The r-circulant Matrices Associated with k-Fermat and k-Mersenne
Numbers
BAHAR KULOĞLU1*, ENGİN ESER2, ENGİN ÖZKAN3
1Department of Mathematics,
Graduate School of Natural and Applied Sciences,
Erzincan Binali Yıldırım University,
Yalnızbağ Campus, 24100, Erzincan,
TURKEY
2Department of Mathematics,
Erzincan Binali Yıldırım University,
Faculty of Art and Sciences,
Erzincan,
TURKEY
3Department of Mathematics,
Faculty of Arts and Sciences,
Erzincan Binali Yıldırım University,
Yalnızbağ Campus, 24100, Erzincan,
TURKEY
*Corresponding Author
Abstract: - In this study, the main goal is to investigate the -circulant matrices of -Fermat and -Mersenne
numbers, then to find eigenvalues, determinants of these matrices, to evaluate their different norms (Spectral
and Euclidean) and finally to find the right and skew-right circulant matrices.
Key-Words: - -Fermat number, -Merssenne number, Norm (Spectral and Euclidean), Eigenvalues,
Circulant matrices
Received: December 11, 2022. Revised: May 27, 2023. Accepted: June 19, 2023. Published: July 24, 2023.
1 Introduction
Prime numbers are numbers that have remained a
mystery throughout human history. Since prime
numbers are generator numbers, people find the
formula for these numbers. The most well-known of
these are Pierre de Fermat and Marin Mersenne.
Fermat numbers included pseudoprimes. Firstly,
Fermat conjectured that all numbers which are
produced by ( is a non-negative integer)
are prime numbers ( must be the power of 2). One
can see easily that the first 5 numbers are prime but
when it comes to the 6th number there is a problem.
Later Euler proved that this number has factors. So,
it was a composite number. Fermat made a
computational mistake. Now it is an open question
that are there any other numbers like this?
Mersenne numbers have the form ( is a
positive integer). These numbers were studied in
ancient times because of their connection to perfect
numbers. Euclid-Euler theorem asserts this
connection.
Later Francois Proth studied Fermat numbers.
He found, [1], the numbers which are a generalized
form of Fermat numbers. They are of form
. Proth numbers are known as -
Fermat numbers. There are restrictions on these
numbers as , where and is odd
numbers. Here it can be easily seen that the general
forms of Proth numbers without any restrictions are
-Fermat numbers. The first terms of these
numbers are as follows:
3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449,
577, 641, 673, 769, 929, 1153, 1217, 1409, 1601,
2113, 2689, 2753, 3137, 3329, 3457, 4481, 4993,
6529, 7297, 7681, 7937, 9473, 9601, 9857 (Proth
numbers are referenced in the On-Line
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.59
Bahar Kuloğlu, Engi
n Eser, Engi
n Özkan
E-ISSN: 2224-2880
531
Volume 22, 2023
Encyclopedia of Integer Sequences in OEIS
as A080076, [2].)
In our study we examine -Fermat numbers and
-Mersenne numbers, [3]. -Fermat number
sequence, denoted by , is defined by
-Mersenne number sequence denoted by is
defined by
Recurrence relations of these numbers respectively,
with
(1.1)
with
The first terms of these numbers are shown in Table
1.
Table 1. Some values for and .
0
1
2
3
…
2
3
…
0
1
…
Now we introduce circulant and -cirulant
matrices. Among the subjects of algebra, especially
in linear algebra, these matrices draw attention.
These matrices are used in a number of operations
such as signaling, coding, etc. They are also used in
the field of chemistry, [4]. The study, [5], presented
the visualization of breathing tense while reflecting
atomic velocities on eigenvectors of the circulant
matrix. In, [6], the author's circulant matrices are
used in Hadamard transform spectroscopy and in the
construction of the optimal chemical design. In [7],
the authors presented the quantum optics effects in
quasi-one-dimensional and two-dimensional carbon
materials by the circulant matrix method. In [8], [9],
the authors gave the lower and upper bounds for the
spectral norms of r -circulant matrices and obtained
some bounds related to the spectral norms of
Hadamard and Kronecker products of these matrices
with the Fibonacci and Lucas numbers. Also, they
presented the spectral norms with - Fibonacci and
-Lucas numbers. Circulant matrices are of great
interest among the subjects of algebra, especially in
the field of linear algebra as Algebraic Geometry,
Number Theory, Topleitz operator, etc.
circulant matrices are shown by or
where represents the size of matrix. Their inverse,
conjugate transposes, sums, and multiplications can
be malleable. For any complex number without
zero, we can define those matrices. We can examine
its eigenvalues, Euclidean norm, spectral norm,
determinants, and inverse. is determined by its
first-row element and . Given and
in recurrence relation we obtain its eigenvalues and
determinant.
Circulant matrices and -circulant matrices
including Fibonacci, Pell, Pell-Lucas, etc. numbers
have been of great interest. In several studies,
eigenvalues, determinants, norms, bounds, and
inverses for these matrices are found. For instance,
[10], presented eigenvalues and the determinant of
the right circulant matrices with Pell numbers and
Pell-Lucas numbers. The study, [11], presented
norms for circulant matrices including Fermat and
Mersenne numbers. In, [12], the authors presented
the exact inverse of circulant matrices with Fermat
and Mersenne numbers. In, [13], the author solved
the determinants of these matrices using matrix
decomposition. In, [14], the authors studied the
properties of the -circulant matices involving
Mersene and Fermat numbers. In this study, we
present circulant matrices by using Fermat
and Mersenne numbers. This study is
constructed of three sections. In the first section, we
introduce the Euclidean norm, Hadamard product,
eigenvalues, and determinants of circulant
matices. In second section we define the
circulant matrices involving - Fermat numbers.
We find the eigenvalues, determinants, sum
identities, norms, and the bound for the spectral
norm for the -Fermat -circulant matrices. In the
last section we study -circulant matrices involving
-Mersenne numbers.
Lemma 1.1, [15], [16]. Let be a matrix.
The Frobenius or Euclidean norm of is defined as
The column norm of is defined as
The row norm of is defined as
The spectral norm of a matrix defined as
where denote the eigenvalues of
and is the conjugate transpose of .
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.59
Bahar Kuloğlu, Engi
n Eser, Engi
n Özkan
E-ISSN: 2224-2880
532
Volume 22, 2023
Matrix has the relationship between the norm
values given below:
(1.2)
Lemma 1.2, [13]. Let
is the Hadamard product of A
and B, then we get
(1.3)
where
and
Definition 1.3, [17]. For a matrix is
said to be -circulant matrix if is of the form
and it is denoted by , where
is the first-row vector. For
and , the right and skew-right circulant
matrices are desired, respectively.
Lemma 1.4, [17]. Let be -circulant matrices
then its eigenvalues
where is the th root of unity and is the th root
of r.
Lemma 1.5, [16]. The Euclidean norm of -
circulant matrix is given by
(1.4)
Lemma 1.6, [10]. For any and , we get
(1.5)
Lemma 1.7, [18]. Determinant of the circulant
matrix is
where the entry is equal to the entry
for and
are the nth
roots of unity. where are th root of .
Using the above lemmas we can calculate the
eigenvalues, the determinant, Euclidean norms, and
bounds for spectral norms of -circulant matrices
involving -Fermat and -Mersenne numbers with
arithmetic indices. We present many new identities
for -Fermat and -Mersenne numbers.
2 -Fermat -circulant Matrix
and be non-negative integers and .
The -circulant matrices -Fermat numbers are
denoted by and defined as follows:
Definition 2.1 The -Fermat -circulant matrix is
defined as where first row
vector is
i.e., matrix of the form
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.59
Bahar Kuloğlu, Engi
n Eser, Engi
n Özkan
E-ISSN: 2224-2880
533
Volume 22, 2023
(2.1)
Theorem 2.2 The eigenvalues of -Fermat -
circulant matrices are
where
Proof.
for
for
and for
as desired.
Corollary 2.3 For and , we get
eigenvalues for the -Fermat right and skew-right
circulant matrices as follows:
and
where is the nth. the root of -1.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.59
Bahar Kuloğlu, Engi
n Eser, Engi
n Özkan
E-ISSN: 2224-2880
534
Volume 22, 2023
Theorem 2.4 For a positive integer n, we have
Proof. To get the desired result, we need to know
that the trace of any given square matrix is equal to
the sum of the eigenvalues of that matrix. In that
case
according to the expression given above, this sum is
equal to . In that case
as desired.
Theorem 2.5 Determinant of is given by
Proof. From Lemma 1.7, we get
For ,
From Lemma 1.6
Corollary 2.6 The determinants of the -Fermat
right circulant and skew-right circulant matrices are
given as, respectively
On setting and in Theorem 2.2,
the following sum identities are verified for the -
Fermat numbers.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.59
Bahar Kuloğlu, Engi
n Eser, Engi
n Özkan
E-ISSN: 2224-2880
535
Volume 22, 2023
3 Norm of -Fermat -circulant
Matrices
When we take and , we get
From Lemma 3.1, the sum of the squares of the -
Fermat numbers will be used to obtain the norms of
different matrices.
Lemma 3.1 The finite sum of squares of the -
Fermat numbers is given by
Proof.
as desired.
Theorem 3.2 The Euclidean norm for the -Fermat
-circulant matrices is given by
Proof. By Eq. (1.4) we get
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.59
Bahar Kuloğlu, Engi
n Eser, Engi
n Özkan
E-ISSN: 2224-2880
536
Volume 22, 2023
as desired.
Theorem 3.3 The bound for the spectral norm of the
-Fermat -circulant matrices is:
for
and for
Proof. By Eq (1.4), the Euclidean norm is given as
Here, we need to examine the proof in 2 stages
according to the state of r.
State 1. If , then from Lemma 3.1 we get
which implies
and from Eq. (1.2) we get
Now to obtain the upper bound for the spectral
norm, we write
in the form of the Hadamard
product of two matrices.
and
then clearly
, where denotes the
Hadamard product. Now,
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.59
Bahar Kuloğlu, Engi
n Eser, Engi
n Özkan
E-ISSN: 2224-2880
537
Volume 22, 2023
Thus, by Lemma 1.2, we get
Hence, we have
State 2. If , then from Eq. (1.4) and Lemma
3.1 we get
and from Eq. (1.2) we get
We calculate the upper bound for the spectral norm
of
.
Let
and
then clearly
, where denotes the
Hadamard product. So,
Hence, by Lemma 1.2, we have
Thus
as desired.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.59
Bahar Kuloğlu, Engi
n Eser, Engi
n Özkan
E-ISSN: 2224-2880
538
Volume 22, 2023
Now we give the relations above for the -
Mersenne numbers.
4 -Mersenne -circulant matrix
and be non-negative integers and .
The -circulant matrices -Mersenne numbers are
denoted by and defined as follows:
Definition 4.1 The -Mersenne -circulant matrix is
defined as where the first-row
vector is
i.e.,
matrix of the form
(4.1)
Theorem 4.2 The eigenvalues of -Mersenne -
circulant matrices are
where
Proof.
for
for
and for
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.59
Bahar Kuloğlu, Engi
n Eser, Engi
n Özkan
E-ISSN: 2224-2880
539
Volume 22, 2023
as desired.
Corollary 4.3 For and we get
eigenvalues for the -Mersenne right and skew-right
circulant matrices as follows:
and
where is the nth. the root of -1.
Theorem 4.4 For a positive integer n, we have
Proof. To get the desired result, we need to know
that the trace of any given square matrix is equal to
the sum of the eigenvalues of that matrix. In that
case,
According to the expression given above, this sum
is equal to . So
as desired.
Theorem 4.5 Determinant of is given by
Proof. We can prove it like Theorem 2.5.
Corollary 4.6 The determinants of the -Mersenne
right circulant and skew-right circulant matrices are
given as
On setting and in Theorem 4.2,
the following sum identities are verified for the -
Mersenne numbers.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.59
Bahar Kuloğlu, Engi
n Eser, Engi
n Özkan
E-ISSN: 2224-2880
540
Volume 22, 2023
5 Norm of -Mersenne -circulant
matrices
When we take and
From Lemma 5.1, the sum of the squares of the -
Mersenne numbers will be used to obtain the norms
of different matrices.
Lemma 5.1 The finite sum of squares of the -
Mersenne numbers is given by
Proof. The proof is similar to that of Lemma 3.1.
Theorem 5.2 The Euclidean norm for the -
Mersenne -circulant matrices is given by
Proof. By Eq. (1.4) we get
Theorem 5.3 The bound for the spectral norm of the
-Mersenne -circulant matrices is:
For
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.59
Bahar Kuloğlu, Engi
n Eser, Engi
n Özkan
E-ISSN: 2224-2880
541
Volume 22, 2023
and for
Proof. We can prove it like Theorem 3.3.
6 Conclusion
Based on the Fermat and Mersenne number
sequences with similar recurrences, which have
been studied less, the r-circulant matrices of these
sequences were created. Euclidean, row, and
spectral norms, which are eigenvalues,
determinants, and some special norm values, are
discussed depending on this matrix. In addition, the
lower and upper bounds of these matrices for the
spectral norm were examined in closed form. In
addition, right circulant and skew-right circulant
matrices were examined for 1 and -1 values of
depending on eigenvalues. Finally, some interesting
results and sum properties were given.
More interesting results can be expected by
working with the lesser-known Fermat equation
and the more general form of this equation,
.
References:
[1] Weisstein, Eric W. (2019) "Proth
Number". mathworld.wolfram.com.
[2] Sloane, N.J.A. (1973) A Handbook of Integer
Sequences, Academic Press, New York,
https://oeis.org
[3] Mourad Chelgham and Ali Boussayoud
(2021). On the k-Mersenne–Lucas numbers,
Notes on Number Theory and Discrete
Mathematics, DOI:
10.7546/nntdm.2021.27.1.7-13.
[4] Min, W., English, B.P., Luo, G. P., Cherayil,
B.J., Kou, S.C., Xie, X.S. (2005)Fluctuating
enzymes: lessons from single-molecule
studies. Acc. Chem. Res. 38, 923–931.
[5] Van Houteghem, M., Verstraelen, T., Van
Neck, D., Kirschhock, C., A. Martens, J.,
Waroquier, M., & Van Speybroeck, V.
(2011). Atomic velocity projection method: a
new analysis method for vibrational spectra in
terms of internal coordinates for a better
understanding of zeolite nanogrowth. Journal
of Chemical Theory and Computation, 7(4),
1045-1061.
[6] Balasubramanian, K. (1995). Computational
Strategies for the Generation of Equivalence
Classes of Hadamard Matrixes. Journal of
chemical information and computer
sciences, 35(3), 581-589.
[7] Yerchuck, D., Dovlatova, A. (2012) Quantum
optics effects in quasi-one-dimensional and
two- dimensional carbon materials. J. Phys.
Chem. 116, 63–80.
[8] Shen, S., & Cen, J. (2010). On the bounds for
the norms of r-circulant matrices with the
Fibonacci and Lucas numbers. Applied
Mathematics and Computation, 216(10),
2891-2897.
[9] Shen, S. Q., & Cen, J. M. (2010). On the
spectral norms of r-circulant matrices with the
k-Fibonacci and k-Lucas numbers. Int. J.
Contemp. Math. Sciences, 5(12), 569-578.
[10] Bueno, A. C. F. (2014). On the Eigenvalues
and the Determinant of the Right Circulant
Matrices with Pell and Pell–Lucas
Numbers. International Journal of
Mathematics and Scientific Computing, 4(1),
19-20.
[11] Solak, S. (2005). On the norms of circulant
matrices with the Fibonacci and Lucas
numbers. Applied Mathematics and
Computation, 160(1), 125-132.
[12] Zheng, Y., & Shon, S. (2015, January). Exact
inverse matrices of Fermat and Mersenne
circulant matrix. In Abstract and Applied
Analysis (Vol. 2015). Hindawi.
[13] Bozkurt, D., & Tam, T. Y. (2012).
Determinants and inverses of circulant
matrices with Jacobsthal and Jacobsthal–
Lucas numbers. Applied Mathematics and
Computation, 219(2), 544-551.
[14] Kumaria, M., Prasada, K., Tantib, J., &
Özkan, E. (2023). On the properties of г-
circulant matrices involving Mersenne and
Fermat numbers. Int. J. Nonlinear Anal. Appl.
In Press, 1, 11.
[15] Simon Foucart, Drexel Üni. Lec. 6. (2023)
http://www.math.drexel.edu/foucart/Teaching
Files/F12/M504Lect6.pdf.
[16] Weisstein, Eric W. (2020) "Matrix
Norm". mathworld.wolfram.com.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.59
Bahar Kuloğlu, Engi
n Eser, Engi
n Özkan
E-ISSN: 2224-2880
542
Volume 22, 2023
[17] Cline, R. E., Plemmons, R. J., and Worm, G.
(1974). Generalized inverses of certain
Toeplitz matrices, Linear Algebra Appl. 8 1,
25-33.
[18] Geller, D., Irwin Kra and Santiago R. S.
(2004) On circulant matrices, Notices
of the American Mathematical society, (59)03,
DOI: 10.1090/noti804
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
All authors contributed to the study's conception and
design. Material preparation, data collection, and
analysis were performed by Bahar KULOĞLU,
Engin ÖZKAN and Engin ESER. The first draft of
the manuscript was written by Engin ÖZKAN, and
all authors commented on previous versions of the
manuscript. All authors read and approved the final
manuscript.
Consent to participate: Accept
Consent for publication: Accept
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funds, grants, or other supports were received.
Conflict of Interest
All authors certify that they have no affiliations with
or involvement in any organization or entity with
any financial interest or non-financial interest in the
subject matter or materials discussed in this
manuscript.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.59
Bahar Kuloğlu, Engi
n Eser, Engi
n Özkan
E-ISSN: 2224-2880
543
Volume 22, 2023
