Ibratjon Aliyev – All sciences. №3, 2023. International Scientific Journal (страница 2)
Now, when the general form for the doubly differentiated case is obtained, it is necessary to return to the primordial ones, because this is the identity, resulting in the following equalities (15—16).
And indeed, this value is close to the most potential value, so this expression can be considered the second kind of writing of the exponential unit. Now, it is possible to proceed to the solution of the Euler equation for the general form of the intentional numbers, having carried out the first substitution and the usual replacement operations at stage (17) and (18) at the beginning.
When the necessary transformations come to an end, and other actions no longer take place, it is also sufficient to differentiate both parts of equality as a valid identity (19).
Differentiating the first part of the equality, we can come to the result in (20), and for the second part, the calculations will continue throughout (21).
Then, applying (22—25), one can come to the form (26).
As a result, it is enough to equalize both results in (20) and (26), since these are two parts of the identity, and then get (27) with the necessary simplification, and already in (28) with additional simplification and differentiation as an identity.
At the same time, the differentiation of the first part of equality is obvious in (29), as well as the second in (30), after which equality and the resulting transformations can be introduced into (31).
As a result, equalities are formed that need to be integrated twice, because their derivatives were taken earlier, getting (32).
Integrating the first part, a separate result is obtained in (33) and integrating the second part in (34).
Thus, it is possible to arrive at equality (35), from where it is possible to arrive at another equality in the same equation.
The result is really quite surprising, but this is equality (35), which came out after substituting the general form of an ingential number into Euler’s formula and the solution for this case is the ingential number (36). Thus, this is the first full-fledged equation, the solution of which was an intentional number.
Although the complex numbers themselves are located on the axis of numbers, this interval can also be expressed on the tangential plane. This coordinate system has an axis starting from infinity as the ordinate, and the abscissa has all real numbers. Thus, all exponential numbers can be represented on such a rectangular coordinate system, in the case of adding complex numbers – already in space.
Used literature
1. I. V. Bargatin, B. A. Grishanin, V. N. Zadkov. Entangled quantum states of atomic systems. Editorial office named after Lomonosov. 2001.
2. G. Kane. Modern elementary particle physics. Publishing house Mir. 1990.
3. S. Hawking. The theory of everything. From singularity to infinity: the origin and fate of the universe. Publishing house AST. 2006.
4. S. Hawking, L. Mlodinov. The supreme plan. A physicist's view of the creation of the world. Publishing house AST. 2010.
5. T. D'amour. The world according to Einstein. From relativity theory to string theory. Moscow Publishing House. 2016.
6. S. Hawking, L. Mlodinov. The shortest history of time. Amphora Publishing House. 2011.
ABOUT RESEARCH ON THE COLLATZ HYPOTHESIS IN THE FACE OF A MATHEMATICAL PHENOMENON
Aliev Ibratjon Khatamovich
2nd year student of the Faculty of Mathematics and Computer Science of Fergana State University
Ferghana State University, Ferghana, Uzbekistan
Аннотация. Когда об этой задаче рассказывают молодым математикам – их сразу предупреждают, что не стоит браться за её решение, ибо это кажется невозможным. Простую на вид гипотезу не смогли доказать лучшие умы человечества. Для сравнения, знаменитый математик Пол Эрдеш сказал: «Математика ещё не созрела для таких вопросов». Однако, стоит подробнее изучить данную гипотезу, что и исследуется в настоящей работе.
Ключевые слова: гипотеза Коллатца, числа-градины, ряды, алгоритм, последовательности, доказательства.
Annotation. When young mathematicians are told about this problem, they are immediately warned that it is not worth taking up its solution, because it seems impossible. A simple-looking hypothesis could not be proved by the best minds of mankind. For comparison, the famous mathematician Paul Erdos said: «Mathematics is not yet ripe for such questions.» However, it is worth studying this hypothesis in more detail, which is investigated in this paper.
Keywords: Collatz hypothesis, hailstone numbers, series, algorithm, sequences, proofs.
In short, its essence is as follows. A certain number is selected and if it is not even, it is multiplied by 3 and 1 is added, if it is even, then divided by 2.
We can give an algorithm of this series for the number 7:
7 – 22 – 11 – 34 – 17 – 52 – 26 – 13 – 40 – 20 – 10 – 5 – 16 – 8 – 4 – 2 – 1
Next, a cycle is obtained:
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.