Number Theory Research Articles

On λ $\lambda $‐backbone coloring of cliques with tree backbones in linear time

AbstractA ‐backbone coloring of a graph with its subgraph (also called a backbone) is a function ensuring that is a proper coloring of and for each it holds that . In this paper we propose a way to color cliques with tree and forest backbones in linear time that the largest color does not exceed . This result improves on the previously existing approximation algorithms as it is ‐absolutely approximate, that is, with an additive error over the optimum. We also present an infinite family of trees with for which the coloring of cliques with backbones requires at least colors for close to . The construction draws on the theory of Fibonacci numbers, particularly on Zeckendorf representations.

MAN[sbnd]C: A masked autoencoder neural cryptography based encryption scheme for CT scan images

Sharing medical images securely is very important towards keeping patients’ data confidential. In this paper we propose MANC: a Masked Autoencoder Neural Cryptography based encryption scheme for sharing medical images. The proposed technique builds upon recently proposed masked autoencoders. In the original paper, the masked autoencoders are used as scalable self-supervised learners for computer vision which reconstruct portions of originally patched images. Here, the facility to obfuscate portions of input image and the ability to reconstruct original images is used an encryption-decryption scheme. In the final form, masked autoencoders are combined with neural cryptography consisting of a tree parity machine and Shamir Scheme for secret image sharing. The proposed technique MANC helps to recover the loss in image due to noise during secret sharing of image.•Uses recently proposed masked autoencoders, originally designed as scalable self-supervised learners for computer vision, in an encryption-decryption setup.•Combines autoencoders with neural cryptography - the advantage our proposed approach offers over existing technique is that (i) Neural cryptography is a new type of public key cryptography that is not based on number theory, requires less computing time and memory and is non-deterministic in nature, (ii) masked auto-encoders provide additional level of obfuscation through their deep learning architecture.•The proposed scheme was evaluated on dataset consisting of CT scans made public by The Cancer Imaging Archive (TCIA). The proposed method produces better RMSE values between the input the encrypted image and comparable correlation values between the input and the output image with respect to the existing techniques.

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 and 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.

Torsion primes for elliptic curves over degree 8 number fields

Let d≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 1$$\\end{document} be an integer and let p be a rational prime. Recall that p is a torsion prime of degree d if there exists an elliptic curve E over a degree d number field K such that E has a K-rational point of order p. Derickx et al. (Algebra Number Theory 17(2):267–308, 2023) have computed the torsion primes of degrees 4, 5, 6 and 7. We verify that the techniques used in Derickx et al. (Algebra Number Theory 17(2):267–308, 2023) can be extended to determine the torsion primes of degree 8.

Scaling spectrum of a class of self-similar measures with product form on ℝ

Abstract Let p, q, N ≥ 2 {N\geq 2} be three positive integers and let D = { 0 , 1 , … , N - 1 } ⊕ N p ⁢ { 0 , 1 , … , N - 1 } {D=\{0,1,\ldots,N-1\}\oplus N^{p}\{0,1,\ldots,N-1\}} be a product form digit set. It is well known that if q ∤ p {q\nmid p} , then the self-similar measure μ N q , D {\mu_{N^{q},D}} generated by the iterated function system { ( N q ) - 1 ⁢ ( x + d ) } d ∈ D , x ∈ ℝ {\{(N^{q})^{-1}(x+d)\}_{d\in D,x\in\mathbb{R}}} is a spectral measure with a spectrum Λ ⁢ ( N q , C ) = { ∑ i = 0 finite c i ⁢ N q ⁢ i : c i ∈ C } , \Lambda(N^{q},C)=\Bigg{\{}\sum_{i=0}^{\text{finite}}c_{i}N^{qi}:c_{i}\in C% \Bigg{\}}, where C = N q - p - 1 ⁢ D {C=N^{q-p-1}D} . In this paper, based on the properties of cyclic groups in number theory, we give some conditions on real number t under which the scaling set t ⁢ Λ ⁢ ( N q , C ) {t\Lambda(N^{q},C)} is also a spectrum of μ N q , D {\mu_{N^{q},D}} .

Из опыта преподавания. XIII. Имя кристаллического полиэдра. К 130-летию со дня рождения А. Ф. Лосева и 100-летию «Философии имени»

The article is timed to the 130th anniversary of the birth of the outstanding Russian philosopher A. F. Losev (1893—1988) and the 100th anniversary of his work «The Philosophy of Name». Against the background of his ideas, the article considers the nomenclature of polyhedral crystalline simple forms, developed by the scientific school of the «Fedorov Institute» under the leadership of A. K. Boldyrev in the walls of the Leningrad Mining Institute. A general algorithmic approach to the nomenclature of convex polyhedra, convenient for computer data processing, is proposed. It opens an extensive research programme on the frontier of combinatorial theory of convex polyhedra, linear algebra and number theory. The aim of the article is to improve the teaching of crystallography in Russian universities and to broaden the outlook of students in related fields of knowledge.

Paired fins in vertebrate evolution and ontogeny.

The origin of paired appendages became one of the most important adaptations of vertebrates, allowing them to lead active lifestyles and explore a wide range of ecological niches. The basic form of paired appendages in evolution is the fins of fishes. The problem of paired appendages has attracted the attention of researchers for more than 150 years. During this time, a number of theories have been proposed, mainly based on morphological data, two of which, the Balfour-Thacher-Mivart lateral fold theory and Gegenbaur's gill arch theory, have not lost their relevance. So far, however, none of the proposed ideas has been supported by decisive evidence. The study of the evolutionary history of the appearance and development of paired appendages lies at the intersection of several disciplines and involves the synthesis of paleontological, morphological, embryological, and genetic data. In this review, we attempt to summarize and discuss the results accumulated in these fields and to analyze the theories put forward regarding the prerequisites and mechanisms that gave rise to paired fins and limbs in vertebrates.

On Sequences of Geophine Triples Involving Padovan and Bernstein Polynomial with Propitious Property

Objective: To bring forth a new conception in the time-honoured field of Diophantine triples, namely “Geophine triple”. To examine the feasibility of proliferating an unending sequence of Geophine triples from Geophine pairs with the property comprising Padovan and Bernstein polynomial. Method: Established Geophine triples employing Padovan and Bernstein polynomial by the method of polynomial manipulations. Findings: An unending sequences of Geophine triples and with the property and are promulgated from Geophine pairs, precisely involving Padovan and Bernstein polynomials and few numerical representation of the sequences are computed using MATLAB. Novelty: This article carries an innovative approach of determining this definite type of triples using Geometric mean and thereby, two infinite sequences of Geophine triples with the property are ascertained. Also, few numerical representations of the sequences utilizing MATLAB program are figured out, thus broadening the scope of computational Number Theory. Keywords: Polynomial Diophantine triple, Geophine triple, Bernstein polynomial, Padovan polynomials, Pell’s equation, Special Polynomials

Aspects of Aperiodic Order

The theory of aperiodic order expanded and developed significantly since the discovery of quasicrystals, and continues to bring many mathematical disciplines together. The focus of this workshop was on harmonic analysis and spectral theory, dynamical systems and group actions, Schrödinger operators, and their roles in aperiodic order – with links into a full range of problems from number theory to operator theory.

A quantum-secure certificateless aggregate signature protocol for vehicular ad hoc networks

Vehicular ad hoc networks (VANETs) have revolutionized communication between vehicles and infrastructure, notably enhancing traffic management and passenger safety. However, VANETs are vulnerable to security threats, especially regarding data authenticity. Aggregate signature is a powerful technique that reduces computational and communication burdens by aggregating multiple signatures from different signers into a single signature. Traditional aggregate signature schemes, based on large prime number decomposition and the discrete logarithm problem, cannot effectively resist quantum attacks. This paper introduces a novel quantum secure certificateless aggregate signature (QSCLAS) scheme designed to enhance data security and privacy in VANETs. Our proposed scheme employs the number theory research unit (NTRU) algorithm. As a lattice-based cryptographic algorithm, NTRU is renowned for its security against quantum computer attacks, making it an essential component of our quantum-secure solution. By eliminating the need for expensive bilinear pairing operations, our proposed scheme achieves high efficiency and practicality in resource-limited VANETs environments. The security analysis demonstrates our scheme's resilience against both Type-I and Type-II adversaries in the random oracle model under the small integer solution (SIS) problem on the NTRU lattice. Furthermore, compared with existing approaches, the results illustrate that our proposed scheme offers significant advantages in signature generation and verification cost, as well as lower transmission overhead than other lattice-based schemes, thereby making it highly suitable for VANETs environments.

A Detailed Proof of the Strong Goldbach Conjecture Based on Partitions of a New Formulation of a Set of Even Numbers

The Strong Goldbach's conjecture, a fundamental problem in Number Theory, asserts that every even integer greater than 2 can be expressed as the sum of two prime numbers. Despite significant efforts over centuries, this conjecture remains unproven, challenging the core of mathematics. The known algorithms for attempting to prove or verify the conjecture on a given interval [a,b] consist of finding two sets of primes Pi and Pj such that Pi+Pj cover all the even numbers in the interval [a,b]. However, the traditional definition of an even number as 2n for n ∈ ℕ (where ℕ is the set of natural numbers), has not provided mathematicians with a straightforward method to obtain all Goldbach partitions for any even number of this form. This paper introduces a novel approach to the problem, utilizing all odd partitions of an even number of a new formulation of the form Eij = ni + nj + (nj - ni)n or alln ∈ ℕ. By demonstrating that there exist at least a pair of prime numbers in these odd partitions, the fact that the sum of any two prime numbers is even and there exists infinitely many prime numbers, this paper provides a compelling proof of the conjecture. This breakthrough not only solves a long-standing mathematical mystery but also sheds light on the structure of prime numbers.

Closed geodesics and Frøyshov invariants of hyperbolic three-manifolds

Frøyshov invariants are numerical invariants of rational homology three-spheres derived from gradings in monopole Floer homology. In the past few years, they have been employed to solve a wide range of problems in 3- and 4-dimensional topology. In this paper, we look at connections with hyperbolic geometry for the class of minimal L -spaces. In particular, we study relations between Frøyshov invariants and closed geodesics using ideas from analytic number theory. We discuss two main applications of our approach. First, we derive explicit upper bounds for the Frøyshov invariants of minimal hyperbolic L -spaces purely in terms of volume and injectivity radius. Second, we describe an algorithm to compute Frøyshov invariants of minimal L -spaces in terms of data arising from hyperbolic geometry. As a concrete example of our method, we compute the Frøyshov invariants for all spin ^{c} structures on the Seifert–Weber dodecahedral space. Along the way, we also prove several results about the eta invariants of the odd signature and Dirac operators on hyperbolic three-manifolds which might be of independent interest.

Post‐quantum attack resilience blockchain‐assisted data authentication protocol for smart healthcare system

AbstractThe smart healthcare system (SHS), a significant medical domain underpinning the Internet of Things (IoT), which collects and analyzes health data from many sources to provide better medical treatment. The smart healthcare system is a combination of hardware and software used in the medical care field, providing remote diagnosis and treatment via a patient‐based health data‐sharing system. To increase security, a large variety of authenticated techniques have been developed over the past several decades, most of which are based on conventional number‐theoretic assumptions such as discrete logarithms and integer factorization problems. However, Shor's method is capable of solving number‐theory‐based problems. As a result, Shor's technique might be used to resolve challenging number theory problems on a quantum computer effectively. Therefore, this article presents blockchain‐based healthcare record solutions with lattice RLWE‐based key exchange protocol using a smart card. Blockchain applications may correctly detect errors, including those that are risky, in the medical industry. It can also improve the efficiency, security, and transparency of transferring medical data throughout the healthcare protocol. The formal security of this protocol is shown under the ROM (random oracle model), and the informal security is also given in this article against well‐known attacks. The presented protocol outperforms related earlier mechanisms in terms of communication and computational cost overheads, according to the performance study.

The Basic k-ϵ Model and a New Model Based on General Statistical Descriptions of Anisotropic Inhomogeneous Turbulence Compared with DNS of Channel Flow at High Reynolds Number

Predictions are presented of mean values of statistical variables of large-scale turbulent flow of the widely used basic k-ϵ model, and of a new model, which is based on general statistical descriptions of turbulence. The predictions are verified against published results of direct numerical simulations (DNSs) of Navier–Stokes equations. The verification concerns turbulent channel flow at shear Reynolds numbers of 950, 2000, and 104. The basic k-ϵ model is largely based on empirical formulations accompanied by calibration constants. This contrasts with the new model, where descriptions of leading statistical quantities are based on the general principles of statistical turbulence at a large Reynolds number and stochastic theory. Predicted values of major output variables such as turbulent viscosity, diffusivity of passive admixture, temperature, and fluid velocities compare well with DNS for the new model. Significant differences are seen for the basic k-ϵ model.

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.

On the solutions of some Lebesgue–Ramanujan–Nagell type equations

Denote by [Formula: see text] the class number of the imaginary quadratic field [Formula: see text] with [Formula: see text] prime. It is well known that [Formula: see text] for [Formula: see text]. Recently, all the solutions of the Diophantine equation [Formula: see text] with [Formula: see text] were given by Chakraborty et al. in [Complete solutions of certain Lebesgue–Ramanujan–Nagell type equations, Publ. Math. Debrecen 97(3–4) (2020) 339–352]. In this paper, we study the Diophantine equation [Formula: see text] in unknown integers [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. To do this, we use the known results from the modularity of Galois representations associated with Frey–Hellegoaurch elliptic curves, the symplectic method and elementary methods of classical algebraic number theory. The aim of this paper is to extend the above results of Chakraborty et al.

The least primary factor of the multiplicative group

Let [Formula: see text] denote the least primary factor in the primary decomposition of the multiplicative group [Formula: see text]. We give an asymptotic formula, with order of magnitude [Formula: see text], for the counting function of those integers [Formula: see text] for which [Formula: see text]. We also give an asymptotic formula, for any prime power [Formula: see text], for the counting function of those integers [Formula: see text] for which [Formula: see text]. This group-theoretic problem can be reduced to problems of counting integers with restrictions on their prime factors, allowing it to be addressed by classical techniques of analytic number theory.

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.

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.