The Pell equation

Abstract

The so-called Pell equation x² − ny² = 1 (wrongly attributed to Pell by Euler) is one of the oldest equations in mathematics and it is fundamental to the study of quadratic Diophantine equations. The Greeks studied the special case x² − 2y² = 1 because they realized that its natural number solutions throw light on the nature of $\sqrt{2}$. There is a similar connection between the natural number solutions of x² − ny² = 1 and $\sqrt{n}$ when n is any nonsquare natural number.