THE FERMAT'S LAST THEOREM FROM THE EYE OF PHYSICIST

: the article is about the fact that the extraordinary beauty and concise-ness 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.

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 of physicist.
If a triple of integers a n + b n ≡ c n 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 a n is equal to the difference between the volumes c n -b n . 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 c n -b n into a small cube a n and vice-versa. Here below a layer is defined as a set of points of a multidimensional spaces of real numbers R n between successively following hypercubes with integer edges S i = e i+1 \e i . The layer, like the whole n-dimensional figure, consists of elementary hypercubes 1 n in whole number space denoted as Z n .
The designed construction of three nested hypercubes can be filled of layers stepby-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 more 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: (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 i k 1 n-k -i.e. cuboid. «At the destination» volumes 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 1 n .
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 right 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 Z n , 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. There is no parallax effect.
(The thesis about the piercing (or penetrating) rather than cutting plane of a twodimensional 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 R n . 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].  What follows is a reformulated statement of the theorem a n + b n ≠ c n for integers and degree n > 2, the name and surname of the author of the short proof.

Proof of the opposite
Let's compare the expression a n + b n = c n 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 = {a n }, C = {c n \ b n } and VA = VC, or cardinality |A| = |C| take place, then elementary unit cubes 1 n 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 n for n > 2. It can be assumed sign '≡' in the expression above That means independence from the scale and set partition of our construction The word Reduce! -[try] has been thrown as a challenge. Reduction is prohibited for n > 2 The layer Si = ei+1 \ei is defined as the set of points in R n , obtained by the operation of the difference of sets in the form of successive hypercubes with integer edges ei +1 = (i + 1) n 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 1 n or 2 n , depending on the parity of the partition (it does not matter). The chain of sets forms a large hypercube c n , 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 Sj in the Z n , n >2. => 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 where the first hypercube e 0 = 1 n or 2 n 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 a n is the set of layers from 1 to k, the b-Medium b n is the set of layers from k+1 to k + l and the c-Large c n is the set of layers from k+l+1 to k + l + m. The layer is defined as the subset difference S i = e i \ e i-1 , i > 1. The first hypercube e 0 denote 1 n or 2 n , 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.   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
Why does Pythagorean triples exists specifically for the two-dimensional case, i.e. a n + b n = c n 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 caseon triangles. Any arbitrary layers are commensurate, as well as Definition of the layer as a set of points in R n obtained by the operation of the difference of sets Si = ei+1\ei or algebraic expression (i+1) ni n via the Newton's binomial theorem: here the coefficients of C k n 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. Содержимое доступно по лицензии Creative Commons Attribution 4.0 license (CC-BY 4.0) From the point of view of physics, we compare hypercubes with integer edges: a n = c n -b n . 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 V A ↔ V C 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 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. Edge of the cube -72 mm, thickness of plywood -4 mm.

In a few words
If a triple of integers a n + b n ≡ c n 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 |a n | is equal to the difference between the volumes |c n \ b n |. Because of equivalence of volumes (measures) there should exist continuous bijection function f: {c n \b n } → {a n } so single layer from the set {c n \ b n } is mapped to a set of layers into |a n |. But such funtion in Z n , n > 2 maintaining the fundamental properties of the construction: central symmetry and continuous succession of layers based on a series of natural numbers N 1 .
The construction of three nested hypercubes with integer edges is not exists in a space of whole numbers Z n , 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