Improving regulator verification and compact representations in real quadratic fields

The 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.

Name thesis:degree_name

Doctor of Philosophy (PhD)

Discipline thesis:degree_discipline

Mathematics and Statistics

Grantor dc:publisher.institution

Graduate Studies

Year dc:date.issued

2013

Author dc:creator

• Silvester, Alan

Advisors dc:contributor.advisor

• Williams, Dr. Hugh C.
• Jacobson, Dr. Michael J. Jr.

• Mathematics

Statement dc:rights

• University 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.

Language dc:language.iso

eng

DOI

http://dx.doi.org/10.11575/PRISM/26493

OAI identifier oai:identifier

oai:ucalgary.scholaris.ca:11023/396