On the Solutions of Pell and Negative Pell Equations

Abstract

In this paper, we use multiple techniques to find the complete set of solutions to Pell and negative Pell equations. We prove the existence of a nontrivial solution to the former using Dirichlet's Approximation Theorem and then find all solutions from the minimal positive solution. We briefly mention the generalized Pell equation and how to generate infinitely many solutions from just one. In addition, we find all solutions to both Pell and negative Pell equations given the minimal solution to the latter. We then utilize the method of continued fractions as developed by Joseph-Louis Lagrange to represent solutions to both equations as convergents to the continued fraction of $\sqrt{d}$. We provide the proofs of multiple obscure theorems related to continued fractions, such as the Law of Best Approximations, Lagrange's Theorem, and Galois' Theorem. We end by providing resources on generalized Pell equations and their applications. All in all, this paper thoroughly describes the solutions of Pell and negative Pell equations and gives an in-depth introduction to continued fractions.

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