Large Positive Integer Research Articles

FROBENIUS NUMBERS ASSOCIATED WITH DIOPHANTINE TRIPLES OF $x^2+y^2=z^3$

Abstract We give an explicit formula for the Frobenius number of triples associated with the Diophantine equation $x^2+y^2=z^3$ , that is, the largest positive integer that can only be represented in p ways by combining the three integers of the solutions of $x^2+y^2=z^3$ . For the equation $x^2+y^2=z^2$ , the Frobenius number has already been given. Our approach can be extended to the general equation $x^2+y^2=z^r$ for $r>3$ .

On Bh[1]-sets which are asymptotic bases of order 2h

Let h,k≥2 be integers. A set A of positive integers is called asymptotic basis of order k if every large enough positive integer can be written as the sum of k terms from A. A set of positive integers A is said to be a Bh[g]-set if every positive integer can be written as the sum of h terms from A at most g different ways. In this paper we prove the existence of Bh[1] sets which are asymptotic bases of order 2h by using probabilistic methods.

Frobenius Numbers Associated with Diophantine Triples of x2-y2=zr

We give an explicit formula for the p-Frobenius number of triples associated with Diophantine Equations x2−y2=zr (r≥2), that is, the largest positive integer that can only be represented in p ways by combining the three integers of the solutions of Diophantine equations x2−y2=zr. This result is also a generalization because if r=2 and p=0, the (0-)Frobenius number of the Pythagorean triple has already been given. To find p-Frobenius numbers, we use geometrically easy to understand figures of the elements of the p-Apéry set, which exhibits symmetric appearances.

A Novel Scalable Quantum Protocol for the Dining Cryptographers Problem

This paper presents an innovative entanglement-based protocol to address the Dining Cryptographers problem, utilizing maximally entangled |GHZn⟩ tuples as its core. This protocol aims to provide scalability in terms of both the number of cryptographers n and the amount of anonymous information conveyed, represented by the number of qubits m within each quantum register. The protocol supports an arbitrary number of cryptographers n, enabling scalability in both participant count and the volume of anonymous information transmitted. While the original Dining Cryptographers problem focused on a single bit of information—whether a cryptographer paid for dinner—the proposed protocol allows m, the number of qubits in each register, to be any arbitrarily large positive integer. This flexibility allows the transmission of additional information, such as the cost of the dinner or the timing of the arrangement. Another noteworthy aspect of the introduced protocol is its versatility in accommodating both localized and distributed versions of the Dining Cryptographers problem. The localized scenario involves all cryptographers gathering physically at the same location, such as a local restaurant, simultaneously. In contrast, the distributed scenario accommodates cryptographers situated in different places, engaging in a virtual dinner at the same time. Finally, in terms of implementation, the protocol accomplishes uniformity by requiring that all cryptographers utilize identical private quantum circuits. This design establishes a completely modular quantum system where all modules are identical. Furthermore, each private quantum circuit exclusively employs the widely used Hadamard and CNOT quantum gates, facilitating straightforward implementation on contemporary quantum computers.

Primes with a missing digit: Distribution in arithmetic progressions and an application in sieve theory

AbstractWe prove Bombieri–Vinogradov type theorems for primes with a missing digit in their ‐adic expansion for some large positive integer . The proof is based on the circle method, which relies on the Fourier structure of the integers with a missing digit and the exponential sums over primes in arithmetic progressions. Combining our results with the semi‐linear sieve, we obtain an upper bound and a lower bound of the correct order of magnitude for the number of primes of the form with a missing digit in a large odd base .

A novel study for solving systems of nonlinear fractional integral equations

In this study, we explore the solution of a nonlinear system of fractional integro-differential equations based on the operational matrix method. We have modified the operational matrix method to accommodate such systems and have streamlined the resulting algebraic system in a practical manner. Instead of dealing with a system of equations, we have simplified the problem to finding the solution for a set of nonlinear equations. Here, m represents the number of block pulse functions, which is typically a large positive integer. This approach significantly reduces computational costs, especially for large values of m. We provide proofs for both the existence and uniqueness of the solution for this system. Additionally, we investigate two types of stability: Ulam–Hyers stability and generalized Ulam–Hyers stability. We have evaluated the effectiveness of the proposed method by applying it to three different problems and comparing our results with existing literature. The outcomes of these tests highlight the efficiency and efficacy of our proposed method.

Multi-twisted additive self-orthogonal and ACD codes are asymptotically good

Multi-twisted (MT) additive codes over finite fields constitute an important class of additive codes and are generalizations of cyclic and constacyclic additive codes. In this paper, we employ probabilistic methods and results from groups and geometry to study the asymptotic behavior of the rates and relative Hamming distances of two special subclasses of MT additive codes over finite fields, viz. MT additive self-orthogonal codes and MT additive codes with complementary duals (MT ACD codes) with respect to ordinary, Hermitian, and ⁎ trace bilinear forms. More precisely, let p be an odd prime, q be a prime power coprime to p, v be the multiplicative order of q modulo p, and let η be the largest positive integer such that pη|(qv−1). Here we establish the existence of asymptotically good infinite sequences of MT additive self-orthogonal and ACD codes of length pαℓ and block length pα over Fqt with pα→∞, relative Hamming distance at least δ and rates vpηℓt and 2vpηℓt with respect to the aforementioned trace bilinear forms, where ℓ≥1, t≥2 are integers, Fqt is the finite field of order qt and δ is a positive real number satisfying hqt(δ)<12−12ℓt, (here hqt is the qt-ary entropy function). This shows that MT additive self-orthogonal and ACD codes over finite fields are asymptotically good. As a special case, we deduce that constacyclic additive self-orthogonal and ACD codes over finite fields are asymptotically good.

On the restricted sumsets containing powers of an integer

In 2008, Lev proved that for any sufficiently large positive integer n, if A⊆[1,n] with gcdA=1 and |A|>6n/19, then there is a power of 2 that can be represented as the sum of at most 5 distinct elements of A. In this paper, as a corollary of our main results, we prove that for any ɛ>0 and any sufficiently large positive integer n, if A⊆[1,n] with gcdA=1 and |A|>(1/6+ɛ)n, then there is a power of 2 that can be represented as the sum of at most 36 distinct elements of A. On the other hand, there are infinitely many positive integers n, for each of which there exists B⊆[1,n] with gcdB=1 and |B|>n/6 such that no power of 2 can be represented as the sum of distinct elements of B.

Optimal Strategies Trajectory with Multi-Local-Worlds Graph

This paper constructs a non-cooperative/cooperative stochastic differential game model to prove that the optimal strategies trajectory of agents in a system with a topological configuration of a Multi-Local-World graph would converge into a certain attractor if the system’s configuration is fixed. Due to the economics and management property, almost all systems are divided into several independent Local-Worlds, and the interaction between agents in the system is more complex. The interaction between agents in the same Local-World is defined as a stochastic differential cooperative game; conversely, the interaction between agents in different Local-Worlds is defined as a stochastic differential non-cooperative game. We construct a non-cooperative/cooperative stochastic differential game model to describe the interaction between agents. The solutions of the cooperative and non-cooperative games are obtained by invoking corresponding theories, and then a nonlinear operator is constructed to couple these two solutions together. At last, the optimal strategies trajectory of agents in the system is proven to converge into a certain attractor, which means that strategies trajectory are certainty as time tends to infinity or a large positive integer. It is concluded that the optimal strategy trajectory with a nonlinear operator of cooperative/non-cooperative stochastic differential game between agents can make agents in a certain Local-World coordinate and make the Local-World payment maximize, and can make the all Local-Worlds equilibrated; furthermore, the optimal strategy of the coupled game can converge into a particular attractor that decides the optimal property.

Density of the union of positive diagonal binary quadratic forms

Let $X$ be a sufficiently large positive integer. We prove that one may choose a subset $S$ of primes with cardinality $O(\log X)$, such that a positive proportion of integers less than $X$ can be represented by $x^2 + p y^2$ for at least one of $p \in S$.

Conjectures of Sun About Sums of Polygonal Numbers

In this paper, we consider representations of positive integers as sums of generalized m-gonal numbers, which extend the formula for the number of dots needed to make up a regular m-gon. We mainly restrict to the case where the sums contain at most four distinct generalized m-gonal numbers, with the second m-gonal number repeated twice, the third repeated four times, and the last is repeated eight times. For a number of small choices of m, Sun conjectured that every positive integer may be written in this form. By obtaining explicit quantitative bounds for Fourier coefficients related to theta functions which encode the number of such representations, we verify that Sun’s conjecture is true for sufficiently large positive integers. Since there are only finitely many choices of m appearing in Sun’s conjecture, this reduces Sun’s conjecture to a verification of finitely many cases. Moreover, the bound beyond which we prove that Sun’s conjecture holds is explicit.

Dense sumsets of Sidon sequences

Let k≥2 be an integer. We say a set A of positive integers is an asymptotic basis of order k if every large enough positive integer can be represented as a sum of k terms from A. A set of positive integers A is called Sidon set if all the two-term sums formed by the elements of A are different. Many years ago P. Erdős, A. Sárközy and V. T. Sós asked whether there exists a Sidon set which is an asymptotic basis of order 3. In this paper we prove the existence of a Sidon set A with positive lower density of the three fold sumset A+A+A by using probabilistic methods.

On an Equation with Prime Numbers Close to Squares

Let [ · ] be the fioor function. In this paper, we show that when 1 &lt; c &lt; 37/36, then every sufficiently large positive integer N can be represented in the formwhere p1, p2, p3 are primes close to squares.

Leaper Tours

Let $p$ and $q$ be positive integers. The $(p, q)$-leaper $L$ is a generalised knight which leaps $p$ units away along one coordinate axis and $q$ units away along the other. Consider a free $L$, meaning that $p + q$ is odd and $p$ and $q$ are relatively prime. We prove that $L$ tours the board of size $4pq \times n$ for all sufficiently large positive integers $n$. Combining this with the recently established conjecture of Willcocks which states that $L$ tours the square board of side $2(p + q)$, we conclude that furthermore $L$ tours all boards both of whose sides are even and sufficiently large. This, in particular, completely resolves the question of the Hamiltonicity of leaper graphs on sufficiently large square boards.

Sylvester power and weighted sums on the Frobenius set in arithmetic progression

Let a1,a2,…,ak be positive integers with gcd(a1,a2,…,ak)=1. Frobenius number is the largest positive integer that is NOT representable in terms of a1,a2,…,ak. When k≥3, there is no explicit formula in general, but some formulae may exist for special sequences a1,a2,…,ak, including, those forming arithmetic progressions and their modifications. In this paper, we give formulae for the power and weighted sum of nonrepresentable positive integers. As applications, we show explicit expressions of these sums for a1,a2,…,ak forming arithmetic progressions.

Reliability measure of the n-th cartesian product of complete graph K4 on h-extra edge-connectivity

As a measurement parameter of the reliability about interconnection networks of parallel and distributed systems, the h-extra edge-connectivity λh(G) is a better alternative compared with the classical Menger's theorem of the edge-connectivity. Recently, Li and Yang (2013) [7] determined the values of the h-extra edge-connectivity of hypercube Qn for each h≤2⌊n2⌋. Because of easy scalability, the interconnection networks based on cartesian product operation are extensively investigated. This paper focuses on the h-extra edge-connectivity of the n-th cartesian product of complete graph K4 with exponentially many faulty links. For a sufficiently large positive integer n, about 60 percent of positive integers h in the interval 1≤h≤2⋅4n−1 corresponding h-extra edge-connectivity of K4n, λh(K4n), presents a concentration phenomenon, that is, these exact values of λh(K4n) concentrate on 3⋅4n−1 and 4n for each ⌈3⋅4n−1/5⌉≤h≤4n−1 and ⌈6⋅4n−1/5⌉≤h≤2⋅4n−1, respectively. And the lower and upper bounds of h are sharp. Furthermore, the values of λh(K2n) also have this phenomenon. We obtain λh(K4n)=32λh(K22n)=3⋅4n−1 or λh(K4n)=2λh(K22n)=4n in the subintervals where the concentration phenomenon occurs simultaneously.

Speeding Up Fermat’s Factoring Method using Precomputation

The security of many public-key cryptosystems and protocols relies on the difficulty of factoring a large positive integer n into prime factors. The Fermat factoring method is a core of some modern and important factorization methods, such as the quadratic sieve and number field sieve methods. It factors a composite integer n=pq in polynomial time if the difference between the prime factors is equal to ∆=p-q≤n^(0.25) , where p&gt;q. The execution time of the Fermat factoring method increases rapidly as ∆ increases. One of the improvements to the Fermat factoring method is based on studying the possible values of (n mod 20). In this paper, we introduce an efficient algorithm to factorize a large integer based on the possible values of (n mod 20) and a precomputation strategy. The experimental results, on different sizes of n and ∆, demonstrate that our proposed algorithm is faster than the previous improvements of the Fermat factoring method by at least 48%.

The balanced metrics and cscK metrics on certain holomorphic ball bundles

In this paper, we use the well-known Calabi ansatz, further generalized by Hwang-Singer, to study the existence of balanced metrics and constant scalar curvature Kähler (cscK for short) metrics on certain holomorphic ball bundles M which are locally expressed as M={(z,u,w)∈Ω×Cd1×Cd2:eλ1ϕ(z)‖u‖2+eλ2ϕ(z)‖w‖2<1}. Let g be the Kähler metric on M associated with the Kähler forms locally expressed as ω=−1∂∂‾(ϕ(z)+F(eλ1ϕ(z)‖u‖2+eλ2ϕ(z)‖w‖2)). Firstly, we obtain sufficient and necessary conditions for g to be cscK metrics. Secondly, using this result, we obtain necessary and sufficient conditions for mg to be balanced metrics for all sufficiently large positive integer numbers m. Finally, we obtain complete cscK metrics and balanced metrics on the ball bundles over simply connected Riemann surfaces. The main contribution of this paper is the explicit construction of complete, non-compact cscK metrics and balanced metrics.

СПЕЦІАЛІЗОВАНИЙ ПРОЦЕСОР ДЛЯ УЩІЛЬНЕННЯ ДАНИХ

One of the effective approaches to data compression is the approach based on the use of optimizing properties of Fibonacci numbers. The essence of the approach is that in the process of compaction the block of digital data is considered as a large positive integer, given in the form of a linear Fibonacci form. The implementation of data compression methods based on the linear form of Fibonacci software requires a lot of time, which is associated with calculations over large numbers (up to 8000 binary digits). For some applications, such time is unacceptable, so there is a need to create a specialized processor that will speed up the process of data compression. The development of mathematical and structural models of a specialized processor and its components is carried out using a functional-structural approach to the design of digital devices. Based on the generalized model of the process of adaptive data compression based on the linear Fibonacci form, the main functions to be implemented by a specialized processor are identified. This processor is part of a computer system and is in some way connected to the computer's CPU. Because the files to be compressed and the compressed files are stored in computer memory, the CPU is expected to read and write the file, generate P and P* sequences, and implement a sequence-level optimization function. The specialized processor is responsible for calculations over large numbers. To implement a set of all functions, it is proposed to build more than one operating machine, and to decompose it into machines, each of which implements the corresponding function. Mathematical models and structures of such modules of the specialized processor are considered: modeling of a data source, coding, decoding, optimization at the level of blocks, formation of structure of sequence P*. Hardware implementation of calculations over large numbers and the ability to implement basic functional transformations of individual modules in the pipeline mode provides acceleration of the data compression process compared to software implementation.

On products of primes and almost primes in arithmetic progressions

Let $q$ be a large positive integer and let $(a,q)=1$. We prove that there exist primes $p_1,p_2\le q$ and a number $n\le q$ with at most two prime factors such that $p_1p_2n\equiv a\pmod {q}$. This improves upon a result of Shparlinski (2018).