Pell's equation

Definition

Let $d$ be a positive integer that is not a perfect square. Pell's equation for $d$ is the Diophantine equation $x^2 - d y^2 = 1$, with unknowns $x, y \in \mathbb{Z}$. A solution is a pair $(x, y)$ of integers that satisfies the equation.

Remarks

The pairs $(\pm 1, 0)$ are the trivial solutions. For $d = 2$, the pair $(3, 2)$ is a solution, since $9 - 2 \cdot 4 = 1$. A solution $(x, y)$ with $y \neq 0$ gives the rational approximation $x / y$ of $\sqrt{d}$. The name comes from an error of Euler, who attributed a solution by William Brouncker to John Pell.

Explanations

L1 human

Pell's equation asks for integers $x$ and $y$ such that $x^2$ is exactly one more than $d y^2$. For $d = 2$, the solutions $(3, 2)$, $(17, 12)$ and $(577, 408)$ give fractions very close to $\sqrt{2}$.