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maybe

one should add this :There are infinitely many twin primes if and only if there are infinitely many natural numbers that are not of the form 6nm+/-n +/-m. Proof: Every number that is not a multiply of 2 or 3 is of the form 6N+/-1. So the only couples that are not divisible by 2 or 3 are (6N-1,6N+1) for any N.Now are there infinitely many such prime couples (twin primes)?. If the number 6N-1 is prime it should not be written as a product of some numbers 6n+1 ,6m-1 for any n,m. So (6n+1)(6m-1)=6(6nm-n+m)-1 which means that N should not be of the form 6nm-n+m for any n,m. Similarly if 6N+1 is a prime it should not be a product of some numbers (6n-1)(6m-1) =6(6nm-n-m)+1 , or (6n+1)(6m+1) =6(6nm+n+m)+1 .Which means that we have a prime couple of the form (6N-1,6N+1) if and only if N is not of the form 6nm+/-n+/-m for any n,m. http://mathoverflow.net/questions/49647/finite-or-infinite-closed
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maybe one should add this :There are infinitely many twin primes if and only if there are infinitely many natural numbers that are not of the form 6nm+/-n +/-m. Proof: Every number that is not a multiply of 2 or 3 is of the form 6N+/-1. So the only couples that are not divisible by 2 or 3 are (6N-1,6N+1) for any N.Now are there infinitely many such prime couples (twin primes)?. If the number 6N-1 is prime it should not be written as a product of some numbers 6n+1 ,6m-1 for any n,m. So (6n+1)(6m-1)=6(6nm-n+m)-1 which means that N should not be of the form 6nm-n+m for any n,m. Similarly if 6N+1 is a prime it should not be a product of some numbers (6n-1)(6m-1) =6(6nm-n-m)+1 , or (6n+1)(6m+1) =6(6nm+n+m)+1 .Which means that we have a prime couple of the form (6N-1,6N+1) if and only if N is not of the form 6nm+/-n+/-m for any n,m.so if i am not wrong you can have a road of conditioned ,less strong types of the twin prime conjecture and why not a way to the conjecture.for a start you can have a look to my first question and ask the commentator Pete Ln,m. Clark for a proof for a=100 http://mathoverflow.net/questions/49647/finite-or-infinite-closed

i am working in generalizations for primes who have distance 4 etc.
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maybe one should add this :There are infinitely many twin primes if and only if there are infinitely many natural numbers that are not of the form 6nm+/-n +/-m. Proof: Every number that is not a multiply of 2 or 3 is of the form 6N+/-1. So the only couples that are not divisible by 2 or 3 are (6N-1,6N+1) for any N.Now are there infinitely many such prime couples (twin primes)?. If the number 6N-1 is prime it should not be written as a product of some numbers 6n+1 ,6m-1 for any n,m. So (6n+1)(6m-1)=6(6nm-n+m)-1 which means that N should not be of the form 6nm-n+m for any n,m. Similarly if 6N+1 is a prime it should not be a product of some numbers (6n-1)(6m-1) =6(6nm-n-m)+1 , or (6n+1)(6m+1) =6(6nm+n+m)+1 .Which means that we have a prime couple of the form (6N-1,6N+1) if and only if N is not of the form 6nm+/-n+/-m for any n,mn,m.so if i am not wrong you can have a road of conditioned ,less strong types of the twin prime conjecture and why not a way to the conjecture.for a start you can have a look to my first question and ask the commentator Pete L. Clark for a proof for a=100 http://mathoverflow.net/questions/49647/finite-or-infinite-closed

i am working in generalizations for primes who have distance 4 etc.http://mathoverflow.net/questions/49730/twin-primes-etc-closed
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