Related papers: Integer Algorithms to Solver Diophantine Linear Eq…
The paper proposes artificial intelligence technique called hill climbing to find numerical solutions of Diophantine Equations. Such equations are important as they have many applications in fields like public key cryptography, integer…
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
Let E_n={x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. For each integer n \geq 13, J. Browkin defined a system B_n \subseteq E_n which has exactly b_n solutions in integers x_1,...,x_n, where b_n \in N\{0} and the sequence…
We develop a numerical method for solving a system of nonlinear integral equations involving two integral terms: at the current time t, one integral is taken from 0 to t, and a different integral is taken from t to infinity. We prove the…
The concept of a universal algorithm is discussed. Examples of this kind of algorithms are presented. Software implementations of such algorithms in C++ type languages are discussed together with means that provide for computations with an…
In this paper we generalize the involutive methods and algorithms devised for polynomial ideals to differential ones generated by a finite set of linear differential polynomials in the differential polynomial ring over a zero characteristic…
We propose a protocol for solving systems of linear algebraic equations via quantum mechanical methods using the minimal number of qubits. We show that $(M+1)$-qubit system is enough to solve a system of $M$ equations for one of the…
A general formalism to solve nonlinear differential equations is given. Solutions are found and reduced to those of second order nonlinear differential equations in one variable. The approach is uniformized in the geometry and solves…
In this paper, we solve Diophantine equation in the tittle in nonnegative integers m,n, and a. In order to prove our result, we use lower bounds for linear forms in logarithms and and a version of the Baker-Davenport reduction method in…
Motivated by a certain molecular reconstruction methodology in cryo-electron microscopy, we consider the problem of solving a linear system with two unknown orthogonal matrices, which is a generalization of the well-known orthogonal…
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…
Motivated by problems in optimization we study the sparsity of the solutions to systems of linear Diophantine equations and linear integer programs, i.e., the number of non-zero entries of a solution, which is often referred to as the…
An infinite set is orbit-finite if, up to permutations of the underlying structure of atoms, it has only finitely many elements. We study a generalisation of linear programming where constraints are expressed by an orbit-finite system of…
A perfect cuboid is a rectangular parallelepiped whose all linear extents are given by integer numbers, i. e. its edges, its face diagonals, and its space diagonal are of integer lengths. None of perfect cuboids is known thus far. Their…
In this paper, we give a specific way of describing positive integer solutions of a Diophantine equation $(x+y)^2+(y+z)^2+(z+x)^2=12xyz$ and introduce a generalized cluster pattern behind it.
In this paper, we prove the finiteness of the number of integer solutions of the decomposable form inequalities. We also study the number of integer solutions of a sequence of decomposable form inequalities.
The subject matter of this work are the linear, three variable diophantine equation ax+by+cz=d (1), and the diophantine system ax+by+cz=d (2) ex+fy+gz=h with the coefficients a,b,c,d,e,f,g,h being integers. Introductory number theory books,…
In this paper, we prove a theorem about the integer solutions to the Diophantine equation $x^{4}-q^{4}=py^{r}$, extending previous work of K.Gy\H ory, and F.Luca and A.Togbe, and of the author.
A three-point iterative method for solving scalar non-linear equations was selected and then adapted to solve systems of non-linear equations. Subsequently, by applying Taylor's theorem to functions of $\R^{n}$ in $\R^{n}$, it is shown that…
This paper deals with exploiting symmetry for solving linear and integer programming problems. Basic properties of linear representations of finite groups can be used to reduce symmetric linear programming to solving linear programs of…