pi-digits

Compute nth digit of Pi using an asymptotic formula from Plouffe (2022).

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 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 as follows:

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

Usage

Make sure to have mpmath installed.

pip install mpmath

from pi_python import pi_nth_digit
pi_nth_digit(10) # 5

To run tests:

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. The paper also discusses a way to calculate the nth digit of pi using Euler numbers.

Big thanks to Martin Bauer for the inspiration and his illustration of this formula.