https://github.com/carlolepelaars/pi-digits

Compute nth digit of pi in Python
https://github.com/carlolepelaars/pi-digits

mathematics

Last synced: about 20 hours ago
JSON representation

Compute nth digit of pi in Python

• Host: GitHub
• URL: https://github.com/carlolepelaars/pi-digits
• Owner: CarloLepelaars
• License: mit
• Created: 2023-11-18T15:40:20.000Z (about 1 year ago)
• Default Branch: main
• Last Pushed: 2023-11-18T21:05:49.000Z (about 1 year ago)
• Last Synced: 2023-11-18T22:21:42.621Z (about 1 year ago)
• Topics: mathematics
• Language: Python
• Homepage:
• Size: 13.7 KB
• Stars: 0
• Watchers: 1
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • License: LICENSE

Awesome Lists containing this project

README

        
# pi-digits

Compute nth digit of Pi using an asymptotic formula from [Plouffe (2022)](https://arxiv.org/abs/2201.12601). 

Implemented in the following programming languages:

- Python (Python 3.11+)

## Explanation

For $n \geq 3$, the nth digit of $\pi$ is obtained by first calculating $\pi_n$.

$$\pi_n = \left( \frac{2 (-1)^{n+1} (2n)!}{2^{2n} B_{2n} (1 - 2^{-2n}) (1 - 3^{-2n}) (1 - 5^{-2n}) (1 - 7^{-2n}) (1 - 11^{-2n})} \right)^{\frac{1}{2n}}$$

where $B_{2n}$ is the [Bernoulli number](https://en.wikipedia.org/wiki/Bernoulli_number) of $2n$.

Then the nth digit of $\pi$ is given by:

$$d_n = \text{int} \left( 10 \text{frac} \left( 10^{n-1} \pi_{n-1} \right) \right)$$

The Bernoulli number can be obtained using the [Zeta function](https://en.wikipedia.org/wiki/Riemann_zeta_function) as follows:

$$B_{2n} = -2n * \zeta(1 - 2n)$$

## Usage

Make sure to have `mpmath` installed.

`pip install mpmath`

```python

from pi_python import pi_nth_digit

pi_nth_digit(10) # 5

```

To run tests:

```bash

pip install pytest

test_pi_python.py -s

```

## Credits

This asymptotic formula for calculating the nth digit of pi was discovered by [Simon Plouffe in 2022](https://arxiv.org/abs/2201.12601). The paper also discusses a way to calculate the nth digit of pi using [Euler numbers](https://en.wikipedia.org/wiki/Euler_numbers).

Big thanks to [Martin Bauer](https://twitter.com/martinmbauer/status/1614571838721622022?s=20&t=IznMtorWVeNbjlX-A5obNw) for the inspiration and his illustration of this formula.