Note this relies on floating point numbers. You can do it entirely with integer arithmetic.

Can you give me reference for doing it using integer arithmetic completely?

"Quadratic irrationality" section of continued fraction article has it:

# compute the continued fraction of sqrt(n)
def sqrt(n):
    n0 = math.floor(math.sqrt(n))
    x, y, z = 1, 0, 1
    a = []
    def step(x, y, z):
        a.append((x * n0 + y) // z)
        t = y - a[-1]*z
        x, y, z = -z*x, z*t, t**2 - n*x**2
        g = math.gcd(x, math.gcd(y, z))
        return x // g, y // g, z // g

    used = dict()
    for i in range(n):
        used[x, y, z] = i
        x, y, z = step(x, y, z)
        if (x, y, z) in used:
            return a

included with a link after non-integer example

I would actually strongly suggest to replace the whole section to only show integer calculation, as floating-point calculation might be unstable.

Do not include this

.gitgnore just ignores files to not include them when we do git add .
So, that won't affect anything

I think we should remove this, unless it is something that is really needed for the project at large, rather than specific developer.

Why is it here again 🤔

This equation looks like it just comes out of nowhere. As in, it isn't connected with the paragraphs that are immediately adjusted to it. I would add some explanation on what are we doing here exactly.

Also consider wrapping this in $$...$$ instead of $...$.

You use the term "norm" here, but it's not defined. We should add a definition + an explanation why we care about it (primarily because it is multiplicative).

We defined the smallest solution by saying that it is the solution with the smallest positive $x$ (but we actually want $x > 1$). It also means that $x + y \sqrt d$ is the smallest possible, of course, but we should mention it explicitly abouve.

Let's add a link to continued fractions article. Ideally, I would also add explanation on why it is a solution at all, and why is it the smallest possible.

Also, the paragraph says that "the smallest solution is given by ...", but doesn't tell that, specifically, the solution is $p_n^2 - q_n^2 d = 1$. Instead, you just write below that $p_n / q_n$ is the solution.

This looks complicated. Are there specific reasons to do it over continued fractions variant? I find the exposition very hard to read (largely because of misrenders, though).

Unless it's in some way better than continued fractions method, I'd consider dropping the section altogether.