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MathGroup Archive 2004

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Question about shape of histogram of minimal prime partition p's

  • To: mathgroup at smc.vnet.net
  • Subject: [mg49440] Question about shape of histogram of minimal prime partition p's
  • From: gilmar.rodriguez at nwfwmd.state.fl.us (Gilmar Rodr?guez Pierluissi)
  • Date: Tue, 20 Jul 2004 07:53:31 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

The following algorithm:

MGPPP[n_] := Module[{p, q},{m = n/2; If[(Element[m, Primes]),
             {(p = m),(q = m)}, {k = PrimePi[m];
              Do[If[Element[(n - Prime[i]), Primes],
              {hit = i, Break[]}], {i, k, 1, -1}],
               p = Prime[hit], q = (n - p)}]}; {p,q}]

calculates the Minimal Goldbach Prime Partition Point corresponding
to n, for n Even, and n >= 4.

Here the word "Minimal" means that, 

(1.) MGPPP[n] = {n/2, n/2}, if n = 2*p, with p a prime, or

(2.) If n =/= 2*p then MGPPP[n] is the point with the shortest
     perpendicular distance to the point {n/2, n/2},
     (here of course, we are assuming that such a point exists;
     since otherwise we would be famous)
     among all prime partition points {p, q}, with 2 =< p < n/2,
     and n/2 < q < (n - 2) resting on the line y = - x + n.

Examples: MGPPP[14] = {7,7} and MGPPP[100]= {47, 53}.

A slight modification of the above algorithm given by:

MGp[n_] := Module[{p},{m = n/2; If[(Element[m, Primes]),
            {(p = m), (q = m)},{k = PrimePi[m];
             Do[If[Element[(n - Prime[i]), Primes],
             {hit = i, Break[]}], {i, k, 1, -1}],
              p = Prime[hit], q = (n - p)}]}; p]

gives the "minimal p" corresponding to n.

Examples: MGp[14] = 7 and MGp[100]= 47.

The first 500,000 minimal p's can be calculated via:
      data = Table[MGp[n],{n, 4, 10^6 , 2}];

If you now call:

<<Graphics`Graphics`

and do:

Histogram[data]

you get an essentially FLAT histogram, with the highest bars
not exceeding a frequency value of 600 (or so).

My questions are: is this histogram correct, and if it is correct
how do you interpret it from a statistical point of view?
Is this an example of n-mode distribution?  Please, elaborate.

Thank you!


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