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Chudnovsky algorithm

Fast method for calculating the digits of π

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated August 28, 2026

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, it was used to calculate π to a billion decimal places.

It was used in the world record calculations of approx. 2.7 trillion digits of π in December 2009, 10 trillion digits in October 2011, around 22.4 trillion digits in November 2016, 31.4 trillion digits in September 2018-January 2019, 50 trillion digits on January 29, 2020, 62.8 trillion digits on August 14, 2021, 100 trillion digits on March 21, 2022, 105 trillion digits on March 14, 2024, and 202 trillion digits on June 28, 2024. Recently, the record was broken yet again on November 18, 2025 with 314 trillion digits of pi. This was done through the usage of the algorithm on y-cruncher.

01Algorithm

The algorithm is based on the negated Heegner number d=-163, the j-function j\left({\tfrac {1+i{\sqrt {163}}}{2}}\right)=-640320^{3}, and on the following rapidly convergent generalized hypergeometric series:{\frac {1}{\pi }}={\frac {\sqrt {10005}}{4270934400}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}(6k)!(545140134k+13591409)}{(3k)!(k!)^{3}(640320)^{3k}}}

This identity is similar to some of Ramanujan's formulas involving π, and is an example of a Ramanujan-Sato series.

The time complexity of the algorithm is O\left(n(\log n)^{3}\right), where n is the number of digits desired. Each term produces around 14 correct decimal digits of π.

02Optimizations

The optimization technique used for the world record computations is called binary splitting.

Watch videos about Chudnovsky algorithmExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Chudnovsky algorithm”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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