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January 15th, 2012, 02:37 PM   #1
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Additive theory of number and prime twins numbers

Hi all
I propose this conjecture in additive number theory
Consider the n-th 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
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January 15th, 2012, 02:38 PM   #2
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Re: Additive theory of number and prime twins numbers

Quote:
Originally Posted by ibougueye
Hi all
I propose this conjecture in additive number theory
Consider the n-th 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

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 ...
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January 15th, 2012, 02:39 PM   #3
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Re: Additive theory of number and prime twins numbers

Quote:
Originally Posted by ibougueye
Quote:
Originally Posted by ibougueye
Hi all
I propose this conjecture in additive number theory
Consider the n-th 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

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 ...

So basically it is a mathematical induction
We check for the first couple from {17, 19}.
We assume that this is true for the n-th pair
And show that if this is true for the n-th, will be the case for the (n +1)-th pair of twin primes.
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January 15th, 2012, 04:08 PM   #4
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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 Zhi-Wei Sun if you'd like to research this.
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January 16th, 2012, 02:15 AM   #5
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Re: Additive theory of number and prime twins numbers

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Originally Posted by CRGreathouse
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 Zhi-Wei Sun if you'd like to research this.
I attach below a graph showing, according to n (for the 1500 first smaller twins), the number of representations of n-th smallest first of a pair of twins in the first sum of two other (also smaller of a pair of twins) to which is added 1.
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January 16th, 2012, 06:33 AM   #6
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Re: Additive theory of number and prime twins numbers

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Originally Posted by ibougueye
If we prove that the limit of this fonction in infinite is infinit, we can say that the conjectur of triw primes numbers is true.
Actually, that would show that it has only finitely many counterexamples.

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.
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January 18th, 2012, 04:42 PM   #7
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Re: Additive theory of number and prime twins numbers

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Quote:
Originally Posted by ibougueye
If we prove that the limit of this fonction in infinite is infinit, we can say that the conjectur of triw primes numbers is true.
Actually, that would show that it has only finitely many counterexamples.

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.
Good evening all
Jointed a paper abou twin primes and additive theory of numbers
Thanks a lot!
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February 7th, 2012, 03:00 AM   #8
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Re: Additive theory of number and prime twins numbers

Quote:
Originally Posted by ibougueye
Quote:
Originally Posted by CRGreathouse
Quote:
Originally Posted by ibougueye
If we prove that the limit of this fonction in infinite is infinit, we can say that the conjectur of triw primes numbers is true.
Actually, that would show that it has only finitely many counterexamples.

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.
Good evening all
Jointed a revised paper about twin primes and additive theory of numbers
Thanks a lot!
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February 7th, 2012, 03:33 PM   #9
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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 2.5-page 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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