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. 323.
GENERAL IDEA OF THE CONCEPT AND USE OF DIFFERENTIAL EQUATIONS IN THE DESCRIPTION OF SOME DYNAMIC PHENOMENA
Aliev Ibratjon Khatamovich
2nd year student of the Faculty of Mathematics and Computer Science of Fergana State University
Ferghana State University, Ferghana, Uzbekistan
Аннотация. Как некогда сказал Стивен Строгац: «Со времён Ньютона человечество пришло к осознанию того, что физики выражаются на языке дифференциальных уравнений». Разумеется, что данный язык используется далеко за пределами физики и талант использовать его, ровно, как и воспринимать, даёт новые краски при изучении окружающего мира. В настоящей работе описывается общее представление об этом методе и сам процесс его изучения.
Ключевые слова: дифференциальные уравнения, исчисления, алгоритмы, математическая физика.
Annotation. As Stephen Strogatz once said: «Since the time of Newton, mankind has come to realize that physicists are expressed in the language of differential equations.» Of course, this language is used far beyond physics and the talent to use it, exactly as to perceive it, gives new colors when studying the surrounding world. This paper describes a general idea of this method and the process of studying it.
Keywords: differential equations, calculus, algorithms, mathematical physics.
By themselves, differential equations arise every time it is easier to describe a change than absolute values. For example, it is easier to describe the nature of an increase or decrease in the growth or decline of the population or population of a particular species than to describe certain values at a certain point in time. In physics, more precisely in Newtonian mechanics, motion is described by force, and force is determined by constant mass and changing acceleration, which is a statement of change.
Differential equations are divided into 2 large categories ordinary differential equations or ODES involving functions with one variable, most often in the face of time and partial differential equations with several variables. If partial differential equations describe more complex characteristics, for example, temperature changes at different points in space, then ordinary differential equations describe more static characteristics that change over time.
As a good example, we can consider the process of falling of an object. As you know, the gravitational acceleration is 9.81 m/s2, which means that if you analyze the position of the body at every second and translate this state into vectors, they will accumulate an additional downward acceleration of 9.81 m/s2 every second. This gives an example of the simplest differential equation, the solution of which is the function y (t), the derivative of which gives the vertical component, and the velocity gives the vertical component of acceleration (1).
This equation can be solved by allocating (2) for speed and (3) for path.
Another interesting moment is when it is possible to describe the movement of celestial objects on this scale due to the force of gravity. So, two bodies are given whose attraction is directed towards each other with a force inversely proportional to the square of the distance between them (4).
It is known that the derivative of the coordinate is velocity, the derivative of velocity is acceleration and it is necessary to obtain a function for motion, but according to equation (4), only the equation for acceleration (5) is known.
It may be strange here that the derivative is equal to the same function, but this is a common phenomenon when the derivative of the first or higher orders is determined by the values of themselves. But in practice, it is more often necessary to work with second-order differential equations, as can be seen in the previous examples.
However, there are also differential equations with third (6) or fourth (7) derivatives or higher (8) derivatives, which are considered higher-order differential equations.
In a way, it turns out that you need to find infinitely many numbers, one for each moment of time, but in general this coincides with the description of the function. And most often, even if in many cases it is possible to apply the classical description, then to a greater extent the use of the technology of ordinary mathematical transformations no longer meets the requirements. The usual description of the characteristics of a mathematical pendulum can be a proof of this.
Considering the real and idealized case, it can be noted that idealization works only at small angles of deflection of the pendulum, but when the angle becomes large enough, for example, equal to a semicircle, then the graph describing its oscillations as a whole ceases to be similar to the graphs of sine or cosine. The reason for this is the need to describe its motion exclusively using not partial, but general equations of harmonic oscillations with second-order differential equations.