In 1657, amateur mathematician Pierre de Fermat wrote to mathematicians William Brouncker and John Wallis asking them if they could find positive integers x and y such that x^2 = 1 + 61y^2. Through a series of letters, they worked out a general theory of finding such integers for a more general equation in the form x^2 = 1 + dy^2. Today, these are mistakenly called ``Pell's Equations'', after Leonhard Euler learned about them in a book on the subject by Johann Rahn and John Pell. In this talk, we'll explain the theory of Pell's Equations, exploring the theory via ideas by Joseph-Louis Lagrange. Along the way, we explore the properties of Continued Fractions.
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