No description available.

No description available.

This dissertation consists of three parts. In the first part we consider the equidistribution of roots of quadratic congruences. The roots of quadratic congruences are known to be equidistributed. However,we establish a bound for the discrepancy of this sequence using a spectral method involvingautomorphic forms, especially Kuznetsov's formula, together with an Erdős-Turán inequality. Then we discuss the implications of our discrepancy estimate for the reducibility problem of arctangents of integers. In the second and third part of this dissertation we consider some aspects of Farey fractions. The set of Farey fractions of order at most [mathematical formula] is, of course, a classical object in Analytic Number Theory. Our interest here is in certain sumsets of Farey fractions. Also, in this dissertation we study Farey fractions by working in the quotient group Q/Z, which is the modern point of view. We first derive an identity which involves the structure of Farey fractions in the group ring of Q/Z. Then we use these identities to estimate the asymptotic magnitude of the size of the sumset [mathematical formula]. Our method uses results about divisors in short intervals due to K. Ford. We also prove a new form of the Erdős-Turán inequality in which the usual complex exponential functions are replaced by a special family of functions which are orthogonal in L²(R/Z). / text

What constitutes an adequate solution to a mathematical problem? When is an adequate solution a 'good' solution? In this thesis I consider these questions in relation to two Diophantine equations, namely, x$ sp2$ + k = y$ sp3$ and 6y$ sp2$ = x(x + 1)(2x + 1). The first dates back to Diophantus himself (c. 250 AD) while the second can be traced to a puzzle proposed by Edouard Lucas in 1875. Each of these equations has attracted a number of solutions and each solution reveals something about its era. An examination and comparison of these solutions will give us an opportunity to reflect on some of the criteria used for judging proofs in mathematics. In particular, we shall see that contemporary computer technology has made a certain kind of solution to these equations acceptable which might have seemed pointless, incomplete or inelegant to the mathematicians who first studied them. Included among these 'computer solutions' is my own solution to 6y$ sp2$ = x(x + 1)(2x + 1).

Thesis (Ph. D.)--Illinois State University, 2006. / Title from title page screen, viewed on June 7, 2007. Dissertation Committee: Dissertation Committee: Sharon S. McCrone, Beverly S. Rich (co-chairs), James F. Cottrill, Lucian L. Ionescu. Includes bibliographical references (leaves 155-159) and abstract. Also available in print.

A remarkable inequality among integer numbers is given. Easily deduced from Fermat's Last Theorem, it would be nevertheless very difficult to establish through other means.

No description available.

No description available.

Master of Science / Department of Mathematics / Christopher G. Pinner / This thesis will look at Quadratic Diophantine Equations. Some well known proofs, including how to compute all Pythagorean triples and which numbers can be represented by the sum of two and four squares will be presented. Some concepts that follow from these theorems will also be presented. These include how to compute all Pythagorean Quadruples, which number can be represented by the difference of two squares and the Crossed Ladders problem. Then, Ramanujan's problem of finding which positive integers, a,b,c and d which allow aw^2+bx^2+cy^2+dz^2 to represent all natural numbers will be shown. The paper will conclude with a lengthy discussion of Uspensky's proof on which numbers can be represented by the sum three squares.

A thesis submitted to the Faculty of Science, University of the Witwatersrand and to the University Cheikh Anta Diop of Dakar(UCAD) in fulfillment of the requirements for a Dual-degree for Doctor in Philosophy in Mathematics. November 6th, 2017. / This thesis is about the study of Diophantine equations involving binary recurrent sequences with arithmetic functions. Various Diophantine problems are investigated and new results are found out of this study. Firstly, we study several questions concerning the intersection between two classes of non-degenerate binary recurrence sequences and provide, whenever possible, effective bounds on the largest member of this intersection. Our main study concerns Diophantine equations of the form '(jaunj) = jbvmj; where ' is the Euler totient function, fungn 0 and fvmgm 0 are two non-degenerate binary recurrence sequences and a; b some positive integers. More precisely, we study problems involving members of the recurrent sequences being rep-digits, Lehmer numbers, whose Euler’s function remain in the same sequence. We prove that there is no Lehmer number neither in the Lucas sequence fLngn 0 nor in the Pell sequence fPngn 0. The main tools used in this thesis are lower bounds for linear forms in logarithms of algebraic numbers, the so-called Baker-Davenport reduction method, continued fractions, elementary estimates from the theory of prime numbers and sieve methods. / LG2018