On Fermat’s Last Theorem and Galaxies of Sequences of Positive Integers

We construct sequences of positive integers which are solutions of the equation z . We introduce Mouanda’s choice functions which allow us to construct galaxies of sequences of positive integers. We give many examples of galaxies of numbers. We show that the equation has no integer solutions. We prove that the equation has no solutions in  . We introduce the notion of the planetary representation of a galaxy of numbers which allow us to predict the structure, laws of the universe and life in every planet system of every galaxy of the universe. We show that every multiverse contains a finite number of

We introduce Mouanda's choice functions which allow us to construct galaxies of sequences of positive integers. We give many examples of galaxies of numbers. We show that the equation

Introduction and Main Result
It is well that there are many solutions in positive integers to the equation + = x y z , for instance (3,4,5); (5,12,13). Around 1500 B.C., the Babylonians were aware of the solution (4961, 6480, 8161) and the Egyptians knew the solutions (148, 2736, 2740) and (514, 66,048, 66,050). Also Greek mathematicians were attracted to the solutions of this equation. We notice that this equation has sequences of complex number solutions ( ) 2 In 1637, Pierre de Fermat wrote a note in the margin of his copy of Diophantus Arithmetica [1] stating that the equation Weil [13]; Jean Pierre Serre (1985Serre ( -1986 [14] [15] [16], who gave an interested formulation and (with J. F. Mestre) tested numerically a precise conjecture about modular forms and Galois representations mod p and proved how a small piece of this conjecture the so called epsilon conjecture together Modularity Conjecture would imply Fermat's Last Theorem; Kennedy Ribet (1986) [17], who proved Serre's epsilon conjecture, thus reducing the proof of Fermat's Last Theorem; Barry Mazur (1986), who introduced a significant piece of work on the deformation of Galois representations [18] [19]. However, no final proof was given to this Theorem. This Theorem was unsolved for nearly 350 years. In 1995, using Mazur's deformation theory of Galois representations, recent results on Serre's conjecture on the modularity of Galois representations, and deep arithmetical properties of Hecke algebras, Andrew Wiles with Richard Taylor succeeded in proving that all semi-stable elliptic curves defined over the rational numbers are modular. This result is less than the full Shimura-Taniyama conjecture. This result does imply that the elliptic curve given above is modular.
This Theorem has many applications in Cryptography. It is well known that Fermat's Last Theorem is true. Now we know that only the equation 2 2 2 + = x y z admits positive integer solutions. Perhaps it will be a good idea of investigating the properties of these solutions and found out the applications of these solutions. For more than 3500 years, no sequences of triples of positive integers solutions of this equation were introduced before. This paper is structured as follows. In Section 2, we show that the universe We also prove that the equation x y z n has no positive integer solutions. Our study generates problems which are still unsolved. Since the creation, humans had a big strangle to understand the universe. This strangle did lead to the invention of telescopes. This instrument did allow humans to have clear pictures of galaxies of the universe. It becomes so crucial to create other tools which could lead to a better understanding of the structure and laws of the universe. Perhaps a mathematical tool will be cheaper. In Section 8, we introduce the main reasons why the Fermat Last Theorem is true. In Section 9, we show that every multiverse of triples of complex numbers contains a finite number of universes. This result was predicted in 2018 by Stephen Hawking [26]. We also identify every triple ( ) , , x y z of ( )   n as the continuous function , sin cos , sin sin , 2 cos 3 .
The graph of the function ( ) ( ) This planetary representation is the mathematical tool which allows us to have a clear understanding of the structure and laws of the universe. This mathematical tool allows us to predict the structure, laws of the universe. In Section 11, we predict life in every planet system of every galaxy of the universe.

Triples of Positive Integers Solutions of the Equation
Define the complex polynomial x n x m (2.7) Finally, if we assume that There are several ways of constructing sequences ( ) 0 , , A simple calculation shows that • Once again, let now choose Again, we have the following: The elements of the sequence of triples Perhaps it is possible to construct another model of sequences of triples of positive integers which depends on two parameters. Let us choose 2 0 The triples The construction of this sequence of triples allows us to set up our second model of sequences which satisfy the Equation (2.55). Let us replace 3 in The elements of the sequence of triples We can now claim that the natural universe ( ) 2   has no power element.

Mouanda's Choice Functions
Denote by , the set of complex functions over  . Let : leads to the construction of sequences of triples of positive (or negative) integers which satisfy the equation :    :  Mouanda's choice function  f allows us to construct galaxies of numbers.
For instance, the model • The elements of the galaxy

Construction of the Galaxy of Sequences of Positive Integers of Order 4 and Proof of the Main Result
In this section, we prove our main result and we show that there are galaxies which are bigger than the galaxy The elements of the galaxy , , , , , , , , , , . β λ β λ β λ The order of the galaxy 2,3, 2,3 2,3, 2,3 2,3, 2,3 , .
We can now give the characterization of the solutions of the equation , , ,

The Main Reasons Why Fermat's Last Theorem Is True
The main reasons why the Fermat Last Theorem is true are: n n x y z x y z x y z (8.4)

Problems
Our study generates several problems: The resolution of these problems will lead to the birth of a new theory called Galaxies of Numbers Theory.

What We Learn
We learn so far that the equation

Disjoint Multiverses of Complex Numbers
A multiverse (or parallel universes) is the collection of alternate universes that share a universal hierarchy. The idea of the existence of the multiverse has been around for long time. An idea which many theoretical physicists have been trying to prove by using string theory which is a branch of theoretical physics that attempts to reconcile gravity and general relativity with quantum physics. In 2018, Stephen Hawking on his paper entitled "A smooth exit from eternal inflation?" predicted that there are not infinite parallel universes in the multiverse, but instead a limited number and these universes would have laws of physics like our own [26]. Perhaps our strangle of understanding the existence of the multiverse and its structure is coming from the absence of a strong mathematical tool capable of representing our entire universe in terms of triples of numbers. , , x y z of complex numbers of a universe and our universe is made of an infinite number of galaxies.

n th Root of a Complex Number
It is well known that de Moivre's formula can be used to compute roots of complex numbers. Assume that n is a positive integer and ω is the n th root of the complex number z denoted by

Multiverse (or Parallel Universes) of Complex Numbers
In this section, we show that every multiverse contains a finite number of un-  x y z f ) of the associated universe.
In this model of representation, the temperature, the mass, the radius, the orbital period, the surface area, the pressure, the distance between stars (or planets), the speed of rotation and the age limit of a star (or a planet) are functions of ( )