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The Chudnovsky algorithm is a fast method for calculating the digits of π, based on Ramanujan's π formulae.Published by the Chudnovsky brothers in 1988, [1] it was used to calculate π to a billion decimal places.
These are counted by the double factorial 15 = (6 − 1)‼. In mathematics, the double factorial of a number n, denoted by n‼, is the product of all the positive integers up to n that have the same parity (odd or even) as n. [1] That is,
It produces about 14 digits of π per term [134] and has been used for several record-setting π calculations, including the first to surpass 1 billion (10 9) digits in 1989 by the Chudnovsky brothers, 10 trillion (10 13) digits in 2011 by Alexander Yee and Shigeru Kondo, [135] and 100 trillion digits by Emma Haruka Iwao in 2022. [136]
is the number of collisions made (in ideal conditions, perfectly elastic with no friction) by an object of mass m initially at rest between a fixed wall and another object of mass b 2N m, when struck by the other object. [1] (This gives the digits of π in base b up to N digits past the radix point.)
Start by setting [4] = = = + Then iterate + = + + = (+) + + = (+ +) + + + Then p k converges quadratically to π; that is, each iteration approximately doubles the number of correct digits.The algorithm is not self-correcting; each iteration must be performed with the desired number of correct digits for π 's final result.
Every sequence of digits, in any base, is the sequence of initial digits of some factorial number in that base. [ 60 ] Another result on divisibility of factorials, Wilson's theorem , states that ( n − 1 ) ! + 1 {\displaystyle (n-1)!+1} is divisible by n {\displaystyle n} if and only if n {\displaystyle n} is a prime number . [ 52 ]
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In 1995, the Borweins collaborated with Yasumasa Kanada of the University of Tokyo to compute π to more than four billion digits.. Borwein has developed an algorithm that applies Chebyshev polynomials to the Dirichlet eta function to produce a very rapidly convergent series suitable for high precision numerical calculations, which he published on the occasion of the awarding of an honorary ...