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Chudnovsky algorithm

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The Chudnovsky algorithm is a fast method for calculating the digits of π, based on Ramanujan's π formulae. In 1988, it was published by the Chudnovsky brothers,[1] and used to calculate π to a billion decimal places.[2]

It has since been used in the world record calculations of approx. 2.7 trillion digits of π in December 2009,[3] 10 trillion digits in October 2011,[4][5] around 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 November 18, 2025 with 314 trillion digits of pi.[13][14] This was done through the usage of the algorithm on y-cruncher.

Algorithm#edit

The algorithm is based on the negated Heegner number {\displaystyle d=-163}, the j-function {\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 ={\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,[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)}, where n is the number of digits desired.[16] Each term produces around 14 correct decimal digits of π.[17]

Optimizations#edit

The optimization technique used for the world record computations is called binary splitting.[18]

See also#edit

References#edit

  1. ^ Chudnovsky, David; Chudnovsky, Gregory (1988), Approximation and complex multiplication according to Ramanujan, Ramanujan revisited: proceedings of the centenary conference
  2. ^ Warsi, Karl; Dangerfield, Jan; Farndon, John; Griffiths, Johny; Jackson, Tom; Patel, Mukul; Pope, Sue; Parker, Matt (2019). The Math Book: Big Ideas Simply Explained. New York: Dorling Kindersley Limited. p. 65. ISBN 978-1-4654-8024-8.
  3. ^ Baruah, Nayandeep Deka; Berndt, Bruce C.; Chan, Heng Huat (2009-08-01). "Ramanujan's Series for 1/π: A Survey". American Mathematical Monthly. 116 (7): 567–587. doi:10.4169/193009709X458555.
  4. ^ Yee, Alexander; Kondo, Shigeru (2011), 10 Trillion Digits of Pi: A Case Study of summing Hypergeometric Series to high precision on Multicore Systems, Technical Report, Computer Science Department, University of Illinois, hdl:2142/28348
  5. ^ Aron, Jacob (March 14, 2012), "Constants clash on pi day", New Scientist
  6. ^ "22.4 Trillion Digits of Pi". www.numberworld.org.
  7. ^ "Google Cloud Topples the Pi Record". www.numberworld.org.
  8. ^ "The Pi Record Returns to the Personal Computer". www.numberworld.org.
  9. ^ "Pi-Challenge - Weltrekordversuch der FH Graubünden - FH Graubünden". www.fhgr.ch. Retrieved 2021-08-17.
  10. ^ "Calculating 100 trillion digits of pi on Google Cloud". cloud.google.com. Retrieved 2022-06-10.
  11. ^ Yee, Alexander J. (2024-03-14). "Limping to a new Pi Record of 105 Trillion Digits". NumberWorld.org. Retrieved 2024-03-16.
  12. ^ Ranous, Jordan (2024-06-28). "StorageReview Lab Breaks Pi Calculation World Record with Over 202 Trillion Digits". StorageReview.com. Retrieved 2024-07-20.
  13. ^ "StorageReview Sets New Pi Record: 314 Trillion Digits on a Dell PowerEdge R7725". StorageReview.com. Retrieved 2026-01-02.
  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". Tom's Hardware. Retrieved 2026-01-02.
  15. ^ a b Baruah, Nayandeep Deka; Berndt, Bruce C.; Chan, Heng Huat (2009), "Ramanujan's series for 1/π: a survey", American Mathematical Monthly, 116 (7): 567–587, doi:10.4169/193009709X458555, JSTOR 40391165, MR 2549375
  16. ^ "y-cruncher - Formulas". www.numberworld.org. Retrieved 2018-02-25.
  17. ^ The limiting ratio between two consecutive terms, using Stirling's approximation, is {\displaystyle {\frac {640320^{3{1728; and {\displaystyle \log _{10}({\frac {640320^{3{1728)\approx 14.18}.
  18. ^ Brent, Richard P.; Zimmermann, Paul (2010). Modern Computer Arithmetic. Vol. 18. Cambridge University Press. doi:10.1017/CBO9780511921698. ISBN 978-0-511-92169-8.
Categories Pi algorithms

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