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

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

Note that 545140134 = 163 x 3344418 and,

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

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