Related papers: A Polynomial Diophantine Generator Function for In…
In this paper, we establish two mean value theorems for the number of solutions of the Diophantine equation $\frac{a}{n}=\frac{1}{x}+\frac{1}{y}$, in the case when $a$ is fixed and $n$ varies and in the case when both $a$ and $n$ vary.
In this paper, we refine the method introduced by Izadi and Baghalaghdam to search integer solutions to the Diophantine equation $X_1^5+X_2^5+X_3^5=Y_1^3+Y_2^3+Y_3^3$. We show that the Diophantine equation has infinitely many positive…
We study some Diophantine problems related to triangles with two given integral sides. We solve two problems posed by Zolt\'an Bertalan and we also provide some generalization.
Divide-and-conquer functions satisfy equations in F(z),F(z^2),F(z^4)... Their generated sequences are mainly used in computer science, and they were analyzed pragmatically, that is, now and then a sequence was picked out for scrutiny. By…
This article focuses on automatically generating polynomial equations that are inductive loop invariants of computer programs. We propose a new algorithm for this task, which is based on polynomial interpolation. Though the proposed…
Four integer parametrizations for the bi-orthogonal monoclinic Diophantine parallelepiped are given.
We provide bounds on the size of polynomial differential equations obtained by executing closure properties for D-algebraic functions. While it is easy to obtain bounds on the order of these equations, it requires some more work to derive…
This is a survey of a connection between the distribution of certain power residues modulo $p$, $p$ a prime, and relative class numbers. The focus lies on quadratic residues and sixth power residues. Dirichlet's class number formula yields…
The number of parts in the partitions (resp. distinct partitions) of $n$ with parts from a set were considered. Its generating functions were obtained. Consequently, we derive several recurrence identities for the following functions: the…
We develop an algebraic method of studying of Diophantine quadratic equations in three variables over the ring of Gaussian integers.
Let $u_n$ be a fixed non-degenerate binary recurrence sequence with positive discriminant, $w$ a fixed non-zero integer and $p_1,p_2,\dots,p_s$ fixed, distinct prime numbers. In this paper we consider the Diophantine equation $u_n+u_m=w…
An averaged generating function for coloured hard-dimers is being investigated by proving estimates for the latter. Furthermore, two different enumerating problems and their distributions are studied numerically.
In this paper, we gave solutions of the Diophantine equations 16^{x}+p^{y}=z^{2}, 64^{x}+p^{y}=z^{2} where p is an odd prime, n is a positive integer and x,y,z are non-negative integers. Finally we gave a generalization of the Diophantine…
In this article, we will analyze the existence of Peregrine type solutions for the fractional diffusion reaction equation by applying Splitting-type methods. These functions that have two main characteristics, they are direct sum of…
A redundant generating function is a generating function having terms which are not part of the solution of the original problem. We use redundant generating functions to study two path problems. In the first application we explain a…
Given a Fractional Repetition code, finding the reconstruction and repair degree in a distributed storage system is an important problem. In this work, we present algorithms for computing the reconstruction and repair degree of fractional…
In this paper, we study cut generating functions for conic sets. Our first main result shows that if the conic set is bounded, then cut generating functions for integer linear programs can easily be adapted to give the integer hull of the…
This paper presents a new generating function for Hermite polynomials of one variable in the form of $g(x,t)=\sum_{n=0}^{\infty }t^{n}H^{e}_{n}(x)$ and reveals its connection with incomplete gamma function.
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.
Starting from Ritt's classical theorems, we give a survey of results in functional decomposition of polynomials and of applications in Diophantine equations. This includes sufficient conditions for the indecomposability of polynomials, the…