Mersenne Prime Research Articles

Global Generalized Mersenne Numbers: Characterization and Distribution of Composites

Global Generalized Mersenne Numbers: Characterization and Distribution of Composites

Electronic Fourier–Galois Spectrum Analyzer for the Field GF(31)

A scheme for the Fourier–Galois spectrum analyzer for the field GF(31) is proposed. It is shown that this analyzer allows for solving a wide enough range of problems related to image processing, in particular those arising in the course of experimental studies in the field of physical chemistry. Such images allow digital processing when divided into a relatively small number of pixels, which creates an opportunity to use Galois fields of relatively small size. The choice of field GF(31) is due to the fact that the number 31 is a Mersenne prime number, which considerably simplifies the algorithm of calculating the Fourier–Galois transform in this field. The proposed scheme of the spectrum analyzer is focused on the use of threshold sensors, at the output of which signals corresponding to binary logic are formed. Due to this fact, further simplification of the proposed analyzer scheme is achieved. The constructiveness of the proposed approach is proven using digital modeling of electronic circuits. It is concluded that when solving applied problems in which an image can be divided into a relatively small number of pixels, it is important to take into account the specificity of particular Galois fields used for their digital processing.

On Mersenne Power GCD Matrices

This paper explores the 𝑛 × 𝑛 Mersenne power GCD matrices defined on sets of positive integers, focusing on factor-closed and gcd-closed sets. By employing the form 𝑓(𝑡𝑖 ,𝑡𝑗) = 2 (𝑡𝑖 ,𝑡𝑗 ) − 1, we investigate the 𝑟 𝑡ℎ power Mersenne GCD matrix (𝑀𝑟 ) and provide comprehensive insights into its factorizations, determinants, reciprocals, and inverses. Building upon previous research, particularly Chun's work on power GCD matrices, we extend the analysis to Mersenne numbers, offering a thorough understanding of their properties. The study contributes to the broader understanding of arithmetic functions and their applications in matrix theory.

On a solvable difference equations system of second order its solutions are related to a generalized Mersenne sequence

Abstract In this paper, we consider a class of two-dimensional nonlinear difference equations system of second order, which is a considerably extension of some recent results in the literature. Our main results show that class of system of difference equations is solvable in closed form theoretically. It is noteworthy that the solutions of aforementioned system are associated with generalized Mersenne numbers. The asymptotic behavior of solution to aforementioned system of difference equations when a = b and p = 0 are also given. Finally, numerical examples are given to support the theoretical results presented in this paper.

Non-Homogeneous Binary Cubic Equation a(x-y)<sup>3</sup>=8bxy

Polynomial equations an interesting subject in theory of numbers, occupy a pivotal role in the realm of mathematics and have a wealth of historical significance. The theoretical importance of polynomial equations of third degree in two unknowns having integral coefficients is great as they are closely connected with many problems of number theory. Specifically, the third degree polynomial equations having two unknowns in connection with elliptic curves occupy a pivotal role in the region of mathematics. This paper discusses on finding many solutions in integers to a typical third degree equation having two variables expressed as a(x-y)<sup>3</sup>=8bxy. The substitution strategy is employed in obtaining successfully different choices of solutions in integers. Some of the special fascinating numbers, namely, Pyramidal numbers, Polygonal numbers, Centered pyramidal numbers, Centered polygonal numbers, Thabit ibn Qurra numbers, Star numbers, Mersenne numbers and Nasty numbers (numbers expressed as product of two numbers in two different ways such that the sum of the factors in one set equals to the difference of factors in another set) are discussed in properties. These special numbers are unique. and have attractive characterization that set them apart from other numbers. The process of formulating second order Ramanujan numbers with base numbers as real integers is illustrated through examples.

The generalized order (k,t)-Mersenne sequences in groups

The purpose of this paper is to determine the algebraic properties of finite groups via a Mersenne-like sequence. Firstly, we introduce the generalized order $(k,t)$-Mersenne number sequences and study the periods of these sequences modulo $m$. Then, we get some interesting structural results. Furthermore, we expand the generalized order $(k,t)$-Mersenne number sequences to groups and we give the definition of the generalized order $(k,t)$-Mersenne sequences, $MQ_k^t(G,X)$, in the $j$-generator groups and also, investigate these sequences in the non-Abelian finite groups in detail. At last, we obtain the periods of the generalized order $(k,t)$-Mersenne sequences in some special groups as applications of the results produced.

Algebra of quaternions and octonions involving higher order Mersenne numbers

Algebra of quaternions and octonions involving higher order Mersenne numbers

Global Generalized Mersenne Numbers: Definition, Decomposition, and Generalized Theorems

A new generalized definition of Mersenne numbers is proposed of the form an−a−1n, called global generalized Mersenne numbers and noted GMa,n with base a and exponent n positive integers. The properties are investigated for prime n and several theorems on Mersenne numbers regarding their congruence properties are generalized and demonstrated. It is found that for any a, GMa,n−1 is even and divisible by n, a and a−1 for any prime n>2, and by aa−1+1 for any prime n>5. The remaining factor is a function of triangular numbers of a−1, specific for each prime n. Four theorems on Mersenne numbers are generalized and four new theorems are demonstrated, showing first that GMa,n≡1or7mod12 depending on the congruence of amod4; second, that GMa,n−1 are divisible by 10 if n≡1mod4 and, if n≡3mod4, GMa,n≡1or7or9mod10, depending on the congruence of amod5; third, that all factors ci of GMa,n are of the form 2nfi+1 such that ci is either prime or the product of primes of the form 2nj+1, with fi,j natural integers; fourth, that for prime n>2, all GMa,n are periodically congruent to ±1or±3mod8 depending on the congruence of amod8; and fifth, that the factors of a composite GMa,n are of the form 2nfi+1 with fi≡umod4 with u=0, 1, 2 or 3 depending on the congruences of nmod4 and of amod8. The potential use of generalized Mersenne primes in cryptography is shortly addressed.

Mersenne Matrix Operator and Its Application in $p-$Summable Sequence Space

In this study, it is introduced the regular Mersenne matrix operator which is obtained by using Mersenne numbers and examined sequence spaces described as the domain of this matrix in the space of $p$-summable sequences for $1\leq p \leq \infty$. After that, it investigated some properties and inclusion relations, established the Schauder basis, and stated $\alpha-$, $\beta-$, and $\gamma-$duals of the aforementioned spaces. Additionally, it is characterized by the matrix classes from newly described spaces to classical sequence spaces. Finally, we studied the compactness of matrix operators on related sequence spaces.

Mersenne Numbers in Generalized Lucas Sequences

Mersenne Numbers in Generalized Lucas Sequences

New sequences from the generalized Pell $ p- $numbers and mersenne numbers and their application in cryptography

<abstract><p>This paper presents the generalized Pell $ p- $numbers and provides some related results. A new sequence is defined using the characteristic polynomial of the Pell $ p- $numbers and generalized Mersenne numbers. Two algorithms for Diffie-Hellman key exchange are given as an application of these sequences. They are illustrated via numerical examples and shown to be secure against attacks. Thus, these new sequences are practical for encryption and constructing private keys.</p></abstract>

Origin of Sexy Prime Numbers, Origin of Cousin Prime Numbers, Equations from Supposedly Prime Numbers, Origin of the Mersenne Number, Origin of the Fermat Number

Origin of Sexy Prime Numbers, Origin of Cousin Prime Numbers, Equations from Supposedly Prime Numbers, Origin of the Mersenne Number, Origin of the Fermat Number

On some new families of $k$-Mersenne and generalized $k$-Gaussian Mersenne numbers and their polynomials

In this paper, we define the generalized k-Mersenne numbers and give a formula of generalized Mersenne polynomials and further, we study their properties. Moreover, we define Gaussian Mersenne numbers and obtain some identities like Binet Formula, Cassini's identity, D'Ocagne's Identity, and generating functions. The generalized Gaussian Mersenne numbers are described and their relation with the classical Mersenne numbers are explained. We also introduce a generalization of Gaussian Mersenne polynomials and establish some properties of these polynomials.

Generalizing the eight levels theorem: a journey to Mersenne prime discoveries and new polynomial classes

Mersenne primes, renowned for their captivating form as 2 p − 1 , have intrigued mathematicians for centuries. In this paper, we embark on a captivating quest to unveil the intricate nature of Mersenne primes, seamlessly integrating methods with the Eight Levels Theorem. Initially, we extend the Eight Levels Theorem and introduce an innovative approach that harmoniously combines arithmetic and differential techniques to compute the coefficients of the polynomial expansions of x n + y n in terms of binary quadratic forms. This endeavor leads us to the genesis of novel polynomial sequences as we scrutinize the coefficients within this expansion. Our research unearths previously uncharted connections between Mersenne numbers and the derivatives of specific polynomial sequences. By forging this linkage, we not only enhance our understanding of Mersenne primes but also bridge the divide between well-established sequences in number theory and differential equations. This broadens the applicability of our findings across diverse scientific domains, revealing fresh avenues for exploration in number theory and beyond. These polynomial bridges serve as conduits between these sequences, unveiling exciting prospects for future research. This interdisciplinary exploration opens up exciting possibilities for the broader implications of Mersenne primes, extending their significance beyond the realm of pure mathematics. In our paper, we also delve into the intriguing influence of the Golden Ratio constant, which unveils segments reminiscent of the beauty found in nature, adding an unexpected dimension to the world of arithmetic. In this harmonious journey, the allure of Mersenne primes resonates through the symphony of mathematical discovery.

Unified algorithms for generalized new mersenne number transforms

Unified algorithms for generalized new mersenne number transforms

Features of digital signal processing algorithms using Galois fields GF(2n+1).

An alternating representation of integers in binary form is proposed, in which the numbers -1 and +1 are used instead of zeros and ones. It is shown that such a representation creates considerable convenience for multiplication numbers modulo p = 2n+1. For such numbers, it is possible to implement a multiplication algorithm modulo p, similar to the multiplication algorithm modulo the Mersenne number. It is shown that for such numbers a simple algorithm for digital logarithm calculations may be proposed. This algorithm allows, among other things, to reduce the multiplication operation modulo a prime number p = 2n+1 to an addition operation.

Radix-22 Algorithm for the Odd New Mersenne Number Transform (ONMNT)

This paper introduces a new derivation of the radix-22 fast algorithm for the forward odd new Mersenne number transform (ONMNT) and the inverse odd new Mersenne number transform (IONMNT). This involves introducing new equations and functions in finite fields, bringing particular challenges unlike those in other fields. The radix-22 algorithm combines the benefits of the reduced number of operations of the radix-4 algorithm and the simple butterfly structure of the radix-2 algorithm, making it suitable for various applications such as lightweight ciphers, authenticated encryption, hash functions, signal processing, and convolution calculations. The multidimensional linear index mapping technique is the conventional method used to derive the radix-22 algorithm. However, this method does not provide clear insights into the underlying structure and flexibility of the radix-22 approach. This paper addresses this limitation and proposes a derivation based on bit-unscrambling techniques, which reverse the ordering of the output sequence, resulting in efficient calculations with fewer operations. Butterfly and signal flow diagrams are also presented to illustrate the structure of the fast algorithm for both ONMNT and IONMNT. The proposed method should pave the way for efficient and flexible implementation of ONMNT and IONMNT in applications such as lightweight ciphers and signal processing. The algorithm has been implemented in C and is validated with an example.

The Forgery Attack on the Post-Quantum AJPS-2 Cryptosystem and Modification of the AJPS-2 Cryptosystem by Changing the Class of Numbers Used as a Module

In recent years, post-quantum (quantum-resistant) cryptography has been actively researched, in particular, due to the National Institute of Standards and Technology's (NIST) Post-Quantum Cryptography Competition (PQC), which has been running since 2017. One of the participants in the first round of the competition is the Mersenne-756839 key encapsulation mechanism based on the AJPS-2 encryption scheme. The arithmetic modulo Mersenne number is used to construct the cryptoprimitives of the AJPS family. In this paper, we propose a forgery attack on the AJPS-2 cryptosystem using an active eavesdropper, and two modifications of the post-quantum AJPS-2 cryptosystem, namely, the modification of AJPS-2 using the arithmetic modulo generalized Mersenne number and Crandall number. Moreover, new algebraic problems are defined, on the complexity of which the security of the created modifications is based. The advantages of these modifications are the extension of the number class used as a module in the cryptosystem and the security against the forgery attack with the active eavesdropper, which was successful in the original AJPS-2.

A Study on the Norms of Toeplitz Matrices with the Generalized Mersenne Numbers

In this article presents findings related to Toeplitz matrices with Mersenne numbers. We started by creating Toeplitz matrices whose components are Mersenne numbers. We then calculated the Euclidian, row and column norms of these matrices and established both lower and upper bounds for their spectral norms. Additionally, we determined the upper bounds for the Frobenius and spectral norms of the Kronecker and Hadamard product matrices of the Toeplitz matrices with Mersenne numbers.

Computers As a Novel Mathematical Reality: III. Mersenne Numbers and Sums of Divisors

Computers As a Novel Mathematical Reality: III. Mersenne Numbers and Sums of Divisors