https://github.com/sbplat/pell

Find the first # of smallest positive integer pairs (x, y) that satisfy Pell's equation: x^2 - n * y^2 = 1.
https://github.com/sbplat/pell

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Find the first # of smallest positive integer pairs (x, y) that satisfy Pell's equation: x^2 - n * y^2 = 1.

• Host: GitHub
• URL: https://github.com/sbplat/pell
• Owner: sbplat
• Created: 2022-07-05T14:44:29.000Z (over 2 years ago)
• Default Branch: main
• Last Pushed: 2024-01-13T17:24:40.000Z (10 months ago)
• Last Synced: 2024-01-14T02:50:45.557Z (10 months ago)
• Language: JavaScript
• Homepage: https://sbplat.github.io/pell/
• Size: 8.79 KB
• Stars: 0
• Watchers: 1
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README.md

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README

        
# Pell's Equation Solver

Find the first # of smallest positive integer pairs $(x, y)$ that satisfy Pell's equation: $x^2 - ny^2 = 1$. Try it out [here](https://sbplat.github.io/pell/)!

## How does it work?

The hardest part of this problem is finding the primitive solution (not the trivial one). Luckily, there is an algorithm known as the [Chakravala method](https://en.wikipedia.org/wiki/Chakravala_method) that can be used to find the primitive solution to Pell's equation.

Once we've found the primitive solution, we can generate all the other solutions really easily (in $O(1)$ time!).

Notice that given a solution $(x_i, y_i)$, we have

$$x_{i+1}+\sqrt{n}y_{i+1}=(x_{i}+\sqrt{n}y_{i})(x_0+\sqrt{n}y_0)$$

where $(x_0, y_0)$ is the primitive solution and $(x_{i+1}, y_{i+1})$ is the next solution. This can be proven by induction. Now, by comparing the coefficients of $1$ and $\sqrt{n}$, we get the following recurrence relations:

```math

\begin{align*}

    x_{i+1} &= x_0x_{i} + n y_0y_{i}\\

    y_{i+1} &= x_0y_{i} + y_0x_{i}

\end{align*}

```

We can use this to generate any finite number of solutions to Pell's equation.

## Why can't $n$ be a perfect square?

Assume $x\neq 1$ and $y\neq 0$ since this is a trivial solution.

Let's also assume $n$ is a perfect square, so $n = k^2$ for some positive integer $k$. We can rewrite Pell's equation as

$$x^2 - k^2y^2 = 1\implies x^2 - (ky)^2 = 1$$

It's easy to see that the only solution to this is $x = 1$ and $ky = 0$. But this contradicts our first assumption that $x\neq 1$ and $y\neq 0$, so $n$ cannot be a perfect square.

## How do I run this from source?

1. First, clone the repository and change into the directory:

```sh

git clone https://github.com/sbplat/pell.git

cd pell

```

2. Open `index.html` with your browser.