Pell equation

A Pell equation is a type of diophantine equation in the form $x^2-Dy^2 = 1$ for a natural number $D$ . Generally, $D$ is taken to be square-free, since otherwise we can "absorb" the largest square factor $d^2 | D$ into $y$ by setting $y' = dy$ .

Notice that if $D = d^2$ is a perfect square, then this problem can be solved using difference of squares. We would have $x^2 - Dy^2 = (x+dy)(x-dy) = 1$ , from which we can use casework to quickly determine the solutions.

Alternatively, we would like to find the set of solutions to $z = x + y\sqrt{D} \in \mathbb{Z}[\sqrt{D}]$ such that the norm of $z$ , $\|z\| = z \cdot \overline{z} = (x + y \sqrt{D}) \cdot (x - y\sqrt{D}) = 1$ .

Family of solutions

Given a smallest solution $z$ , then all solutions are of the form $\pm z^n$ for natural numbers $z$ .

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Continued fractions

The solutions to the Pell equation when $D$ is not a perfect square are connected to the continued fraction expansion of $\sqrt D$ . If $a$ is the period of the continued fraction and $C_k=P_k/Q_k$ is the $k$ th convergent, all solutions to the Pell equation are in the form $(P_{ia},Q_{ia})$ for positive integer $i$ .