In this thesis I will outline the impact of Pell's equation on various branches of number theory, as well as some of the history. I will also discuss some recently discovered properties of the solutions of Pell's equation.

Thesis (M.S.)--Boise State University, 2009. / Includes abstract. Includes bibliographical references (leaf 82).

Thesis (M.S.)--Boise State University, 2009. / Title from t.p. of PDF file (viewed June 15, 2010). Includes abstract. Includes bibliographical references (leaf 82).

Equations of the form 𝑎𝑥³ + 𝑏𝑦³ = 1, where the constants 𝑎 and 𝑏 are integers of some number field such that 𝑎𝑥³ + 𝑏𝑦³ is irreducible, are a particularly significant class of cubic Thue equations that notably includes the cubic Pell equation. For a positive cubefree rational integer 𝑑, we consider the family of equations of the form 𝑚𝑥³ − 𝑑𝑛𝑦³ = 1 where 𝑚 and 𝑛 are squarefree. We define an 𝐿-function associated to 𝑑 whose nonvanishing coefficients correspond to the nontrivial solutions of those equations. That definition uses expressions related to the cubic theta function Q (􏰇√ 􏰈-), and we study that 𝐿-function’s analytic properties by using a method generalizing the approach used by Takhtajan and Vinogradov to derive a trace formula using the quadratic theta function for Q. We construct its meromorphic continuation and determine the locations and orders of its poles. Specifically, the poles occur at the eigenvalues of the Laplacian for the Maass forms 𝑢_𝑗 , 𝑗 = 1, 2, 3, · · · in the discrete spectrum, with a double pole at 𝑠 = ½ and possible simple poles at 𝑠=𝑠_𝑗,1−𝑠_𝑗,where𝜆𝑗 =2𝑠_𝑗(2−2𝑠_𝑗)istheLaplaceeigenvalueof𝑢𝑗 and𝜆𝑗 ≠1.

Submitted by Cássia Santos (cassia.bcufg@gmail.com) on 2015-10-27T14:48:51Z No. of bitstreams: 2 Dissertação - Ronaldo Pires da Silva - 2015.pdf: 1653286 bytes, checksum: 63a72d8fbcc7390f80fb41dbadaaa9fe (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-10-27T14:53:07Z (GMT) No. of bitstreams: 2 Dissertação - Ronaldo Pires da Silva - 2015.pdf: 1653286 bytes, checksum: 63a72d8fbcc7390f80fb41dbadaaa9fe (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-10-27T14:53:07Z (GMT). No. of bitstreams: 2 Dissertação - Ronaldo Pires da Silva - 2015.pdf: 1653286 bytes, checksum: 63a72d8fbcc7390f80fb41dbadaaa9fe (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-07-06 / Our work had as main objective to study the intersection of integer sequences, denominated polygonal numbers, through Pell's equation. In this context, the solution of two equations will be treated: x2 􀀀 Dy2 = 1 and x2 􀀀 Dy2 = N, jNj > 1. For the rst one we have used results from the theory of continued fractions. For the last one, we have used the method of solution delineated in literature. Besides, propositions referring to the intersection of polygonal numbers for some particular cases are presented and demonstrated. Also, the proposition of the general case is presented and demonstrated. Finally, we have performed the solution of some of Pell's equations in order to determine the intersection of some polygonal numbers. / Nosso trabalho teve como objetivo central estudar a interseção de sequências de inteiros, denominadas números geométricos, através da equação de Pell. Neste contexto, a resolução de duas equações serão tratadas: x2 􀀀 Dy2 = 1 e x2 􀀀 Dy2 = N com jNj > 1. Para a primeira utilizamos importantes resultados presentes na teoria das frações contínuas. Para última, utilizamos o método de resolução delineado na literatura. Além disso, proposições referentes a interseção de números geométricos para alguns casos particulares são apresentadas e demonstradas. Também a proposição do caso geral é apresentada e demonstrada. Por m, realizamos a resolução de algumas equações de Pell para determinarmos a interseção de alguns números geométricos.

Ce mémoire s'emploie à étudier les corps quadratiques réels ainsi qu'un élément particulier de tels corps quadratiques réels : l'unité fondamentale. Pour ce faire, le mémoire commence par présenter le plus clairement possible les connaissances sur différents sujets qui sont essentiels à la compréhension des calculs et des résultats de ma recherche. On introduit d'abord les corps quadratiques ainsi que l'anneau de ses entiers algébriques et on décrit ses unités. On parle ensuite des fractions continues puisqu'elles se retrouvent dans un algorithme de calcul de l'unité fondamentale. On traite ensuite des formes binaires quadratiques et de la formule du nombre de classes de Dirichlet, laquelle fait intervenir l'unité fondamentale en fonction d'autres variables. Une fois cette tâche accomplie, on présente nos calculs et nos résultats. Notre recherche concerne la répartition des unités fondamentales des corps quadratiques réels, la répartition des unités des corps quadratiques réels et les moments du logarithme de l'unité fondamentale. (Le logarithme de l'unité fondamentale est appelé le régulateur.) / This memoir aims to study real quadratic fields and a particular element of such real quadratic fields : the fundamental unit. To achieve this, the memoir begins by presenting as clearly as possible the state of knowledge on different subjects that are essential to understand the computations and results of my research. We first introduce quadratic fields and their rings of algebraic integers, and we describe their units. We then talk about continued fractions because they are present in an algorithm to compute the fundamental unit. Afterwards, we proceed with binary quadratic forms and Dirichlet's class number formula, which involves the fundamental unit as a function of other variables. Once the above tasks are done, we present our calculations and results. Our research concerns the distribution of fundamental units in real quadratic fields, the disbribution of units in real quadratic fields and the moments of the logarithm of the fundamental unit. (The logarithm of the fundamental unit is called the regulator.)

Ce mémoire s'emploie à étudier les corps quadratiques réels ainsi qu'un élément particulier de tels corps quadratiques réels : l'unité fondamentale. Pour ce faire, le mémoire commence par présenter le plus clairement possible les connaissances sur différents sujets qui sont essentiels à la compréhension des calculs et des résultats de ma recherche. On introduit d'abord les corps quadratiques ainsi que l'anneau de ses entiers algébriques et on décrit ses unités. On parle ensuite des fractions continues puisqu'elles se retrouvent dans un algorithme de calcul de l'unité fondamentale. On traite ensuite des formes binaires quadratiques et de la formule du nombre de classes de Dirichlet, laquelle fait intervenir l'unité fondamentale en fonction d'autres variables. Une fois cette tâche accomplie, on présente nos calculs et nos résultats. Notre recherche concerne la répartition des unités fondamentales des corps quadratiques réels, la répartition des unités des corps quadratiques réels et les moments du logarithme de l'unité fondamentale. (Le logarithme de l'unité fondamentale est appelé le régulateur.) / This memoir aims to study real quadratic fields and a particular element of such real quadratic fields : the fundamental unit. To achieve this, the memoir begins by presenting as clearly as possible the state of knowledge on different subjects that are essential to understand the computations and results of my research. We first introduce quadratic fields and their rings of algebraic integers, and we describe their units. We then talk about continued fractions because they are present in an algorithm to compute the fundamental unit. Afterwards, we proceed with binary quadratic forms and Dirichlet's class number formula, which involves the fundamental unit as a function of other variables. Once the above tasks are done, we present our calculations and results. Our research concerns the distribution of fundamental units in real quadratic fields, the disbribution of units in real quadratic fields and the moments of the logarithm of the fundamental unit. (The logarithm of the fundamental unit is called the regulator.)