The Pell equation
This chapter is devoted to one of the oldest diophantine equations, the Pell equation. John Pell was an English mathematician who lived in the seventeenth century but he had nothing to do with this equation; it was Euler who mistakenly attributed a solution method to Pell which in fact was found by Pell’s contemporaries Wallis and Lord Brouncker. The history of the Pell equation dates back at least to the ancient Greeks. Its main importance lies in the role it plays in the arithmetic of quadratic number ﬁelds. The solution of the Pell equation relies on the theory of diophantine approximation, and so it is our ﬁrst non-trivial example for the close relation between diophantine approximations and diophantine equations. Nevertheless, the Pell equation still contains some unsolved interesting questions.