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~