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MathGroup Archive 2004

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Follow up: Help wanted ... bounding function is pierced for n even > 10^7.

  • To: mathgroup at smc.vnet.net
  • Subject: [mg49733] Follow up: Help wanted ... bounding function is pierced for n even > 10^7.
  • From: gilmar.rodriguez at nwfwmd.state.fl.us (Gilmar Rodr?guez Pierluissi)
  • Date: Thu, 29 Jul 2004 07:45:42 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Thank you Mathematica Group for all your excellent feedback through e-mail!
Keep in mind that I welcome comments long after the date when this message
was posted.

(1.) Many of you pointed out that the first sentence in my original
presentation should read as:

"The function B[n] = (360.874) / (Exp[2 / Log[n]] - 1) is a bounding
functionfor all Minimal Goldbach Prime Partition Points ("MGPPP's"
for short),for all even integers n between (and including) 4 and 10^7"

i.e. not 10^8 and this was indeed a typing error.  Sorry!

(2.) Others pointed out that one spike seems to pierce the bounding
function in the last plot (i.e. Show[Plt3,Plt1]) near the (vertical)
line N = 2*10^6. This is only an optical illusion caused by the way
that we built Plt1 and Plt3, and put them together, without using
a lot of computer memory.

To convince yourself that the data is well separated from its bounding
function (for large even N) please, either:

(1.) download and evaluate the following notebook
found in:
http://gilmarlily.netfirms.com/download/data&bf.nb
or
(2.) add the following input lines to the original notebook:
http://gilmarlily.netfirms.com/download/bf(4,10to7).nb

In[14]: B[n_] = (360.874) / (Exp[2 / Log[n]] - 1)

In[15]: bfdata = Table[BF[data[[i]][[1]]], {i, 1, Length[data]}];

In[16]: Plt4=ListPlot[bfdata, AspectRatio -> 1\/GoldenRatio,
 PlotJoined -> True, PlotLabel ->"MINIMAL GOLDBACH PRIME
 PARTITION POINTS FOR EVEN INTEGERS N BETWEEN 4 and 10^7
 (DEPICTED IN RED).", FrameLabel -> {"BOUNDING FUNCTION BF[N]
  =360.874/(Exp[2/Log[N]] - 1)","BOUNDING FUNCTION (DEPICTED IN BLUE).",
  "PLANE ROTATED CLOCKWISE BY AN ANGLE THETA = PI/4 RADIANS ABOUT
  THE ORIGIN."}, Frame -> True, PlotStyle -> {Hue[.7],{RGBColor[1,1,0]},
  Thickness[.001]},Background -> RGBColor[1,1,0], ImageSize -> 800,
  PlotRange -> All]

In[17]: Show[Plt1,Plt4]

If you just want to take a quick look at the plot without
evaluating neither notebook, please visit:
http://gilmarlily.netfirms.com/download/data&bf.htm

Notice that the the data set (in red) is well separated from
its bounding function(in blue) for large (even) N.  Building
Plt4 increased the size of my Mathematica notebook to a whopper
332209 KB's, and I this is why I didn't include inputs[14]
to [17] in my original presentation.  Thanks again!
P.S.  I still have not received an e-mail from anyone claiming
that the bounding function was pierced for N > 10^7.


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