Number Theory Research Articles

Positivity-hardness results on Markov decision processes

This paper investigates a series of optimization problems for one-counter Markov decision processes (MDPs) and integer-weighted MDPs with finite state space. Specifically, it considers problems addressing termination probabilities and expected termination times for one-counter MDPs, as well as satisfaction probabilities of energy objectives, conditional and partial expectations, satisfaction probabilities of constraints on the total accumulated weight, the computation of quantiles for the accumulated weight, and the conditional value-at-risk for accumulated weights for integer-weighted MDPs. Although algorithmic results are available for some special instances, the decidability status of the decision versions of these problems is unknown in general. The paper demonstrates that these optimization problems are inherently mathematically difficult by providing polynomial-time reductions from the Positivity problem for linear recurrence sequences. This problem is a well-known number-theoretic problem whose decidability status has been open for decades and it is known that decidability of the Positivity problem would have far-reaching consequences in analytic number theory. So, the reductions presented in the paper show that an algorithmic solution to any of the investigated problems is not possible without a major breakthrough in analytic number theory. The reductions rely on the construction of MDP-gadgets that encode the initial values and linear recurrence relations of linear recurrence sequences. These gadgets can flexibly be adjusted to prove the various Positivity-hardness results.

Joint Moments of Higher Order Derivatives of CUE Characteristic Polynomials I: Asymptotic Formulae

Abstract We derive explicit asymptotic formulae for the joint moments of the $n_{1}$-th and $n_{2}$-th derivatives of the characteristic polynomials of Circular Unitary Ensemble random matrices for any non-negative integers $n_{1}, n_{2}$. These formulae are expressed in terms of determinants whose entries involve modified Bessel functions of the first kind. We also express them in terms of two types of combinatorial sums. Similar results are obtained for the analogue of Hardy’s $Z$-function. We use these formulae to formulate general conjectures for the joint moments of the $n_{1}$-th and $n_{2}$-th derivatives of the Riemann zeta-function and of Hardy’s $Z$-function. Our conjectures are supported by comparison with results obtained previously in the number theory literature.

ПРО ОДИН ПІДХІД ДО ПЕРЕТВОРЕННЯ ЧИСЕЛ У СИСТЕМАХ ЗАЛИШКОВИХ КЛАСІВ ІЗ РІЗНИМИ МОДУЛЯМИ

The purpose of the study is an analytical consideration of the system of residual classes for the implementation of the operation of converting numbers from one system of residual classes to another. System analysis, number theory, and the Chinese remainder theorem are tools of the research methodology. The method uses the representation of the number both by its remainders and in the polyadic code. The methodology is based on determining the positional characteristics for this module on the basis of the received positional characteristics for the remaining modules of the original system, with the subsequent construction on their basis of the residuals for the modules of the sought system. The theoretical justification of the approach for obtaining an effective solution of this non-modular operation is performed. The considered solutions have high speed and can be effective in the development of modular computing structures for promising information technologies.

The resolvent kernel on the discrete circle and twisted cosecant sums

In this paper we present a general unifying principle for computing finite trigonometric sums of types that arise in physics and number theory. We obtain formulas that are more general than previous expressions and deduce linear recursions, which are computationally more efficient than the degree two recursions proved by Zagier. As an application, we provide an answer to a question recently posed by Xie-Zhao-Zhao concerning special values of Dirichlet L-functions. The proofs use the combinatorial Laplacian on cyclic graphs and their twisted coverings. The techniques therefore connect the trigonometric sums to spectral invariants of graphs and open up for future investigations.

Prime number factorization with light beams carrying orbital angular momentum

We point out a link between orbital angular momentum (OAM) carrying light beams and number theory. The established link makes it possible to formulate and implement a simple and ultrafast protocol for prime number factorization by employing OAM endowed beams that are modulated by a prime number sieve. We are able to differentiate factors from non-factors of a number by simply measuring the on-axis intensity of light in the rear focal plane of a thin lens focusing on a source beam. The proposed protocol solely relies on the periodicity of the OAM phase distribution, and hence, it is applicable to fully as well as partially coherent fields of any frequency and physical nature—from optical or x-ray to matter waves—endowed with OAM. Our experimental results are in excellent agreement with our theory. We anticipate that our protocol will trigger new developments in optical cryptography and information processing with OAM beams.

Nilai Islam Dalam Media (Analisis Wacana Kritis Film Hati Suhita)

This study discusses the Islamic values contained on the media in the form of the film Hati Suhita. with a relatively large audience of 507,167 in 2023. The aim of this study is to reveal the da'wah discourse and the Islamic values contained in the film. There are a number of theories used, including the da'wah theories popularized by Ansari and Asmuni Syukur. The theories discuss "maddah" or material and "wasilah" or media. Another theory used in this study is about the basic Islamic values which were popularized by Hadi Yasin and Muhammad Quraish Shihab. The Islamic values discussed in the theory regarding aqidah, sharia, and akhlak. This study is qualitative research with Norman Fairclough's critical discourse analysis approach. Data collection was carried out by text analysis, observing the production team's statements, as well as extracting audience opinions through surveys and interviews. The results of this study show, 1) strengthening the Islamic message in the film was carried out from the preparation period in the production process; 2) Islamic messages about aqidah, sharia and morals can be received directly by the audience; 3) da'wah discourse with the main values of Islamic teachings packaged with narratives of relationships among characters.

About one approach for solving bounary-value problems of differential equation of pantograph type

Pantograph equations have been studied for a long time. In 1940, K. Mahler introduced functional differential equations of this type into number theory. In 1971, J. Ockendon used a functional differential equation with a transformed argument, , to describe the dynamics of the pantograph of an electric locomotive. Subsequently, Pantograph-type equations were studied in the works of many authors. This article considers a boundary value problem for a differential equation of the pantograph type. To solve the posed boundary value problem, the parameterization method proposed by Professor D. Dzhumabaev is used. To do this, we denote the value of the function at the initial point of the segment under consideration by a parameter and perform a change of variable Then the solvability of the original boundary value problem is reduced to studying the solvability of the resulting Cauchy problem for the original equation and to a linear algebraic equation to determine the introduced parameter. Next, using the method of successive approximations, we find solutions to the Cauchy problem for the Pantograph type equation. The convergence of the resulting sequence and the convergence of its solution to the solution of the Cauchy problem of Pantograph type are proved. By requiring continuity of the free term, we establish its unique solvability. Substituting the resulting solution into a linear algebraic equation to determine the entered parameter, we calculate the values of the entered parameter using the original data. Substituting the resulting expressions into we find the solution to the original problem . And assuming the unique solvability of a linear algebraic equation, we establish the solvability of the boundary value problem for an equation of Pantograph type.

Genuinely nonabelian partial difference sets

AbstractStrongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of genuinely nonabelian PDSs, that is, PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which—one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil—are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research.

Smallest denominators

AbstractWe establish higher dimensional versions of a recent theorem by Chen and Haynes [Int. J. Number Theory 19 (2023), 1405–1413] on the expected value of the smallest denominator of rational points in a randomly shifted interval of small length, and of the closely related 1977 Kruyswijk–Meijer conjecture recently proved by Balazard and Martin [Bull. Sci. Math. 187 (2023), Paper No. 103305]. We express the distribution of smallest denominators in terms of the void statistics of multidimensional Farey fractions and prove convergence of the distribution function and certain finite moments. The latter was previously unknown even in the one‐dimensional setting. We furthermore obtain a higher dimensional extension of Kargaev and Zhigljavsky's work on moments of the distance function for the Farey sequence [J. Number Theory 65 (1997), 130–149] as well as new results on pigeonhole statistics.

Open Problems Column

This Issues Column! This issue's Open Problem Column is by P.G. Walsh and is Integral Points on Elliptic Curves: A Journey into Speculative Number Theory. Request for Columns! I invite any reader who has knowledge of some area to contact me and arrange to write a column about open problems in that area. That area can be (1) broad or narrow or anywhere inbetween, and (2) really important or really unimportant or anywhere inbetween.

Legal Dogmatic Questions about the Impact of the European Union’s Digital Legislation on Hungarian Contract Law

The complexity and flexibility of contract law, and its ability to meet various social, economic and technical-technological needs, are indicated by a number of theories (approaches) that offer a new approach to the processing of contracts. Among the predominant theories one might include the following: overview of contracts from a constitutional and human rights approach2; deriving from this the contracts related to private and family life (intimate contracts)3; by connecting the concepts of contract law and property rights, exploring the specific characteristics of existing contracts4; filling the term "digital contract type" with content5; classification of different kinds of interconnection of contracts (complex contracts).6 The author dedicates the study to the topic of whether Hungarian contract law can meet the challenges created by digitalization, which have not yet been identified in all its details, and what are the critical points that require consideration and action as soon as possible. After the summary of the digital legislation of the European Union the author identifies the effects of digitalization in relation to the Hungarian contract law and the special contracts resulting from digitalization. Finally, the author makes de lege lata and de lege ferenda conclusions in light of this topic.

Stability risk early warning for mine goaf: Based on D-RES and asymmetric fuzzy connection cloud model

To prevent and mitigate safety risks in the mining goaf and achieve risk prevention and shifting the control frontier forward, a novel goaf stability risk early warning model is developed based on D-RES and asymmetric fuzzy association cloud. Taking into consideration the intrinsic characteristics of the goaf, an indicator system covering 12 indicators from three aspects, namely goaf morphology, rock mass physical-mechanical parameters, and goaf development status, is constructed to assess goaf stability. To address the vagueness and randomness issues in the classification criteria for goaf stability early warning, a fuzzy interval-based model is proposed to quantify warning indicators and determine their fuzzy interval range and optimal boundary information. The RES theory is introduced to analyze the interaction relationships among warning indicators, and the D numbers theory is utilized to comprehensively synthesize opinions from multiple experts, establishing a D-RES coupled weight calculation model for collaborative decision-making. Additionally, the asymmetric fuzzy association cloud is introduced to represent the uncertainties and vagueness among different risk levels, considering the finite interval and asymmetric distribution features of warning indicator levels. By integrating the D-RES-derived coupled weights, early warning of goaf stability is achieved. The validation on mining goafs demonstrates that the warning results of the model are consistent with the actual safety conditions, indicating the high applicability and accuracy of the proposed early warning model. The developed model provides a theoretical foundation for formulating safety management policies relevant to mining goafs.

Density questions in rings of the form 𝒪K[γ] ∩ K

We fix a number field [Formula: see text] and study statistical properties of the ring [Formula: see text] as [Formula: see text] varies over algebraic numbers of a fixed degree [Formula: see text]. Given [Formula: see text], we explicitly compute the density of [Formula: see text] for which [Formula: see text] and show that this does not depend on the number field [Formula: see text]. In particular, we show that the density of [Formula: see text] for which [Formula: see text] is [Formula: see text]. In a recent paper [Singhal and Lin, Primes in denominators of algebraic numbers, Int. J. Number Theory (2023), doi:10.1142/S1793042124500167], the authors define [Formula: see text] to be a certain finite subset of [Formula: see text] and show that [Formula: see text] determines the ring [Formula: see text]. We show that if [Formula: see text] satisfy [Formula: see text], then the events [Formula: see text] and [Formula: see text] are independent. As [Formula: see text], we study the asymptotics of the density of [Formula: see text] for which [Formula: see text].

Fast Multivariate Multipoint Evaluation Over All Finite Fields

Multivariate multipoint evaluation is the problem of evaluating a multivariate polynomial, given as a coefficient vector, simultaneously at multiple evaluation points. In this work, we show that there exists a deterministic algorithm for multivariate multipoint evaluation over any finite field \(\mathbb {F} \) that outputs the evaluations of an m -variate polynomial of degree less than d in each variable at N points in time \[ (d^m+N)^{1+o(1)}\cdot {\rm poly}(m,d,\log |\mathbb {F}|) \] for all \(m\in \mathbb {N} \) and all sufficiently large \(d\in \mathbb {N} \) . A previous work of Kedlaya and Umans (FOCS 2008, SICOMP 2011) achieved the same time complexity when the number of variables m is at most d o (1) and had left the problem of removing this condition as an open problem. A recent work of Bhargava, Ghosh, Kumar and Mohapatra (STOC 2022) answered this question when the underlying field is not too large and has characteristic less than d o (1) . In this work, we remove this constraint on the number of variables over all finite fields, thereby answering the question of Kedlaya and Umans over all finite fields. Our algorithm relies on a non-trivial combination of ideas from three seemingly different previously known algorithms for multivariate multipoint evaluation, namely the algorithms of Kedlaya and Umans, that of Björklund, Kaski and Williams (IPEC 2017, Algorithmica 2019), and that of Bhargava, Ghosh, Kumar and Mohapatra, together with a result of Bombieri and Vinogradov from analytic number theory about the distribution of primes in an arithmetic progression. We also present a second algorithm for multivariate multipoint evaluation that is completely elementary and in particular, avoids the use of the Bombieri–Vinogradov Theorem. However, it requires a mild assumption that the field size is bounded by an exponential tower in d of bounded height . More specifically, our second algorithm solves the multivariate multipoint evaluation problem over a finite field \(\mathbb {F} \) in time \[ (d^m+N)^{1+o(1)}\cdot {\rm poly}(m,d,\log |\mathbb {F}|) \] for all \(m\in \mathbb {N} \) and all sufficiently large \(d\in \mathbb {N} \) , provided that the size of the finite field \(\mathbb {F} \) is at most (exp(exp(exp(⋅⋅⋅(exp( d ))))), where the height of this tower of exponentials is fixed.

Corrigendum to “The discriminant of compositum of algebraic number fields”

We point out that there is an error in the proof of Theorem 1.1 in [The discriminant of compositum of algebraic number fields, Int. J. Number Theory 15 (2019) 353–360]. We also prove that the result of this theorem holds with an additional hypothesis. However, it is an open problem whether the result of the theorem is true in general or not.

Polynomial extension of some symmetric partition regular structures

The study of symmetric structures is a new trend in combinatorial number theory. Recently in [9], Di Nasso proved symmetrized versions of some classical theorems like Hindman's theorem, Van der Waerden's theorem and Deuber's theorem. This opens the question of which other classical theorems can be symmetrized, as well as if other symmetric operations allow for such generalizations. Here we give positive answers to both questions, by showing that symmetrization of the polynomial extension of Van der Waerden's and Deuber's Theorem is possible.

Lookup Table-Based Design of Scalar Multiplication for Elliptic Curve Cryptography

This paper is aimed at using a lookup table method to improve the scalar multiplication performance of elliptic curve cryptography. The lookup table must be divided into two polynomials and requires two iterations of point doubling operation, for which negation operations are needed. It is well known that an inversion operation requires a lot of multiplication time. The advantage of this paper is that we are able to reduce one inverse element calculation for this problem and also improve the basic operations of finite fields through segmentation methods. If the normal basis method is used in the design of the inverse element operation, it must be converted to the normal basis through the standard basis. However, the conversion process requires a lot of matrix operations. Though the anti-element operation has good speed performance, it also increases the computational complexity. Using number theory and grouping methods will greatly improve the performance of inverse element operations. With application of the two-time point doubling operation in the hardware implementation, the developed approach reduces the computing time by 48% as compared with the conventional approach. The computational time of the scalar multiplication using the presented method is further improved by 67% over the traditional algorithm with only an area increase of 12%. Finally, the proposed lookup table-based technique can be utilized for software and hardware implementation, as the developed arithmetic operations are simple and are consistent in their execution.

On the distribution of αp^2 + β modulo one for primes p such that p + 2 has no more two prime divisors

A classical problem in analytic number theory is to study the distribution of frac- tional part αp^k + β, k ≥ 1 modulo 1, where α is irrational and p runs over the set of primes. We consider the subsequence generated by the primes p such that p + 2 is an almost-prime (the existence of infinitely many such p is another topical result in prime number theory) and prove that its distribution has a similar property.

A Logical Proof of the Polignac's Conjecture Based on Partitions of an Even Number of a New Formulation

Polignac's Conjecture, proposed by Alphonse de Polignac in the 19th century, is a captivating hypothesis that extends the notion of twin primes to a broader context. It posits that for any even positive integer 2k, there exist infinitely many pairs of consecutive prime numbers whose difference is 2k. This conjecture is a natural generalization of the Twin Prime Conjecture, which focuses solely on pairs of primes differing by two. The conjecture has significant implications for our understanding of the distribution of prime numbers and the nature of their gaps and its exploration serves as a testament to the enduring fascination and mystery surrounding prime numbers and their properties. However, despite extensive efforts by mathematicians over the years, Polignac's Conjecture remains unproven, standing as one of the many unsolved problems in number theory.   This study utilizes a set of all odd partitions generated from an even number of a new formulation, and we show that from this set of all pairs of odd numbers, there exist proper subsets containing infinitely many pairs of prime numbers whose differences is a fixed even gap. Finally, using these results and the facts that the difference of any two prime numbers is even and there exist infinitely many prime numbers, a logical proof of the Polignac's Conjecture is provided.

Hrotsvit of Gandersheim and Philosophy

Hrotsvit of Gandersheim sees music and arithmetic as parts of philosophy. Presenting the motives for engaging in literary creativity and describing her writing process, she says that by glorifying God with her work, she incorporated threads and pieces of the ancient mantle of philosophy into it. The paper aims to provide insight into Hrotsvit’s knowledge of philosophy and the specificities and significance of her approach to the contents of the disciplines of arithmetic and music (in particular to musica humana and the harmony of a world) in her plays Paphnutius and Sapientia, to analyze different elements of her authorial self-perception and to emphasize the similarities between her and Augustine’s understanding of music, and the role of certain types of writing. The paper concludes that Hrotsvit incorporated her knowledge and understanding of ancient and medieval theories of music and numbers into her plays uniquely, showing originality in their interpretation and merging them with everyday moral life and education. In this way, the rarely treated philosophical elements of her oeuvre are presented and evaluated.