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. Published by the Chudnovsky brothers in 1988,cite-ref-1[1] it was used to calculate π to a billion decimal places.cite-ref-2[2]
It was used in the world record calculations of 2.7 trillion digits of π in December 2009,cite-ref-3[3] 10 trillion digits in October 2011,cite-ref-4[4]cite-ref-5[5] 22.4 trillion digits in November 2016,cite-ref-6[6] 31.4 trillion digits in September 2018–January 2019,cite-ref-7[7] 50 trillion digits on January 29, 2020,cite-ref-8[8] 62.8 trillion digits on August 14, 2021,cite-ref-9[9] 100 trillion digits on March 21, 2022,cite-ref-10[10] 105 trillion digits on March 14, 2024,cite-ref-11[11] and 202 trillion digits on June 28, 2024.cite-ref-12[12] Recently, the record was broken yet again on April 2nd 2025 with 300 trillion digits of pi.cite-ref-13[13]cite-ref-14[14] This was done through the usage of the algorithm on y-cruncher.
Contents
• See also
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Algorithm
The algorithm is based on the negated Heegner number d = − − 163 {\displaystyle d=-163} , the j-function j ( 1 + i − − 163 2 ) = − − 640320 3 {\displaystyle j\left({\tfrac {1+i{\sqrt {-163}}}{2}}\right)=-640320^{3}} , and on the following rapidly convergent generalized hypergeometric series:cite-ref-baruah-15-0[15] 1 π π = 12 ∑ ∑ k = 0 ∞ ∞ ( − − 1 ) k ( 6 k ) ! ( 545140134 k + 13591409 ) ( 3 k ) ! ( k ! ) 3 ( 640320 ) 3 k + 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,cite-ref-baruah-15-1[15] and is an example of a Ramanujan–Sato series.
The time complexity of the algorithm is O ( n ( log n ) 3 ) {\displaystyle O\left(n(\log n)^{3}\right)} , where n is the number of digits desired.cite-ref-16[16]
Optimizations
The optimization technique used for the world record computations is called binary splitting.cite-ref-17[17]
See also
References
cite-note-11. ↑ citerefchudnovskychudnovsky1988Chudnovsky, David; Chudnovsky, Gregory (1988), Approximation and complex multiplication according to Ramanujan, Ramanujan revisited: proceedings of the centenary conference
cite-note-22. ↑ citerefwarsidangerfieldfarndongriffiths2019Warsi, 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.
cite-note-55. ↑ citerefaron2012Aron, Jacob (March 14, 2012), "Constants clash on pi day", New Scientist
cite-note-66. ↑ "22.4 Trillion Digits of Pi". www.numberworld.org.
cite-note-77. ↑ "Google Cloud Topples the Pi Record". www.numberworld.org/.
cite-note-88. ↑ "The Pi Record Returns to the Personal Computer". www.numberworld.org/.
cite-note-99. ↑ "Pi-Challenge - Weltrekordversuch der FH Graubünden - FH Graubünden". www.fhgr.ch. Retrieved 2021-08-17.
cite-note-1010. ↑ "Calculating 100 trillion digits of pi on Google Cloud". cloud.google.com. Retrieved 2022-06-10.
cite-note-1111. ↑ citerefyee2024Yee, Alexander J. (2024-03-14). "Limping to a new Pi Record of 105 Trillion Digits". NumberWorld.org. Retrieved 2024-03-16.
cite-note-1212. ↑ citerefranous2024Ranous, Jordan (2024-06-28). "StorageReview Lab Breaks Pi Calculation World Record with Over 202 Trillion Digits". StorageReview.com. Retrieved 2024-07-20.
cite-note-1313. ↑ "News (2024)". www.numberworld.org. Retrieved 2025-05-16.
cite-note-1414. ↑ citereflinus-tech-tips2025Linus Tech Tips (2025-05-16). This World Record took YEARS (and a Million dollars..). Retrieved 2025-05-16 – via YouTube.
cite-note-1616. ↑ "y-cruncher - Formulas". www.numberworld.org. Retrieved 2018-02-25.
cite-note-1717. ↑ citerefbrentzimmermann2010Brent, Richard P.; Zimmermann, Paul (2010). Modern Computer Arithmetic. Vol. 18. Cambridge University Press. doi:10.1017/CBO9780511921698. ISBN 978-0-511-92169-8.