### Records on computing the value of \(\pi\)

Up to the moment these lines were
written (year 2000),
the record for computing decimal places of \(\pi\) was obtained by **Takahashi** and **Kanada** on September 20, 1999.

For this, a supercomputer with 128 parallel processors **HITACHI SR8000**
from the Information Technology Center, Computer Centre Division from the University
of Tokyo was used.

For the calculation it was used two different algorithms which generated **\(3\times2^{36}
= 206 \,158 \,430 \,208\)** decimal digits.

The main program used the **Gauss**-**Legendre** algorithm,
required 865GB of memory and spent **37h 21m and 4s**
to complete the computations.

The verification program used the 4th order
**Borwein** algorithm, required 817GB of memory and completed the calculations after **46h 7m and 10s**.

Comparing the generated sequences, it was found that these coincide to the \(206 \,158 \,430 \,163\) significant digits, differing only in the last \(45\) digits.
The new record with **\(206 \,158 \,430 \,000\)** significant digits for the value of \(\pi\) was then announced.

The value of \(\pi\) presented
at Matemática Viva display module had
**\(1 \,073 \,741\, 000\)** significant digits.

For its computation the program PiFast, version
3.2, from **Xavier Gourdon** was used and the process ran on a computer Pentium II
400Mhz, with 256MB of memory and 20GB of dedicated hard disk space. It computed **\(2^{30}=1\, 073 \,741 \,824\)**
digits of \(\pi\)
by the **Chudnovsky** brothers algorithm and it took **2d
13h 18m 5.64s**.

Several other unsuccessful attempts were also made, until we achieved a value for \(\pi\)
with **\(2\, 147\, 483 \,000\)** digits.

In this case the 3.3 version of PiFast program was used. With the aid of a Pentium III 600Mhz with 256MB of memory and 30GB of dedicated hard disk space,
**\(2^{31}=2\, 147\, 483\, 648\)** digits were computed in
**5d 1h 41m 38.09s**.

For this computation the **Chudnovsky** brothers algorithm was also used, which is based on the following formula, known by **Chudnovsky** formula\[\frac{426880\sqrt{10005}}{\pi}=\sum_{k=0}^{\infty}\frac{(6k)!(545140134k+13591409)}{(k!)^{3}(3k!)(-640320)^{3k}}.\]

The same program also allows to compute \(\pi\) using the expression of **Ramanujan**\[\frac{1}{\pi}=2\sqrt{2}\sum_{k=0}^{\infty}\frac{(4k)!(1103+26390k)}{4{}^{4k}(k!)99{}^{4k+2}}.\]