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Re: "Gilmar's Postulate"
*To*: mathgroup at smc.vnet.net
*Subject*: [mg59172] Re: "Gilmar's Postulate"
*From*: "Scout" <not at nothing.net>
*Date*: Sun, 31 Jul 2005 01:30:44 -0400 (EDT)
*References*: <dcccqj$3gq$1@smc.vnet.net>
*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!
>
I hope that this help you!
I've written these 2 short functions:
In[1]:=
MyListOfPrimes[n_]:=Select[Range[IntegerPart[n/4]+1,IntegerPart[n/2]],PrimeQ];
In[2]:=
GoldbachPairs[n_]:=Cases[MyListOfPrimes[n], p_/;PrimeQ[n-p]->{n-p,p}];
Regards,
~Scout~
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