Diophantine Equations Research Articles

Efficient Approaches to Solving Quadratic Diophantine Equations and their Time Complexity

In this paper, we present an efficient approach to solving quadratic Diophantine equations and analyze their time complexity. We propose a deterministic polynomial-time algorithm that provides an upper bound on the elementary operations required to solve such equations. We also present a non-deterministic polynomial-time algorithm for the construction of quadratic non-resiude modulo d, which is a more efficient alternative to the deterministic approach.

Exceptional biases in counting primes over function fields

AbstractWe study how often exceptional configurations of irreducible polynomials over finite fields occur in the context of prime number races and Chebyshev's bias. In particular, we show that three types of biases, which we call “complete bias,” “lower order bias,” and “reversed bias,” occur with probability going to zero among the family of all squarefree monic polynomials of a given degree in as , a power of a fixed prime, goes to infinity. The bounds given extend a previous result of Kowalski, who studied a similar question along particular one‐parameter families of reducible polynomials. The tools used are the large sieve for Frobenius developed by Kowalski, an improvement of it due to Perret–Gentil and considerations from the theory of linear recurrence sequences and arithmetic geometry.

The resolution of three exponential Diophantine equations in several variables

We find all solutions of three exponential Diophantine equations, arising from certain quadratic, cubic and quartic identities. The first identity comes from a painting of the famous Russian painter Nikolay Bogdanov-Belsky, highlighted by Ja. I. Perelman. The equations have five, four and six terms, respectively, so they cannot be handled by classical tools based upon Baker's method. To solve the equations we use our method developed earlier, which is based upon Skolem's conjecture, local considerations and a computational approach.

Reverse engineered Diophantine equations

We answer a question of Samir Siksek, asked at the open problems session of the conference “Rational Points 2022”, which, in a broader sense, can be viewed as a reverse engineering of Diophantine equations. For any finite set S of perfect integer powers, using Mihăilescu’s theorem, we construct a polynomial fS∈Z[x] such that the set fS(Z) contains a perfect integer power if and only if it belongs to S. We first discuss the easier case where we restrict to all powers with the same exponent. In this case, the constructed polynomials are inspired by Runge’s method and Fermat’s Last Theorem. Therefore we can construct a polynomial–exponential Diophantine equation whose solutions are determined in advance.

Observer-based HOFA predictive cooperative control for networked multi-agent systems under time-variant communication constraints

This research focuses on a cooperative control problem for networked multi-agent systems (NMASs) under time-variant communication constraints (containing time-variant communication delays and time-variant data losses) in the forward and feedback channels. From the perspective of high-order fully actuated (HOFA) system theory, a HOFA system model is adopted to describe the NMAS, which is called the networked HOFA multi-agent system (NHOFAMAS). Because of complicated working scenarios over the network, the states of NMASs are immeasurable and the communication constraints are always present, such that an observer-based HOFA predictive control (OB-HOFAPC) method is designed to implement the cooperative control when existing the immeasurable states and time-variant communication constraints. In this method, a HOFA observer is established to estimate the immeasurable states for constructing a consensus control protocol. Then, an incremental prediction model (IPM) in a HOFA form is developed via a Diophantine equation to take the place of a reduced-order prediction model. Through this IPM, multi-step output ahead predictions are derived to optimize the cooperative control performance and compensate for time-variant communication constraints in real-time. The depth discussion gives a sufficient and necessary criterion to analyze the simultaneous consensus and stability for closed-loop NHOFAMASs. The capability and advantage of OB-HOFAPC method are illustrated via numerical simulation and experimental verification on a cooperative flying-around task of three air-bearing spacecraft simulators.

The formal solutions of Diophantine equation agy = bx + c

We develop a novel method to completely solve the 3-term partial exponential Diophantine equation that represents a generalization of the standard discrete logarithm problem. Our method not only reveals the internal structure of the equation's solution and yields a numerical algorithm to solve it systematically, but also provides an alternative approach to the discrete logarithm problem.

A Proof for Fermat’s Last Theorem

Fermat’s proposition during 1637 that the Diophantine equation xn + yn = zn where x, y, z and n are integers has no solution for n > 2 has come to be known as Fermat’s Last Theorem. Taking the proofs of the theorem by Fermat and Euler for the index n = 4 and n = 3, it would suffice to prove the theorem for the index p which is any prime >3. We consider the equation rp + sp = tp to prove the theorem. We take another auxiliary equation x3 + y3 = z3 to substantiate the proof. Both equations have been combined by means of equivalent equations, into which we have employed the Ramanujan-Nagell equation. Solving the equivalent equations using the Ramanujan-Nagell Equation we prove the theorem.

A note on the Diophantine equations [formula omitted] and application to odd perfect numbers

Let N be an odd perfect number. Then, Euler proved that there exist some integers n,α and a prime q such that N=n2qα, q∤n, and q≡α≡1mod4. In this note, we prove that the ratio σ(n2)qα is neither a square nor a square times a single prime unless α=1. It is a direct consequence of a certain property of the Diophantine equation 2ln2=1+q+⋯+qα, where l denotes one or a prime, and its proof is based on the prime ideal factorization in the quadratic orders Z[1−q] and the primitive solutions of generalized Fermat equations xβ+yβ=2z2. We give also a slight generalization to odd multiply perfect numbers.

Diophantine imaging reveals the broken symmetry of sums of integer cubes

We introduced a novel method for visualizing large diophantine datasets and in particular found that mapping the known integer triplets {a,b,c} solving either equations of the type a^3+b^3+c^3=d or a^3+b^3+c^3=d^3 on certain proper subgroups of the circle group exposed a very clear breaking in their symmetry and a strongly non-ergodic distribution of the solutions of sums of three cubes that had never been described before. This method could be further applied to a larger diversity of diophantine problems, informing both number-theoretical conjectures and novel methods in computer sciences on the way, along with paving the road for specific uses of machine learning in exploring diophantine datasets with possible applications in cryptography among others.

An Extensive Review of the Literature Using the Diophantine Equations to Study Fuzzy Set Theory

Every field in mathematics has made significant progress in research with fuzzy sets. Numerous application fields were discovered in both empirical and theoretical investigations, ranging from information technology to medical technology, from the natural sciences to the physical sciences, and from technical education to fine arts education. However, it has limitations of its own and has not been able to function in real‐world situations. An interdisciplinary approach of fuzzy theory with number theory, especially Diophantine equations, needs to be accomplished to overcome this problem. A thorough literature study of the Diophantine equations, fuzzy sets, and the combination known as the linear Diophantine fuzzy set (LDFS) is accomplished in the present study. New forms of LDFSs have been added recently, and these additions have found use in a variety of fields, including the disciplines of pharmacology, power, healthcare, goods, and finance. The genesis of these expansions is also examined in this study of the literature. Hence, in the present work, some applications of LDFS are described in detail. Further in the present study, the existing primary constraints in the research on LDFS are highlighted. Also, the last section of the review is dedicated to outlining some future directions for the study of LDFS.

SOLUTION OF THE EXPONENTIAL DIOPHANTINE EQUATION 10^X + 400^Y=Z^2

Diophantine equations are so important in solving important real-world problems like network flow problems, pole placement problems, business investment problems, and data privacy problems that researchers are becoming more interested in developing new techniques for analyzing the nature and solutions of the various Diophantine equations. In the present study, authors examined the Diophantine equation〖10〗^X+〖400〗^Y=Z^2, where X,Y,Zare non-negative integers, for the non-negative integer solution of this equation. A result of the present study shows that it has no non-negative integer solution.

Studies of Positive Integer Solutions of the Diophantine Equation x2 − ay2 − bx − cy − d = 0 by the Transformation Method

Solving the Diophantine equation has fascinated mathematicians from various civilizations. In this paper, we propose the resolution of quadratic Diophantine equations with integer coefficients. Our contribution consists of generalizing certain results which have already been developed in the literature. This paper also proposes the criterion on the solvability of the quadratic Diophantine equation here studied.

An Effective Algorithm for Solving Weak Fuzzy Complex Diophantine Equations in Two Variables

Weak fuzzy complex numbers are defined as , with as an extension of real numbers with . This paper is dedicated to studying weak fuzzy complex linear Diophantine equations in two weak fuzzy complex variables, by transforming the weak fuzzy complex Diophantine equation to a classical equivalent Diophantine system and going directly from the solutions of classical system into the desired equation. Algorithms for generating solutions of the previous equation will be presented in terms of theorems with many related examples that clarify the validity of our work.

On certain Diophantine equations concerning the areas and perimeters of rational triangles

On certain Diophantine equations concerning the areas and perimeters of rational triangles

Solving Quadratic Diophantine EquationforIntegral Powers of 37

Diophantine Equations named after ancient Greek mathematician Diophantus, plays a vital role not only in number theory but also in several branches of science. In this paper, we have solved an quadratic Diophantine equations where the right hand side are positive integral powers of 37 and provide its integer solutions. The method adopted to solve the given equation is using the concept of polar form of a particular complex number. This concept can be generalized for solving similar equations.

On the Non-Commutative Logical Rings As Novel Extensions of Neutrosophic Rings

This paper uses some logical algebraic elements to extend any ring into a non-commutative ring containing the original ring with many generalized substructures and special elements. On the other hand, we study the substructures of non-commutative logical rings such as AH-homomorphisms and AH-ideals with many examples that explain their algebraic validity. Also, we discuss the possibility of solving a linear Diophantine equation with two variables in the non-commutative logical ring of integers, where we present an easy algorithm to solve this kind of generalized Diophantine equation.

Producing new semi-orthogonal decompositions in arithmetic geometry

Статья посвящена построению новых допустимых подкатегорий и полуортогональных разложений из исходных. Пусть $\mathcal{T}$ и $\mathcal{T}'$ - триангулированные подкатегории некоторой категории $\mathcal{D}$, а $(\mathcal{A},\mathcal{B})$ - полуортогональное разложение $\mathcal{T}$; мы ищем или такое разложение $(\mathcal{A}',\mathcal{B}')$ категории $\mathcal{T}'$, что нет ненулевых $\mathcal{D}$-морфизмов из $\mathcal{A}$ в $\mathcal{A}'$ и из $\mathcal{B}$ в $\mathcal{B}'$, или такое разложение $(\mathcal{A}_{\mathcal{D}},\mathcal{B}_{\mathcal{D}})$ категории $\mathcal{D}$, что $\mathcal{A}_{\mathcal{D}}\cap \mathcal{T}=\mathcal{A}$ и $\mathcal{B}_{\mathcal{D}}\cap \mathcal{T}=\mathcal{B}$. Доказываются несколько общих теорем существования (они также обобщаются на полуортогональные разложения произвольной длины); они применяются к различным производным категориям когерентных пучков на схеме $X$, собственной над спектром нeтерова кольца $R$. Это дает взаимно однозначное соответствие между полуортогональными разложениями категорий $D_{\mathrm{perf}}(X)$ и $D^{\mathrm{b}}(\operatorname{coh}(X)) $; последние распространяются на $D^-(\operatorname{coh}(X))$, $D^+_{\mathrm{coh}}(\operatorname{Qcoh}(X))$, $D_{\mathrm{coh}}(\operatorname{Qcoh}(X))$ и $D(\operatorname{Qcoh}(X))$ (если выполнены очень слабые дополнительные предположения). В частности, доказывается широкое обобщение некоторой теоремы Дж. Кармазина, А. Кузнецова и Е. Шиндера. Для получения этих результатов применяются недавние результаты Неемана, выражающие категории $D^{\mathrm{b}}(\operatorname{coh}(X))$ и $D^- (\operatorname{coh}(X))$ через $D_{\mathrm{perf}}(X)$. Также доказывается аналогичная новая теорема, связывающая $D^+_{\mathrm{coh}}(\operatorname{Qcoh}(X))$ и $D_{\mathrm{coh}}(\operatorname{Qcoh}(X))$ (это некоторые модификации ограниченной снизу и неограниченной производной категории когерентных пучков на $X$) с гомологическими функторами $D_{\mathrm{perf}}(X)^{\mathrm{op}}\to R-\operatorname{mod}$. Мы также изучаем применение этой теорем к построению некоторых сопряженных функторов. Библиография: 30 названий.

Producing new semi-orthogonal decompositions in arithmetic geometry

This paper is devoted to constructing new admissible subcategories and semi-orthogonal decompositions of triangulated categories out of old ones. For two triangulated subcategories $\mathcal{T}$ and $\mathcal{T}'$ of some category $\mathcal{D}$ and a semi-orthogonal decomposition $(\mathcal{A},\mathcal{B})$ of $\mathcal{T}$ we look either for a decomposition $(\mathcal{A}',\mathcal{B}')$ of $\mathcal{T}'$ such that there are no nonzero $\mathcal{D}$-morphisms from $\mathcal{A}$ into $mathcal{A}'$ and from $\mathcal{B}$ into $\mathcal{B}'$, or for a decomposition $(\mathcal{A}_{\mathcal{D}},\mathcal{B}_{\mathcal{D}})$ of $\mathcal{D}$ such that $\mathcal{A}_{\mathcal{D}}\cap \mathcal{T}=\mathcal{A}$ and $mathcal{B}_{\mathcal{D}}\cap \mathcal{T}=\mathcal{B}$. We prove some general existence statements (that also extend to semi-orthogonal decompositions of arbitrary length) and apply them to various derived categories of coherent sheaves over a scheme $X$ that is proper over the spectrum of a Noetherian ring $R$. This produces a one-to-one correspondence between semi-orthogonal decompositions of $D_{\mathrm{perf}}(X)$ and $D^{\mathrm{b}}(\operatorname{coh}(X))$; the latter extend to $D^-(\operatorname{coh}(X))$, $D^+_{\mathrm{coh}}(\operatorname{Qcoh}(X))$, $D_{\mathrm{coh}}(\operatorname{Qcoh}(X))$ and $D(\operatorname{Qcoh}(X))$ under very mild assumptions. In particular, we obtain a broad generalization of a theorem of Karmazyn, Kuznetsov and Shinder. These applications rely on some recent results of Neeman that express $D^{\mathrm{b}}(\operatorname{coh}(X))$ and $D^-(\operatorname{coh}(X))$ in terms of $D_{\mathrm{perf}}(X)$. We also prove a rather similar new theorem that relates $D^+_{\mathrm{coh}}(\operatorname{Qcoh}(X))$ and $D_{\mathrm{coh}}(\operatorname{Qcoh}(X))$ (these} are certain modifications of the bounded below and the unbounded derived category of coherent sheaves on $X$ to homological functors $D_{\mathrm{perf}}(X)^{\mathrm{op}}\to R-\mathrm{mod}$. Moreover, we discuss an application of this theorem to the construction of certain adjoint functors. Bibliography: 30 titles.

On The Diophantine 3-Cyclic Refined Neutrosophic Roots of Unity

The 3-cyclic refined neutrosophic roots of unity are exactly the solutions of the Diophantine equation in the 3-cyclic refined neutrosophic ring of integers . This paper is dedicated to finding all 3-cyclic refined neutrosophic Diophantine roots of unity, where it proves that there exist only three solutions for the case of odd order (n), and twelve different solutions for the case of even order (n). On the other hand, the group generated from all solutions will be classified as a finite abelian group with direct products of finite cyclic groups.

Algebraic tools in autotuning design

Autotuners represent a combination of a relay feedback identification test and some control design method. In this contribution, models with up to three parameters are estimated by means of a single asymmetrical relay experiment. Then a stable low order transfer function with a time delay term is identified by a relay experiment. Autotuning principles then combine asymmetrical relay feedback tests with a control synthesis. Two algebraic control syntheses then are presented in this paper. The first one is based on the ring of proper and stable rational functions RPS. The second one utilizes a special ring RMS, a set of RQ-meromorphic functions. In both cases, controller parameters are derived through a general solution of a linear Diophantine equation in the appropriate ring. A final controller can be tuned by a scalar real parameter m>0. The presented philosophy covers a generalization of PID controllers and the Smith-like control structure. This contribution deals with the implementation of proposed autotuners and presents some illustrative examples.