Williams, Dr. Hugh C.Jacobson, Dr. Michael J. Jr.Silvester, Alan2013-01-082013-06-152013-01-082012Silvester, A. (2013). Improving regulator verification and compact representations in real quadratic fields (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca. doi:10.11575/PRISM/26493http://hdl.handle.net/11023/396The study of Diophantine equations, and in particular the erroneously-named Pell equation, has a long and intriguing history. In this work, we investigate solutions to the Pell equation and to a closely related quantity called the fundamental unit. Though it is somewhat simple to show these equations have non-trivial solutions, finding unconditionally correct solutions and being able to express them is extremely challenging, both from a theoretical and a computational perspective. We develop improvements to the algorithm presented by de Haan, Jacobson, and Williams [21, 22] which unconditionally verifies the regulator of a real quadratic number field and refinements to the concept of a compact representation of a quadratic integer, originally given by Buchmann, Thiel, and Williams [12]. In addition, we consider the well-known applications of this theory to principal ideal testing, finding integer points on elliptic curves, finding solutions to Diophantine equations, and some particular cryptographic applications.engUniversity of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission.MathematicsMathematicsdoctoral thesis10.11575/PRISM/26493