Fundamental solution of Pell's equation

Definition

Let $d$ be a positive integer that is not a perfect square. The fundamental solution of Pell's equation $x^2 - d y^2 = 1$ is the solution $(x_1, y_1)$ with $x_1 > 1$ and $y_1 > 0$ such that $x_1$ is as small as possible.

Remarks

The fundamental solution exists by the existence of nontrivial solutions. Among the solutions with $x > 1$ and $y > 0$, it also has the smallest $y$ and the smallest value of $x + y \sqrt{d}$. For example, the fundamental solution for $d = 2$ is $(3, 2)$, and for $d = 7$ it is $(8, 3)$. It appears among the convergents of the continued fraction of $\sqrt{d}$.

Explanations

L1 human

The fundamental solution is the smallest solution with positive entries. All other solutions come from it by taking powers of $x_1 + y_1 \sqrt{d}$.