Related papers: On the smallest open Diophantine equations

We find all solutions of Diophantine equation x^{2}+11^{2k} = y^{n} where x>=1, y>=1, n>=3 and k is natural number. We give p-adic interpretation of this equation.

Letting $L_{n}(N, u)$ denote a polylogarithm ladder of weight $n$ and index $N$ with $u$ as an algebraic number, there is a rich history surrounding how mathematical objects of this form can be constructed for a given weight or index. This…

In this paper we present a new method of solving certain quartic and higher degree homogeneous polynomial diophantine equations in four variables. The method can also be extended to solve simultaneous homogeneous polynomial diophantine…

The study of several naturally arising "nearest neighbours" random walks benefits from the study of the associated orthogonal polynomials and their orthogonality measure. I consider extensions of this approach to a larger class of random…

The minimization problem for propositional formulas is an important optimization problem in the second level of the polynomial hierarchy. In general, the problem is Sigma-2-complete under Turing reductions, but restricted versions are…

We study the discrete semiclassical orthogonal polynomials of class s=1. By considering all possible solutions of the Pearson equation, we obtain five canonical families. We also consider limit relations between these and other families of…

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…

Recently, W. M. Schmidt and L. Summerer introduced a new theory which allowed them to recover the main known inequalities relating the usual exponents of Diophantine approximation to a point in $\mathbb{R}^n$, and to discover new ones. They…

For relatively prime natural numbers $a$ and $b$, we study the two equations $ax+by = (a-1)(b-1)/2$ and $ax+by+1= (a-1)(b-1)/2$, which arise from the study of cyclotomic polynomials. Previous work showed that exactly one equation has a…

These are the notes of my lectures at the 1996 European Congress of Mathematicians. {} Polynomials appear in mathematics frequently, and we all know from experience that low degree polynomials are easier to deal with than high degree ones.…

This work investigates a new approach to find closed form analytical approximate solution of linear initial value problems. Classical Bernoulli polynomials have been used to derive a finite set of orthonormal polynomials and a finite…

This paper focuses on the study of a mathematical program with equilibrium constraints, where the objective and the constraint functions are all polynomials. We present a method for finding its global minimizers and global minimum using a…

Let $d\ge 2$ and $n\ge d$ with $(d,n)\notin \{(2,2),(3,3)\}$. We consider homogeneous Diophantine equations of degree $d$ in $n+1$ variables and whether they have solutions in the primes. In particular, we show that a certain local-global…

For the bi-orthogonal polynomials with the third degree polynomial potential functions, the 3 x 3 matrix Riemann-Hilbert problem is explicitly constructed. The developed approach admits an extension to the bi-orthogonal polynomials with…

Let $\mathfrak{p}=(\mathfrak{p}_1,...,\mathfrak{p}_r)$ be a system of $r$ polynomials with integer coefficients of degree $d$ in $n$ variables $\mathbf{x}=(x_1,...,x_n)$. For a given $r$-tuple of integers, say $\mathbf{s}$, a general local…

Let k => 1, m => 1 be small fixed integers, gcd(k, m) = 1. This note develops some techniques for proving the existence of infinitely many primes solutions x = p, and y = q of the linear Diophantine equation y = mx + k.

We present a new probabilistic algorithm to find a finite set of points intersecting the closure of each connected component of the realization of every sign condition over a family of real polynomials defining regular hypersurfaces that…

For a positive integer n, let {\theta}(n) denote the smallest positive integer b such that for each system S \subseteq {x_i \cdot x_j=x_k, x_i+1=x_k: i,j,k \in {1,...,n}} which has a solution in positive integers x_1,...,x_n and which has…

This paper is a continuation of [1], in which I studied Harvey Friedman's problem of whether the function f(x,y) = x^2 + y^3 satisfies any identities; however, no knowledge of [1] is necessary to understand this paper. We will break the…

We define the notion of infimum of a set of points with respect to the second order cone. This problem can be showed to be equivalent to the minimum ball containing a set of balls problem and to the maximum intersecting ball problem, as…