Chudnovsky algorithm

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Fast method for calculating the digits of π

title: "Chudnovsky algorithm" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["pi-algorithms"] description: "Fast method for calculating the digits of π" topic_path: "technology/algorithms" source: "https://en.wikipedia.org/wiki/Chudnovsky_algorithm" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Fast method for calculating the digits of π ::

The Chudnovsky algorithm is a fast method for calculating the digits of , based on Ramanujan's formulae. Published by the Chudnovsky brothers in 1988, it was used to calculate to a billion decimal places.

It was used in the world record calculations of 2.7 trillion digits of in December 2009, 10 trillion digits in October 2011, 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 23rd, 2025 with 314 trillion digits of pi. This was done through the usage of the algorithm on y-cruncher.

Algorithm

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:{{citation | last1 = Baruah | first1 = Nayandeep Deka | last2 = Berndt | first2 = Bruce C. | last3 = Chan | first3 = Heng Huat | doi = 10.4169/193009709X458555 | issue = 7 | journal = American Mathematical Monthly | jstor = 40391165 | mr = 2549375 | pages = 567–587 | title = Ramanujan's series for 1/: a survey | volume = 116 | year = 2009}} \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 , 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.

Optimizations

The optimization technique used for the world record computations is called binary splitting. |last1=Brent |first1=Richard P. |author-link=Richard P. Brent |last2=Zimmermann |first2=Paul |author2-link=Paul Zimmermann (mathematician) |year=2010 |title=Modern Computer Arithmetic |volume=18 |publisher=Cambridge University Press |isbn=978-0-511-92169-8 |doi=10.1017/CBO9780511921698

References

References

  1. (1988). "Approximation and complex multiplication according to Ramanujan".
  2. Warsi, Karl. (2019). "The Math Book: Big Ideas Simply Explained". [[Dorling Kindersley Limited]].
  3. Baruah, Nayandeep Deka. (2009-08-01). "Ramanujan's Series for 1/π: A Survey". American Mathematical Monthly.
  4. (2011). "10 Trillion Digits of Pi: A Case Study of summing Hypergeometric Series to high precision on Multicore Systems". Computer Science Department, University of Illinois.
  5. Aron, Jacob. (March 14, 2012). "Constants clash on pi day". [[New Scientist]].
  6. "22.4 Trillion Digits of Pi".
  7. "Google Cloud Topples the Pi Record".
  8. "The Pi Record Returns to the Personal Computer".
  9. "Pi-Challenge - Weltrekordversuch der FH Graubünden - FH Graubünden".
  10. "Calculating 100 trillion digits of pi on Google Cloud".
  11. Yee, Alexander J.. (2024-03-14). "Limping to a new Pi Record of 105 Trillion Digits".
  12. Ranous, Jordan. (2024-06-28). "StorageReview Lab Breaks Pi Calculation World Record with Over 202 Trillion Digits".
  13. "StorageReview Sets New Pi Record: 314 Trillion Digits on a Dell PowerEdge R7725".
  14. OBrien, Kevin. (2025-12-25). "Pi calculation world record shattered at 314 trillion digits with a four-month run on a single server — StorageReview retakes the crown, thanks to storage bandwidth".
  15. "y-cruncher - Formulas".

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