Date of Award

4-2-2006

Document Type

Thesis

Department

Mathematics, Physics and Statistics

First Advisor

Dr. Lamarr Widmer

Abstract

This paper is an investigation of Pell Equations-equations of the form x2 - dy2 = k where d is a nonsquare, positive integer, k is an integer, and we are looking for integer solutions in x and y. We will provide motivation, both algebraic and geometric, for this definition of Pell Equations. Next, we will examine the k = 1 case, proving not only that it is solvable, but also that infinitely many solutions can be obtained easily from the fundamental solution. We will classify some Pell Equations as solvable or unsolvable when k [does not equal] 1, examining in detail the k = 4 case. After explaining several patterns that appear when k = 4 and d = 5 (mod 8), we will prove the existence of a fundamental solution for these cases. Finally, we will briefly examine how a computer may be used to find solutions, especially the fundamental solution, for Pell Equations.

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