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September 24th, 2013, 11:48 PM   #1
ogh
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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 ?
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September 25th, 2013, 01:11 AM   #2
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Re: Something about prime numbers

Quote:
Originally Posted by ogh
if higher primes also appear only on 5, 7, 11 and 13 column then it makes twin prime conjecture kind of "visible"/
Yes, that's what it was conjectured -- since it was "visible"; so I don't understand what "opinion" you exactly want about this.
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September 25th, 2013, 01:28 AM   #3
ogh
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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
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September 25th, 2013, 04:18 AM   #4
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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.
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September 25th, 2013, 04:37 AM   #5
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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.
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September 25th, 2013, 08:45 AM   #6
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Re: Something about prime numbers

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Originally Posted by gelatine1
Conclusion all primes are of the form 5+12k,7+12k,11+12k and 13+12k.
Just for pedantry's sake, that should be all primes > 3.

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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