Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatum in duos ejusdem nominis fas est dividere: cujes rei demonstrationem mirabilem sane detexi. Hane marginis exiguitas non caperet.

I. INTRODUCTION AND MAIN RESULT

It is well known that there are many solutions in integers to the equation $x^2+y^2=z^2$, 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, 66048, 66050). Also Greek mathematicians were attracted to the solutions of this equation. We notice that this equation has sequences of complex number solutions

and matrix solutions

In 1637, Pierre de Fermat wrote a note in the margin of his copy of Diophantus Arithmetica [1] stating that the equation

has no integer solutions. This is the Fermat Last Theorem. He claimed that he had found the proof of this Theorem. The only case Fermat actually wrote down a proof is the case $n=4$. In his proof, Fermat introduced the idea of infinite descent which is still one the main tools in the study of Diophantine equations. He proved that the equation $x^4+y^4=z^2$ has no solutions in relatively prime integers with $xyz\ne 0$. Solutions to this equation correspond to rational points on the elliptic curve $v^2=u^3-4u$. The proof of the case $n=3$ was given first by Karl Gauss. In 1753, Leonhard Euler gave a different prove of Fermat's Last Theorem for $n=3[2,3]$. In 1823, Sophie Germain proved that if $l$ is a prime and $2l+1$ is also prime, the equation $x^l+y^l=z^l$ has no solutions (x,y,z) with $xyz\ne 0(\mathrm{mod}l)$. The case $n=5$ was proved simultaneously by Adrien Marie Legendre in 1825 [4, 5] and Peter Lejeune Dirichlet [6] in 1832. In 1839, Gabriel Lame proved the case $n=7$ [7, 8, 9, 10]. Between 1840 - 1843, V. A. Lebesque worked on Fermat's Last Theorem [11, 12]. Between 1847 and 1853, Ernst Eduard Kummer published some masterful papers about this Theorem. Fermat's Last Theorem attracted the attention of many researchers and many studies have been developed around this Theorem. For example the work of Arthur Wieferich (1909), Andre Weil (1940), John Tate (1950), Gerhard Frey (1986), who was the first to suggest that the existence of a solution of the Fermat equation might contradict the modality conjecture of Taniyama, Shimura and Weil [29]; Jean Pierre Serre (1985 - 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. Therefore, proving Fermat's Last Theorem [20, 21]. Many mathematicians are still heavenly involved on studying Fermat's Last Theorem [22, 23, 24]. In 2021, Nag introduced an elementary proof of Fermat's

Last Theorem for epsilons[25]. In 2022, Mouanda constructed the galaxies of sequences of triples of positive integers solutions of the equation $x^2+y^2=z^2$. The unique characterization of the solutions of this equation allowed him to provide an elementary analytic proof of Fermat's Last Theorem [26]. The Fermat Last Theorem for positive integers has been extended over some number fields. In 1966, Domiaty proved that the equation $X^4+Y^4=Z^4$ is solvable in $M_2(\mathbb{Z})$ [27]. Let $G{L}_n(\mathbb{Z})$ be the group of units of ring $M_n(\mathbb{Z})$. Denote by

In 1989, Vaserstein investigated the question of the solvability of the equation

for matrices of the group $G{L}_2(\mathbb{Z})$ [28]. In 1993, Frejman studied the solvability of the equation (1.3) in the set of positive integer powers of a matrix $A$ with elements $a_{11}=0$, $a_{12}=a_{21}=a_{22}=1$ [29]. In 1995, the same case was studied by Grytczuk [30]. The same year, Khazanov proved that in $G{L}_3(\mathbb{Z})$ solutions of the equation (1.3 do not exist if $n$ is a multiple of either 21 or 96, and in $S{L}_3(\mathbb{Z})$ solutions do not exist if $n$ is a multiple of 48 [31]. In 1996, Qin gave another proof of Khazanov's result on the solvability of the equation (1.3) in $S{L}_2(\mathbb{Z})$ [32]. In 2002, Patay and Szakacs described the periodic elements in $G{L}_2(\mathbb{Z})$ and gave the answer to some problems concerning the equation (1.3) in matrix groups and in irreducible elements of matrix rings [33]. In 2021, Mao-Ting and Jie proved that Fermat's matrix equation has many solutions in a set of 2-by-2 positive semi-definite integral matrices, and has no nontrivial solutions in some classes including 2-by-2 symmetric rational and stochastic quadratic field matrices [34]. Fermat's Last Theorem has been extended to the field of complex polynomials of one variable [35].

This Theorem has many applications in Cryptography.

In this paper, we are mainly concerned with Fermat's Last Theorem for circulant matrices with positive integers as entries. Firstly, we focus our attention on the construction of the galaxies of sequences of triples of circulant matrices with positive integers as entries solutions of the equation $X^2+Y^2=Z^2$. In particular, Mouanda's matrix choice function allows us to construct practical examples of such galaxies. The elementary characterization of these matrix solutions allows us to prove Fermat's Last Theorem for circulant matrices with positive integers as entries.

Theorem 1.1. The equation

has no circulant matrix with positive integers as entries solutions.

We construct a galaxy of sequences of eigenvalues of circulant matrices and we prove Fermat's Last Theorem for eigenvalues of circulant matrices. Also, we construct a galaxy of sequences of complex polynomials over the unit disk $\mathbb{D}$ associated to circulant matrices and we prove Fermat's Last Theorem for complex polynomials over $\mathbb{D}$.

II. PRELIMINARIES

Definition 2.1. Let $\mathbb{A}$ be a unital Banach algebra. We say that $a\in \mathbb{A}$ is invertible if there is an element $b\in \mathbb{A}$ such that $ab=ba=1$. In this case $b$ is unique and written $a^{-1}$. The set

is a group under multiplication. If $a$ is an element of $\mathbb{A}$, the spectrum of $a$ is defined as

and its spectral radius is defined to be

Let $V=\{a_0,a_1,\dots ,a_{m-1}\}\subset \mathbb{C}$ be a subset of the set of complex numbers, denote by $C_V$ the following Toeplitz matrix:

This matrix is called a $m\times m$ -complex circulant matrix or a complex circulant matrix of order $m$. Denote by $C_m(\mathbb{C})$ the commutative algebra of $m\times m$ -complex circulant matrices. Let $ϵ=e^{\frac{2\pi i}{m}}$ be a primitive $m$ -th root of unity. Let us denote by $\mathrm{U}$ the following matrix:

This matrix is called Vandermonde matrix. It is well known that this matrix has the following properties:

U is non-singular, unitary, $U^{-1}={\bar{U}}^T$, $U^T=U$ and $U^{-1}=\bar{U}=U^{\ast }$. It is well known that all the elements of $C_m(\mathbb{C})$ are simultaneously diagonalized by the same unitary matrix $U$, that is, for $A$ in $C_m(\mathbb{C})$, one has

with $D_A$ is a diagonal matrix with diagonal entries given by the ordered eigenvalues of A: ${\lambda }_1^A,{\lambda }_2^A,\dots ,{\lambda }_m^A$. The factorization $U^{\ast }AU=D_A$ is called the spectral factorization of A [36, 37, 38, 39]. It is possible to write the matrix $C_V$ as one variable complex polynomial. Indeed, let $P$ be the cyclic permutation $m\times m$ -matrix given by

It is simple to see that

Let $\mathbb{D}=\{z\in \mathbb{C}:|z|\le 1\}$ be the unit disk. The complex polynomial

over $\mathbb{D}$ is called the associated complex polynomial of the matrix $C_V=f_V(P)$. It follows that if

is a $m\times m$ -complex matrix, then

is a $m\times m$ -upper complex triangular Toeplitz matrix. The complex polynomial

is also called the associated complex polynomial of the matrix $f_V(X)$.

III. THE UNIVERSE OF AN ALGEBRA

Definition 3.1. Let $x,y,z\in \mathbb{C}$ be complex numbers. Denote by

The triple $(x^n,y^n,z^n)$ is called the triple $(x,y,z)$ to the power $n$.

Definition 3.2. Let $x,y,z\in \mathbb{C}$ be complex numbers. Denote by

Definition 3.3. A universe of degree $\frac{p}{q}$ of the algebra $\mathcal{B}$ is the set ${\mathbb{F}}_{\frac{p}{q}}(\mathcal{B})$ of triples $(x,y,z)$ of elements of $\mathcal{B}$ which satisfy the law of stability

The element $(x,y,z)$ is called a star (or a planet) of the universe ${\mathbb{F}}_{\frac{p}{q}}(\mathcal{B})$. Every sequence $(x_k,y_k,z_k{)}_{n\ge 0}$ of elements of the universe ${\mathbb{F}}_{\frac{p}{q}}(\mathcal{B})$ is called a planet system of elements of $\mathcal{B}$.

The set

is called the complex cicutant universe of degree $\frac{p}{q}$. In particular, the set

is called the natural circulant universe of degree $n$. We are going to show that the universe ${\mathbb{F}}_2(C_m(\mathbb{N}))$ is not empty. Fermat's Last Theorem for circulant matrices is equivalent to say that

In other words, there are matrix complex universes which don't have triples of matrices of positive integers as entries elements.

IV. MOUANDA'S CHOICE FUNCTION FOR MATRICES

Denote by $C_{\ast }(C_m(\mathbb{C}))=\{h/h:C_m(\mathbb{C})⟶C_m(\mathbb{C})\}$, the set of complex functions over $\mathbb{C}$. Let

be the set of all subsets of ${\mathbb{F}}_2(C_m(\mathbb{C}))$. Theorem 2.5 of [26] allows us to claim that the appropriate choice of the values of $m_0(k)$ and $n_0(k)$ such that

leads to the construction of sequences of triples of circulant matrices with positive (or negative) integers as entries which satisfy the equation

Let $f_{\mathbb{M}}:C_{\ast }(C_m(\mathbb{C}))\times C_{\ast }(C_m(\mathbb{C}))⟶\mathrm{\Omega }({\mathbb{F}}_2(C_m(\mathbb{C})))$ be the function defined by