September 21st, 2010, 09:14 PM  #1 
Member Joined: Nov 2007 From: New Zealand Posts: 38 Thanks: 0  Twin Prime Conjecture
After sieving with 2 and 3 using the Erathostenes Sieve we get the basic pattern for all twin primes above 5 which is: 57 1113 1719 2325 2931 3537 4143 4749 5355 5961 6567 .... This sequence contains all future primes, but not all of them are primes because they are declared composites when we continue sieving with higher prime numbers. When we sieve with 5 all multiples of 5 are removed from the sequence which means single primes are produced when one number of the above twins is declared composite.From this pattern follows that all primes can be descsribed by 6n1 (low primes) or 6n+1 (high primes) and that the center of a twin prime is multiple of 6. After sieving with a prime number all numbers in the remaining pattern up to the prime square are primes and the numbers above the prime's square are a mixture of composites and primes. There is a special group of twins that cannot be removed in a sieving pattern because the multiples of all primes involved create a composite dividable by 6. Here is an example: Sieving with 2 creates the composite 30 coming from 28 and continuing to 32 Sieving with 3 creates the composite 30 coming from 27 and continuing to 33 Sieving with 5 creates the composite 30 coming from 25 and continuing to 35 This means that there is an infinite number of surviving twins in the sieving pattern of 5 with the centers n*30. In other words  sieving with 2, 3 and 5 cannot remove twins with the center positions 30, 60, 90, 120 and so on. Now we look at higher sieving numbers: When we sieve with 7 twins at the multiples of 2*3*5*7 cannot be removed When we sieve with 11 twins at the multiples of 2*3*5*7 cannot be removed and so on. This means that in each sieving pattern there is an INFINITE number of surviving twins with the center position P(1)*P(2)*P(3)*...P(n) when the latest sieving number is P(n). To turn any of these twins into single primes or to remove them both has to be left to higher sieving numbers. Now we assume that there is a highest number N after which the twin primes stop. Because N is finite and the number of primes is infinite we can always obtain a prime P(n) which square is higher than N and look at the sieving pattern of that prime. Up to P(n)^2 there will be primes and twin primes and AFTER P(n)^2 there will be an INFINITE number of twins with the centers P(1)*P(2)*P(3)*...P(n). It is now the task of higher sieving primes P(m) to turn ALL of these twins into single primes or remove them completely from the sieving pattern of P(n). Now comes the point  This task cannot be carried out by higher sieving numbers, because EACH ONE of them cannot remove the INFINITE number of twins with the centers P(1)*P(2)*P(3)*...P(m) in THEIR OWN sieving pattern. This means that we can set N to any number we like  which in turn means that the number of twin primes must be INFINITE. 
September 21st, 2010, 09:35 PM  #2 
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: Twin Prime Conjecture
Your proof is incorrect. You have proven that, for any N, there are infinitely many pairs (n, n+2) that have no prime divisors smaller than N, but this does not prove that there are infinitely many twin primes.

September 24th, 2010, 08:43 PM  #3 
Member Joined: Nov 2007 From: New Zealand Posts: 38 Thanks: 0  Re: Twin Prime Conjecture
After thinking for a while about your answer I think it is best to divide the full proof into a number of little proofs to find the exact point at which we start to disagree. 1. After sieving with 2 and 3 we get the using the Erathostenes Sieve we get the basic pattern for all twin primes (pairs) above 5 which is: 57 1113 1719 2325 2931 3537 4143 4749 5355 5961 6567 .... Do you agree ? 2. After sieving with 5 we get all primes up to 25 and an ifinite number of pairs with the center n*30=n*2*3*5 above 25. X7 1113 1719 23X _ 2931 5961 8991 119121 .... If I interpret your answer right, you agree with this, but to be sure I want to ask again. Do you agree ? The last twin prime in this pattern is 1719. 3. 1719 can ONLY be the last twin prime when future sieving removes at least one number from ALL pairs with a center n*30 above 25. X7 1113 1719 23X _ 29X 59X 89X 119X .... For simplicity I have just removed the upper ones. Do you agree ? Now we inverse this statement to produce a contradiction like in Euclid's proof that the number of primes is infinite. 4. The pair 1719 CANNOT be the last twin prime when future sieving removes at least one number from ALL pairs with a center n*30 above 25. Do you agree ? 5. We CANNOT remove at least one number from ALL pairs with a center n*30 regardless how high we sieve. Explanation in the original proof. Do you agree ? 6. When we CANNOT remove at least one number from ALL pairs with a center n*30 regardless how high we sieve ANY twin prime CANNOT be last twin prime. Do you agree ? I have spent some days reading through many of your other comments  just to get a feeling how you think. In one of the posts you mentioned that you collect wrong proofs of famous conjectures. We should make this mandatory, because we can learn more from them then from the right ones. I do the same in physics by studying Perpetual Mobiles. I have another proof of the twin prime conjecture which is different to this one. It exploits the property that sieving patterns sieve forward as well as backwards it ends like this. 7. If there exists a sieving pattern which removes all twin primes above N it will also remove all twin primes below N. Do you agree ? My hope is of course that my proofs will not land in your collection. What interests me is how do you decide that they are wrong ? Do you go through every proof yourself or do you just accept what others say ? 
September 25th, 2010, 03:59 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: Twin Prime Conjecture Quote:
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If this means "there are finitely many twin primes only if there is some prime p such that there are only finitely many integers of the form (2 * 3 * 5 * ... * p)n ± 1", then I disagree. If this means something else, please be more precise. Quote:
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September 25th, 2010, 05:58 PM  #5 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Twin Prime Conjecture
FYI Macky, the "C" in CRGreathouse stands for "Chuck Norris".

September 28th, 2010, 01:18 AM  #6  
Member Joined: Nov 2007 From: New Zealand Posts: 38 Thanks: 0  Re: Twin Prime Conjecture Quote:
I am from New Zealand  maybe it is used only in the US.  
September 28th, 2010, 04:57 AM  #7 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Twin Prime Conjecture
Please excuse my failed attempt at humo(u)r! Chuck Norris is something of a lower deity in these parts. 
September 28th, 2010, 07:26 AM  #8  
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: Twin Prime Conjecture Quote:
 
September 28th, 2010, 11:39 AM  #9  
Member Joined: Nov 2007 From: New Zealand Posts: 38 Thanks: 0  Re: Twin Prime Conjecture Quote:
 

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