All sciences. №3, 2023. International Scientific Journal

1 – 4 – 2 – 1 etc.

This leads to the hypothesis that if you take any positive integer, if you follow the algorithm, it necessarily falls into the cycle 4, 2, 1. The hypothesis is named after Lothar Collatz, who is believed to have come to this hypothesis in the 30s of the last century, but this problem has many names, it is also known as the Ulam hypothesis, Kakutani's theorem, Toitz's hypothesis, Hass's algorithm, the Sikazuz sequence, or simply as "3n+1".

How did this hypothesis gain such fame? It is worth noting that in the professional environment, the fame of such a hypothesis is very bad, so the very fact that someone is working on this hypothesis may lead to the fact that this researcher will be called crazy or ignorant.

The numbers themselves that are obtained during this transformation are called hailstones, because, like hail in the clouds, the numbers then fall, then rise, but sooner or later, all fall to one, at least so it is believed. For convenience, we can make an analogy that the values entered into this algorithm are altitude above sea level. So, if you take the number 26, then it first sharply decreases, then rises to 40, after which it drops to 1 in 10 steps. Here you can give a series for 26:

26 – 13 – 40 – 20 – 10 – 5 – 16 – 8 – 4 – 2 – 1

However, if we take the neighboring number 27, it will jump at a variety of heights, reaching the mark of 9,232, which, continuing the analogy, is higher than Mount Everest, but even this number is destined to collapse to the Ground, although it will take 111 steps to reach 1 and get stuck in the same loop. The same interesting numbers can be numbers 31, 41, 47, 54, 55, 62, 63, 71, 73, 82 and others. For comparison, we can analyze the table (Table 1) and the graph (Fig. 1) for these interesting numbers.

Table 1. A series of long numbers for interesting values of numbers-granules (the first line is the original value)

Table 1. A series of long numbers for interesting values of numbers-granules (the first line is the original value)

Table 1. A series of long numbers for interesting values of numbers-granules (the first line is the original value)

Table 1. A series of long numbers for interesting values of numbers-granules (the first line is the original value)

Table 1. A series of long numbers for interesting values of numbers-granules (the first line is the original value)

Table 1. A series of long numbers for interesting values of numbers-granules (the first line is the original value)

Table 1. A series of long numbers for interesting values of numbers-granules (the first line is the original value)

Table 1. A series of long numbers for interesting values of numbers-granules (the first line is the original value)

Table 1. A series of long numbers for interesting values of numbers-granules (the first line is the original value)

Table 1. A series of long numbers for interesting values of numbers-granules (the first line is the original value)

Table 1. A series of long numbers for interesting values of numbers-granules (the first line is the original value)

Table 1. A series of long numbers for interesting values of numbers-granules (the first line is the original value)

Table 1. A series of long numbers for interesting values of numbers-granules (the first line is the original value)

Fig. 1. Graph of values for interesting numbers-granules of the algorithm

When the path of one number is so much different even from the neighboring one, how do you even approach the proof of such a hypothesis? Of course, all mathematicians were at a loss and absolutely no one could solve this problem. So Jeffrey Lagarias is a world expert on this problem, and he said that no one should take up this problem if he wants to become a mathematician. A large-scale work was carried out and a huge number of hailstones were studied, trying to find a pattern. Here it can be argued that all values come to one, however, what can be said about the path that all numbers take? The interesting thing is that this path is absolutely random.

For example, we can give a graph of all the values of this algorithm from 1 to 100 (Fig. 2).

Fig. 2. Graph of values for numbers-granules from 1 to 100

As you can see, most often growth begins initially and after a sharp decline, while the value of the number is simply not considered, however, if you make the graph logarithmic, there is a downward trend in its fluctuations. It can also be observed on the stock market on the day of the collapse, which is not accidental, because these are examples of geometric Brownian motion, that is, if you take logarithms and calculate the linear component, the fluctuations seem random, as if a coin was thrown at each step. And if we consider this function analysis as part of mathematical analysis, then there begins to be an obvious connection with probability theory. From where it turns out that when heads are obtained, the line goes up, and when tails go down, from where a special graph is obtained.

If we consider this chart when compared with the same exchange, then it is more likely in a short-term analysis, although in the long term, stocks are still growing, and "3x+1" is falling. You can also pay attention to the highest digit of the gradient numbers – this means a histogram, which is obtained by counting the number of digits from which the numbers begin in a number of granules for a particular number of the algorithm. If you add these values each time, for 1, 2, 3, etc., more and more data is obtained, while the ratio of the height of the columns becomes more and more ordered.

So for the first billion sequences, the most frequent value is one, 29.94% of all cases, 2 – 17,47%, 3 – 12,09%, 4 – 10,63%, 5 – 7,94%, 6 – 6,16%, 7 – 5,76%, 8 – 5,31%, 9 – 4,7% and the larger the figure, the less often it turns out to be ahead.

This arrangement is typical not only for hailstone numbers, there are many examples, these are the population of countries, the value of companies, all physical constants or Fibonacci numbers, and much more. This law is called Benford's law. Surprisingly, if you trace the violation of the Benford law in tax returns, you can even determine the fact of fraud. This law also helps to identify anomalies in the counting of votes in elections or many other things.

The best effect of this law occurs when the numbers entered in it have a spread of several orders of magnitude, as in this case, but Benford's law, unfortunately, cannot tell whether all the numbers end up in the 4-2-1 cycle. To do this, you need to use a different method. Initially, it is strange that this algorithm reduces all numbers to 1, given that even and non-even numbers equally and non-even numbers increase by more than 3 times, and even numbers decrease by 2 times.

This suggests the conclusion that all sequences should, in theory, go up, not down. But it is worth paying attention to the fact that whenever an operation is performed with a non-even number, that is, when it is multiplied by 3 and 1 is added, it necessarily turns into an even number, therefore, the next step it will always be divided by 2. It turns out that non-even numbers are not tripled, but multiplied by (3x+1) /2 or more precisely by 1.5, because 0.5 for large numbers can be ignored. So the maximum growth from this is exactly 1.5.

A graph has already been given for all numbers from 1 to 100, but it is worth considering a small case for all non-even numbers. As you know, in the second step they turn into even values, and then exactly half of them are immediately reduced, after dividing again to non-even. But every 4 numbers will have to be divided by 2 twice, which means that these non-even numbers are ? of the previous one. Every 8th number will have to be divided by 2 three times to get an even number. Every 16 four times, etc.

So taking the geometric mean, you can see that in order to get from one non-even number to another through all even numbers, you need to multiply it by ?, which is less than one, hence it turns out that statistically, this sequence decreases more often than it grows.

Let's give an example for a large number, for example 341. Its row looks like this:

341 – 1024 – 512 – 256 – 128 – 64 – 32 – 16 – 8 – 4 – 2 – 1.

He had only one non-even and all even numbers, which is why this series is remarkable. However, they can be depicted both in the form of graphs and in the form of trees, showing how one of the numbers is connected with the next in its sequence, creating a graph.

And if the hypothesis is correct, then any number should be in this huge graph, consisting of an infinite number of "streams" forming 4-2-1 cycles in one stream. There is an interesting visualization of such a graph, which uses an algorithm that on non—even numbers, it rotates clockwise at the selected angle, and counterclockwise on even numbers.

As a result, an interesting curved structure is obtained, more often in one direction. Resembling coral, algae or a tree in the wind. But this is only for a small number of numbers, for huge arrays, changing the angles of rotation, you can create huge and dazzlingly beautiful figures, as if generated by nature.

The hypothesis seems to be incorrect only in 2 cases:

1. If a number is found that will give infinity in the algorithm, that is, for some unknown reason, this "force of attraction" to 4-2-1 should not act on it;

2. Somewhere there is a sequence that would form its own closed loop, and all the numbers in it should be outside the main graph.

However, none of these options has been found yet, although all numbers up to 2 to the 68th power have already been checked by a simple search, which equals 295,147,905,179,352,825,856 numbers. It is known for sure that all the numbers from these values come to the 4-2-1 cycle. Moreover, based on these data, it is calculated that even if there is such a special data cycle, it should consist of at least 186 billion numbers. And it turns out that all the works indicate that the hypothesis is true, but still does not prove it.

Another way was also chosen. A scattering graph was constructed by taking the numbers themselves on one axis and the values on the other. If it can be proved that in any sequence of the algorithm there is a number smaller than the original one, it is possible to confirm the Collatz hypothesis. But any initial number is reduced to a smaller number, which in its own sequence will lead to a number even smaller, etc., up to 1.

That is, the only possible outcome for this particular case is the 4-2-1 cycle, but it has not yet been possible to prove this.

Although in 1976, Riho Terras showed that almost all sequences include values below the original one. In 1979, it was shown that the values would be less than the original ones by these values raised to the power of 0.869. Later, in 1994, the degree became more precise – 0.7925. Here, almost all the numbers mean that when the initial values tend to infinity, the proportion of the limiting function tends to 1. In 2019, mathematician Terry Tao was able to prove that this algorithm obeys even stricter restrictions.

He managed to show that all numbers will be less than the values of the function at any point, provided that the limit of the function, when the variable tends to infinity, will be equal to infinity. In this case, the function can grow arbitrarily slowly, the same logarithm, or logarithm logarithm, or logarithm logarithm, etc. This allows us to assert that arbitrarily small numbers exist in the series of any initial number. And as it was said in 2020, only a direct proof of the hypothesis can be better than this.

Used literature

1. Hayes, Brian. The ups and downs of hailstone numbers. American. – 1984. – No. 3. – pp. 102-107.

2. Stewart, Ian. The greatest mathematical problems. – M.: Alpina non-fiction, 2015. – 460 p.

3. Jeff Lagarias. The 3x+1 and its generalizations. American Mathematical Monthly. – 1985. – Vol. 92. – P. 3—23.

GENERAL IDEA OF THE CONCEPT AND USE OF DIFFERENTIAL EQUATIONS IN THE DESCRIPTION OF SOME DYNAMIC PHENOMENA

Aliev Ibratjon Khatamovich