ A32. Because A32 < A , this theorem

holds.(4) If A2 is even, A2122 ⇒ A3222. By repeating this analysis process, we

obtain A3222 ⇒ Ak+1 because the result should finally is even in sometime k. Since Ak+1 < A, this theorem holds.

Therefore, the edge A← A1 is in a loop.

Lemma 5.6. For any given positive integer A, there exists a path from A to 1.

Lemma 5.7. In model MS, Ωs = N+.

Definition 5.8 (The removable edge). In Fig. 2 (model MS), when an edge e isremoved, Ωs does not change, then the edge e is said removable.

Theorem 5.9 (The de-looping theorem). For any given edge e, if e belongs to theedge set E1 indicated by x−1

3if x mod 6 ≡ 1, then e is removable.

Proof. According to Theorem 5.5, and notice that the edge e has a different direc-tion to the other path is the loop, therefore, e is removable.


6. THE MODEL M0

The graph of M0 is a directed graph. Compared with the model MS, modelM0 eliminates all the edges indicated by the action F .

Lemma 6.1 (The Collatz conjecture). For any given positive integer A, A is reach-able in M0.

Proof. According to Lemma 5.7, all positive integers which exist in Fig. 2 (modelMS) still exist in Fig. 1 (model M0).

According to the de-looping theorem in model MS, all these edges in E1 can beremoved one by one starting from 7→ 2.

After the removals of the edge set E1, all the edges belonging to the edge set E4,which is indicated by x−1

3if x mod 6 ≡ 4, would be abundant (have no successions

nodes) in MS, and hance are removable.After the removals of the edge sets E1 and E4, MS becomes M0. Therefore, the

Collatz conjecture holds.

Theorem 6.2. There is only one circle 4→ 2→ 1→ 4 in M0.

Proof. Because every node in M0 has only an out-link, according to graph theoryand Fig. 1 (modelM0) is a connected graph, only the positive integer 1 has an extraout-link, and it out-links to the positive integer 4, so only one circle 4→ 2→ 1→ 4exists.

7. CONCLUSIONS

This paper proves the 3x + 1 problem. The result shows that all the positiveintegers can be transformed to 1 by the iteration of f . Equivalently, there are noother cycles other than 4→ 2→ 1→ 4 cycle and there is no divergent trajectories.

The result in this paper would be useful to the research of chaos [18, 17], com-puter science [6], complex systems and so on.

Acknowledgements. The authors are grateful to Dr. CL Zhou, Prof. YXLi, Prof. J Qin, Dr. BB Wang, Dr. WW Wang, Mr. QF Wang, Mr. GC Tangand Mr. JW Zheng for discussions and inspirations. The authors thank to thesupports from the State Key Laboratory of Networking and Switching Technology(No. SKLNST-2010-1-04) and the State Key Laboratory of Software Engineering(No. SKLSE2012-09-15) and the Fundamental Research Funds for the Central Uni-versities (No. CZY13010). This paper is also supported by the China ScholarshipCouncil.

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the 3n+ 1 problem, Acta Arithmetica 117 (2005), no. 1, 51–70.15. Matti K. Sinisalo, On the minimal cycle lengths of the Collatz sequences, preprint, Univ. of

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Abstract. The 3x + 1 problem, also called the Collatz conjecture, is a veryinteresting unsolved mathematical problem related to computer science. Thispaper generalized this problem by relaxing the constraints, i.e., generalizingthis deterministic process to non-deterministic process, and set up three mod-els. This paper analyzed the ergodicity of these models and proved that theergodicity of the Collatz process in positive integer field holds, i.e., all the pos-itive integers can be transformed to 1 by the iterations of the Collatz function.

College of Computer Science, South-Central University for Nationalities, Wuhan

430074, China

Current address: School of Informatics and Computing, Indiana University, Bloomington, IN47408, USA

E-mail address: [email protected]

College of Computer Science, South-Central University for Nationalities, Wuhan

430074, China

Current address: College of Computer Science, South-Central University for Nationalities,Wuhan 430074, China

The State Key Laboratory of Software Engineering, Wuhan University, 430074,

China

Current address: The State Key Laboratory of Software Engineering, Wuhan University,430074, China

E-mail address: [email protected]

The State Key Laboratory of Software Engineering, Wuhan University, 430074,

China

Current address: The State Key Laboratory of Software Engineering, Wuhan University,430074, China

Documents

arXiv:1311.5690v1 [cs.CC] 22 Nov 2013 · 2013-11-25 · arxiv:1311.5690v1 [cs.cc] 22 nov 2013 journal of the american mathematical society volume 00, number 0, pages 000–000 s 0894-0347(xx)0000-0