Number Theory Research Articles

Collatz Conjecture

This paper presents an analysis of the number of zeros in the binary representation of natural numbers. The primary method of analysis involves the use of the concept of the fractional part of a number, which naturally emerges in the determination of binary representation. This idea is grounded in the fundamental property of the Riemann zeta function, constructed using the fractional part of a number. Understanding that the ratio between the fractional and integer parts of a number, analogous to the Riemann zeta function, reflects the profound laws of numbers becomes the key insight of this work. The findings suggest a new perspective on the interrelation between elementary properties of numbers and more complex mathematical concepts, potentially opening new directions in number theory and analysis.This analysis has allowed to understand that the Collatz sequence initially tends towards a balanced symmetric arrangement of zeros and ones, and then it collapses, realizing the scenario of the Collatz conjecture.

Congratulations to our Editorial Board Members Prof. Taekyun Kim, Dr. Mladen Vassilev-Missana and Prof. Krassimir Atanassov for their anniversaries in 2024!

[EDITORIAL] Journal “Notes on Number Theory and Discrete Mathematics” congratulates its Editorial Board Members Prof. Taekyn Kim and Dr. Mladen Vassilev-Missana, as well as its Editor-in-Chief Prof. Krassimir Atanassov on the occasion of their anniversaries in 2024!

Number theory based modern cryptography: RSA and Diffie-Hellman algorithms

Abstract. In the twenty-first century, the Internet has grown rapidly. Cryptography, with the protection of number theory as an important guarantee for the Internet, is critical to study, and algorithms must be updated to keep up with the evolution of the digital world. With the rapid spread of the Internet, security flaws in certain algorithms have been exposed, generating widespread alarm. This literature review focuses on the introductions of the Rivest-Shamir-Adleman (RSA) and Diffie-Hellman algorithms, integrating and summarizing previous research. The content of this literature review includes an introduction to algorithms and number theory, specific applications of these algorithms, and security issues associated with them. Finally, this literature review concludes that RSA and Diffie-Hellman are inefficient and insecure in some sectors of application, necessitating the use of alternative algorithms or the replacement of some unique encryption methods to achieve the desired level of safety.

On the existence of certain Lehmer numbers modulo a prime

A Lehmer number modulo an odd prime numberp is a residue class a∈Fp× whose multiplicative inverse ā has opposite parity. Lehmer numbers that are also primitive roots are called Lehmer primitive roots. Analogously, in this article we say that a residue class a∈Fp× is a Lehmer non-primitive root modulop if a is Lehmer number modulo p which is not a primitive root. We provide explicit estimates for the difference between the number of Lehmer non-primitive roots modulo a prime p and their “expected number”, which is p−1−ϕ(p−1)2. Similar explicit bounds are also provided for the number of k-consecutive Lehmer numbers modulo a prime, and k-consecutive Lehmer primitive roots We also prove that for any prime number p>3.05×1014, there exists a Lehmer non-primitive root modulo p. Moreover, we show that for any positive integer k≥2 (respectively, k≥5) and for all primes p≥exp(122k3) (respectively, p≥exp(6.87k)), there exist k consecutive Lehmer numbers modulo p (respectively, k consecutive Lehmer primitive roots modulo p). For large primes p, these theorems generalize two results which were proven in a paper by Cohen and Trudgian appeared in the Journal of Number Theory in 2019.

Generalized NTRU Algorithms on Algebraic Rings

The NTRU (Number Theory Research Unit) is a prominent post-quantum public key cryptography algorithm and a current focus of research. Although many NTRU variants have been proposed, a comprehensive generalization of these variants is still lacking. To address this issue, we propose the G-NTRU (Generalized NTRU), an algorithmic framework for NTRU variants on algebraic rings. This framework generalizes specific ring extensions to more general algebraic structures, allowing the unification of various NTRU variants. We then analyze the key properties of the G-NTRU, including correctness, lattice-based characteristics, and security. The semantic security of the G-NTRU is demonstrated through the introduction of the G-AGCD (Generalized Approximate Greatest Common Divisor). To validate the generality of the G-NTRU framework, we introduce the CNTRU (Complex Number NTRU), a variant of the NTRU over the ring of complex numbers. The CNTRU shares the same properties as the G-NTRU, further confirming the versatility of the G-NTRU in studying NTRU variants over algebraic rings.

Symmetries of a General Polygon

Summary Most basic texts on group theory include a brief discussion of the symmetry group of a regular polygon in the Euclidean plane, but few go beyond this. Here we discuss the symmetry group of a general polygon by using ideas from geometry and number theory.

Statistical Analysis and Distribution of Fermat Pseudoprimes Within the Given Interval

Prime numbers are natural numbers that can only be divided by one and the original number. There is more than one of them. Error-correcting codes used in telecommunications are generated using prime numbers. They guarantee automatic message correction both during transmission and reception. Algorithms used in public-key cryptography are built upon primes. They're also employed in the production of pseudorandom numbers. Mathematicians and many other scientific and technological communities have always been fascinated by prime numbers. Additionally, computer engineers can use it to tackle a wide range of real-world problems. The analysis of prime numbers is very important for finding their applications in different fields. The statistical analysis of pseudoprimes within a given interval is carried out in the presented paper and an algorithm of Python program to find the distribution of pseudoprimes has also been generated, which is used to find their distribution with different bases within the given intervals. The data analysis process made use of graphical depiction. The discovery will surely open up new avenues for future number theory study and applications outside of mathematics.

Euler’s Formula and Its Applications in Modern Mathematics

This article examines the various uses of Euler‘s formula in complex analysis, topology, number theory, and other mathematical fields. Renowned for its mathematical elegance, exponential and trigonometric functions are closely related according to Euler‘s formula. acting as a fundamental basis for various key developments in these fields. By conducting a thorough analysis, the study reveals the extensive significance and lasting impact of Euler’s formula in both theoretical and applied mathematics. The research employs a comprehensive literature review, enhanced by rigorous mathematical derivations and practical examples, to demonstrate the widespread applicability and versatility of Euler’s formula in a wide range of contexts. The findings underscore that Euler’s formula not only holds a central position in theoretical mathematics but also plays a crucial role in engineering, quantum mechanics, signal processing, and physics. This study contributes to a deeper understanding of the intrinsic relationships within mathematical formulas, emphasizing their far-reaching practical relevance and potential for future research and technological innovation.

A Survey on Ring Theory for its Mathematical Foundation and Applications

The origin of the ring theory can be dated to the early 19th century. With the rapid development of technology, more and more complex engineering problems and computer problems need to obtain support and results from more basic mathematics theory. Hence, based on this demand, the researcher discovered that ring theory is a very noteworthy math theory for solving the complicated problems above. This paper investigated the ring theory and related articles based on number theory and corresponding applications, which produces and results in a literature review in this particular direction.

The artificial intelligence in auditing: Corporate behaviour and technological adoption in an emerging market

The objective of this research is to offer an empirically grounded assessment of the intention of the auditing profession to adopt "disruptive" technologies. This study investigates, using data collected from employees of the Big Four firms in Tunisia, the determinants that drive auditors to adopt blockchain technology (BT). To achieve this objective, in 2022, 53 auditors from the "big four" enterprises in Tunisia, including both certified auditors and auditing students in training, participated in a survey. The study employs statistical methods and ordinary least squares (OLS) regression to identify the associations between the intention to implement BT and the five variables under investigation. The study examines perceived utility, ease of use, trust, support cost (SC), and facilitating condition (FC) while drawing on a number of theories, including the Technology Acceptance Model (TAM). Results show that two factors, particularly perceived utility for auditing (PUA) practice and trust, drive auditing professionals' intention among Big Four companies to use BT. This study results illuminate the factors motivating Big Four companies to embrace BT, enabling the development of strategies to expedite the adoption and utilization of this technology in developing accounting and auditing firms. The study fills a gap in the literature about adopting BT in emerging economies by concentrating on this specific setting. This study contributes to the knowledge of technology adoption and provides valuable recommendations for accelerating the uptake and application of BT in this particular setting.

Role of Linear Diophantine Equations in RSA Encryption

Abstract. Diophantine equations are mathematical equations that include only integer solutions and two or more unknown variables. To illustrate, the most elementary form of Diophantine equations are linear Diophantine equations when two different one-degree monomials added up to a constant value. Such equations were given the name of the extraordinary Greek mathematician Diophantus who made great contribution to algebra and number theory. In addition, they help in defining concepts in algebraic geometry and contribute to the development of algorithms for cryptography. They work in the same way as public keys do in cryptocurrencies within blockchain technology to curb cyber threats and scams targeting public systems. They serve as a means of securing transactions on a computer network when using cryptocurrencies such as bitcoins. These encryption techniques also permit cryptographers and mathematicians to create new ideas in relation to these number systems, thus making it possible for further cryptographic techniques to be put into place.

Windmills of the Minds: A Hopping Algorithm for Fermat’s Two Squares Theorem

Fermat’s two squares theorem asserts that a prime one more than a multiple of 4 is a sum of two squares. There are many proofs of this gem in number theory, including a remarkable one-sentence proof by Don Zagier based on two involutions on a finite set built from such a prime. Applying the two involutions alternatively leads to an iterative algorithm to find the two squares for the prime. Moreover, a detailed analysis of the computation reveals that it is possible to jump through the iteration nodes, leading to a better hopping algorithm. Here is a formalisation of Zagier’s proof, deriving the involutions using windmill patterns. Theories developed for the formal proof are used to establish the correctness of both algorithms.

Imaginary latent variables: Empirical testing for detecting deficiency in reflective measures

Imaginary latent variables are variables with negative variances and have been used to implement constraints in measurement models. This article aimed to advance this practice and rationalize the imaginary latent variables as a method to detect possible latent deficiencies in measurement models. This rationale is based on the theory of complex numbers used in the measurement process of common factor model–based structural equation modeling. Modeling an imaginary latent variable produces a potential deficiency within its relative reflective measures through a considerable reduction in common variance indicating the most affected indicator(s).

Tighter bound for generalized multiple discrete logarithm problem via MDS matrix method

Discrete logarithm problem (DLP) is one of the fundamental hard problems used in cryptography. For 1≤k≤n, solving the k-out-of-n DLP instances is an important problem emerging in certain scenarios in public-key cryptography. Ying and Kunihiro (ACNS 2017) pioneered in studying k-out-of-n instance solutions of DLP, which is a generalized version of multiple DLP. By reducing the multiple DLP to the generalized version, they established lower bounds on the computational complexity of k-out-of-n DLP for different parameter values of k.In this paper, we further reduce the reduction complexity presented in Ying and Kunihiro's work and increase the range of k and n for the tight lower bound of k-out-of-n DLP in the generic group model, which has applications in related cryptographic schemes. To achieve the goal, the key technique is to utilize a variant of fast multipoint evaluation. We divide the discussion into two cases. In the special case when n divides p−1, by leveraging Number Theory Transform (NTT) technique, we expand k and n to a larger range. In the general case, by using a variant of fast multipoint evaluation, we increase k and n to a moderately larger range.

Optimality of the Howard-Vala T-gate in stabilizer quantum computation

In a remarkable work [Phys. Rev. A 86 022316 (2012)], Howard and Vala introduced a qudit version of the qubit T-gate (i.e., π/8-gate) for any prime dimensional system. This non-Clifford gate is a key ingredient of the paradigm ‘Clifford +T’, which are widely employed in the stabilizer formalism of universal and fault-tolerant quantum computation. Considering the applications and significance of the T-gate, it is desirable to characterize it from various angles. Here we prove that in any prime dimensional system, the Howard-Vala T-gate is optimal, among all diagonal gates, for generating magic resources from stabilizer states when the magic is quantified via the L 1-norm of characteristic functions (Fourier transforms) of quantum states. The quadratic Gaussian sum in number theory plays a key role in establishing this optimality. This highlights an extreme feature of the Howard-Vala T-gate. We further reveal an intrinsic relation between the Howard-Vala T-gate and the Watson-Campbell-Anwar-Browne T-gate [Phys. Rev. A 92 022312 (2015)] for any prime dimensional system.

Design of Time Series Fuzzy Segmentation Algorithm for Communication Data Classification

In order to solve the problem that the imbalance of communication data sets leads to a significant increase in classification difficulty, a classification algorithm for fuzzy segmentation of time series is proposed. The principal component analysis method is used to obtain the eigenvector with the largest eigenvalue. The time series of data is established according to the interval number theory. The segmentation target of the communication data time series is characterized by the Langmuir distance measurement function between data and categories. The classification result of fuzzy segmentation is obtained based on the judgment relationship between the difference of fuzzy classification matrix and the closing condition. The experimental results show that the number of data in the three classification situations of the algorithm in this paper is always at the corresponding ideal level, with high accuracy and low error and failure.

Multiscale Sieve for Smart Prime Generation and Application in Info-Security, IoT and Blockchain

The huge computational cost required to test whether a number is prime and the inefficiency of the known sieving algorithms for extremely large inputs have posed significant challenges in computational number theory. Traditional deterministic prime generation methods struggle to maintain performance when the input sizes increase exponentially. In this work, we show that, through multiscale distribution and deterministic prime number generation, it is possible to create a multiscale sieve with drastically better performance than the deterministic algorithms known to date, providing a more efficient solution for large-scale prime number generation, demonstrated by several benchmarks that highlight the potential of our approach. Consequently, we can gain some advantages in cryptography and in info-security, such as in IoT and blockchain environments.

Cyclotomic Generating Functions

It is a remarkable fact that for many statistics on finite sets of combinatorial objects, the roots of the corresponding generating function are each either a complex root of unity or zero. These and related polynomials have been studied for many years by a variety of authors from the fields of combinatorics, representation theory, probability, number theory, and commutative algebra. We call such polynomials cyclotomic generating functions (CGFs). With Konvalinka, we have studied the support and asymptotic distribution of the coefficients of several families of CGFs arising from tableau and forest combinatorics. In this paper, we survey general CGFs from algebraic, analytic, and asymptotic perspectives. We review some of the many known examples of CGFs in combinatorial representation theory; describe their coefficients, moments, cumulants, and characteristic functions; and give a variety of necessary and sufficient conditions for their existence arising from probability, commutative algebra, and invariant theory. As a sample result, we show that CGFs are "generically" asymptotically normal, generalizing a result of Diaconis on $q$-binomial coefficients using work of Hwang-Zacharovas. We include several open problems concerning CGFs.

De Bruijn’s Short Route to Rényi’s Parking Constant

Rényi’s parking constant describes the mean density of randomly parked cars in a long street. We present a short elementary derivation of the exact formula for Rényi’s constant, which is inspired by de Bruijn’s research in number theory and differential equations.

A risk assessment model of spontaneous combustion for sulfide ores using Bayesian network combined with grounded theory

Sulfide ores, collectively referred to as a class of minerals with extensive applications, are increasingly susceptible to spontaneous combustion incidents as the mining environment evolves. The spontaneous combustion of sulfide ores is a complex nonlinear process, and the mechanisms of which remain not fully understood. Consequently, it is challenging to delineate the accident trajectory through traditional accident causation models and subsequently construct a probabilistic model for spontaneous combustion risk. To address this issue, this paper proposed a risk assessment model based on grounded theory and Bayesian network. The advantage of this model is that it employs grounded theory to categorize and organize factors influencing the occurrence of accidents, thereby forming a clear accident trajectory and providing a framework and basis for constructing a probabilistic assessment model. Furthermore, the model presented in this paper introduces the D number theory and the Noisy-OR model to handle uncertainties in the risk assessment process and to establish more rational conditional probability tables. The findings suggest that environmental temperature, heat dissipation conditions, safety awareness, and safety skills are the most influential factors in spontaneous combustion incidents when considering a particular point in time. However, when considering the entire process of sulfide ore pile spontaneous combustion, the temperature of the ore pile is the most critical accident factor to monitor. Therefore, the key to suppressing the spontaneous combustion of sulfide ores lies in reducing environmental temperatures, enhancing heat dissipation, and elevating the safety awareness of relevant personnel.