Related papers: A systematic approach to Diophantine equations: op…
The main idea of this article is simply calculating integer functions in module. The algebraic in the integer modules is studied in completely new style. By a careful construction, a result is proven that two finite numbers is with unequal…
An open problem about two new families of orthogonal polynomials was posed by Alhaidari. Here we will identify one of them as Wilson polynomials. The other family seems to be new but we show that they are discrete orthogonal polynomials on…
This paper aims to employ a cluster-theoretic approach to provide a class of Diophantine equations whose solutions can be obtained by starting from initial solutions through mutations. We establish a novel framework bridging cluster theory…
We demonstrate how connections between graph theory and Diophantine approximation can be used in conjunction to give simple and accessible proofs of seemingly difficult results in both subjects.
In this paper, we deal with the quartic Diophantine equation $X^4-Y^4=R^2-S^2$ to present its infinitely many integer solutions.
In this paper we show a way to generalize the linear Diophantine equation a1x1+a2x2+...+anxn=d . We deal with the nonlinear Diophantine equation det|A X|=+-d , which generalizes the linear one, and we give a necessary and sufficient…
We provide a simple method to recognize classical orthogonal polynomials on lattices defined only by their coefficients of the three term recurrence relation.
New formulae are presented for the number $P(b)$ of non-negative integer solutions of a Diophantine equation $\sum_{i=1}^{n}a_ix_i=b$ and for the number $Q(b)$ of non-negative integer solutions of the Diophantine inequality…
In this paper, the elliptic curves theory is used for solving the Diophantine equations $\sum_{i=1}^n a_ix_{i} ^6+\sum_{i=1}^m b_iy_{i} ^3= \sum_{i=1}^na_iX_{i}^6\pm\sum_{i=1}^m b_iY_{i} ^3$, where $n$, $m$ $\geq 1$ and $a_i$, $b_i$, are…
Polynomials related to rational solutions of Painleve' equations satisfy certain difference equations. Conditions are given to acertain that all solutions really are polynomials.
This paper is a short introduction to orthogonal polynomials, both the general theory and some special classes. It ends with some remarks about the usage of computer algebra for this theory.
Finding integer solutions to norm form equations is a classical Diophantine problem. Using the units of the associated coefficient ring, we can produce sequences of solutions to these equations. It is known that these solutions can be…
We present a method for the solution of polynomial equations. We do not intend to present one more method among several others, because today there are many excellent methods. Our main aim is educational. Here we attempt to present a method…
We discuss a problem initially thought for the Mathematical Olympiad but which has several interpretations. The recurrence sequences involved in this problem may be generalized to recurrence sequences related to a much larger set of…
In this article, we consider the equation x^2+19^{m}=y^n, n>2, m>0. We find the solutions of the title equation for not only 2 \mid m but also 2\notdividesm.
In this paper we obtain a new parametric solution of the problem of finding two triads of biquadrates with equal sums and equal products.
We prove new results, related to the Littlewood and Mixed Littlewood conjectures in Diophantine approximation.
This note describes some open problems that can be examined with the purpose of gaining additional insight of how to solve the problem of finding a general classification of geodetic graphs
The main purpose of this note is to pose a couple of problems which are easily formulated thought some seem to be not yet solved. These problems are of general interest for discrete mathematics including a new twig of a bough of theory of…
A unified approach, for solving a wide class of single and many-body quantum problems, commonly encountered in literature is developed based on a recently proposed method for finding solutions of linear differential equations. Apart from…