Pell's Equation

Definition
The Diophantine equation:
 * $x^2 - n y^2 = 1$

is known as Pell's equation.

Solution
Let the continued fraction of $\sqrt n$ have a cycle whose length is $s$.

Let $\dfrac {p_n} {q_n}$ be a convergent of $\sqrt n$.

Then:
 * $p_{r s}^2 - n q_{r s}^2 = \left({-1}\right)^{rs}$ for $r = 1, 2, 3, \ldots$

and all solutions of:
 * $x^2 - n y^2 = \pm 1$

are given in this way.

Proof
First note that if $x = p, y = q$ is a positive solution of $x^2 - n y^2 = 1$ then $\dfrac p q$ is a convergent of $\sqrt n$.

The continued fraction of $\sqrt n$ is periodic from Continued Fraction Expansion of Irrational Square Root and of the form:
 * $\left[{a \left \langle{b_1, b_2, \ldots, b_{m - 1}, b_m, b_{m - 1}, \ldots, b_2, b_1, 2 a}\right \rangle}\right]$

or
 * $\left[{a \left \langle{b_1, b_2, \ldots, b_{m - 1}, b_m, b_m, b_{m - 1}, \ldots, b_2, b_1, 2 a}\right \rangle}\right]$

For each $r \ge 1$ we can write $\sqrt n$ as the (non-simple) finite continued fraction:
 * $\sqrt n \left[{a \left \langle{b_1, b_2, \ldots, b_2, b_1, 2 a, b_1, b_2, \ldots, b_2, b_1, x}\right \rangle}\right]$

which has a total of $r s + 1$ partial quotients.

The last element $x$ is of course not an integer.

What we do have, though, is:

The final three convergents in the above FCF are:
 * $\dfrac {p_{r s - 1}} {q_{r s - 1}}, \quad \dfrac {p_{r s}} {q_{r s}}, \quad \dfrac {x p_{r s} + p_{r s - 1}} {x q_{r s} + q_{r s - 1}}$

The last one of these equals $\sqrt n$ itself.

So:
 * $\sqrt n \left({x q_{r s} + q_{r s - 1} }\right) = \left({x p_{r s} + p_{r s - 1} }\right)$

Substituting $a + \sqrt n$ for $x$, we get:
 * $\sqrt n \left({\left({a + \sqrt n}\right) q_{r s} + q_{r s - 1} }\right) = \left({\left({a + \sqrt n}\right) p_{r s} + p_{r s - 1}}\right)$

This simplifies to:
 * $\sqrt n \left({a q_{r s} + q_{r s - 1} - p_{r s} }\right) = a p_{r s} + p_{r s - 1} - n q_{r s}$

The of this is an integer while the  is $\sqrt n$ times an integer.

Since $\sqrt n$ is irrational, the only way that can happen is if both sides equal zero.

This gives us:

Multiplying $(1)$ by $p_{r s}$, $(2)$ by $q_{r s}$ and then subtracting:
 * $p_{r s}^2 - n q_{r s}^2 = p_{r s} q_{r s - 1} - p_{r s - 1} q_{r s}$

By Difference between Adjacent Convergents of Simple Continued Fraction, the of this is $\left({-1}\right)^{r s}$.

When the cycle length $s$ of the continued fraction of $\sqrt n$ is even, we have $\left({-1}\right)^{r s} = 1$.

Hence $x = p_{r s}, y = q_{r s}$ is a solution to Pell's Equation for each $r \ge 1$.

When $s$ is odd, though:
 * $x = p_{r s}, y = q_{r s}$ is a solution of $x^2 - n y^2 = -1$ when $r$ is odd
 * $x = p_{r s}, y = q_{r s}$ is a solution of $x^2 - n y^2 = 1$ when $r$ is even.