Title
Streaming Down the Stern-Brocot Tree: Finding and Expressing Solutions to Pell's Equation in SL(2,Z)
Date of Award
Spring 5-7-2021
Document Type
Thesis
Degree Name
Master of Science (MS) in Mathematics
Department
Mathematical, Computing & Information Sciences
Committee Chair
Dr. Jaedeok Kim
Abstract
This paper explores and elaborates on a method of solving Pell’s equation as introduced by Norman Wildberger. In the first chapters of the paper, foundational topics are introduced in expository style including an explanation of Pell’s equation. An explanation of continued fractions and their ability to express quadratic irrationals is provided as well as a connection to the Stern-Brocot tree and a convenient means of representation for each in terms of 2×2 matrices with integer elements. This representation will provide a useful way of navigating the Stern-Brocot tree computationally and permit us a means of computing continued fractions without the tedium of unraveling nested denominators. The paper also introduces simple unary operations for describing select permutations on continued fractions and, more importantly, their matrix-product counterparts. In the latter chapters of the paper, interesting symmetries appear as a result of using the Wildberger Algorithm. Quadratic forms and the subset of balanced quadratic forms will be shown to act as SL2(Z)-sets. Using this language we explore solutions to the generalized Pell equation and demonstrate a generalization for Norman Wildberger’s algorithm.