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January 15th, 2012, 02:37 PM  #1 
Newbie Joined: Jan 2012 Posts: 26 Thanks: 0  Additive theory of number and prime twins numbers
Hi all I propose this conjecture in additive number theory Consider the nth pair of twin primes. I conjecture that for the smallest element of (n +1) th pairs of twin primes simply combine two by two the smallest elements of the first n pairs of twin primes and adding the unit (ie 1). At least one combination matches. Someone there against an example? AD can we afford to continue here? Although all God bless you all 
January 15th, 2012, 02:38 PM  #2  
Newbie Joined: Jan 2012 Posts: 26 Thanks: 0  Re: Additive theory of number and prime twins numbers Quote:
My approach is to "build" the first twins from the first three pairs {3, 5}, {5, 7} and {11.13}. It is noted that from 17, the smallest element of any pair of twin primes is equal to the sum (a + b +1) such that: a is the smallest element of a pair of twin primes b is the smallest element of another pair of twin primes examples: 17 = 11 +5 +1, 29 = 17 +11 +1, 41 = 29 +11 +1, 59 = 41 +17 +1, 71 = 59 +11 +1 = 41 +29 +1 etc ...  
January 15th, 2012, 02:39 PM  #3  
Newbie Joined: Jan 2012 Posts: 26 Thanks: 0  Re: Additive theory of number and prime twins numbers Quote:
So basically it is a mathematical induction We check for the first couple from {17, 19}. We assume that this is true for the nth pair And show that if this is true for the nth, will be the case for the (n +1)th pair of twin primes.  
January 15th, 2012, 04:08 PM  #4 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Additive theory of number and prime twins numbers
As best I can tell, you're suggesting that for each member p of A001359 greater than or equal to 17, there are two smaller members q and r such that p = q + r + 1. This is almost certain to be true, but it's unlikely to be provable within current technology. I don't know who was the first to conjecture this; see A152126 and the papers of ZhiWei Sun if you'd like to research this.

January 16th, 2012, 02:15 AM  #5  
Newbie Joined: Jan 2012 Posts: 26 Thanks: 0  Re: Additive theory of number and prime twins numbers Quote:
 
January 16th, 2012, 06:33 AM  #6  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Additive theory of number and prime twins numbers Quote:
Also remember that it's not enough to show that the lim sup is infinite  you need the lim inf to even get the result above.  
January 18th, 2012, 04:42 PM  #7  
Newbie Joined: Jan 2012 Posts: 26 Thanks: 0  Re: Additive theory of number and prime twins numbers Quote:
Jointed a paper abou twin primes and additive theory of numbers Thanks a lot!  
February 7th, 2012, 03:00 AM  #8  
Newbie Joined: Jan 2012 Posts: 26 Thanks: 0  Re: Additive theory of number and prime twins numbers Quote:
 
February 7th, 2012, 03:33 PM  #9 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Additive theory of number and prime twins numbers
If you're looking for comments on the paper: References should not be translated. Brun's paper was published in French, so the title should remain " "La série 1/5+1/7+1/11+1/13+1/17+1/19+1/29+1/31+1/41+1/43+1/59+1/61+..., où les dénominateurs sont nombres premiers jumeaux est convergente ou finie". Similarly with Clement's paper  I assume the original is French since the title you give contains a common French > English translation error. You spend far too long on preliminaries, 2 pages out of a 2.5 page paper. A 2.5page paper should have at most half a page on introduction, summary, and abstract. At the moment the paper consists of your conjecture and a statement that you have not found any counterexamples (that's the term in English, by the way: "counterexample" not "example against"). Can you add more material? 

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