Ramanujan Formulae for Pi

Perhaps this man takes the title for the most mystical of mathematicians: one who would "simply know" that a complex result in number theory is true, and then be able to prove it. An excellent short biography of Ramanujan is available from St Andrews University in Scotland.

There are several formulae for pi (actually the reciprocal of pi) created by Ramanujan which converge extremely quickly. One of these was used by the Chudnovsky brothers, as reported in the "New Yorker" in 1993 or thereabouts, to compute two billion digits from surplus computing equipment hacked together in their apartment in Manhattan.

One of the fomulae is this one:

which produces eight more decimal palces with each term. Incidentally, the large integers are in an arithmetic progression.