The Diophantine equation from the eye of physicist

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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 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 [2]. But there is also a brief proof to the contrary:

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 constuction 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.

Picture 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. 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 [3]. 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

$S_i={(i+1)}^n-i^n=\sum _{k=0}^{k=n-1}\left(\begin{array}{c}n\\ k\end{array}\right)i^k{1}^{n-k}$ (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 planes 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^k1^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 principle 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 integers, 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 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.

Picture 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 is required [1].

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 $\sqrt[n]{2}$.

The case of negative numbers can be considered by moving term into another part of the equation and substitution of variables - it is enough prove 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 N₁, the centers which coincide with the origin of coordinates, and the faces are perpendicular to the axes coordinates Hypercubes e_i with edges i based on a series natural numbers inscribed in each other form an increasing chain sets and the inclusion relation in the set U which is understood as large hypercube with edge c:

e₀ ⊆ e₁ . . . ⊆ e_k ⊆ e_k+1 . . . e_k+l ⊆ e_k+l+1 … ⊆ e_k+l+m ⊆ U

1ⁿ ∪ S₁ ∪ S₂ … ∪ S_k ∪ S_k+1 . . . ∪ S_k+l ∪ S_k+l+1 … ∪ S_k+1+m ⊆ U     (3)

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 Picture 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 S_i = e_i \ e_i-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. The postulates of Euclid in the Digital epoch

The linear segments of length √2 and 1 are incommensurable. From these positions, each layer S_i is incommensurable with another S_j 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: ∄ scale and set partition and natural i, j for which the measure |S_i| = |S_j |± |S_j-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 are being prohibited. So formula (3) describing structure of hypercube and understanding measure axiom for S_i in Zⁿ are enough for proof.

If ∃ the function F map the set of hypercube elements from the large hypercube cⁿ (the set U) into it F = { (y, z) | ∀ y ∈ U ∃! x ∈ U} then: ∀ F = G * H where: H = {(x, y) | x ∈ {Si} ∧ y ∈ {Si} } are equivalent relations within and G = {(x, y) | x ∈ {Si} ∧ y ∈ {Sj} : i ≠j } between different layers. Let us focus on the restriction of the relation G to one specific layer G|_Si = {(x, y)| x ∈ {S_i} ∧ {S_i} ∈ cⁿ \ bⁿ, y ∈ a-Small}.

Rejecting options that violate the central symmetry of the constuction, one can get either a set of layers in a-Small, or the entire subset of it as a whole, depending on the ratio of powers |S_i| and |a-Small| subsets. By scaling and decreasing the thickness of the layers, it is possible to achieve a situation where a single layer from cⁿ \ bⁿ is mapped to a set of layers into a-Small. Obviously ∄ equivalence function F in Zⁿ, n > 2 maintaining the fundamental properties of the our construction: central symmetry and continuous succession of layers (because the function G should transfer pairwise disjoint equivalence classes of the elements i^k1^n-k – cuboid, but 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):

j^n-1 = i^n-1 + (i-1)^n-1  + . . . ( two or more terms)

j^n-2 = i^n-2 + (i-1)^n-2 + . . . (two or more terms)    (4)

     . . .         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.

Picture 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  ∃ G, thanks to one equivalence class: comparison of trapezoid square is possible for ∀ i , j: ∃ h_i and h_j such as S_i * h_i = S_j*h_j 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” (https://clck.ru/32nv3t). The algorithmic unsolvability 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 [4]. From a philosophical standpoint, formula (1) has a contradiction between form (central symmetry) and content (volume) for n >2.

About the authors

M. Avdyev

Author for correspondence.
Email: red@nvsu.ru
ORCID iD: 0000-0002-3283-3467
Russian Federation, Surgut

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2. Picture 1. The figure of one layer (left) and set of layers in the octant (+, +, -)

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3. Picture 2. Three nested hypercubes. Piercing by a two-dimensional plane. There is no parallax effect

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4. Picture 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)

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