An overview of the proof of the Collatz conjecture

0. Proof of the Collatz Conjecture

　By making a good correspondence between the operations of the Collatz conjecture and the properties of the Möbius loop, we can show that only one loop will appear in the operations of the Collatz conjecture. Since the Collatz conjecture is "only one loop will appear due to the operation", the Collatz conjecture is proved. 1.

What is the Collatz Conjecture?

　Just recently, the Collatz conjecture became a 100 million yen bounty issue. In commemoration of this, I thought I would come up with this proof.

　First, let me explain what the Collatz conjecture is. Please see image 1.

(Image 1)

　For an odd number, multiply it by three and add one, and for an even number, divide it by two. If we repeat this process, any natural number should eventually arrive at 1, and then enter a loop of 4→2→1→4... That is the claim of the Collatz conjecture.

Let me briefly check the Collatz conjecture.

Since we're here, let's consider the edit dates 1) 7 and 2) 716.

1) For 7

7→22→11→34→17→52→26→13→40→20→10→5→16→8→4→2→1

2) For 716

716→358→179→538→269→808→404→202→101→304→152→76→38→19→58→29→88→44→22→(Abbreviation. Same as above) 3.

3. 'False' proof of the Collatz conjecture

　Often, you will find papers and websites (including this one) claiming that the Collatz conjecture has been solved. Most of them are not sufficiently peer-reviewed and cannot be relied upon for sufficient credibility. So, perhaps a complete proof of the Collatz conjecture has already been given, but it just hasn't been properly evaluated.

　On the other hand, some of the 'proofs' are clearly false. They can be categorized as follows

The first one assumes that there is only one loop, and claims that all natural numbers can be represented by only one operation by doing the opposite operation of the Collatz conjecture (doubling at any given time, or subtracting 1 and dividing by 3) under this assumption.

The second is not a proof of the Collatz conjecture, but a program to efficiently find the order of operations as described in Chapter 2, or a rough estimate of the number of operations of the Collatz conjecture.

　In each of these cases, we are assuming at the beginning of the discussion that the Collatz conjecture is correct, that there will be only one loop in the method shown in Image 1. This would be a fallacy. 4.

4.1 Consideration of the Möbius loop

4.1 Considerations for Entering Data into a Möbius Loop

Summary of this section

Cauchy's integral formula depends on the information of the zeros contained inside the closed curve.

A closed surface cannot be created by a Möbius wheel.

A closed surface cannot be created by a Möbius circle. - A Möbius circle can only input data into itself.

　There is a Cauchy's integral formula. It says that regardless of the shape of the closed curve to be integrated around, the result of the integration is known by the zeros contained in its interior.

　In other words, inputting data into a closed surface on the complex plane is equivalent to creating a zero point inside it that corresponds to the data.

　So what if the path of integration is a Möbius circle? A Möbius circle has no front or back, so even if there is a Möbius circle in the plane, it will not create a closed surface.

(Image 2) The person in the middle with a different color is the thief.

　It is a little difficult. Take a look at image 2. Suppose there is a policeman standing in the integral path and a thief in the circle. If the integral path is twisted like a Möbius circle, then for one policeman, the thief is right in front of him, and for another policeman, the thief is behind him. Now, the thief has escaped from the circle. Then, to one cop, the thief is in front of him and behind him, and to another cop, the thief is in front of him and behind him.

　Let's get this straight. Before the thief ran away, to one officer, the thief was right in front of him; to another officer, the thief was behind him. This information has been passed on to headquarters. Now, after the thief escaped, for one officer, the thief was in front of him, and for another officer, the thief was behind him. Headquarters continues to be informed of this information. So what do you think? The thief can easily escape.

　Thus, using the Mobius circle, it is not possible to create a closed surface.

(Image 3)

　So where should we trap the data? The answer is inside a circle. Now look at image 3. The thieves are lined up among the cops. (You've already caught him, haven't you?). At this point, if the thief escaped, the two policemen could inform the headquarters that the thief beside them has moved.

4.2 How to represent the information of all natural numbers in a finite form

　In 4.1, we saw that we can input data into the Möbius circle itself. By the way, in order to consider the Collatz conjecture, we need to consider all natural numbers as the target. So, we introduce p-decimal numbers (p is a prime number). To be honest, I don't know much about p-decimal numbers, but this time, simply think of it as a modulo operation that can classify all natural numbers into p numbers (1~p), with p as the law.

　Thus, to consider all the natural numbers, we only need to consider the set consisting of 1~p.

4.3 Properties of the Möbius Wheel

The Möbius wheel is known to have the following four amazing properties One twist means that the front and back can be seen from the same perspective for the first time at the connection. One turn of incision refers to the method of finishing the cut as it approaches the point where it started, without changing it, while two turns of incision refers to the method of finishing the cut as it approaches the point where it started, once avoiding it, making another turn, and then returning to the point where it started.

Also, for the sake of convenience, we will assume that the torsion is more than two times. This is to maintain property 3.

1) If the torsion is odd and the incision is one round, there will be one circle; if the incision is two rounds, there will be two rings.

2) An even number of twists and one turn of the incision will result in two rings. In this case, the incision cannot be made two times in a row.

3) No matter how you twist, no matter how you cut the Möbius circle, if there are multiple rings, they will always have intervals where they touch each other. This property is not satisfied if and only if the torsion is a single one, so we are removing it this time.

4) No matter what state the Möbius wheel is in, the sum of the surface areas is always constant.

3) can be found by the backward method.

　Suppose there are two rings, A and B, whose relationship does not satisfy 3). If there exists another circle C that satisfies the condition of 3), it will interfere with the original circle A and B, making it impossible to return them.

　Assuming that there is no such wheel C, A and B are independent of each other, but it is not possible to create multiple wheels independent of a Mobius wheel that has been twisted more than once by an incision.

　From the above, 3) is correct.

4.4 Notes on Entering Data into a Möbius Wheel

For the sake of convenience, we will now define the data storage rule using the storage rule in 4.3(4). The definition is as follows "The ratio of the surface area of the initial Möbius loop to the total amount of data is preserved for any Möbius loop that appears in subsequent operations."

Suppose, for example, that the original Möbius wheel was twisted three times and cut into one round. At this point, a single circle is created, but it has twice the circumference length and 1/2 times the average height than the original. By definition, this Möbius circle will have the same total amount of data as the original Möbius circle. So the data is properly stored.

　Let's also suppose that the original Möbius wheel was twisted twice and cut into one circle. At this time, two rings are created. The ratio of the length of the circumference is 1:2 and the average height is both 1/3 times the original. By definition, this Möbius circle can hold 2/3 times as much data as the original Möbius circle on one side and 1/3 times as much on the other. The total amount of data will still be the same as the original Möbius wheel. The data is properly stored.

Also, in principle, by cutting the Möbius loop many times, the data capacity may become so small that not even a single number can be stored. Such rings should be assumed to have zero data volume and should be ignored. Also, consider that even if you ignore those rings, the total data volume of the whole is still saved.

4.5 Correspondence between the Collatz conjecture and the data input to the Möbius circle

Here is a Möbius wheel with 1~p data input. Let's assume there are more than two twists.

There are two possible operations. The first is to choose whether to increase the number of torsions by one or leave it as it is. The second is to choose whether to add one or two rounds of incision. However, when the number of twists is even, it is sufficient to select one round of infeed.

All other operations can be explained by a combination of the above two operations.

The result of the operations is a new loop with even number of torsions (Case C_(even)) and possibly another loop with odd number of torsions (Case C_(even&odd)).

By the way, in the Collatz conjecture, there are two possible cases: if you multiply by 1/2, you get an odd result and an even result (Case C_(n/2)), while if you multiply by 3 and add 1, you always get an even result (Case C _(3n+1)). This corresponds to C_(even&odd) and C_(even), respectively.

For clarity, the correspondence table is shown below.

(Table 1)

4.6 Proof of the Collatz Conjecture

　Finally, the conclusion. We have seen in 4.7 that there is a correspondence between the Möbius loop and the Collatz conjecture.

We now turn our attention to 4.3(3). According to this, if a Möbius circle is made of rings, then all rings will always have intervals that are tangent to each other. This means that by the way of the Collatz conjecture, there is one loop and all the numbers come back to this loop.

In the Collatz conjecture, such a loop is known as 4→2→1→4..., as already mentioned in 1.

From the above, we can see the following theorem.

"Using the method in the Collatz conjecture, all natural numbers return to only one loop. That loop is 4→2→1→4..."

This means that the Collatz conjecture is correct.

5. end

I've given an overview, but I've talked about it quite passionately. However, a more detailed proof would require more complicated formulas and would be much longer. The important thing is to pursue (pursue?) the essence of the proof. The important thing is to pursue (pursue?) the essence of the proof, so I think an overview is enough. If you still want to think about it rigorously using complicated formulas, please do so.

6. extra

If you would like to think more rigorously about the method of this proof, here are some points that I did not consider.

How do the congruent numbers t and u behave in the Möbius circle?

The trajectory of the Möbius circle for a certain number t. (Which way does it move when it is cut?)

The difference in behavior between p- and q-decimal numbers (p≠q)

Looking at Table 1, we can see that there are two operations A and B, and two sets of A and one set of B are created. If there is an algorithm that can make three kinds of sets with two operations like this for all natural numbers, in principle, it would make us come back to just one loop, just like the Collatz conjecture. Search for such an algorithm.

©︎2021 Yume Isioto