Pell's Equation

# Pell's Equation

 Definition: Pell's Equation is the equation $x^2 - dy^2 = N$ where $d, N \in \mathbb{Z}$.

Of course, we will only be interested in integer solutions $(x, y)$ to the equation.

 Proposition 1: If $d < 0$ then Pell's equation has at most finitely many solutions.
• Proof: Suppose that $d < 0$. Then $-d > 0$. Since $N$ is fixed, and $x^2 - dy^2 \geq 0$, there can be at most finitely many solutions to Pell's equation.
 Proposition 2: If $d > 0$ is a perfect square then Pell's equation has at most finitely many solutions.
• Proof: Suppose that $d > 0$ is a perfect square. Then $\sqrt{d} \in \mathbb{N}$. Furthermore, the lefthand side of Pell's equation can be factored to get:
(1)
\begin{align} \quad (x + \sqrt{d}y)(x - \sqrt{d}y) = N \end{align}
• Which clearly has at most finitely many solutions. $\blacksquare$

Pell's equation only becomes interesting when we consider $d > 0$ and $d$ not a perfect square.