Chudnovsky algorithm

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] Recently, the record was broken yet again on April 2, 2025 with 300 trillion digits of pi.^{[13] [14]} This was done through the usage of the algorithm on y-cruncher.

The algorithm is based on the negated Heegner number ${\displaystyle d=-163}${\displaystyle d=-163}, the j-function ${\displaystyle j\left({\tfrac {1+i{\sqrt {-163}}}{2}}\right)=-640320^{3}}${\displaystyle j\left({\tfrac {1+i{\sqrt {-163}}}{2}}\right)=-640320^{3}}, and on the following rapidly convergent generalized hypergeometric series:^[15] $${\displaystyle {\frac {1}{\pi }}=12\sum _{k=0}^{\infty }{\frac {(-1)^{k}(6k)!(545140134k+13591409)}{(3k)!(k!)^{3}(640320)^{3k+3/2}}}}$${\displaystyle {\frac {1}{\pi }}=12\sum _{k=0}^{\infty }{\frac {(-1)^{k}(6k)!(545140134k+13591409)}{(3k)!(k!)^{3}(640320)^{3k+3/2}}}}

This identity is similar to some of Ramanujan's formulas involving π,^[15] and is an example of a Ramanujan–Sato series.

The time complexity of the algorithm is ${\displaystyle O\left(n(\log n)^{3}\right)}${\displaystyle O\left(n(\log n)^{3}\right)}, where n is the number of digits desired.^[16]

The optimization technique used for the world record computations is called binary splitting.^[17]

• Bailey–Borwein–Plouffe formula
• Borwein's algorithm
• Approximations of π

• Pi algorithms

• Articles with short description
• Short description matches Wikidata