Solving Diophantine Equations Research Articles

A Discussion on the Solution(s) of the Diophantine Equation 3<sup>x</sup> + 15<sup>y</sup> =Z<sup>2</sup>

The absence of a generalized method for solving Diophantine equations having more unknowns than a number of equations is a challenge for researchers in different fields. The presence of the Diophantine equation is reported in the study of the Hydrogen spectrum, quantum Hall effect, chemistry, cryptography, etc. Some special types of Diophantine equations could be addressed with the help of Catalan’s conjecture and Congruence theory. The Diophantine equation 3x + 15y =z2 is addressed in this paper to find the solution(s) in positive integers. It is found that the equation has only two solutions of (x,y,z) as (1,0,2) and (0,1,4) in non-negative integers.

Construction of Perfect Squares and Pythagorean Triples using Continued Fractions

Continued fractions play a major role in Number theory and solving Diophantine equations. With the help of Continued fractions and Pell’s equations, we have proved some interesting results like obtaining Pythagorean triples whose smaller sides differ by some particular integer and also we have provided methods to generate primitive Pythagorean triples satisfying a specific condition. Perfect squares are one of the most well-known class of numbers in number theory. In this paper we introduced ways to produce natural numbers that are related to perfect squares with some specified conditions.

Application of Continued Fraction in Pell's Equation

This paper uses a continued fraction to explain various approaches to solving Diophantine equations. It first examines the fundamental characteristics of continued fractions, such as convergent and approximations to real numbers. Using continued fractions, we can solve the Pell's equation. Certain theorems have also been discussed for how to expand quadratic irrational integers into periodic continued fractions. Finally, the relationship between convergent and best approximations and use of continuous fraction in calendar construction has-been investigated. The analytical theory of continued fractions is a significant generalization of continued fractions and represents a large field for current and future research.

ЭВРИСТИЧЕСКИЙ ГЕНЕТИЧЕСКИЙ АЛГОРИТМ РЕШЕНИЯ ДИОФАНТОВЫХ УРАВНЕНИЙ

The problem of diophantine equations solving is considered in this article. This problem canbe applied in cryptography and cryptanalysis. The description of the genetic algorithm solvingdiophantine equations is stated briefly in the article. The rule of calculation the value of fitnessfunction of chromosome is determined, the coding system in the genetic algorithm is described.The genetic operators used in the algorithm are mentioned and the conditions for their executionare determined. The criterion for stopping the genetic algorithm is described. One of the shortcomingsof the genetic algorithm is analyzed. The shortcoming of the algorithm lies in its attemptsto solve any diophantine equation, including one that has no solutions. A method eliminating thisshortcoming in some cases is proposed. This method is based on number theory. An explanation isgiven in which cases this method will be used. The definition of residue and nonresidue of fixedpower for fixed modulus is given before describing this method. After describing this method theimplementation of the algorithm for solving diophantine equations and systems of them is describedin detail. Then the results of experimental studies of the time and quality of the geneticalgorithm are presented. Then the result of the algorithm is presented for an equation that has nosolutions and for a system of equations that also has no solutions, but in which the total number ofunknowns is too large for the proposed method to work. The algorithm running time is comparedwhen solving an equation and when solving a system of equations. The conclusion is made aboutthe usefulness of the proposed method in solving diophantine equations and systems of diophantineequations.

Alternative Generalized Predictive Control for Output Disturbed Multi-Input Multi-Output Discrete-Time Systems

This article proposes an option to execute conveniently the traditional Model Predictive Control (GPC), called the Alternative Generalized Predictive Control (AGPC). In this AGPC the disturbed discrete-time input-output mapping of controlled plant is utilized directly for the prediction of plant outputs, instead of its transfer function as being done in the conventional approach. Hence, solving Diophantine equations will be avoided. Within this AGPC all recorded values of plant inputs/outputs in the last time-horizon are matched into separate vectors for computing predictive control signals at the next control step, which helps therefore that its implementation becomes more manageable. To verify via virtually real simulation the control performance of this proposed AGPC a Simscape water tank model, which is chosen as the controlled plant, had been created among Thermal Fluids Toolbox. The simulation is carried out for two different circumstances, one by using AGPC and the other by applying conventional PID, for comparison purposes. The simulation demonstrates also how to realize this AGPC in practice.

Diophantine equations with sum of cubes and cube of sum

We solve Diophantine equations of the type $ \, a \, (x^3 + y^3 + z^3 ) = (x + y + z)^3$, where $x,y,z$ are integer variables, and the coefficient $a \neq 0$ is rational. We show that there are infinite families of such equations, including those where $a$ is any ratio of cubes or certain rational fractions, that have nontrivial solutions. There are also infinite families of equations that do not have any nontrivial solution, including those where $1/a = 1 - 24/m$ with certain restrictions on the integer $m$. The equations can be represented by elliptic curves unless $a = 9$ or 1. If $a$ is an integer and two variables are equal and nonzero, there exist nontrivial solutions only for $a=4$ or 9; there are no solutions for $a = 4$ when $xyz \neq 0$. Without imposing constraints on the variables, we find the general solution for $a = 9$, which depends on two integer parameters. These cubic equations are important in particle physics, because they determine the fermion charges under the $U(1)$ gauge group.

Solving Diophantine Equations by Factoring in Number Fields

In this text we provide an introduction to algebraic number theory and show its applications in solving certain difficult diophantine equations. We begin with a quick summary of the theory of quadratic residues, before diving into a select few areas of algebraic number theory. Our article is accompanied by a couple of worked problems and exercises for the reader to tackle on their own.

Classification of Spherical $2$-distance $\{4,2,1\}$-designs by Solving Diophantine Equations

In algebraic combinatorics, the first step of the classification of interesting objects is usually to find all their feasible parameters. The feasible parameters are often integral solutions of some complicated Diophantine equations, which cannot be solved by known methods. In this paper, we develop a method to solve such Diophantine equations in $3$ variables. We demonstrate it by giving a classification of finite subsets that are spherical $2$-distance sets and spherical $\{4,2,1\}$-designs at the same time.

Connectionist based models for solving Diophantine equation

Diophantine equation is an important concept while studying number theory. But there is as such no standard method or procedure to find the solution of the Diophantine equation. In this article, the numerical solution of Diophantine equation has been discussed using the concept of Artificial Neural Network (ANN). A four layer ANN architecture has been constructed for solving Diophantine eqution. Detail ANN procedure with few example problems and their network architectures have been addressed. Convergence plots and/or convergence tables for different example problems have also been included to validate the proposed method.

Robust Simultaneous Stabilization Using Derivative State Feedback

Abstract In simultaneous stabilization of continuous time switched linear system models of inertially coupled dynamic systems with bounded persistent external disturbance and internal parameter variations, the generalized proportional integral derivative (PID) controller suffers from a serious drawback. It loses the ability to guarantee exponential stability of the closed-loop error response, which is the difference between the actual and the desired closed-loop responses. The high value of the hardware gain has no control over the situation. On the contrary, the derivative state feedback (DSF) controller effectively employs the hardware gain in diminishing the effects of external and internal disturbances. These facts are proved and illustrated by suitable examples. The proposed method of simultaneous stabilization is shown to be an easier alternative to solve Diophantine equations encountered. The method for single input single output (SISO) systems is also shown to extend seamlessly to the control of multiple input multiple output (MIMO) square models with equal number of inputs and outputs. The shortcoming of the method is pointed out when the models are not square. The advantages of the proposed methodology over the existing methods are outlined. Finally, an efficient time domain simulation scheme is presented for the numerical study of such models in time and transfer function domain with nonzero initial conditions.

On the Number of Simple K_4 Groups

In this paper, by solving Diophantine equations involving simple K_4-groups, we will try to point out that it is not easy to prove the infinitude of simple K_4-groups. This problem goes far beyond what is known about Dickson’s conjecture at present.

О МЕТОДИКЕ ОБУЧЕНИЯ СОСТАВЛЕНИЮ НЕКОТОРОГО ВИДА АРИФМЕТИЧЕСКИХ ЗАДАЧ

The article is devoted to a current topic related to the development of methods for composing problems in teaching students of pedagogical universities of mathematical faculties. This problem becomes especially important in the context of the need to involve students in independent creative activities to acquire and apply knowledge. The material is presented in relation to the section of the discipline «Algebra and number theory», dedicated to solving Diophantine equations, the main objectives of which are not only mastering the theory and algorithms for solving basic problems, but also obtaining the necessary knowledge and skills for further professional activity. Solving a problem, the student must not only solve it correctly and quickly enough, but also show the creative component of the activity, using it as much as possible for their mathematical development. In this regard, the process of composing problems by students is undoubtedly useful, which reflects the systematic application of the material and elements of mathematical actions based on the laws and methods of mathematics. In addition, the ability to compose problems will be required in future activities related to teaching mathematics. The processes of solving and composing tasks are interconnected and this allows you to increase the efficiency and effectiveness of composing and solving tasks. Therefore, the teacher can give a task to the student with the requirement to compose (fully or partially) and solve the problem. In this paper, examples of tasks for the compilation of indefinite equations solvable in integers are considered, for the solution of which the methods of number theory are used: the study of possible residuals from dividing an algebraic integer expression by a specific integer; finding integer solutions to a linear equation with two variables. The stages of composing Diophantine equations are described in detail, the ways of obtaining equations solvable on a given set of integers or natural numbers are analyzed, and the application of various theoretical propositions used for their solution is shown.

Rank Reduction Processes for Solving Linear Diophantine Systems and Integer Factorizations: A Review

Abaffy, Broyden and Spediacto (ABS) introduced a class of the so-called ABS methods to solve systems of linear equations. The ABS approach was specialized to solve linear Diophantine systems by Esmaeili, Mahdavi-Amiri and Spedicato. The method was extended to systems of certain linear inequalities to provide all solutions. Here, we mainly focus on the theoretical research into rank reduction processes for solving linear Diophantine systems and computing integer factorizations. Basic integer ABS algorithm, integer extended ABS (IEABS) algorithm, rank one perturbed problem, the generalized Rosser’s approach (GRA), the extended integer rank reduction process, the integer Wedderburn rank reduction formula and the associated integer biconjugation process are studied. We present an integrated integer rank reduction process, develop various integer factorizations and show their use in solving Diophantine equations.

Non-existence of Solutions of Diophantine Equations of the Form <img src=image/13490640_01.gif>

Numerous researches have been devoted in finding the solutions , in the set of non-negative integers, of Diophantine equations of type (1), where the values p and q are fixed. In this paper, we also deal with a more generalized form, that is, equations of type (2), where n is a positive integer. We will present results that will guarantee the non-existence of solutions of such Diophantine equations in the set of positive integers. We will use the concepts of the Legendre symbol and Jacobi symbol, which were also used in the study of other types of Diophantine equations. Here, we assume that one of the exponents is odd. With these results, the problem of solving Diophantine equations of this type will become relatively easier as compared to the previous works of several authors. Moreover, we can extend the results by considering the Diophantine equations (3) in the set of positive integers.

Diophantine Geometry in Space E2 and E3

In this paper, we are focusing on solving Diophantine equations with geometry. Integers that are solutions of Diophantine equations are represented in space by a lattice. We will use the fact that equations can be interpreted geometrically as curves or surfaces. To determine the properties of the solutions, we will use tangents (of the surface).

Model Predictive Control of a Multilevel CHB STATCOM in Wind Farm Application Using Diophantine Equations

The finite control set model predictive control (FCS-MPC) for power electronic converters provides high dynamic performance, based on the limited number of inputs and accurate model of the converter. By applying this algorithm to multilevel converters such as a cascaded-H-bridge-based static var compensator (CHB STATCOM), the dynamic performance is degraded, because the optimized input is achieved by searching among a large set of switching combinations and redundancies. This paper proposes an FCS-MPC algorithm, which benefits high dynamic performance for the CHB STATCOM, despite the large set of inputs. The proposed FCS-MPC replaces the time-consuming optimization algorithm by solving Diophantine equations over the large set of switching combinations. The desired switching combination and all its redundancies are determined in a minimum execution time. The proposed FCS-MPC performance is validated by applying to two configurations: 1) a 15-level CHB STATCOM with energy storage capability for a short-term active power smoothing and reactive power compensation of a 10 MW fixed speed wind farm at medium voltage; and 2) an experimental seven-level CHB STATCOM at low voltage.

Balancing Numbers and Application

In this paper, we present about origin of Balancing numbers; It!s connection with Triangular, Pells numbers, and Fibonacci numbers; beginning with connections of balancing numbers with other numbers system, It elaborate the different generating functions of balancing numbers. It also include some amazing recurrence relations; and the application of balancing numbers in solving Diophantine equation.

Effective resolution of Diophantine equations of the form $$u_n+u_m=w p_1^{z_1} \cdots p_s^{z_s}$$ u n + u m = w p 1 z 1 ⋯ p s z s

Let $$u_n$$ be a fixed non-degenerate binary recurrence sequence with positive discriminant, w a fixed non-zero integer and $$p_1,p_2,\ldots ,p_s$$ fixed, distinct prime numbers. In this paper we consider the Diophantine equation $$u_n+u_m=w p_1^{z_1} \ldots p_s^{z_s}$$ and prove under mild technical restrictions effective finiteness results. In particular we give explicit upper bounds for n, m and $$z_1, \ldots , z_s$$ . Furthermore, we provide a rather efficient algorithm to solve Diophantine equations of the described type and we demonstrate our method by an example.

Design of robust fractional‐order lead–lag controller for uncertain systems

In this study, a new method for designing fractional-order robust lead–lag controllers in order to ensure robust stability in uncertain systems with unstructured uncertainty, has been presented. In this method, at first by the use of pole placement and solving Diophantine equation, a fractional-order controller, which has a new form, is designed in order to stabilise the nominal system. Then a constrained optimisation problem is solved, in which, the cost function consists of the numerator and denominator coefficients of the proposed fractional-order controller and the constraints are the robust stability conditions. Finally, the parameters of the robust fractional-order controller are acquired. This method is implemented on an unstable system and simulation results are presented.

Methods for Arriving at Numerical Solutions for Equations of the Type (k+3) & (k+5) Bi-quadratic's Equal to a Bi-quadratic (For Different Values of k)

Different authors have done analysis regarding sums of powers (Ref. no. 1,2 & 3), but systematic approach for solving Diophantine equations having sums of many bi-quadratics equal to a quartic has not been done before. In this paper we give methods for finding numerical solutions to equation (A) given above in section one. Next in section two, we give methods for finding numerical solutions for equation (B) given above. It is known that finding parametric solutions to biquadratic equations is not easy by conventional method. So the authors have found numerical solutions to equation (A) & (B) using elliptic curve theory.