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. It was published by the Chudnovsky brothers in 1988.[1]

It was used in the world record calculations of 2.7 trillion digits of π in December 2009,[2] 10 trillion digits in October 2011,[3][4] 22.4 trillion digits in November 2016,[5] 31.4 trillion digits in September 2018–January 2019,[6] 50 trillion digits on January 29, 2020,[7] 62.8 trillion digits on August 14, 2021,[8] and 100 trillion digits on March 21, 2022.[9]


The algorithm is based on the negated Heegner number , the j-function , and on the following rapidly convergent generalized hypergeometric series:[2]

A detailed proof of this formula can be found here:[10]

There are 3 big integer terms (the multinomial term Mq, the linear term Lq, and the exponential term Xq) that make up the series and π equals the constant C divided by the sum of the series, as below:

, where:

The terms Mq, Lq, and Xq satisfy the following recurrences and can be computed as such:

The computation of Mq can be further optimized by introducing an additional term Kq as follows:

Note that


This identity is similar to some of Ramanujan's formulas involving π,[2] and is an example of a Ramanujan–Sato series.

The time complexity of the algorithm is .[11]

See also[edit]


  1. ^ Chudnovsky, David; Chudnovsky, Gregory (1988), Approximation and complex multiplication according to ramanujan, Ramanujan revisited: proceedings of the centenary conference
  2. ^ a b c 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
  3. ^ 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
  4. ^ Aron, Jacob (March 14, 2012), "Constants clash on pi day", New Scientist
  5. ^ "22.4 Trillion Digits of Pi".
  6. ^ "Google Cloud Topples the Pi Record".
  7. ^ "The Pi Record Returns to the Personal Computer".
  8. ^ "Pi-Challenge - Weltrekordversuch der FH Graubünden - FH Graubünden". Retrieved 2021-08-17.
  9. ^ "Calculating 100 trillion digits of pi on Google Cloud". Retrieved 2022-06-10.
  10. ^ Milla, Lorenz (2018), A detailed proof of the Chudnovsky formula with means of basic complex analysis, arXiv:1809.00533
  11. ^ "y-cruncher - Formulas". Retrieved 2018-02-25.