Turkish Journal of Analysis and Number Theory
Volume 12, 2024 - Issue 1
Website: https://www.sciepub.com/journal/tjant

ISSN(Print): 2333-1100
ISSN(Online): 2333-1232

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Research Article

Open Access Peer-reviewed

Joachim Moussounda Mouanda^{ }

Received January 16, 2024; Revised February 18, 2024; Accepted February 25, 2024

We introduce a new family of Toeplitz matrices called Rare matrices. We construct multiplicative commutative groups of Rare matrices and we establish connections with circulant matrices. We show that, in general, for every positive integer , the equation has an infinite number of matrix triple solutions with the matrices A, B and C do not have a common matrix factor. In other words, Beal's conjecture is not always true for matrix solutions. Mathematics Subject Classification(2010) 15A24, 11D72, 15A16, 11A51.

In 1993, Andrew Beal, a banker and amateur mathematician, while investigating generalizations of Fermat's Last Theorem, conjectured that if

(1.1)

then *a, b *and c have a common prime factor. Similar conjectures have been made before Beal's conjecture. For example, in 1914, Brun stated several similar problems ^{ 1}. In 1995, Darmon and Granville showed that if the positive integers x, y, and z are such that

then there are only finitely many triples of coprime integers A,B, C satisfying

This is known as the Fermat-Catalan Conjecture formulated by Darmon and Granville. For the case

with at least one of them is greater than 2, we have 10 well known examples:

It is straightforward to observe that a,b and c, in each equation, do not have a common prime factor. As we can see, one of the exponents (x,y,z) is 2 while Beal's Conjecture requires all three exponents (x,y,z) to be 3 or greater ^{ 2}. It is well known that the equation (1.1) has an infinite number of positive integer solutions. For example, the solutions

has bases with a common factor of 3, the solution has bases with a common factor of 7, and the solution has bases with a common factor of 2. There are infinitely many more well known solutions which are

and

In 1997, Beal initially offered a prize of 5,000.00 US dollars for a published correct proof or counterexample in an internationally renowned refereed mathematics journal. This prize was raised to 50,000.00 US dollars ^{ 3}. Recently, Andrew Beal changed his mind by raising this prize from 50,000.00 US dollars to 1,000,000.00 US dollars ^{ 4}. Beal's conjecture still remain unsolved. However, partial results have been proved by many authors ^{ 4, 5, 8}. In 2022, for the first time, Mouanda constructed a new family of Toeplitz matrices called Rare matrices which allowed him to show that the Diophantine matrix equation

has an infinite number of matrix solutions in ^{ 6}.

In this paper, we investigate the matrix approach of Beal's conjecture. In section 2, we introduce the new elementary properties of Rare matrices and we show that every positive rational number generates a multiplicative commutative group of Rare matrices. In section 3, we establish the connections between these groups and circulant matrices. In section 4, we discuss the Integer Factorization problem. In section 5, we notice that

with

The matrix structures of the above equation allow us to prove that, in general, Beal's conjecture is not true for matrix solutions.

**Theorem 1.1. **For every positive integer the equation

(1.2) |

has an infinite number of matrix solutions

with the matrices A, B and C do not have a common matrix factor. We have the same observations for the matrix solutions of the equations

(1.3) |

In this section, we introduce the structures and properties of Rare matrices. We also construct multiplicative commutative groups of Rare matrices.

Let

be the set of n-by-n complex matrices. The set

is not commutative.

**Definition 2.1.** A finite matrix is called a Toeplitz matrix if

Each descending diagonal from left to right of a Toeplitz matrix is constant. For instance, the matrix

are Toeplitz matrices with positive integers coefficients.

**Definition 2.****2****.**** **The nxn-Toeplitz matrices of the form

are called Rare matrices of order n and index 1.The index defines the number of non-zero complex coefficients of the matrix different to 1 and 0. For example, the matrix

is called a Rare matrix of order n and index 2. Rare matrices have been first introduced by Mouanda in 2022 during his investigation on the matrix solutions of the Fermat matrix equation [6].

A simple calculation shows that

and

We are interested on the investigation of the physical mutations of the coefficient inside the matrix

After simple calculations, we find out that

and

We can see that the coefficient is keeping moving through diagonals when the value of k increases. This is an interesting phenomenon. Let us investigate more properties of Rare matrices.

**Remark 2.3.** Let

be a Rare matrix of order n and index 1. Then

We can see that

These observations allow us to say that

Therefore,

Suppose that

We can observe that

**Definition 2.4.** A (binary) operation on a non empty set G is a function

An operation on a set G is associative if

for every a, b, c .

**Definition 2.5. **A semigroup (G,*) is a nonempty set G equipped with an associative operation A group is a semigroup G containing an element e such that:

(i) for all ;

(ii) for every , there is an element with

A pair of elements a and b in a semigroup commutes if . A group (or semigroup) is abelian if every pair of its elements commutes.

We can now prove that every positive rational number generates a commutative group of Rare matrices.

**Theorem ****2.6****.** Let

be a positive rational number such that

and let

be a Rare matrix of order n and index 1. Then the set

is a multiplicative commutative group.

**Proof.** Let

be a positive rational number. Remark 2.3 allows us to claim that

Therefore,

is a multiplicative commutative group. This yields us the desired result.

**New Notation:** Denote by

the set of multiplicative commutative groups of nxn-Rare matrices. This set can be considered as the representations classes of positive rational numbers. In this set, every positive rational number can be identified as the group .

We can denote by

, the representation class of in . Also, we can notice that

with is a Rare matrix of order n and index q. Assume that

Theorem 2.6 allows us to say that

In section, we show that the elements of the group allow the construction of circulant matrices of order n. Suppose that and . In this case, we have

Therefore,

is a commutative group of circulant matrices, since

and

We could use the elements of the finite group

to construct circulant matrices of order 5. Indeed, let

be a finite set of complex numbers. A simple calculation shows that

In the same way, we could use the elements of the group

to construct circulant matrices of order n+1. Indeed,

Integer Factorization is the decomposition, when possible, of a positive integer into a product of smaller integers and prime factorization is the decomposition, when possible, of a positive integer into a product of smaller prime numbers. It is well known that when the numbers are sufficiently large no integer factorization algorithm is known. The difficulty of this problem is very important for the algorithms used in cryptography such as RSA public key encryption and RSA digital signature ^{ 9}. If we know the value of the matrix , it is very easy to construct the elements of the group . However, constructing the matrix once we know the value of the element of the group , k for sufficiently large (k has 650 digits), could be a difficult problem. This difficulty could lead to serious studies in cryptography. For example, given

.

Find and construct the commutative group .

A simple calculation shows that

We can now construct . Indeed,

and

Now we can claim that

What we learn in this section is that every element of the group allows us to identify . In other words, we could say that knowing one element of the group allows us to identify . Let us keep in mind that the elements of the group for k sufficiently large make difficult to identify . These elements completely hide the identity of . Another example: Given

. Construct the representation class . Answer:

As we can see, the element allows us to identify the value of *a*. In our case,

Multiplicative commutative groups

of Rare matrices, previously introduced in section 2, are powerful tools which can be used for finding the matrix solutions of the Diophantine equation

.

Let us notice that

with

We can say that every pair (a,b) of positive integers generates a matrix solution of the equation

The above matrix solutions of this equation do not have a common factor. Triples of positive integers which are solutions of the equation

are called Pythagorean triples. Let us notice that

since 9 + 16 = 25. We can deduce that if *a,b* and c are positive integers such that

then

Due to the fact that there is an infinite number of Pythagorean triples ^{ 7} implies that the matrix equation has an infinite number of matrix solutions which do not have common matrix factors. The elementary properties and the structure of the elements of the multiplicative commutative groups and allow us to prove our main result.

**Proof of Theorem 1.1**

Let (a,b,c) be a Pythagorean triple of positive integers. Therefore,

Let

be a triple of Rare matrices of order and n index 1 defined by

and

Remark 2.3 allows us to claim that

and

It follows that

Therefore, the triple

is a matrix solution of the equation . Due to the fact that every Pythagorean triple generates a matrix solution of this equation implies that this equation has an infinite number of matrix solutions. As we can see, the matrices

and

do no have a common matrix factor. We have many similar observations. For instance, the triple

is a matrix solution of the Diophantine equation

Again the matrices

and

do no have a common matrix factor. The same phenomenon can be observed to the matrix triple

which is a solution of the matrix equation

We have the same observation for the matrix solutions of the equation In this case, we could use the matrix triple .

We could even construct sequences of matrix triple solutions of the equation and end up having the same observations. For example, we could use the sequence of matrix triple solutions of this equation. We observe that the matrices and do no have a common matrix factor.

This yields us the desired result.

**Remark ****5.1****.** Let be a positive integer. Then,

Recall that

Then,

with

Finally, the Diophantine equation admits an infinite number of matrix solutions, which do no have a common matrix factor, for every positive integer .

[1] | V. Brun, Über hypothesenbildung, Arc. Math. Naturvidenskab 34 (1914), 1–14. | ||

In article | |||

[2] | H. Darmon and A. Granville, On the equations and, in Bull. London Math. Soc., 27, 513-543(1995). | ||

In article | View Article | ||

[3] | Daniel Mauldin, R. A Generalization of Fermat's Last Theorem: The Beal Conjecture and Prize Problem. Notices of the American Mathematical Society, 44, 1436-1439(1997). | ||

In article | |||

[4] | Castelvecchi, D. "Mathematics Prize Ups the Ante to 1 Million US dollars". June 4, 2013. | ||

In article | |||

[5] | Leandro Torres Di Gregorio. Developments on Beal Conjecture from Pythagoras and Fermat's Equations. Pure and Applied Mathematics Journal. Vol.2, No.5, 149-155(2013). | ||

In article | View Article | ||

[6] | J. Moussounda Mouanda, On Matrix Solutions in of the Diophantine Equation https://www.researchgate.net/publication, 2022. | ||

In article | |||

[7] | J. Moussounda Mouanda, On Fermat's Last Theorem and Galaxies of sequences of positive integers, American Journal of Computational Mathematics, 12(2022), 162-189. | ||

In article | View Article | ||

[8] | R.C. A Proof to Beal's Conjecture. Bulletin of Mathematical Sciences and Applications, 89-93(2014). | ||

In article | |||

[9] | R. Rivest, A. Shamir and L. Adleman, ”A Method for Obtaining DigitalSignatures and Public Key Cryptosystems.” Comm. ACM 21, 120-126,1978. | ||

In article | View Article | ||

Published with license by Science and Education Publishing, Copyright © 2024 Joachim Moussounda Mouanda

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit https://creativecommons.org/licenses/by/4.0/

Joachim Moussounda Mouanda. On Beal's Conjecture for Matrix Solutions and Multiplicative Commutative Groups of Rare Matrices. *Turkish Journal of Analysis and Number Theory*. Vol. 12, No. 1, 2024, pp 1-7. https://pubs.sciepub.com/tjant/12/1/1

Mouanda, Joachim Moussounda. "On Beal's Conjecture for Matrix Solutions and Multiplicative Commutative Groups of Rare Matrices." *Turkish Journal of Analysis and Number Theory* 12.1 (2024): 1-7.

Mouanda, J. M. (2024). On Beal's Conjecture for Matrix Solutions and Multiplicative Commutative Groups of Rare Matrices. *Turkish Journal of Analysis and Number Theory*, *12*(1), 1-7.

Mouanda, Joachim Moussounda. "On Beal's Conjecture for Matrix Solutions and Multiplicative Commutative Groups of Rare Matrices." *Turkish Journal of Analysis and Number Theory* 12, no. 1 (2024): 1-7.

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[1] | V. Brun, Über hypothesenbildung, Arc. Math. Naturvidenskab 34 (1914), 1–14. | ||

In article | |||

[2] | H. Darmon and A. Granville, On the equations and, in Bull. London Math. Soc., 27, 513-543(1995). | ||

In article | View Article | ||

[3] | Daniel Mauldin, R. A Generalization of Fermat's Last Theorem: The Beal Conjecture and Prize Problem. Notices of the American Mathematical Society, 44, 1436-1439(1997). | ||

In article | |||

[4] | Castelvecchi, D. "Mathematics Prize Ups the Ante to 1 Million US dollars". June 4, 2013. | ||

In article | |||

[5] | Leandro Torres Di Gregorio. Developments on Beal Conjecture from Pythagoras and Fermat's Equations. Pure and Applied Mathematics Journal. Vol.2, No.5, 149-155(2013). | ||

In article | View Article | ||

[6] | J. Moussounda Mouanda, On Matrix Solutions in of the Diophantine Equation https://www.researchgate.net/publication, 2022. | ||

In article | |||

[7] | J. Moussounda Mouanda, On Fermat's Last Theorem and Galaxies of sequences of positive integers, American Journal of Computational Mathematics, 12(2022), 162-189. | ||

In article | View Article | ||

[8] | R.C. A Proof to Beal's Conjecture. Bulletin of Mathematical Sciences and Applications, 89-93(2014). | ||

In article | |||

[9] | R. Rivest, A. Shamir and L. Adleman, ”A Method for Obtaining DigitalSignatures and Public Key Cryptosystems.” Comm. ACM 21, 120-126,1978. | ||

In article | View Article | ||