Chudnovsky algorithm

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The Chudnovsky algorithm is a fast method for calculating the digits of π. It was published by the Chudnovsky brothers in 1989,[1] and was used in the world record calculations of 2.7 trillion digits of π in December 2009,[2] 5 trillion digits of π in August 2010,[3] 10 trillion digits of π in October 2011,[4][5] and 12.1 trillion digits in December 2013.[6]

The algorithm is based on the negated Heegner number d = −163, the j-function j(1+−163/2) = −6403203, and on the following rapidly convergent generalized hypergeometric series:[2]

Note that 545140134 = 163 × 3344418 and,

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

See also[edit]


  1. ^ Chudnovsky, David V.; Chudnovsky, Gregory V. (1989), "The Computation of Classical Constants", Proceedings of the National Academy of Sciences of the United States of America, 86 (21): 8178–8182, doi:10.1073/pnas.86.21.8178, ISSN 0027-8424, JSTOR 34831, PMC 298242free to read, PMID 16594075 
  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. ^ Geeks slice pi to 5 trillion decimal places, Australian Broadcasting Corporation, August 6, 2010 
  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 
  5. ^ Aron, Jacob (March 14, 2012), "Constants clash on pi day", NewScientist 
  6. ^ Alexander J. Yee; Shigeru Kondo (28 December 2013). "12.1 Trillion Digits of Pi".