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November 13th, 2010, 10:52 AM  #1 
Senior Member Joined: Nov 2010 Posts: 288 Thanks: 1  new conjecture on prime numbers .... i hope
for any prime number p bigger or equal to 13 ;p>=13 there exists at least one k such that both p6k and p+6k are prime numbers for example : if p=13 k=1 coz 136 =7 and 13+6=19 which r both primes p=31 we have two k : k=2 and k=3 coz 3112=19 and 31+12=43 and 3118=13 and 31+18=47 which r all primes actually i checked it for many cases it always worked u can try urself ....... any ideas how to prove this conjecture thanks in advance... 
November 13th, 2010, 12:40 PM  #2  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 933 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: new conjecture on prime numbers .... i hope Quote:
 
November 13th, 2010, 01:29 PM  #3  
Member Joined: Jun 2010 Posts: 71 Thanks: 0  Re: new conjecture on prime numbers .... i hope Quote:
The GC states that every even integer n = 2m can be expressed as the sum of 2 primes ( at least once ) Example: 30 = 11 + 19 ( 30 is a Goldbach number ) Which could be equivalently restated as follows: for every integer m, there exists 2 primes that are equidistant from m ( mk, m+k ), at least once . Example: for m = 15, 11 and 19 are 2 primes equidistant from m ( where n = 2m = 30 is a Goldbach number ). Islam's statement is: for every prime number p, there exists ( at least ) two primes that are equidistant from p, which is a special case ( m is prime ). Example: p = 13, 6 and 19 are 2 primes equidistant from p ( where n = 2p = 26 is a Goldbach number ). Since all primes ( except 2 and 3 ) are of the form 6k + 1, then the distance between any 2 primes will always be 6k  
November 13th, 2010, 06:14 PM  #4  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 933 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: new conjecture on prime numbers .... i hope Quote:
Quote:
 
November 14th, 2010, 01:05 AM  #5  
Member Joined: Jun 2010 Posts: 71 Thanks: 0  Re: new conjecture on prime numbers .... i hope Quote:
That's what I meant. I was referring to Islam's statement.  
November 14th, 2010, 06:44 AM  #6 
Senior Member Joined: Nov 2010 Posts: 288 Thanks: 1  Re: new conjecture on prime numbers .... i hope
well actually that isnt always true if both primes r of the form 6n +1 or both r of the form 6n1 the distance between them will be of the form 6k however if one is of the form 6n+1 while the other of the form 6n1 then the distance will be of the form 6n+2 for example 195 =14 14 isnt of the form 6k

November 14th, 2010, 07:04 AM  #7  
Member Joined: Jun 2010 Posts: 71 Thanks: 0  Re: new conjecture on prime numbers .... i hope Quote:
You started with any prime p >= 13 ( actually it should be p >= 11 ), so either p = 6k + 1 or p = 6k 1 If p = 6k + 1, then p  6k' = 6(kk') + 1 = 6k'' + 1 & p + 6k' = 6(k+k') + 1 = 6k'' + 1 If p = 6k  1, then p  6k' = 6(kk')  1 = 6k''  1 & p + 6k' = 6(k+k')  1 = 6k''  1  

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