Pell's Equation

In February of 1657, Pierre Fermat issued the following challenge:

Given a positive integer $d$, find a positive integer $y$ such that $dy^2 + 1$ is a perfect square.

In other words, find a solution to $x^2 - dy^2 = 1$ with $y\in\mathbb{N}$.

Note Fermat's emphasis on integer solutions. It is easy to find rational solutions to the equation $x^2 - dy^2 = 1$. Simply divide the relation

$$(r^2+d)^2 - d(2r)^2 = (r^2-d)^2$$

by $(r^2 - d)^2$ to arrive at

$$x = \frac{r^2+d}{r^2-d}, \qquad y = \frac{2r}{r^2-d}.$$

Fermat said: ``Solutions in fractions, which can be given at once from the merest elements of arithmetic, do not satisfy me.''

The equation $x^2 - dy^2 = 1$ is called Pell's equation. This is because Euler (in about 1759) accidently called it ``Pell's equation'' and the name stuck, though Pell (1611-1685) had nothing to do with it.

If $d$ is a perfect square, $d=n^2$, then

$$(x+ny)(x-ny) = x^2-dy^2 = 1$$

which implies that $x+ny = x-ny = 1$, so

$$x = \frac{x+ny+x-ny}{2} = \frac{1+1}{2}=1.$$

We will thus always assume that $d$ is not a perfect square. You can read about Pell's equation in Section 0.6 of Kato-Kurokawa-Saito and on pages 107-111 of Davenport. Pell's equation is best understood in terms of units in real quadratic fields.