The Proof of Collatz Conjecture

[Abstract] This paper redefines Collatz conjecture, and proposes strong Collatz conjecture, the strong Collatz conjecture is a sufficient condition for the Collatz conjecture. Based on the computer data structure–tree, we construct the non-negative integer inheritance decimal tree. The nodes on the decimal tree correspond to non-negative integers. We further define the Collatz-leaf node (corresponding to the Collatz-leaf integer) on the decimal tree. The Collatz-leaf nodes satisfy strong Collatz conjecture. Derivation through mathematics, we prove that the Collatz-leaf node (Collatz-leaf integer) has the characteristics of inheritance. With computer large numbers and big data calculation, we conclude that all nodes at depth 800 are Collatz-leaf nodes. So we prove that strong Collatz conjecture is true, the Collatz conjecture must also be true. And for any positive integer N greater than 1, the minimum number of Collatz transform times from N to 1 is log2 N, the maximum number of Collatz transform times is 800 * （ N-1 ） . The non-negative integer inheritance decimal tree proposed and constructed in this paper also can be used for the proof of other mathematical problems.


1.Introduction
Take any positive integer greater than 1. if it's even, divide it by 2. If it's odd, multiply it by 3 and add 1. Repeat this process, the final result is 1. This is the Collatz conjecture which is quite possibly the simplest unsolved problem in mathematics.
In August 1999, mathematicians from all over the world gathered in Eichstadt, Germany, to hold a two-day international seminar on Collatz conjecture. As a special issue, American Mathematical Society released the "Ultimate Challenge: 3x +1 Problem." "This is a really dangerous problem. People become obsessed with it and it really is impossible," said Jeffrey Lagarias, a mathematician at the University of Michigan and an expert on the Collatz conjecture.
In September 2019, one of the top mathematicians in the world dared to confront the problem and came away with one of the most significant results on the Collatz conjecture in decades.
Dr. Terence Tao posted a proof showing thatat the very leastthe Collatz conjecture is "almost" true for "almost" all numbers. While Tao's result is not a full proof of the conjecture, it is a major advance on a problem that doesn't give up its secrets easily.

2．Redefinition of Collatz conjecture
For the discussion, we make the following definition.
Collatz transform is divided into Collatz even transform and Collatz odd transform.
(Definition 2.2)Collatz even transform：Take any positive integer N greater than 1, if N is even, divide N by 2, i.e. transform N / 2. Now we redefine the above mentioned Collatz conjecture as below: (Conjecture 2.5)Collatz conjecture：Take any positive integer N greater than 1, perform Collatz transforms on N, repeat the process mc times, the final result must be 1.
In the above definition, We define mc as Collatz transform times(Definition 2.6)。 From the Collatz transform result sequences of integers 8 and 11: The Collatz transform times for the integer 8 is 3 and the Collatz transform times for the integer 11 is 10.
Suppose Collatz conjecture is correct, for a positive integer N greater than 1, if its Collatz transform result sequence are all even, the Collatz transform times mcMin is the minimum, and we have: For positive integer 8, its Collatz transform result sequence are all even, so its Collatz transform times satisfies (2-1): From (2-1), when N→+∞，the minimum number of Collatz transform times: mcMin →+∞.
The expression of Collatz conjecture is quite possibly the simplest in unsolved conjecture, it's hard to find a mathematical method to solve the problem. We will seek another expression of the Collatz conjecture to find a solution (Conjecture 2.7)Strong Collatz conjecture：Take any positive integer N greater than 1, perform Collatz transforms on N, repeat the process msc finite times, and the transform result Nm < N.
(Definition 2.8)Strong Collatz transform times: Take any positive integer N greater than 1, perform Collatz transforms on N, repeat the process msc finite times, and the transform result Nm < N; If in the processes of the first msc-1 times, the transform results are all greater than integer N, then the transform times mcs is defined as Strong Collatz transform times Below we will prove that if the strong Collatz conjecture is true, the Collatz conjecture must also be true.
According to the strong Collatz conjecture: take any positive integer N greater than 1, perform mn finite Collatz transforms on N, and the transform result Nm < N. Because of Nm <N, we can assume that Nm takes the largest possible integer N-1; perform mn -1 finite Collatz transforms on N-1, and the transform result is N-2; and so on, we get the following Collatz transform result sequence： The above sequence has N elements, its corresponding sequence of strong Collatz transform times is as below: {mn，mn-1，mn-2 . . . mn-k+1，mn-k，mn-k-1 . . . m3, m2} This sequence has N -1 elements. Let mscMax be the maximum value of the sequence of strong Collatz transform times above.
Then the maximum number of Collatz transform times from N to 1: mcMax = mn + mn-1 + . . . + mn-k+1 + mn-k + mn-k-1 + . . .+ m3 + m2 Therefore, if the strong Collatz conjecture is true, the Collatz conjecture must also be true; the strong Collatz conjecture is a sufficient condition for the Collatz conjecture.

3．Computer Data Structure -Tree
A tree is a data structure for which the running time of most operations is O(logN) on average.
A tree can be defined in several ways. One natural way to define a tree is recursively. A tree is a collection of nodes. The collection can be empty; otherwise, a tree consists of a distinguished node r, called the root. The remaining nodes can be divided into disjoint subsets, and each subset is itself a tree. Figure1 shows a typical tree.

Figure 1 A typical Tree
In the tree of Figure 1, the root is A. Node B has A as a parent and E, F as children. Nodes with no children are known as leaves; the leaf nodes in the tree above are K, L, F, G, M, I, J. Leaf nodes are also called termination nodes.
For any node ni, the depth of ni is the length of the unique path from the root to ni. Thus, the root is at depth 0. For the tree in Figure1, B, C and D are at depth 1; E is at depth 2; K, L and M are at depth 3.
If there is a path from n1 to n2, then n1 is an ancestor of n2 and n2 is a descendant of n1. For the tree in Figure1, B is an ancestor of E, F, K and L. E, F, K and L are descendants of B.
4．Non-negative integer inheritance decimal tree (Definition 4.1)Non-negative integer inheritance decimal tree: A tree that each node corresponds to a non-negative integer, and the depth of the node equals the number of digits of the non-negative integer. Each node has 10 child nodes, the 10 child nodes contain all the numbers of the parent node and add one digit (0, 1, 2, 3, 4, 5, 6, 7, 8 or 9) in front of the parent node. Figure 2 shows a typical Non-negative integer inheritance decimal tree. Figure 2 Non-negative integer inheritance decimal tree.

Basic characteristics of the non-negative integer inheritance decimal tree
(1) The root is at depth 0, and it does not have a number.
(2) Each node corresponds to a non-negative integer, and the depth of the node equals the number of digits of the non-negative integer.
(3) Each node has 10 child nodes, the 10 child nodes contain all the digits of the parent node, and add one digit (0, 1, 2, 3, 4, 5, 6, 7, 8 or 9) in front of the parent node to form these 10 child nodes.
(4) If there is a path from n1 to n2, then n1 is an ancestor of n2 and n2 is a descendant of n1.
The integer corresponding to the descendant node is called descendant integer; The integer corresponding to the ancestor node is called ancestor integer.
(5) A node with the first digit of 0 is defined as a clone node. The clone node is a direct inheritance of its parent.
(6) There are 10 nodes at depth 1, corresponding to 10 one-bit non-negative integers (0 to 9) which includes a clone node 0.
(8) There are 10 n nodes at depth n, corresponding to 10 n n-bit non-negative integers(0 to 10 n -1)which Includes 10 (n-1) clone nodes with the first digit of 0.
Based on the non-negative integer inheritance decimal tree, including the clone nodes, we define the following node.
(Definition 4.1)Collatz-leaf node: Take one node at depth m on the non-negative integer inheritance decimal tree where the corresponding m-bit integer Nmx is greater than 1. If the strong Collatz transform times of Nmx is less than or equal to the depth m, then this node at depth m is defined as Collatz leaf node, and the corresponding m-bit integer Nmx is defined as

Collatz-leaf integer(Definition 4.2).
For the decimal tree in Figure 2, at depth 1, nodes 2, 4, 6, 8 are Collatz-leaf nodes. Because the strong Collatz transform times of any positive even integer is 1, all positive even integers are Collatz-leaf integers.
For the Collatz-leaf node, we have the following important theorem.

(Theorem 4.3)Collatz-leaf node inheritance theorem：The descendant nodes of a Collatz-
leaf node are also Collatz-leaf nodes; The descendant integers of a Collatz-leaf integer are also Collatz-leaf integers; The strong Collatz transform times of the descendant integer of a Collatz-leaf integer is not bigger than that of the Collatz-leaf integer.
Below we will prove the Collatz-leaf node inheritance theorem.
For the decimal tree in Figure 2, take one node Nodemx at depth m. Assume that this node is a Collatz-leaf node, the m-bit positive integer Nmx corresponding to this node must be a Collatzleaf integer. Perform Collatz transforms on Nmx m times and the transform result is Nmxc, Nmxc< Nmx must be true. Take one node Nodenx at depth n. Assume that this node is a descendant of node Nodemx and the corresponding n-bit positive integer is Nnx, perform Collatz transforms on Nnx m times and the transform result is Nnxc. We will prove that Nnxc < Nnx must also be true.
The n-bit positive integer Nnx can be expressed as follows: Divide Nnx into two parts at bit m: Let the high part of the Nnx be Nnm and its low part be nmx: We know that the Collatz transform is divided into Collatz even transform and Collatz odd transform. Perform Collatz transforms on Nnx m times, suppose that the Collatz even transforms are i times, and the Collatz odd transforms are j times (i + j = m; i> 0, j> = 0).
Perform the Collatz even transform on Nnm i times, the ratio coefficient: Perform the Collatz even transform on Nmx i times, the ratio coefficient: Thus, perform the Collatz odd transforms on Nmx j times, the ratio coefficient: Form(3-6),when Nnx takes the Collatz odd transform, Nnm does not take Collatz odd transform. It takes the odd transform：multiply Nnm by 3 and divide it by 2. So Nnm takes odd transform j times, the ratio coefficient:

Now we conclude: If the m-bit integer Nmx is a Collatz-leaf integer, its n-bit descendant positive
integer Nnx must also be Collatz-leaf integer. Correspondingly, if a node at depth m is a Collatzleaf node, then its descendant nodes at depth n must also be Collatz-leaf nodes.
In the above proof, the strong Collatz transform times of Nmx is m, and the strong Collatz transform times of Nnx is also m. So, the strong Collatz transform times of the descendant integer of a Collatz-leaf integer is not bigger than that of the Collatz-leaf integer.
The above proof, we suppose the following is true: Perform the Collatz transforms on Nnx m times, the transform result of its high part Nnm must be an integer. This is also the reason why we set the strong Collatz transform times to be less than or equal to the depth of the node when we define the Collatz-leaf node.

In summary, we have the following conclusions:
(1) If the Collatz conjecture is true, for any positive integer N greater than 1, the minimum number of its Collatz transform times： mcMin = log2 N (2-1) (2) If the strong Collatz conjecture is true, the Collatz conjecture must also be true, and for any positive integer N greater than 1, the maximum number of its Collatz transform times： mcMax < mscMax *(N-1) mscMax is a finite positive integer which is the maximum number of strong Collatz transform times。 (3) By constructing a non-negative integer inheritance decimal tree, we defined the Collatzleaf node on the decimal tree. Furthermore, we proposed and proved the Collatz-leaf node Inheritance theorem(Theorem4.3).
Because of the inheritance of Collatz leaf node, with the increase of node depth, the proportion of Collatz-leaf nodes in the total nodes is also increasing. If all nodes are Collatz-leaf nodes at a limited depth, then all nodes at and below this depth are Collatz leaf nodes. Correspondingly, all positive integers at and below this depth are Collatz leaf integers. Because the clone nodes at this depth correspond to all non-negative integers less than this depth, so we prove that all integers greater than 1 satisfy the strong Collatz conjecture. The strong Collatz conjecture is true, the Collatz conjecture must also be true,

Computer verification and processing
In the rest of this paper, we will prove by computer calculation and processing: All nodes at and below depth 800 are Collatz-leaf nodes. Correspondingly, all the integers greater than or equal to 10 800 are Collatz leaf integers. Because the clone nodes at depth 800 correspond to all non-negative integers less than 10 800 . So we prove that all integers greater than 1 satisfy the strong Collatz conjecture and the maximum number of strong Collatz transform times mscMax is 800.
Strong Collatz conjecture is true, the Collatz conjecture must also be true. From(2-2), for any positive integer N greater than 1, the maximum number of Collatz transform times: mcMax < 800 * (N-1)

Collatz-leaf node inheritance verification
Collatz-leaf node inheritance is the core theorem of this paper. By computer calculation and processing, we will verify the correctness of this theorem. Traversal the nodes at depth 10 (from 0,000,000,002 to 9,999,999,999), get all Collatz-leaf nodes, for each Collatz-leaf node at depth 10, verify that its 9 child nodes at depth 11 are also Collatz-leaf nodes. Figure 3 is the computer main program to verify Collatz-leaf node inheritance theorem.

Strong Collatz conjecture computer verification
According to Collatz-leaf node Inheritance theorem, the descendant nodes of a Collatz-leaf node are also Collatz-leaf nodes. Therefore, we can traverse the whole non-negative integer inheritance decimal tree from the first depth, search and find the Collatz-leaf nodes on the decimal tree, take the Collatz-leaf nodes as the termination nodes with degree 0, get all the Collatz-leaf nodes and connect the neighboring Collatz-leaf nodes in turn, which will form a closed curve. The all nodes at and below the closed curve must be Collatz-leaf nodes. The depth of the lowest Collatz-leaf node on the closed curve is the maximum number of strong Collatz transform times mscMax. Thus, for any positive integer N greater than 1, its maximum number of Collatz transform times: mcMax < mscMax *(N-1) Because of traversing the whole non-negative integer inheritance decimal tree, it needs Supercomputer Center and cloud computing for super large data processing. In this paper, computer program works at nodes based on depth. It traverses the nodes at depths from 1 to 12, each depth is processed separately. For the nodes below depth 12, we select some depths, and carry out random sampling calculation for each depth of nodes.
(Definition 5.1)Collatz-leaf node ratio：The ratio of Collatz-leaf node number to total node number is defined as Collatz-leaf node ratio.
Thus, when Collatz-leaf node ratio is equal to 1, all nodes at this depth are Collatz-leaf nodes. Figure 5 is the computer main program to traverse nodes based on the depth.

Figure 5
Traversal nodes based on depth Figure 6 is the program running interface for traversal nodes based on depth. The program performs traversal processes at depth 10, from 0,000,000,002 to 9,999,999,999. The total nodes are 9.999,999,998 and the Collatz-leaf nodes are 9,374,999,998, so the Collatz-leaf node ratio at tenth depth is 0.937499999.

Figure 6
Program interface of node traversal Running the program from depth 1 to 12, the results are shown in In order to process large numbers, we use 50 unsigned long integers to represent a large number. An unsigned long integer is 64 binary-bit, which can represent a maximum of 20 decimal-bit integer, and 50 unsigned long integers can represent a 1000 decimal-bit positive integer, that is, a large positive integer such as 10 1000 Figure 7 is the program running interface with 50 unsigned long integers. At depth 100, we randomly sample 10 billion nodes and the Collatz-leaf nodes is 9997618831, so the Collatzleaf node ratio at depth 100 depth is 0.9997618831.

Figure 7 Program running interface with 50 unsigned long integers
On the non-negative integer inheritance decimal tree，we randomly sample 10 or 100 billion nodes at depths 50, 100, 200, 300, 400, 600 and 800, respectively. The running results are shown in table 2. At depth 800, the Collatz-leaf node ratio is 1.00000000000. In order to assess the precision of the above random sampling results，we randomly sampled nodes 10 times at depth 300. Each random sample has 10 billion nodes. The Collatz-leaf node ratio is calculated separately, and the 10 results are shown in table 3.
The mean square error is very small, so the results are quite credible.

Conclusions
This paper redefined Collatz conjecture. Furthermore, we proposed the strong Collatz conjecture: For any integer N greater than 1, perform the Collatz transform on N, repeat the process msc finite times, and the transform result Nm < N. We have proved that if strong Collatz conjecture is true, Collatz conjecture must also be true.
Based on the computer data structure -tree, a non-negative integer inheritance decimal tree is constructed, and the nodes on the inheritance decimal tree correspond to non-negative integers. We further defined the Collatz-leaf node (corresponding to the Collatz-leaf integer) on the decimal tree, Collatz-leaf nodes must satisfy strong Collatz conjecture. Derivation through mathematics, we have proved that the Collatz-leaf node (corresponding to Collatz-leaf integer) has the characteristics of inheritance. With computer large numbers and big data traversal and random sampling calculation, we have concluded that all nodes at and below depth 800 are Collatz-leaf nodes. Correspondingly, all the integers greater than or equal to 10 800 are Collatz-leaf integers. Because the clone nodes at depth 800 correspond to all nonnegative integers less than 10 800 . So we prove that all integers greater than 1 satisfy the strong Collatz conjecture and the maximum number of strong Collatz transform times: mscMax < 800 Strong Collatz conjecture is true, Collatz conjecture must also be true. And for any integer N greater than 1, the maximum number of Collatz transform times from N to 1: mcMax < 800 * (N -1)

Next steps and prospects
1. With random sampling calculation, we get the maximum number of strong Collatz transform times: mscMax < 800. In order to calculate the determined value of the maximum strong Collatz transform times, we need Supercomputer Center and cloud computing to traverse the whole non-negative integer inheritance decimal tree.
2. The non-negative integer inheritance decimal tree proposed and constructed in this paper also can be used for the proof of other mathematical problems. For example, the famous Goldbach conjecture: Any even number greater than 2 can be written as the sum of two prime numbers. By combining inheritance decimal tree and mathematical induction, we can use the following proof processes for the Goldbach conjecture: (1)At depth 1 and 2，prove that any even integer greater than 2 can be written as the sum of two prime numbers.
(2)Take any even node at depth m on the inheritance decimal tree, and its corresponding m-bit decimal even integer is Nm, assuming that Nm can be written as the sum of two prime numbers.
Nm= PA + PB PA and PB are prime numbers.
Prove that any of the 9 (m+1)-bit decimal even numbers can also be written as the sum of two prime numbers.
Declarations I declare no competing interests.