Related papers: How to Solve Diophantine Equations
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…
We solve completely the Lebesgue-Nagell equation x^2+D=y^n, in integers x, y, n>2, for D in the range 1 =< D =< 100.
We describe the development of a junior-senior level course for Physics majors designed to teach Mathematica skills in support of their undergraduate coursework, but also to introduce students to modern research level results. Standard…
By finding all integral points on certain elliptic and hyperelliptic curves we completely solve the Diophantine equation $\binom{n}{k}=\binom{m}{l}+d$ for $-3\leq d\leq 3$ and $(k,l)\in\{(2,3),\; (2,4),\;(2,5),\; (2,6),\; (2,8),\; (3,4),\;…
The sufficient conditions for solvability of a linear Diophantine equation $\sum_{i=1}^{n}a_ix_i=b$ (with $a_1,a_2,...,a_n\in \mathbb{N}$) in non-negative integers $x_1,x_2,...,x_n$ are given. The explicit formulas are given for Frobenius…
In Diophantine approximation, inhomogeneous problems are linked with homogeneous ones by means of the so-called Transference Theorems. We revisit this classical topic by introducing new exponents of Diophantine approximation. We prove that…
In this short note we discuss ordinary differential equations which linearize upon one (or more) differentiations. Although the subject is fairly elementary, equations of this type arise naturally in the context of integrable systems.
We present a stable and convergent method for solving initial value problems based on the use of differentiation matrices obtained by Lagrange interpolation. This implicit multistep-like method is easy-to-use and performs pretty well in the…
This article discusses the question - how to estimate the number of solutions of algebraic Diophantine equations with natural coefficients using Circular method developed by Hardy and Littlewood. This paper considers the estimate of the…
We develop an algebraic method of studying of Diophantine quadratic equations in three variables over the ring of Gaussian integers.
In problem solving, understanding the problem that one seeks to solve is an essential initial step. In this paper, we propose computational methods for facilitating problem understanding through the task of recognizing the unknown in…
We state some elementary problems concerning the relation between difference calculus and differential calculus, and we try to convince the reader that, in spite of the simplicity of the statements, a solution of these problems would be a…
This paper gives parametric solutions to quartic equations of the type,(4-3-3),(4-4-4),(4-5-5) and (4-6-6), According to Lander, Parkin, and Selfridge (2) conjecture, there are non-trivial solutions of the quartic…
We study the problem of Diophantine approximation on lines in R^2 with prime numerator and denominator.
The paper is devoted to the problem of estimating the constant of the best Diophantine approximations. The estimates of lower bound $ C_n $ for $ n = 5 $ and $ n = 6 $ was improved. The first chapter gives an overview of the history of…
For two relatively prime positive integers $a, b\in \mathbb{N}$, it is known that exactly one of the two Diophantine equations $$ax + by \ =\ \frac{(a-1)(b-1)}{2}\ \mbox{ and }\ 1 + ax + by \ =\ \frac{(a-1)(b-1)}{2}$$ has a nonnegative…
While there is not much publications, about degree sixteen Diophantine equation we do have an identity given by Ramanujan (ref. #1). Also on the internet even though there are numerical solutions to degree sixteen for eg. (16-7-24) equation…
In this paper we introduce and investigate a new kind of functional (including ordinary and evolutionary partial) differential equations. The main goal of this paper is to explore our new philosophy by some examples on functional ODEs and…
In this work, we accomplish three goals. First, we determine the entire family of positive integer solutions to the three- variable Diophantine equation, xy=z^2; for n=2,3,4,5,6. For n=2, we obtain a 3-parameter family of solutions; for…
We first propose two conjectural estimates on Diophantine approximation of logarithms of algebraic numbers. Next we discuss the state of the art and we give further partial results on this topic.