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"Gilmar's Postulate"

• To: mathgroup at smc.vnet.net
• Subject: [mg59120] "Gilmar's Postulate"
• From: "Gilmar" <gilmar.rodriguez at nwfwmd.state.fl.us>
• Date: Fri, 29 Jul 2005 00:42:02 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

There is a beautiful article appearing in the June-July 2005
issue of the Americam Mathematical Monthly Journal,
(Volume 112, Number 6, page 492), entitled:
" Goldbach's Conjecture implies Bertrand's Postulate",
submitted by Henry J. Ricardo and Yoshihiro Tanaka.
You can download the article via:

http://gilmarlily.netfirms.com/download/maajunjul.jpg

After reading it, one is immediately tempted to test the
converse; namely: Does Bertrand's Postulate imply
Goldbach's Conjecture?  After toying with this question
for a while, I came up with the following assertion
(which I will call for lack of a better name):

"Gilmar's Postulate":

For every positive integer n > 1, there exists a prime p such that
n/4 < p <= n/2.  Moreover; Gilmar's Postulate implies Goldbach's
Conjecture.

I immediately proceeded to get my facts empirically as follows:

n = 4 ; (1 < p <=2) => p=2;  (n - p) = 2, and so 4 = 2 + 2.

n = 6; (1.5 < p <=3) => p = 2, 3; (n - 2) = 0, but (n - 3) = 3,
                                              and so 6 = 3 + 3.

n = 8; (2 < p <=4) => p = 3; (n - 3) = 5, and so 8 = 3 + 5.

n = 10; (2.5 < p <= 4) => p = 3, 5; (n - 3)=7, and so 10 = 3 + 7.
                                    (n - 5) = 5, 10 = 5 + 5.

n = 12; (3 < p <= 6) => p = 5; n - 5 = 7, and so 12 = 5 + 7.

n = 14; (3.5 < p <= 7) => p = 5, 7; (n - 5) = 9 is not prime, but
                                    (n - 7) = 7, and so 14 = 7 + 7.

n = 16; (4 < p <= 8) => p = 5, 7; (n - 5) = 11, and so 16 = 5 + 11.
                                  (n - 7) = 9 is not prime.

Etc.  You get the picture.  It seems that Gilmar's postulate implies
the existence of at least one Goldbach pair {p, n-p}.  Of course;
I want to gather more evidence, and this is where I need your help.

What I need is a program that takes an integer n => 4 and:

(1.) looks into the interval n/4 < p <= n/2, to determine what the
     prime p-sub-i's belonging to this interval might be and,

(2.) makes an assesment of whether the pairs {pi, n - pi}
      are Goldbach pairs or not, and produces a message
      like: "The Goldbach pairs corresponding to n
      are: {p1, n-p1}, {p2, n - p2}... The Regular pairs are:
      {p3, m1}, {p, m2},..."

By "Regular pairs" I mean those pairs for which the m-sub-i's are
composite.  Thank you for your help!