In 1914, the extraordinary mathematician from India, Srinivasa Ramanujan published a set of 14 new formulae ([1]), one of them is famous:
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Each term adds about 8 digits to the result.
In his article, Ramanujan gave very little details of how he found such formulae. To find those relations, he needed modular equations and hypergeometric functions. It's possible to find a complete demonstration of those series in ([2]). In this book the following theorem is given:
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where (aN,bN,gN) are algebraic numbers whenever N is rational and (a)k is the rising factorial : (a)k = a(a+1)(a+2)...(a+k-1). The expression of (aN,bN,gN) as functions of N is rather complex... For example gN is given by :
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where gN is a Ramanujan's invariant.
Ramanujan's formula is given for N = 58, for which g582 = (5+Ö[29])/2 so g58 = 1/992 and a58+kb58 = 2Ö2(1103+ 26390k) ([2], [4]).
Other interesting series are found with (N = 6,10,18,22,58), all given by Ramanujan in his article. Here are the value of gN and gN for the corresponding values of N:
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In the same article, Ramanujan also gave:
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This series adds roughly 2 digits per term. A proof was published in [2].
Much later, a few years ago [3], David and Gregory Chudnovsky found a very powerful series of this type :
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Each term of this series adds about 15 digits...
1985 : Gosper found 17,500,000 decimal digits with formula (*).
1994 : The Chudnovsky brothers found 4,044,000,000 digits with their formula.