# Chudnovsky algorithm

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] and 10 trillion digits of π in October 2011.[4][5]

The algorithm is based on the following rapidly convergent generalized hypergeometric series:[2]

$\frac{1}{\pi} = 12 \sum^\infty_{k=0} \frac{(-1)^k (6k)! (163\cdot 3344418k + 13591409)}{(3k)!(k!)^3 640320^{3k + 3/2}}.\!$

Note that,

$e^{\pi \sqrt{163}} \approx 640320^3+743.99999999999925\dots$

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

## References

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 298242, 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. ^
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.