SOME NEW COMMENTS ON FERMAT 'S THEOREM AND HER PROOF

ABSTRACT

An alternative, new proof of Fermat's theorem is given. To do this, arbitrary integer three points y, x, z are represented in the following form, , . Then, with respect to the unknown number x, an algebraic polynomial equation of degree  is obtained. The absence of an integer natural root value, that is, the absence of the possibility of an integer natural solution for the value  is a proof of the correctness of Fermat's theorem. Three numbers  are numbers involved in Fermat's formula.

The resulting method of indirect proof is sufficient to state it in higher algebra courses for students of pedagogical and technical specialties of colleges and universities.

The deformation of the mapped region by the formula from Fermat's theorem is shown. Which at a simple, general educational level shows that the amplification of the deformation of the map clearly shows the possibility of fulfilling Fermat's theorem for natural numbers.

АННОТАЦИЯ

Приведено альтернативное, новое доказательство теоремы Ферма. Для этого произвольные целые три точки  представлены в следующем виде , , . Тогда относительно неизвестного числа  получается алгебраическое многочленное уравнение степени . Отсутствие целого натурального значения корня, то есть отсутствие возможности целого натурального  решения для значения  является доказательством верности теоремы Ферма. Три числа  –это числа, участвующие в формуле Ферма. Полученный метод косвенного доказательства является достаточным для того, чтобы излагать его в курсах высшей алгебры для студентов педагогических и технических специальностей колледжей и университетов.  Показана деформация отображаемой области по формуле из теоремы Ферма. Которая на простом, общеобразовательном  уровне показывает, что усилиние деформации отображения наглядно показывает возможность выполнения теоремы Ферма для натуральных чисел.   

Keywords: Fermat's theorem, new proof, alternative proof, Nonlinear mapping. Equation of degree n.

Ключевые слова: Теорема Ферма, новое доказательство, альтернативное доказательство, Нелинейное отображение. Уравнение степени n.

Introduction.

An unexpected story prompted the author to show interest in this Fermat theorem. In the 2024/25 academic year, I had a chance to conduct elective classes at the Physics and Technology Lyceum Flagman, Astana. The main goal was to train students to solve problems of increased complexity in physics and teach them how to solve olympiad problems. Classes went as usual. Everyone knows that knowledge is acquired on the basis of hard, hard and time-consuming work on both sides: a student and a teacher. The main thing is that there is a lot of literary material (books, problem books, methodological instructions). Both paper and electronic versions. In addition, video tutorials abound on the Internet. They cover almost all topics of classical physics. And in one of the classes, one student unexpectedly asked me about this very Fermat's theorem?

At first, I answered simply on a general encyclopedic tip. After all, I can't tell the student that I am a technical physicist and therefore I don't know mathematics at all. Then he himself was interested in this question. It became interesting, began to study and delve into the question more deeply.

And that's what came of it. I set forth further the acquired thoughts and knowledge gained as some new comments (and conclusions) to this long-known task.

Theoretical solution.

At the first, at the simplest level, I tried to show the clarity of the problem in a graphic drawing. To make everything clear and intelligible. On the same scale, in the same coordinate system on the plane, I drew three graphs together on one paper: . This is line 1 in Figure 1;  This is line 2;  . This is line 3 in Figure 1. Function graphs were drawn on large millimeter drawing paper. To show the exact correspondence of numerical values   of functions at different degrees. x and y are drawn on the same abscissa axis.

Figure 1. Function graphs drawn on large millimeter drawing paper

Lines for different ;    colored with different colors. On such graphs, you can visually show. That yes, only for a linear and quadratic function, the following Fermat theorem condition holds

                                                        (1)

where  are natural integers.

Function graphs were drawn on large millimeter drawing paper. To accurately, clearly and intelligibly show students the different values   of the function of a given nonlinear map for different values   of arguments. Figure 1 shows a small part of this large and large-scale figure.

Consider, for example, a quadratic function. Take three points along the X axis: 3, 4, 5.

The corresponding z-axis values maps (displayed) are 9, 16, 25. These points lying on the function itself  will be designed on the OZ axis. Then on the coordinate axis OY itself (at the projected points) you can see that it is really 16+9 = 25. The fact that equation (1) is not satisfied for all integer values   of x, y, z and starting from k ≥ 3 seems obvious from the consideration of these graphs. Mentioned above. On line 3, for the same values   X (3, 4, 5), we get the corresponding values   Z: 27, 64 and 125. Hence we see that yes, indeed, starting with k = 3 Fermat's theorem is true. On the other hand, this graphic illustration shows that this mathematical process can be viewed from a new point of view. Namely, at k ≥ 3, the scale of the nonlinear maps (display) is curved

 .

Let for some numbers (x, y) and for values  

.                                                       (2)

As a result of scale distortion, the value of the distance between maps (display) points  increases all the time. That is, the length is stretched. -  is the distance, the difference between two points in the image area of   this display. This can be seen in the above example with respect to the numbers (3, 4, 5); (9, 16, 25) and (27, 64, 125) in Figure 1. For , equation (2) becomes infeasible. As the numerical value k increases, this happens automatically, naturally and logically understandable. Moreover, such an extension, increase (or distortion) of the scale of the nonlinear map (1) occurs literally for all integers (x, y). Which are located on the axis OX. Which initially satisfy equation (1) at k = 1 and k = 2. So Fermat's theorem is correct. The larger the value of k, the steeper the graphs. That is, the nonlinearity of the maps (display) increases. Increasing the value of the number k in Figure 1 we can observe as some "process." Which is available for observation.

However, this is only a visual demonstration of the process

                                                      (3)

Which does not explain the reason, does not prove this theorem (the statement itself). But only visually shows this process (2) and (3). That is, it is just a demonstration of the result on a two-dimensional graph. However, there is a useful point in such a visual demonstration. It shows that only for linear and quadratic mapping is it possible to perform (1). However, for k ≥ 3, condition (1) -is not satisfied. In other words, amplifying the nonlinearity of the mapping greatly alters the scales. This is especially evident for close, adjacent points along the OX axis: .  

 But however, such an explanation, such comments in no way have any evidence. Explaining the nature, the essence of this theorem, this statement. This is just an illustrative display to explain the clarity of this theorem in the picture, in the graph.

The following calculations are still of some particular interest. In other words, they lead to new fundamental thoughts. Let's put it in order.

Equation (1) involves three numbers x, y, z. Let's take the point (number) x in the middle, that is, . And let , . Where -are some integers. That is, we shift by the number  to the left and by the number  to the right of the central number .

Then equation (1) is written as

 .                                            (4)

Applying Newton's binomial formula (4) one can bring to the form

.                                                     (5)

Let's analyze formula (5).

As solution (5) (finding the roots of equations) is known, expressed in terms of specific formulas is possible only for   [1-7]. These formulas must contain radicals of coefficients .

  At , equation (4) is known not to resolve as a function expressed in terms of . In other words, in the literature they say it - there are no solutions. However, this statement is a little inaccurate. Actually (5) may have some solutions. Only they are approximate numerical calculations, such as through Newton's iterative formulas. When they say that there are no exact solutions, they mean that there are no such analytical formulas as, for example, in the form of linear, quadratic and cubic formulas. Which give solutions through the coefficients  [1-7]. These are the following formulas:

,                                                       (6)

 ,                                               (7)

,                                                (8)

  ,                                    (9)

.                                  (10)

,

,                                                     (11)

 .

Formulas (6) - (11) show that in solutions the degree of radical-degree of extraction of the root directly depends on the degree of the algebraic polynomial itself. That is, in the case of the cubic equation, these are radicals of the third and even fifth degree of  , . For equations of degree , such direct solutions using radical formulas are not. However, hypothetically (based on mental fantasy) imagine that if there were such or similar direct solutions of algebraic equations. Expressed through extraction of roots from radicals. Then, in these solutions, the maximum degree of root extraction would correspond to the highest degree of the desired function

.                    (12)

.                  (13)

Hence the conclusion. The solution to  of equation (5) cannot in any way be an integer. Even if we imagine that all coefficients  - will be integers:  . This fact is a new (our) proof of Fermat's theorem. Since according to Fermat's statement at  equation (1) is not feasible, for all integers x, y, z. Therefore, formula (3) takes place.

Formulas (9) to (13) obviously prove and reinforce the above. For any integer values   p and q, the left side of the α and β - will no longer be integers. The well-known rule also comes to the rescue that any rational number that is raised to a rational degree is ultimately either a rational or a real number. It can never be an integer. Let's consider the proof with a simple example

.

By the opposite method, let's say that by chance  turned out to be an integer.  are integers. Then may also be an integer. There is such a probability when performing these operations among many integers. Then there would be . However, this contradicts the original truth that  is an indivisible, irreducible rational fraction, that is,  can only be an irreducible, indivisible rational fraction or a real number! Thus, numerous applications of formula (13) result in   being a non-integer. When the sum of some numbers from (13)   occurs, then it is enough that only one member is not an integer. As a result, the entire amount will not be an integer.

It can be a rational, real, or complex number. This fact proves Fermat's theorem. The fact that the solution of equation (5) has no integer natural root values.

The second option is to prove this statement. From the course of higher algebra we have the following result (Vieta's theorem) [1-6]

,

      (14)

,

.

Here   are different roots of equation (5). In our case , compare formulas (5) and (12).

(14) also shows that the number cannot have n different integer divisors in any way, see formula (14). Or in other words, the system of algebraic equations (14) cannot have only integer solutions, cannot have only integer values   of the roots . They can be (in principle) either rational or real or complex numbers. In other words, the probability of integer values for the roots of equation (5) is zero. This can also be shown (proved) by solving equation (5) or the whole system (14) by computer-based numerical iterative methods.

Thus, the above statements (results) confirm Fermat's theorem. Serves as indirect evidence of it.

Discussion.

In the introduction, it was said that the initial impetus for the study of this issue was precisely the physicists of the engineering and technical direction were the questions asked by the students of the physics and technology scientific circle.

Therefore, it was decided to submit an article for publication in the journal of physical and technical or engineering profile. Since all engineering, physical and applied science relies on and widely uses all the achievements of mathematics. Literally all its sections. The publication of an article in an engineering journal also pursues career guidance goals. Not only "hardware" (that is, engineering products and mechanisms) "live" real engineers. Therefore, we expect that this article will arouse keen interest among young people. Who choose engineering areas of specialization.

The book [7] contains a statement by the famous theoretical physicist, Nobel Prize laureate Academician L. D. Landau. Where it is said that for physicists and specialists of an engineering profile, a special form of presentation of any theoretical and mathematical material is needed. Which would not be carried away by difficult to understand, highly abstract mathematical and theoretical justifications of a fact. The whole book [7] well explains many mathematical and physical results from these positions. Which are distinguished by clarity and simplicity of presentation. It is a pity that this very good book is available only in Russian.

In this article, we did not aim to analyze the available all publications on this topic. Probably their number reaches several hundred articles. Or maybe a thousand will be typed? For one person, one scientist, this is not a feasible, difficult task, to collect all the articles. It's a case and it's a function of AI and Big Date. Therefore, we decided to present only our new idea and our new solution directly, immediately.

Readers will be able to familiarize themselves with the basics of the theory of nonlinear maps [8].

Conclusions.

Fermu's theorem that is, formula (3) can be proved. If arbitrary integer three points represent as follows , , . Then, with respect to the unknown number , an algebraic polynomial equation of degree   obtained.

The absence of an integer root value, that is, the absence of the possibility of an integer natural numerical solution for the value   is a proof of the correctness of Fermat's theorem.

Formula-inequality (3) holds. It holds.

The resulting method of indirect proof is sufficient to state it in higher algebra courses for students of pedagogical and technical specialties of colleges and universities.

Deformation of the displayed area by formula (3) is shown. Which at a simple school level shows that the strengthening of the deformation of the map clearly shows the possibility of fulfilling Fermat's theorem for natural numbers.

References:

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2. Lyapin E. S. (2022). Rurs vysshei algebry [Course of higher algebra]. M.-S-Pb.-Krasnodar: Lan, 368 p. (in Russian)
3. Durakov B. K. (2006). Kratkii kurs vysshei algebry [A short course in higher algebra]. M.: FizMatLit, 230 p. (in Russian)
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