Wednesday, January 9

A googol in pi

In 1999, Mike Keith worked on a base-26 representation of the number π, wherein the digit 0 is replaced with the letter a, the digit 1 with the letter b, .., the digit 25 with the letter z:

π = d.drsqlolyrtrodnlhnqtgkudqgtuirxneqbckbszivqqvgdmelmuexroiqiyalvuz..

First, I will set up indexing terminology so that there is no misunderstanding about exactly where things are. I am partial (dictatorial, because of my work with continued fractions) to dropping, here, the whole number d and indexing the subsequent, fractional drs.. {1,2,3,..}. Thus the expression lol is (begins) at index 5.

Mike and I looked for large words in the base-26 π expansion and number words of all sizes. In 2000, it took me 127 hours to calculate 5 million terms. My present setup accomplishes it in 13 seconds! So it didn't take me long to up the ante and I can now report that I have found a googol at index 454315613. Already appeared by then are the English number words for 1, 4, 2, 6, 10, 5, 9, 0, 7, 3, 50, 40, 8, 60, 11, 90, 80, 12, 30 (first appearances, respectively at indices 10087, 11324, 13463, 14295, 15276, 64838, 175372, 389247, 786244, 1556763, 2300987, 8879098, 9202330, 9946442, 33027856, 126003234, 126348794, 238426469, 389952198). Here are all twenty first appearances, as one would see them in situ:

..datnmvbwezlfygflonebugwcjooatbjbskh..
..wzlwdhgbhphxeqsqfourdjjmmikzqwvivabm..
..pceoeiigwyqtmaustwoamxyatxhqxoyzbno..
..ifnuetcfdxrwipodsixyavsnasweexagllo..
..ectasqgcghjmzirrtenfjiphdkdoumqzpjt..
..dwbfgegkqjwkubjvfivewgjlbihzmwgalifm..
..jyhdeevmsxmbxtbininesuzrnrzsxosctuxm..
..efzoqitnhuljsgztzerovlptejdfwretaara..
..hwdvoynebabsyjkdsevenaabgjthpqgapqnrg..
..cgakthjgxsmoldokthreeapdiuhncqcjgsmnt..
..gfaoforthfmdhgkzfiftyxiboudqllqzqlcmh..
..obptjczwqdirtnvmfortyigfxwmybvztjkhbc..
..hubksmltneixkvpheightmgsxrwaqokrdcpnw..
..ttsybfashnuibvlisixtytdchrjcvvzazzivu..
..wltescttnomsztonelevenfyifyclxdcacsfta..
..qpddtdyuitfganmgninetyhecteviwajvcosrv..
..kvvtnvsdypwsumzveightysstqdeajfbyxrrde..
..iuwxlmueqjjzpbcmtwelveefcsszipcnmifmzp..
..ssoshdnbsnembdkgthirtyqwbfxohiadwgtvya..
..dnihymgxncwxfhsdgoogolxvodlfjhbzlnrucr..

The word google does not appear in the first half billion terms.

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