Continued fractions and their relation to integral binary quadratic forms

Details

Supervisors

Tignol, Jean-Pierre

Degree label

Master [120] en sciences mathématiques

Abstract

Continued fractions and integral binary quadratic forms belong to the classical objects of number theory. At first, both theories have been studied independently. Short time later, mathematicians discovered their natural connection. Unfortunately, there is not a unique way to combine both theories and neither is there a general consent about which approach should be considered. Most of the time, the approaches are chosen depending on the problem to solve. The aim of this Master thesis is first to present one such approach in the modern mathematical language by giving formal definitions and second to highlight that there is indeed a natural connection between integral binary quadratic forms and continued fractions. This fact will be used to prove some classical results of continued fractions. Finally, those results lead to a haracterization of irrational numbers with purely periodic continued fractions. By the natural connection of both domains, the given non-standard approach develops imultaneously some well-known properties of integral binary quadratic forms. In particular the equivalence of integral binary quadratic forms will be studied and cycles of trongly reduced forms will be discovered. Ultimately, all the main results will be combined to deduce in a first instance when a given integral binary quadratic form and its negative are equivalent and to obtain in a second step a well-justified algorithm to compute the class number for a given positive non-square discriminant.