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September 24th, 2013, 11:48 PM  #1 
Newbie Joined: Sep 2013 Posts: 2 Thanks: 0  Something about prime numbers
I made this sieve recently and found some really neat deatils about it. Perfect numbers line up in only one column, as do mersenne primes (its primarly beacuse powers of 2 line up in only 2 columns) Additionally, if higher primes also appear only on 5, 7, 11 and 13 column then it makes twin prime conjecture kind of "visible"/ Can i have your opinion about this ? 
September 25th, 2013, 01:11 AM  #2  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Something about prime numbers Quote:
 
September 25th, 2013, 01:28 AM  #3 
Newbie Joined: Sep 2013 Posts: 2 Thanks: 0  Re: Something about prime numbers
there are two pairs of columns seperated by only one column, that means whenever 2 primes appear in one verse chance is they are going to be twin primes, and if there is infinite amount of prime numbers seeded semi random, they are going to appear in one verse infinitly many times

September 25th, 2013, 04:18 AM  #4 
Senior Member Joined: Mar 2012 Posts: 572 Thanks: 26  Re: Something about prime numbers
You get a similar pattern for any similar array of columns where the number of columns is a primorial or multiple of a primorial. For instance, you get a cleaner version of this array if you use 6 columns (2x3), and a bigger one if you use 30 (2x3x5) or 210 (2x3x5x7). The reasons the primes line up is pretty basic, primes above 2 have to be odd, and can't be 6n+3 (which will always be a multiple of 3 and thus composite > 3), so must be a 6n+1 or 6n+5 number. It doesn't tell us a great deal about the twin prime conjecture other than that it is possible that it is true, so isn't really telling us anything that isn't obvious. 
September 25th, 2013, 04:37 AM  #5 
Senior Member Joined: Mar 2012 From: Belgium Posts: 654 Thanks: 11  Re: Something about prime numbers
The only thing to I can see in that table is that all primes greater than 3 are of the form 5+12k,7+12k,11+12k and 13+12k This can easily be proven. All numbers greater than 1 can be considered as this: 2+12k,3+12k,4+12k,5+12k,6+12k,7+12k,8+12k,9+12k,10 +12k,11+12k,12+12k,13+12k 2+12k is always divisible by 2 and is not a prime. 3+12k is always divisible by 3 and is not a prime. 4+12k is always divisible by 2 and is not a prime. 6+12k is always divisible by 2 and is not a prime. 8+12k is always divisible by 2 and is not a prime. 9+12k is always divisible by 3 and is not a prime. 10+12k is always divisible by 2 and is not a prime. 12+12k is always divisible by 2 and is not a prime. Conclusion all primes are of the form 5+12k,7+12k,11+12k and 13+12k. You made your table such that all those numbers are lined under each other so there is nothing special about that. 
September 25th, 2013, 08:45 AM  #6  
Senior Member Joined: Mar 2012 Posts: 572 Thanks: 26  Re: Something about prime numbers Quote:
Similarly all primes > 5 are of the form 30k+1, +7, +11, +13, +17, +19, or +29. And as a result all twin primes > 7 are of the form 30k+29,+1 30k+11,+13 or 30k +17,+19. And you can go on narrowing down if you look at residues mod 210, 2310 or whatever.  

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