Gaussian Integers Research Articles

Transcendence and Normality of Complex Numbers via Hurwitz Continued Fractions

Abstract We study the topological, dynamical, and descriptive set-theoretic properties of Hurwitz continued fractions. Hurwitz continued fractions associate an infinite sequence of Gaussian integers to every complex number that is not a Gaussian rational. The resulting space of sequences of Gaussian integers $\Omega $ is not closed. Using an iterative procedure, we show that $\Omega $ contains a natural subset whose closure $\overline{\textsf{R}}$ encodes continued fraction expansions of complex numbers that are not Gaussian rationals. We prove that $(\overline{\textsf{R}}, \sigma )$ is a subshift with a feeble specification property. As an application, we determine the rank in the Borel hierarchy of the set of Hurwitz normal numbers with respect to the complex Gauss measure. We also construct a family of complex transcendental numbers with bounded partial quotients.

ON HYPERBOLIC INTEGERS

The purpose of this paper is to investigate an ℓ-ring structure of algebraic integers from an arithmetic point of view. We endow the algebra D of hyperbolic numbers with its standard f-algebra structure [7]. We introduce the ring of hyperbolic integers Zh as a sub f-ring of the ring ZD of integers of D. Next, we prove that Zh is the unique, up to ring isomorphism, Archimedean f-ring of quadratic integers. Our study focuses on arithmetic properties of Zh related to its latticeordered structure. We show that many basic properties of the ring of integers Z such as primes, unique factorization theorem and the notions of floor and ceiling functions can be extended to Zh. A surprising fact is that prime numbers seen as hyperbolic integers are semiprimes. We also obtain some properties of hyperbolic Gaussian integers. As an application, we discuss the Dirichlet divisor problem using hyperbolic intervals.

A graph-theoretic approach to ring analysis: Dominant metric dimensions in zero-divisor graphs

This article investigates the concept of dominant metric dimensions in zero divisor graphs (ZD-graphs) associated with rings. Consider a finite commutative ring with unity, denoted as R, where nonzero elements x and y are identified as zero divisors if their product results in zero (x.y=0). The set of zero divisors in ring R is referred to as L(R). To analyze various algebraic properties of R, a graph known as the zero-divisor graph is constructed using L(R). This manuscript establishes specific general bounds for the dominant metric dimension (Ddim) concerning the ZD-graph of R. To achieve this objective, we examine the zero divisor graphs for specific rings, such as the ring of Gaussian integers modulo m, denoted as Zm[i], the ring of integers modulo n, denoted as Zn, and some quotient polynomial rings. Our research unveils new insights into the structural similarities and differences among commutative rings sharing identical metric dimensions and dominant metric dimensions. Additionally, we present a general result outlining bounds for the dominant metric dimension expressed in terms of the maximum degree, girth, clique number, and diameter of the associated ZD-graphs. Through this exploration, we aim to provide a comprehensive framework for analyzing commutative rings and their associated zero divisor graphs, thereby advancing both theoretical knowledge and practical applications in diverse domains.

SPN based RGB image encryption over Gaussian integers

This research paper proposes a novel approach for constructing substitution boxes (S-boxes) over Gaussian integers, which are complex numbers with integer coefficients. The proposed method is based on the properties of the Gaussian integers and their arithmetic operations and ensures the S-boxes exhibit strong cryptographic properties. Furthermore, the paper demonstrates how these S-boxes can be utilized for image encryption through a substitution-permutation network (SPN) over Gaussian integers. The SPN involves iteratively applying the S-box and a permutation layer to the input image, which effectively scrambles the image data. Experimental results show that the proposed method achieves high security and robustness against various attacks while providing efficient encryption and decryption performance. This research thus provides a promising avenue for developing secure image encryption schemes based on Gaussian integers.

On the metric dimension of extended zero divisor graphs

Let R be a commutative ring with unity, the extended zero divisor graph E(R) is defined by considering the non-zero zero divisors Z(R)∗ as the vertices and two distinct vertices x and y of R are adjacent whenever there exist two positive integers m and n such that xmym=0 with xm≠0 and yn≠ 0. In this paper, we investigate metric dimension, fault tolerant metric dimension and local metric dimension of the extended zero divisor graph for the ring of integers modulo m and the ring of Gaussian integers modulo m.

A Review Study of Prime Period Perfect Gaussian Integer Sequences

Prime period sequences can serve as the fundamental tool to construct arbitrary composite period sequences. This is a review study of the prime period perfect Gaussian integer sequence (PGIS). When cyclic group {1,2,…,N−1} can be partitioned into k cosets, where N=kf+1 is an odd prime number, the construction of a degree-(k + 1) PGIS can be derived from either matching the flat magnitude spectrum criterion or making the sequence with ideal periodic autocorrelation function (PACF). This is a systematic approach of prime period N=kf+1 PGIS construction, and is applied to construct PGISs with degrees 1, 2, 3 and 5. However, for degrees larger than 3, matching either the flat magnitude spectrum or achieving the ideal PACF encounters a great challenge of solving a system of nonlinear constraint equations. To deal with this problem, the correlation and convolution operations can be applied upon PGISs of lower degrees to generate new PGISs with a degree of 4 and other higher degrees, e.g., 6, 7, 10, 11, 12, 14, 20 and 21 in this paper. In this convolution-based scheme, both degree and pattern of a PGIS vary and can be indeterminate, which is rather nonsystematic compared with the systematic approach. The combination of systematic and nonsystematic schemes contributes great efficiency for constructing abundant PGISs with various degrees and patterns for the associated applications.

A Novel Method for Hill Cipher Encryption and Decryption Using Gaussian Integers Implemented in Banking Systems

Cryptographic tools like Hill's encryption algorithm protect digital data. In this work, we present a novel Hill cipher security method that utilizes Gaussian integers from number theory. Using these intriguing mathematical entities to disguise plaintext values dramatically boosts assault resistance and duration. This research includes a three-pass protocol for encryption and decryption without key exchange, ensuring a safe, efficient, and dependable solution. Pandas is used for efficient data processing, and Numpy for computational tasks, notably matrices. Hill cipher-based encryption and decryption can be utilized in real life. It also demonstrates how to save Pandas DataFrame data to Excel. This strategy assures progress in cryptography. Uniquely designed for banking, it emphasizes its applicability and possible influence on present financial systems.

On Standard Forms of the Pairs of Matrices Over the Ring of Gaussian Integers with Respect to the (z,k)-Equivalence

On Standard Forms of the Pairs of Matrices Over the Ring of Gaussian Integers with Respect to the (z,k)-Equivalence

Enhanced Security in Public Key Cryptography: A Novel Approach Combining Gaussian Graceful Labeling and NTRU Public Key Cryptosystem

This research explores an encryption system that combines the Nth-degree Truncated Polynomial Ring Unit (NTRU) public key cryptosystem with Gaussian Graceful Labeling. This process assigns distinct labels to a graph's vertices, resulting in successive Gaussian integers. The NTRU method offers enhanced security and efficient key exchange. The communication encryption process depends on integers P, a, and b, with P being the largest prime number in the vertex labeling. The original receivers are the vertex labeling with the largest prime number coefficient, while all other receivers receive messages from the sender. A polynomial algebraic mixing system and a clustering principle based on the abecedarian probability proposition are used in NTRU encryption and decryption. The choice of relatively prime integers p and q in NTRU plays a role in the construction of polynomial rings used for encryption and decryption, with specific choices and properties designed to ensure scheme security.

АРИФМЕТИЧНІ ФУНКЦІЇ ДЛЯ ҐАУСОВИХ ЧИСЕЛ

Complex numbers whose real and imaginary parts are integers are called Gaussian integers. In this article, the well-known arithmetic functions—the number, sum and product of divisors—are generalized to Gaussian integers. For this functions, the formulae for calculation that use the prime factorization were derived, and some properties of these functions that concern associative or conjugate Gaussian integers were established. Obtained were also the formulae of the number, sum and product of all divisors of a Gaussian integer that are themselves divisible by a certain number. The usefulness of the obtained results are illustrated in the solutions to some problems.

Generating Self-Invertible Matrices by Hill Cipher Algorithm In Gaussian Integers

Generating Self-Invertible Matrices by Hill Cipher Algorithm In Gaussian Integers

Reliability-Based Decoding of Low-Density Lattice Codes Using Gaussian and Eisenstein Integers

Reliability-Based Decoding of Low-Density Lattice Codes Using Gaussian and Eisenstein Integers

On Some Novel Results About Weak Fuzzy Complex Integers

This paper is dedicated to studying for the first time the concept of weak fuzzy complex integers division and units, where we present a full classification of weak fuzzy complex integer units with necessary and sufficient conditions for division in the set of weak fuzzy complex integers. On the other hand, we provide an algorithm for solving weak fuzzy complex linear congruencies with many related examples that explain how algorithms work.

Proof of Sum of Two Squares Theorem using Gaussian Integers

I provide a proof of Fermat’s Sum of Two Squares theorem using Gaussian Integers. The theorem characterizes the integers that can be represented as a sum of two integer squares: n = a2 + b2 for some n, a, b ∈ Z. Few authors have provided proof of the theorem using Thu’s Lemma and Fermat’s Little Theorem. This paper uses Gaussian integers to prove the extended theorem.

Ideal solutions in the Prouhet–Tarry–Escott problem

For given positive integers m m and n n with m > n m>n , the Prouhet–Tarry–Escott problem asks if there exist two disjoint multisets of integers of size n n having identical k k th moments for 1 ≤ k ≤ m 1\leq k\leq m ; in the ideal case one requires m = n − 1 m=n-1 , which is maximal. We describe some searches for ideal solutions to the Prouhet–Tarry–Escott problem, especially solutions possessing a particular symmetry, both over Z \mathbb {Z} and over the ring of integers of several imaginary quadratic number fields. Over Z \mathbb {Z} , we significantly extend searches for symmetric ideal solutions at sizes 9 9 , 10 10 , 11 11 , and 12 12 , and we conduct extensive searches for the first time at larger sizes up to 16 16 . For the quadratic number field case, we find new ideal solutions of sizes 10 10 and 12 12 in the Gaussian integers, of size 9 9 in Z [ i 2 ] \mathbb {Z}[i\sqrt {2}] , and of sizes 9 9 and 12 12 in the Eisenstein integers.

The Factor Ring Structure of Quadratic Principal Ideal Domains

Previous authors have classified the possible factor rings of the Gaussian integers and the Eisenstein integers. Here, we extend this classification to the ring of integers of any quadratic number field, provided the ring has unique factorization. In the case of imaginary quadratic fields, the classification has exactly the same flavor as that of the Gaussian integers and Eisenstein integers. For real quadratic fields, the classification is only slightly more complicated.

H-integral and Gaussian integral normal mixed Cayley graphs

If all the eigenvalues of the Hermitian-adjacency matrix of a mixed graph are integers, then the mixed graph is called H-integral. If all the eigenvalues of the (0,1)-adjacency matrix of a mixed graph are Gaussian integers, then the mixed graph is called Gaussian integral. For any finite group Γ, we characterize the set S for which the normal mixed Cayley graph Cay(Γ,S) is H-integral. We further prove that a normal mixed Cayley graph is H-integral if and only if the mixed graph is Gaussian integral.

Mathematical foundations of complex tonality

Equal temperament, in which semitones are tuned in the irrational ratio of , is best seen as a serviceable compromise, sacrificing purity for flexibility. Just intonation, in which intervals are given by products of powers of 2, 3, and 5, is more natural, but of limited flexibility. We propose a new scheme in which ratios of Gaussian integers form the basis of an abstract tonal system. The tritone, so problematic in just temperament, given ambiguously by the ratios , , , , none satisfactory, is in our scheme represented by the complex ratio . The major and minor whole tones, given by intervals of and , can each be factorized into products of complex semitones, giving us a major complex semitone and a minor complex semitone . The perfect third, given by the interval , factorizes into the product of a complex whole tone and its complex conjugate. Augmented with these supplementary tones, the resulting scheme of complex intervals based on products of low powers of Gaussian primes leads to the construction of a complete system of major and minor scales in all keys.

Fault-Tolerant Routing for Wireless Sensor Networks using a Gaussian Network

Abstract: Wireless Sensor Networks (WSNs) are utilised for sensing various physical and environmental variables like temperature, pressure, motion, and pollution. A component of a network to continue operating even when some sensor nodes fail is regarded as fault tolerance in wireless sensor networks. Due to battery life deadlines and the challenges associated with charging or replacing failing nodes, the deployment of sensor nodes in the target area is done in a dense manner to maximise coverage and connection. This paper implements a hybrid fault-tolerant routing methodology to address the problem of fault tolerance in hierarchical topology for wireless sensor networks (WSNs). The network location is partitioned into small square grids, with a Gaussian integer serving as each grid's cluster head. A Gaussian network is created through communicating these cluster heads together. Discovering the Gaussian network's shortest path and multi-path routing, the project implements a hybrid Fault-tolerant Clustering routing protocol based on Gaussian network for Wireless Sensor Network (FCGW). The purpose of FCGW is to improve fault tolerance and reduce energy consumption for Wireless Sensor Networks. Experimental results using MATLAB shows that FCGW protocol has high data reliability. In addition, the FCGW protocol consumes lesser energy in the network compared to other protocols namely FT-LEACH, HEED and PSO-UFC

A robust hybrid decision making model for Human-Computer interaction in the environment of Bipolar complex picture fuzzy soft Sets

The goal of this paper is to present novel concepts and effective theories for the ability to accurately represent difficult and ambiguous information in real-life problems. In order to accomplish this goal, this paper investigated the original idea of bipolar complex picture fuzzy soft set (BCPFSS), which combination of two distinct theories known as the bipolar complex picture fuzzy set (BCPFS) and the soft set (SS). The capabilities of this new model are to achieve a larger range with the aid of the membership degree (MD), non-membership (NMD), and abstinence degree (AD) with complex integers. The BCPFSS represented two different degrees, known as positive degrees examine positive aspects of the problem and negative degrees examine negative aspects of the problem. Moreover, human–computer interaction (HCI) is an effective human–computer interface that presents a realistic world. HCI is a field that recognizes the significant role of educational applications and platforms in visualizing content, providing students with easy access to education in their homes, and paving the way for the development of alternative and advanced methods for implementing the concept of e-learning. With the technological advancements of HCI, it has become a fundamental tool in every area of activity, revolutionizing work environments and expanding their scope. In order to resolve the challenge of analyzing complex and unreliable data in real-world contexts, we proposed the idea of bipolar complex picture fuzzy soft sets (BCPFSSs). The innovative concept of BCPFSS defines many operations such as complement, extended intersection, and restricted intersection. The idea of BCPFSSs possesses a broad structure as they explore all values of MD, AD, and NMD with both positive (+ve) and negative (−ve) perspectives, making them adept at handling multivariable challenges. In this paper, modeling approaches based on BCPFSSs are proposed for choosing the HCI for effective implementation by formulating and analyzing score functions. Finally, a comparative analysis section illuminates a comprehensive and organized comparison of recent works to the earlier work.