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] 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 $j\big(\tfrac{1+\sqrt{-163}}{2}\big) = -640320^3$, and on the following rapidly convergent generalized hypergeometric series:^[2]

$\frac{1}{\pi} = 12 \sum^\infty_{k=0} \frac{(-1)^k (6k)! (545140134k + 13591409)}{(3k)!(k!)^3 (640320^3)^{k + 1/2}}.\!$

Note that 545140134 = 163 x 3344418 and,

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

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

See also

• Numerical approximations of π
• 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. 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".