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September 21st, 2010, 09:14 PM   #1
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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:

5-7 11-13 17-19 23-25 29-31 35-37 41-43 47-49 53-55 59-61 65-67 ....

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 6n-1 (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.
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September 21st, 2010, 09:35 PM   #2
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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.
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September 24th, 2010, 08:43 PM   #3
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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:

5-7 11-13 17-19 23-25 29-31 35-37 41-43 47-49 53-55 59-61 65-67 ....

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.

X-7 11-13 17-19 23-X _ 29-31 59-61 89-91 119-121 .... 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 17-19.

3. 17-19 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.

X-7 11-13 17-19 23-X _ 29-X 59-X 89-X 119-X .... 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 17-19 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 ?
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September 25th, 2010, 03:59 PM   #4
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Re: Twin Prime Conjecture

Quote:
Originally Posted by Macky
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:

5-7 11-13 17-19 23-25 29-31 35-37 41-43 47-49 53-55 59-61 65-67 ....
Assuming this means "there are infinitely many pairs of integers of the form 6n ± 1, and none of these are divisible by 2 or 3", then I agree.

Quote:
Originally Posted by Macky
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.

X-7 11-13 17-19 23-X _ 29-31 59-61 89-91 119-121 .... If I interpret your answer right, you agree with this, but to be sure I want to ask again.
Assuming this means "there are infinitely many pairs of integers of the form 30n ± 1, and none of these are divisible by 2, 3, or 5", then I agree.

Quote:
Originally Posted by Macky
3. 17-19 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.

X-7 11-13 17-19 23-X _ 29-X 59-X 89-X 119-X .... For simplicity I have just removed the upper ones.
If this means "(17, 19) is the highest twin prime pair implies X" for some statement X, then I agree, since there are higher twin prime pairs. For example, "(17, 19) is the highest twin prime pair implies unicorns exist" is true.

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:
Originally Posted by Macky
4. The pair 17-19 CANNOT be the last twin prime when future sieving removes at least one number from ALL pairs with a center n*30 above 25.
This seems to be the same as #3.

Quote:
Originally Posted by Macky
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.
If this means "for any prime p, there are infinitely many integers of the form (2 * 3 * 5 * ... * p)n ± 1", then I agree.

Quote:
Originally Posted by Macky
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.
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.

Quote:
Originally Posted by Macky
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.
Post away -- ideally in a different thread. I've seen this 'proof' so many times from so many people I'm rather tired of it. A new, different 'proof' would be refreshing.

Quote:
Originally Posted by Macky
7. If there exists a sieving pattern which removes all twin primes above N it will also remove all twin primes below N.
I don't really know what this means, but I suspect that if I could get you to make it precise it would be false for the same reason that your original proof fails. But feel free to make it more precise, in which case I of course reserve the right to agree with it.

Quote:
Originally Posted by Macky
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 ?
I go through every proof myself. Occasionally I will also read others' responses to proofs to help me locate the mistakes, but typically I just find them on my own.
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September 25th, 2010, 05:58 PM   #5
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Re: Twin Prime Conjecture

FYI Macky, the "C" in CRGreathouse stands for "Chuck Norris".
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September 28th, 2010, 01:18 AM   #6
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Re: Twin Prime Conjecture

Quote:
Originally Posted by The Chaz
FYI Macky, the "C" in CRGreathouse stands for "Chuck Norris".
I have no idea what this synonym stands for ! Reading about Chuck Norris wasn't of any help either !

I am from New Zealand - maybe it is used only in the US.
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September 28th, 2010, 04:57 AM   #7
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Re: Twin Prime Conjecture

Please excuse my failed attempt at humo(u)r!
Chuck Norris is something of a lower deity in these parts.
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September 28th, 2010, 07:26 AM   #8
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Re: Twin Prime Conjecture

Quote:
Originally Posted by Macky
Quote:
Originally Posted by The Chaz
FYI Macky, the "C" in CRGreathouse stands for "Chuck Norris".
I have no idea what this synonym stands for ! Reading about Chuck Norris wasn't of any help either !

I am from New Zealand - maybe it is used only in the US.
See, e.g., http://knowyourmeme.com/memes/chuck-norris-facts
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September 28th, 2010, 11:39 AM   #9
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Re: Twin Prime Conjecture

Quote:
Originally Posted by The Chaz
Please excuse my failed attempt at humo(u)r!
Chuck Norris is something of a lower deity in these parts.
You can't be right, because the Bill Gates Theorem shows that we have to click START to stop.
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