https://github.com/sbplat/reciprocal

Find all unique positive integer solutions to a reciprocal equation of the form: 1/x_1 + 1/x_2 + ... + 1/x_n = k.
https://github.com/sbplat/reciprocal

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Find all unique positive integer solutions to a reciprocal equation of the form: 1/x_1 + 1/x_2 + ... + 1/x_n = k.

• Host: GitHub
• URL: https://github.com/sbplat/reciprocal
• Owner: sbplat
• Created: 2022-07-05T14:39:00.000Z (over 2 years ago)
• Default Branch: main
• Last Pushed: 2024-01-13T19:13:16.000Z (10 months ago)
• Last Synced: 2024-01-14T03:24:50.791Z (10 months ago)
• Language: JavaScript
• Homepage: https://sbplat.github.io/reciprocal/
• Size: 12.7 KB
• Stars: 0
• Watchers: 1
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README.md

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README

        
# Reciprocal Equation Solver

Find all unique positive integer solutions to a reciprocal equation of the form: $\frac{1}{x_1}+\frac{1}{x_2}+\ldots+\frac{1}{x_n}=k$. Try it out [here](https://sbplat.github.io/reciprocal/)!

## How does it work?

We use a combination of brute force and narrowing down the search space to find all the solutions.

Let's start with a simple example, with $n=2$ terms and the right hand side $k=1$. We are asked to solve the equation $\frac{1}{x_1}+\frac{1}{x_2}=1$.

Without loss of generality, let's assume $x_2\geq x_1\implies\frac{1}{x_2}\leq\frac{1}{x_1}$.

Then, we have the following inequalities:

```math

\tag{1}

\begin{aligned}

    1=\frac{1}{x_1}+\frac{1}{x_2}\leq\frac{1}{x_1}+\frac{1}{x_1}=\frac{2}{x_1}\implies x_1\leq 2

\end{aligned}

```

```math

\tag{2}

\begin{aligned}

    1=\frac{1}{x_1}+\frac{1}{x_2}\geq\frac{1}{x_2}+\frac{1}{x_2}=\frac{2}{x_2}\implies x_2\geq 2

\end{aligned}

```

Observe that $(1)$ is extremely crucial, while $(2)$ is not. This is because $(1)$ gives us an upper bound on $x_1$, and since $x_1$ is a positive integer, we just need to brute force all the values of $x_1$ in $[1, 2]$. This significantly narrows down the search space and makes it a much easier problem.

Now, what if $k$ is a rational number? We can still follow the same approach as above to narrow it down. There's only one extra step: we need to make sure the last term ($\frac{1}{x_2}$ in this case) is a reciprocal of a positive integer. If it's not, then it's not a solution.

Finally, what if we had $n=3$ terms and the right hand side $k\in\mathbb{Q}$? We can follow a similar approach and reach the conclusion that $x_1\leq\frac{3}{k}$. Recursively following this approach gives us an algorithm to find all the solutions. You can check out the implementation [here](https://github.com/sbplat/reciprocal/blob/main/js/index.js).

## How do I run this from source?

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

```sh

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

cd reciprocal

```

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