Existence of nontrivial solutions of Pell's equation

Statement

Let $d$ be a positive integer that is not a perfect square. Then Pell's equation $x^2 - d y^2 = 1$ has a solution in integers with $y \neq 0$. In fact, it has infinitely many solutions in integers.

Remarks

One proof uses Dirichlet's approximation theorem to find infinitely many $x/y$ with $|x^2 - d y^2|$ bounded. Another proof uses the periodic continued fraction of $\sqrt{d}$. The smallest solution can be very large. For $d = 61$, it is $(1766319049, 226153980)$.

History

Indian mathematicians, such as Brahmagupta, studied the equation many centuries before European mathematicians. Lagrange proved that it always has infinitely many solutions.

Explanations

L1 human

For each non-square $d$, there are integers $x$ and $y \neq 0$ with $x^2 - d y^2 = 1$. They may be hard to find by trial: for $d = 61$, the smallest $x$ has ten digits.