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Recently on the Marilyn vos Savant discussion board:
From robert 46:
Calculating a new projection of w(x) to 10^20, I get a "c" close to Don's modified "c" and much closer to JeffJo's, and a smaller difference which does not go negative until 10^20. However, this is entirely contingent on getting the exact w(x) for the extended range to confirm because my method is very stylized to a first approximation.
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10^20 proj. c=0.640362749200832756806961203683 d=0.25
x w(x) Me: B(x)=((c*x)^0.5-d)^2=
0.64036274920083*x-0.40011334306694*x^0.5
Don: B(x)=0.64036275065348*x-0.40011254844227*x^0.5
10^01 3 5 Me 2 Don 5
10^02 57 60 3 60
10^03 622 627 5 628
10^04 6357 6363 6 6364
10^05 63889 63909 20 63910
10^06 639946 639962 16 639963
10^07 6402325 6402362 37 6402362
10^08 64032121 64032273 152 64032274
10^09 640349979 640350096 117 640350098
10^10 6403587409 6403587480 71 6403587495
10^11 64036148166 64036148393 227 64036148539
New w(x) projection can have cumulative errors
12 640362348368 640362349087 719
Don 640362350541 2173
13 6403626224466 6403626226738 2272
Don 6403626241270 16804
14 64036270911767 64036270918949 7182 1.1215e-8%
15 640362736525473 640362736548137 22664 3.5392e-9%
16 6403627451925812 6403627451996993 71181 1.1115e-9%
17 64036274793336150 64036274793556327 220177 3.4383e-10%
18 640362748800072316 640362748800719413 647097 1.0105e-10%
19 6403627490741503411 6403627490743058081 1554670 2.4277e-11%`
20 64036274916082142250 64036274916082142250 0 0 %
proj. w(x) old new old-new
10^12 640362357070 640362348368 8702
10^13 6403626301352 6403626224466 76886
My previous projections for w(10^12) and w(10^13) appear to have been too high.
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Interval, c=
09-10, 0.64036279308256413820600614015348
10-11, 0.64036274919577461010703035810577
11-12, 0.64036274919923682886172155070800
12-13, 0.64036274920120138100322749426044
13-14, 0.64036274920090069952061219210571
14-15, 0.64036274920090499181953768636223
15-16, 0.64036274920090464640446214680386
16-17, 0.64036274920090468133015417893125
17-18, 0.64036274920090465700536517390538
18-19, 0.64036274920090465678836848025334
19-20, 0.64036274920090465649018966009506
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