## Journey into PI

### 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$\frac{1}{\pi}=2\sqrt{2}\sum_{k=0}^{\infty}\frac{(4k)!(1103+26390k)}{4{}^{4k}(k!)99{}^{4k+2}}.$