**9. Fermat's paradox**

Consider two points that people have been pondering over for centuries,

*Diagrams of processes in asynchronous electric drive model with speed correction and discrete elements.*

One of the most famous paradoxes of philosophy is the Zeno's Paradox about

It states that Achilles will never catch up with a hulking turtle if she begins her

For several centuries this paradox was a "horror" of philosophers and theoretical scientists. And right now the explanations that are offered to ordinary people are

Meanwhile, it seems to me that everything is easily explained if we analyze the discreteness of time that Zeno offers and that which his interlocutors understood. Let's try to figure out the details of Zeno's reasoning. His main position: Achilles will not catch up with the turtle, because in the interval of time for which he will reach her position, she will go further. Zeno suggests that the interlocutors consider the whole movement as a sequence of states and intervals at those points where the turtle has already visited. These intervals will be shorter and shorter until they become infinitely small. However, Zeno did not apply such concepts. Actually, turning to the concepts of infinitesimal ones, he remains himself and leaves his interlocutors in terms of finite time intervals … and space too. And he comes and leads the rest of the participants in the conversation to a clear contradiction. He tells them: "I show you my time and space, which I interrupt at any time when I want and as many times as I want. You too can tear your time, in which Achilles easily catches up with a turtle, on as many sites as you like. So our times are the same, but in mine Achilles is

All ordinary people understand the discreteness of time as the same and fall into the "trap." In their head, time is unbroken, infinite, and the "time of Zeno" is only that time in which the tortoise is ahead of Achilles, and its division into an infinite number of sections—intervals. He "equates" it with the time of the interlocutor infinite time. Zeno says "ALWAYS," but it is in his time, and he evens out times with the number of intervals. But the intervals for Zeno and his interlocutors are different.

forever behind the turtle. So in yours, he will not catch up with her."

unaware that the whole thing is in very small detail.

Achilles and the tortoise. The paradox goes like this:

**8. Achilles paradox**

*Control Theory in Engineering*

movement before him.

very vague.

**342**

**Figure 18.**

One of the founders of modern science is Pierre Fermat, the author of many important decisions and discoveries. But he is best known for 400 years thanks to the paradox or "Fermat's theorem," which is a very vivid illustration of the possibilities of discretization of variables of mathematical quantities, since it is precisely the discreteness of four independent variables in Fermat's theorem that leads one equation to four unknowns for a condition that cannot be fulfilled.

Fermat's theorem states that there are no positive integers that would be a solution to the equation Xn + Yn = Z<sup>n</sup> for n greater than 2.

If any positive values of X, Y, or Z (or at least one of them) were allowed, then an equation with three unknowns for any degree would have an infinite number of solutions. This is undeniable and understandable.

But here is what happens if discreteness is introduced into an indisputable and understandable statement. It turns out that with such discreteness it is impossible to find at least one combination of three numbers and a degree corresponding to the solution of the Fermat equation.

Let us try to formulate; the theorem is a paradox with an "emphasis" on the discreteness of variables:

The sum of the natural degrees of two natural numbers is unequal to the same degree, starting from the third, no natural number.

For the first degree, this condition is not fulfilled, that is, for any two positive integers there is a third for the equality to be fulfilled.

For the second degree, there are solutions to the equation but not for any pair of numbers.

But for the third degree is no longer. Rather, there are, but some very large ones that mathematicians find once every hundred years. It is very difficult to check if there are such numbers yet. Even if there are very good computers.

About 30 years proof of Fermat's theorem was found [20]. Only reasonably good specialists, mathematicians, can understand it. And for all other people, this is not a solution to the original paradox: the simplest mathematical paradox connecting the simplest expressions to the simplest numbers.

It is as if helicopter pilots would win in mountaineering competitions. No one argues with the proof, or almost no one ... But questions remained.

What is the essence of Fermat's paradox? It may be that the discreteness of numbers turns an expression with several degrees of freedom (one equation with three unknowns) into a practically unsolvable expression. In other words, only a rigid definition of the variables involved in this condition turns excessive freedom into nonexistence.

Moreover, natural numbers are what most people see in their practical lives. And all the others are fractions. Complex vectors were for many years a "fabrication of

scientists" that had no connection with reality. And these natural numbers set up such a trick. If you look at Fermat's theorem from this position, a whole series of questions will arise.


Is it a coincidence that for the second degree, three variables still give solutions in our three-dimensional space, and in the third degree there are no solutions already?

And so many others ...

So, to summarize this replica, it can be argued that this condition (in Fermat's theorem) connects three independent variables and their nonlinear transformations defined by the fourth variable, the degree, and it is the discreteness of all variables that makes this simple equation (or formula) impossible.

This relationship of the dimension of equations and the nonlinearity of transformations with the discreteness of variables is, in the opinion of the author of the article, the main meaning of Fermat's paradox and one more confirmation of the fundamental concept of discreteness [21].
