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Thursday, September 22, 2016

Entry 1

One of the most famous results of Ramanujan is his formula for π as 1π=229801k=0(4k)!k!4297013k+1103(3964)k
Note that eπ58=3964104.00000017
The terms have been factored to show some interesting connections to Pell equations. If we define the fundamental unit U29=5+292, then
(U29)3=70+1329,thus70229132=1
(U29)6=9801+182029,thus980122918202=1
26((U29)6+(U29)6)2=3964
A similar situation happens with a pi formula using, of all things, the golden ratio. The fact that these are so has to do with the Dedekind eta function, but that's another story.

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