The Chudnovsky algorithm is a fast method for calculating the digits of π, based on Ramanujan's π formulae.
Published by the Chudnovsky brothers in 1988,[1] it was used to calculate π to a billion decimal places.
[2] It was used in the world record calculations of 2.7 trillion digits of π in December 2009,[3] 10 trillion digits in October 2011,[4][5] 22.4 trillion digits in November 2016,[6] 31.4 trillion digits in September 2018–January 2019,[7] 50 trillion digits on January 29, 2020,[8] 62.8 trillion digits on August 14, 2021,[9] 100 trillion digits on March 21, 2022,[10] 105 trillion digits on March 14, 2024,[11] and 202 trillion digits on June 28, 2024.
[12] The algorithm is based on the negated Heegner number
, the j-function
, and on the following rapidly convergent generalized hypergeometric series:[13]
A detailed proof of this formula can be found here: [14]
This identity is similar to some of Ramanujan's formulas involving π,[13] and is an example of a Ramanujan–Sato series.
The time complexity of the algorithm is
n ( log n
[15] The optimization technique used for the world record computations is called binary splitting.
can be taken out of the sum and simplified to
1 π
, and substitute that into the sum.
1 π
can be simplified to
from the original definition of
This definition of
is not defined for
, so compute the first term of the sum and use the new definition of
1 π
1 π
π =
can never be computed, so instead compute
π
π =
lim
From the original definition of
π =
lim