The Negative Pell's Equation in Positive Characteristic

Abstract

The Pell’s equation and the negative Pell’s equation are two of the most well-studied topics in number theory. While the solvability of the Pell’s equation over the integers is well known for centuries, the solvability problem of the negative Pell’s equation over the integers, especially the density problem of how likely the negative Pell’s equation is solvable, was not fully answered until recently. In this thesis, we consider these equations over the polynomial ring F[t] where F is a finite field whose characteristic is greater than 2. In Chapter 3 of this thesis, two different well-known proofs of the solvability of the Pell’s equation over F[t] are presented. In Chapter 4, we present conditions and examples of when the negative Pell’s equation is solvable and the function field analogue of the density problem on the negative Pell’s equation.