This paper will be devoted to an examination of several general and specific equations and systems of equations of the diophantine type. Only algebraic equations with integral coefficients, not all zero, will considered. The elementary properties of the integers will be assumed.

In 1974, H. Ruderman posed the following question: If $(2^m-2^n)|(3^m-3^n)$, then does it follow that $(2^m-2^n)|(x^m-x^n)$ for every integer $x$? This problem is still open. However, in 2011, M. R. Murty and V. K. Murty showed that there are only finitely many $(m,n)$ for which the hypothesis holds. In this thesis, we examine two generalizations of this problem. The first is replacing 2 and 3 with arbitrary integers $a$ and $b$. The second is to replace 2 and 3 with arbitrary algebraic integers from an imaginary quadratic field. In both of these cases we have shown that there are only finitely many $(m,n)$ for which the hypothesis holds. To get the second result we also generalized a result by Bugeaud, Corvaja and Zannier from the integers to imaginary quadratic fields. In the last half of the thesis we use the abc conjecture and some related conjectures to study some exponential Diophantine equations. We study the Pillai conjecture and the Erd\"{o}s-Woods conjecture and show that they are implied by the abc conjecture and that when we use an effective version, very clean bounds for the conjectures are implied. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2012-04-13 12:04:14.252

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: The aim of this thesis is to study some methods used in solving exponential Diophan- tine equations. There is no generic method or algorithm that can be used in solving all Diophantine equations. The main focus for our study will be solving the exponential Dio- phantine equations using the modular approach and the linear forms in two logarithms approach. / AFRIKAANSE OPSOMMING: Die doel van hierdie tesis is om sommige metodes te bestudeer om sekere Diophantiese vergelykings op te los. Daar is geen metode wat alle Diophantiese vergelykings kan oplos nie. Die fokus van os studie is hoofsaaklik om eksponensiele Diophantiese vergelykings op te los met die modul^ere metode en met die metode van line^ere vorms in twee logaritmes.

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Given natural numbers n and k, with n>k, the Prouhet-Tarry-Escott (PTE) problem asks for distinct subsets of Z, say X={x_1,...,x_n} and Y={y_1,...,y_n}, such that x_1^i+...+x_n^i=y_1^i+...+y_n^i\] for i=1,...,k. Many partial solutions to this problem were found in the late 19th century and early 20th century. When k=n-1, we call a solution X=(n-1)Y ideal. This is considered to be the most interesting case. Ideal solutions have been found using elementary methods, elliptic curves, and computational techniques. This thesis focuses on the ideal case. We extend the framework of the problem to number fields, and prove generalizations of results from the literature. This information is used along with computational techniques to find ideal solutions to the PTE problem in the Gaussian integers. We also extend a computation from the literature and find new lower bounds for the constant C_n associated to ideal PTE solutions. Further, we present a new algorithm that determines whether an ideal PTE solution with a particular constant exists. This algorithm improves the upper bounds for C_n and in fact, completely determines the value of C_6. We also examine the connection between elliptic curves and ideal PTE solutions. We use quadratic twists of curves that appear in the literature to find ideal PTE solutions over number fields.

Given natural numbers n and k, with n>k, the Prouhet-Tarry-Escott (PTE) problem asks for distinct subsets of Z, say X={x_1,...,x_n} and Y={y_1,...,y_n}, such that x_1^i+...+x_n^i=y_1^i+...+y_n^i\] for i=1,...,k. Many partial solutions to this problem were found in the late 19th century and early 20th century. When k=n-1, we call a solution X=(n-1)Y ideal. This is considered to be the most interesting case. Ideal solutions have been found using elementary methods, elliptic curves, and computational techniques. This thesis focuses on the ideal case. We extend the framework of the problem to number fields, and prove generalizations of results from the literature. This information is used along with computational techniques to find ideal solutions to the PTE problem in the Gaussian integers. We also extend a computation from the literature and find new lower bounds for the constant C_n associated to ideal PTE solutions. Further, we present a new algorithm that determines whether an ideal PTE solution with a particular constant exists. This algorithm improves the upper bounds for C_n and in fact, completely determines the value of C_6. We also examine the connection between elliptic curves and ideal PTE solutions. We use quadratic twists of curves that appear in the literature to find ideal PTE solutions over number fields.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 87-88). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 87-88). Also available on the Internet.

We study the size of sets containing sums of two integer cubes such that their representation is unique and also fit between two consecutive integer cubes. We will try to write algorithms that efficiently calculate the size of these sets and also implement these algorithms in PythonTM. Although we will fail to find a non-iterative algorithm, we will find different ways of approximating the size of these sets. We will also find that techniques used in our failed algorithms can be used to calculate the number of integer lattice points inside a circle.

This thesis is an educational text for high school students. It aims to teach them how to solve typical problems concerning divisibility found in mathematical correspondence seminars and mathematical olympiad. Basic notions from the theory of divisibility are recalled (e.g. prime numbers, divisors, multiples). Criteria of divisibility by 2 to 20 are introduced, as well as diophantine equations and practical applications of prime numbers in real life. One whole chapter is dedicated to problems and exercises. Powered by TCPDF (www.tcpdf.org)