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16. Гиперкуб деревянный 3D на сборной подставке с доказательством Великой теоремы Ферма [Электронный ресурс]. - Режим доступа: https://clck.ru/33KZpY (дата обращения: 09.02.23).
17. Белова Л.Ю. Элементы теории множеств и математической логики. Теория и задачи: учебное пособие / Л.Ю. Белова // Ярославский госуниверситет. - 2012 - С. 26-27. - ISBN 978-5-8397-0878.
18. Виро О.Я. Элементарная топология / О.Я. Виро, О.А. Иванов, Н.Ю. Нецветаев [и др.]. - М.: МЦНМО, 2010. - 352 с. -ISBN 978-5-94057-587-0.
19. Начала Евклида / Д.Д. Мордухай-Болтовский; пер. с греч.; под ред. М.Я. Выгодского, И.Н. Веселовского. - М.: ГТТИ, 1948. - С. 123-142.
20. Скулачёв Д.П. Корреляция данных по анизотропии реликтового излучения в экспериментах WMAP и «Реликт-1» / Д.П. Ску-лачёв. - М.: УФН, 2010. - С. 389-392.
УДК 511
DOI 10.21661/r- 559647
Авдыев М.А.
Великая теорема Ферма с точки зрения физика
Аннотация
В статье речь идёт о том, что необычайная красота и лаконичность формулировки последней теоремы Ферма заставляют нас искать ее визуальное решение. Давайте попробуем рассмотреть теорему Ферма глазами физика. Возможно, с этих позиций Пьер де Ферма нашел решение, основные идеи которого схематично уместились бы на довольно широких полях книги, в нескольких рисунках. Скептики продолжают считать, что Пьер де Ферма, вероятно, ошибся. Между тем последовательное применение основных принципов физики, геометрии и инженерного дела заставляет нас мыслить по-другому.
Ключевые слова: Ферма, симметрия, теория чисел, изотропия, однородность.
Avdyev M.A.
The Fermat's last theorem from the eye of physicist
Аннотация
The article is about the fact that the extraordinary beauty and conciseness of the formulation of Fermat's Last Theorem make us look for its visual solution. Let's try to consider Fermat's theorem from the eyes of physicist. Perhaps from this positions Pierre de Fermat found a solution whose main ideas would fit schematically in the fairly wide margins of the book, in a few drawings. Skeptics continue to believe that Pierre de Fermat was probably mistaken. Meanwhile, consistent application of the basic principles of physics, geometry, and engineering make us think differently.
Ключевые слова: Fermat's symmetry, number theory, isotropy, homogeneity.
Introduction
Fermat's Last Theorem was formulated by Pierre de Fermat in 1672, it states that the Diophantine equation:
an + bn = cn (1)
has no solutions in integers, except for zero values, for n > 2. The case degree of two is known in the school course under the name theorem Pythagoras. Euler in 1770 proved Theorem (1) for n=3, Dirichlet and Legendre in 1825 - for n = 5, Lame - for n = 7. In 1994 Prof. Princeton University Andrew Wiles [1] proved (1), for all n, but this proof, contains over
one hundred and forty pages, understandable only to high qualified specialists in the field of number theory.
But there is also a brief proof to the contrary the eyes ofphysicist.
If a triple of integers an + bn = cn exists, then it can map three nested integer edges hypercubes into each other (the centers of the nested hypercubes are aligned with the origin coordinates) while the volume of the small hypercube an is equal to the difference between the volumes cn - bn. Here the identity sign '=' means independence from the scale and set partition of our construction, i.e., a triple of integers in meters, decimeters, centimeters, millimeters. It is easy to prove that the condition for the equality of volumes and the properties of the central symmetry, continuity of
the formed construction mutually exclude each other. To understand this let's mentally move the layer from set of points in space described by the formula cn - bn into a small cube an and vice-versa.
Fig. 1. The figure of one layer (left) and set of layers in the octant (+, +, -) Here below a layer is defined as a set of points of a multidimensional spaces of real numbers Rn between successively following hypercubes with integer edges Si = ei+1 \ei. The layer, like the whole n-dimensional figure, consists of elementary hypercubes 1n in whole number space denoted as Zn.
The designed construction of three nested hypercubes can be filled of layers step-by-step from the periphery to the center and vise-versa like building a frame house. This is the method used Euclid's Elements [2]. A layer from the c-Large hypercube must fit an integer number of times in the a-Small hypercube (due to the excess of large over small - two or mors times), otherwise the central symmetry of the construction or the continuity of the ordered layers will be lost.
Here understanding the structure of the layer gives the following formula:
5i = (i + ir-i- = si:;-l(;)iir-i (2)
The formula above is convenient to use for figure three inscribed in each other hypercubes,»origin of coordinate placed in vertices». Another view is «origin of coordinate placed in centers of the hypercubes». Both geometrical constructions are transformed into each other due to reflections from hyperplanes perpendicular to each of the n coordinate axes, or by cutting the figure and scaling.
Each layer of hypercube have elements of dimensions n-1, n-2,... 1 (hyper)faces and edges such elements is described by formula ik1n-k - i.e. cuboid. «At the destination» volumes
Scanning the faces
of elements of each dimension must be identically equal the volume of the corresponding moved element, by virtue of the principal incompressibility of the volume of a solid body and the equivalence of the quantity elementary hypercubes 1n. These conditions lead to a system of n-1 equations that is not solvable for n > 2 not only in rational, but also in real numbers. To understand this, we recall impossibility of constructing a light triangle, in which the hypotenuse is equal to the sum of the lengths of the legs. It is easy to verify that for these conditions, one of the legs will necessarily be equal to zero. Consequently, the construction of three nested hypercubes with integer edges is not exists in a space of whole numbers Zn, n > 2 (aporia in terms of Ancient Greek philosophy), and there is no such triplet of numbers that would violate the Fermat's Last theorem.
Fig. 2. Three nested hypercubes.
Piercing by a two-dimensional plane.
There is no parallax effect.
(The thesis about the piercing (or penetrating) rather than cutting plane of a two-dimensional hypercube is easy to understand the basics of linear algebra AX = B (matrix form). It follows from the Kronecker-Capelli theorem that the set of solutions X to a system of linear equations forms a hyperplane of dimension n - rank A in Rn. For example, for a three-dimensional space and a two-dimensional intersection plane: dim (X) = 3 - 2 = 1. For 4-dimensional space and more dim(X) = 4 - 2 = 2 and so on. Therefore, a two-dimensional probe can be covered by a closed loop in a plane orthogonal to the piercing one, and it is appropriate to speak of piercing rather than intersection.)
In XVII century described physical approach was enough for proof, but not in XXI. More formal approaches are required [3].
Table 1
of the 3D Cube
Face of Cube
Comment
Ffefwvcrt s Last Theortm for b citions
vas fannitla№d П I 6J7, i Ьвсп irovni in Э020
an ■+■ bn * cn
l-'of wboïc aimibciï А О
- jmeiä n =» z
О .ЧлслС Avdyäv SCM 2-ОЙ Г?
Product name translated into Russian The Fermat's Last Theorem for billions referred to the registration of the database for computers, certificate number: RU 2020621077 Rospatent. Published 2020 Application number: 2020620372 registration: 11.03.2020 publication: 30.06.2020 «Proof of Fermat's theorem for Billions based on school knowledge». The theorem was formulated by Pierre de Fermat (pictured) in 1637 and proved by Marat Avdiyev in an original way in 2020. What follows is a reformulated statement of the theorem an + b" ^ cn for integers and degree n > 2, the name and surname of the author of the short proof.
Proof of the opposite
Continuation of table 1
Face of Cube
Comment
Let's compare the expression an + b11 = cn with the construction of three nested hypercubes having a common center at the origin with integer edges a, b, c. If the condition of equality of volumes in the discrete space of sets A = {an }, C = {cn \ b"} and VA = VC, or cardinality |A| = |C| take place, then elementary unit cubes 1n can freely circulate between the layers of this symmetric construction, since the uniformity of space is postulated in physics. Here below it is easy to make sure that these two conditions mutually exclude each other in the space of integers denoted as Z" for n > 2.
It can be assumed sign '=' in the expression above That means independence from the scale and set partition of our construction
S where
o r1 - Integer - 2'or 1
— o™
ceOS, .r.ustùs^., AJS*,,
»I -
tefupel
The word Reduce! - [try] has been thrown as a challenge. Reduction is prohibited for n > 2
The layer Si = e1+1 W is defined as the set of points in Rn, obtained by the operation of the difference of sets in the form of successive hypercubes with integer edges ei+1 = (i + 1)" based on a series of natural numbers - an «empty box» with a thickness equal to 1 unit. The unit depend on partition (scale q). As a result, a chain of sets is formed - «empty boxes», nested. into each other. In the center of the whole structure is a hypercube of 1n or 2n, depending on the parity of the partition (it does not matter). The chain of sets forms a large hypercube cn, or universal set U. This formula does not allow for layer-by-layer reduction (VA = VC) for our centrally symmetrical construction of homogeneous material. Each layer Si is incommensurable
with another S. in the Zn, n >2.
j '
=> Fermat's theorem is proved
The set Theory and binary relations approaches It should be noted without change generality that the natural numbers in formula (1) are related as a < b < c, and the situation of equality of edges a = b is excluded due to the irrationality of y' 2. The case of negative numbers can be considered by moving term into another part of the equation and substitution of variables - it is enough proving the theorem for the case of natural numbers a, b, c and generalize the result to whole numbers Z.
Let's consider inscribed hypercubes with edges, obtained from a series of consecutive natural numbers Np the centers which coincide with the origin of coordinates, and the faces are perpendicular to the axes coordinates Hypercubes ei with edges i based on a series natural numbers inscribed in each other form an increasing chain set and the inclusion relation in the set U which is understood as large hypercube with edge c: e0 c e1... c ek c ek+1... ek+l c ek+l+1 ... c ek+l+m cU e0 US1 U S2 ... U Sk USk+1... U Sk+l U Sk+l+1 ...
U Sk+1+m c U (3)
where the first hypercube e0 = 1n or 2n depending on parity In the reasoning below, this does not matter. A set partition one can see above. On the other hand, this formula describes a one-dimensional probe penetrating three nested hypercubes through a common center. The result of the Cartesian Product of two orthogonal probes can be seen in Figure 2 above, so the researcher can obtain a two-dimensional plane regardless of the space dimension. There is no parallax effect.
As mentioned in (3) the a-Small n-cube an is the set of layers from 1 to k, the b-Medium bn is the set of layers from k+1 to k + l and the c-Large cn is the set of layers from k+l+1 to k + l + m. The layer is defined as the subset difference Si = ei \ ei-1, i > 1. The first hypercube e0 denote 1n or 2n, in parity, but given the enclosures below, this detail is not leads to qualitative differences. The mathematicians of ancient Greece introduced the concept of incommensurability of linear segments.
Table 2
The postulates of Euclid in the Digital epoch
Figure in Euclidean space (Rn) provided central symmetry Analogue in Zn set of hypercubes provided central symmetry Dimension
A dot 1n 0 and at the same time n depending on the situation
A linear segment i11n-1 set of hypercubes cardinality = i lined up in a row or column one 1
A plane i21n-2 set of hypercubes cardinality = i2 ordered in a square 2
The linear segments of length and 1 are incommensurable. From these positions, each layer Si is incommensurable with another & in the Zn, n >2. It is easy to see that the analogous is true for sets of continuously following layers. The «uniqueness» of a layer can be formed by the condition: /3 scale and set partition and natural i, j for which the measure |SJ = |Sj |± |Sj-1| +... for n > 2, where |S| is cardinality i.e., quantity of 1n in the investigated set. The axiom of defining the measure (volume in in terms of physics) over the set is violated. The measures of the set of layers S are not possess the additivity property in whole number multidimensional space Zn for n > 2. The operations of addition, subtraction, reduction, other comparison of different layers is being prohibited. So, formula (3) describing structure of hypercube and understanding measure axiom for Si in Zn are enough for proof.
Let's define a continuous bijective function f U ^ U maintaining the fundamental properties of our construction: central symmetry and continuous succession of layers. If 3 f: C ^ A, where f is a bijective function mapping a subset C (c-Large) to a subset A (a-Small), then this means |C| = |A|.
Any function f is a superposition f = go h, where g -bijective function within a layer h - bijective function between different layers. Let us focus on the restriction of the relation g to one specific layer Si. What is the product of g|Si? By partition decreasing the thickness of the layers, it is possible to achieve a situation where a single layer Si from C is mapped to a set of layers {... S....}. According to definition of layer S| =ei+1\ei (i+l)n-inandits structure = y^I,"" c'^i"-1 layer S = U dk (where index k runs through values from 1 to n-1) pairwise disjoint equivalence classes of the elements ik1n-k because of equivalence property the function f should transfer pairwise disjoint equivalence classes of the elements ik1n-k (factor set) separately. To ensure the simultaneous matching of the elements of the layer more than to one class is impossible due to the unsolvability for n > 2 of the stipulated below system of n-1 equations):
jn-1 = in-1 + (i-1)n-1 +... ( two or more terms) jn-2 = in-2 + (i-i)n-2 +... (two or more terms) (3)
this series of equations continues from n-1 to 1 power. (The observing construction has been filling of layers from the periphery to the center.) So /3 equivalence function F in Zn, n > 2 maintaining the fundamental properties of our construction: central symmetry and continuous succession of layers.
Fig. 3. /3 equivalence function F in Zn, n > 2 maintaining the fundamental properties of the Construction: central symmetry and continuous succession of layers except
two-dimensional case (trapezoid) For the special case Z2 3 G, thanks to one equivalence class: comparison of trapezoid square is possible for V i, j: 3 hi and h such as Si * h = Sj*hj in Q numbers and by virtue of scaling for Z.
In the middle of the 20th century French mathematician Claude Chabauty in 1938 defended his doctoral dissertation on number theory and algebraic geometry, actively applied the methods of symmetry of subspaces in analysis of Diophantine equations. Minhyong Kim a mathematician from the University of Oxford, researching hidden arithmetic symmetry of the Diophantine equations, said: «It should be possible to use ideas from physicists to solve problems in number theory, but we haven't thought carefully enough about how to set up such a framework» [4]. The algorithmic insolvability of Hilbert's Tenth Problem was proved by Yuri Vladimirovich Matiyasevech in 1970 at the St. Petersburg branch of the Mathematical Institute. V. A. Steklov RAS [5]. From a philosophical standpoint, formula (1) has a contradiction between form (central symmetry) and content (volume) for n >2.
Table 3
Scanning the faces of the 3D Cube (continues)
Face of Cube
Comment
l-ür V IIЯ|XljLU-liJs i...
(,,h S,-.! 3 i,. j iltjlH sc»1 с u
n n . n a
Hyptitulw-'ïn I с gu I « hyper pynuikLd
c" is boiibOBeieous in Zr for a=>2 but for п > a
Why does Pythagorean triples exists specifically for the two-dimensional case, i.e. an + b11 = cn for integers and degree n = 2? Based on the central symmetry of the construction of three nested hypercubes, we consider only one axis. Rays drawn from the origin to the vertices of one face dissect the hypercube into 2n regular hyperpyramids. In the particular two-dimensional case - on triangles. Any arbitrary layers are commensurate, as well as sets of layers {... Si...} and {... Sj...} are trapezoids of height hj and h2. - by the number of layers in the set. For arbitrary averages by the line of the trapezoid Si, Sj, you can choose the corresponding. the number of layers and make the volumes VA = VC. equal with respect to symmetry and homogeneity of space.
и*» --
^ = j^-Hj-i r-4-_ . ,
UctrcG/jivable ca R fee Hi p- 2
fsy^syi^j S^ŒS,
c* is inboïTKïgeftecnis in Z*, fbrn > ?
Definition of the layer as a set of points in Rn obtained by the operation of the difference of sets Si = ei+1\e or algebraic expression (i+1 f-i" via the Newton's binomial theorem: ^ = ^ir' i'1-' 1" "1 here the coefficients of Ckn are the same for any layer i. Therefore, an identical comparison of the volumes (capacities of sets) Si and Sj, regardless of the partition and scale q (see above), means an element-by-element comparison of each dimension k separately (equiva-lence class). This leads to an unsolvable system of equations even in real numbers R, not to mention integers, for n > 2. Incommensurability of layers means heterogeneity of space. This conclusion contradicts to physics and axiom of measure in math. As a result, a logical contradiction was revealed.
Socîi a figure doesn t exist?
cn - bu Ф a0 in Z,:
Jt set uf t-tувтч djm ть1 j-
'&-d.!rn пчдЛк; >
11» io^ic of ii^ihLtrtiid: 3-d
CdUmJ symmetry of S. XOk coniiuiösiiy for a
From the point of view of physics, we compare hypercubes with integer edges: an = cn - bn. On the left is a homogeneous, isotropic, symmetrical figure of dimension n, and on the right is a set of layers of dimension n-1 that is either asymmetric or inhomogeneous, depending on the methods of construction and partitioning (scale). From the logical principle of the exclusion of the third follows: a centrally symmetric construction of three nested (hyper) cubes with integer edges a, b, c doesn't exist in reality -there is aporia. => The theorem is proved.
Conclusion
In the XVII century, there was still no separation between physics and mathematics in science, and it can be assumed that Pierre de Fermat used an interdisciplinary approach. In modern physical cosmology, the fundamental principle is the idea that the spatial distribution of matter in the Universe is homogeneous and isotropic when viewed on a sufficiently large scale, as a result of the evolution of
matter, laid down by the Big Bang. The assumption of free circulation of hypercubes from VA ^ VC and vice versa corresponds to the physical phenomenon of diffusion.
Since Pierre de Fermat claimed that «he found truly wonderful evidence, but the fields are too narrow to fit it.» Solving cumbersome equations is the wrong way to find evidence. From these positions, Fermat's Great Theorem is proved by careful consideration with just one glance, as in
Edge of the cube - 72 mm, thickness of plywood - 4 mm.
In a few words
If a triple of integers an + bn = cn exists, then it can map three nested integer edges hypercubes into each other (the centers of the nested hypercubes are aligned with the origin coordinates) while the volume (cardinality of the set) of the small hypercube |an | is equal to the difference between the volumes |cn \ bn|. Because of equivalence of volumes (measures) there should exist continuous bijection function f: {cn \bn} ^ {an} so single layer from the set {cn \ bn } is mapped to a set of layers into |an |. But such funtion in Zn, n > 2 maintaining the fundamental properties of the construction: central symmetry and continuous succession of layers based on a series of natural numbers Nr The construction of three nested hypercubes with integer edges is not exists in a space of whole numbers Zn, n > 2 (aporia).
It turned out that Pierre de Fermat's statement is not a figure of speech, that it should be taken literally. The mathematician did not lie at all when talking about the possibility of recording the main ideas of the proof in the fields of Diophantus arithmetic. At least six faces of the wooden cube were enough. From a philosophical standpoint, Fermat's Last theorem contains an irremediable conflict between form and content.
Литература
1. Nigel Boston University of Wisconsin-Madison Proof Fermat's Last The theorem, 2003, pp. 4-140.
2. Euclid, M.L. (1948) Beginnings Books, Books II. Translated from Greek and comments by D.D. Mordukhai-Boltovsky, ed. by M.Ya. Vygodsky and I.N. Veselovsky, GTTI, pp. 123-142.
3. Avdiyev, M.A. Education and upbringing of children and adolescents: from theory to practice / ed. by A.Y. Nagornova. -Ulyanovsk: Zebra, 2020, pp.330-348
4. Hartnett, K. (2017) Secret Link Uncovered Between Pure Math and Physics / K. Hartnett. [Электронный ресурс]. - Режим доступа: https://www.quantamagazine.org/secret-link-uncovered-between-pure-math-and-physics-20171201/ (дата обращения: 01.05.2023).
5. Matiyasevich, Yu.V. (1970) Diophantine property of enumerable sets. - Reports of the Academy of Sciences of the USSR, 191, №2, p. 279-282.
the ancient Indian treatises on mathematics, where the proof in one drawing was accompanied by only one word: Look! Perhaps through insight the attentive reader will be able to see the layers, the lack of additive property of equality of their volumes, the inevitable violation of the symmetry of the figure during the circulation of hypercubes.
Fig. 4. Above of author's 3D wooden hypercube with the proof Fermat's Last Theorem.
