Loading [MathJax]/jax/output/HTML-CSS/jax.js

Friday, September 23, 2016

Entry 10

The Chudnovsky algorithm is 1π=12k=0(1)k(6k)!k!3(3k)!(163×3344418k+13591409)(6403203)k+1/2 Recall the famous eπ163=6403203+743.99999999999925 The formula (1) was inspired by Ramanujan's work and is used to calculate world records for the digits of pi. Now why didn't he discover this? Actually, he almost did. In his list of 17 formulas, there were two that belong to this family, one of which is 1π=1838585n=0(12)n(16)n(56)nn!3(133n+8)(485)3n where (a)n is a Pochhammer symbol. However, this can be translated into the form of (1) namely 1π=162k=0(6k)!k!3(3k)!(133k+8)(2553)k+1/2 Similarly,eπ28=2553744.01 where the "excess" 744 indicates that the j-function is involved. In fact, in Ramanujan's Lost Notebook, he had calculations involving Eisenstein series using the primes d=11,19,43,67,163 which is precisely the family which (1) belongs to. Thus, if only Ramanujan lived longer, he would almost have surely found the Chudnovsky algorithm.

No comments:

Post a Comment