Re:Prime number patterns
by Retric on Tuesday July 17, 2012 @01:50PM
(#4256842)
It does give you a good intuition as to why twin primes aka N, N+2 are so common.
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how so ?
The number between them have a high number of small factors including 2 and 3.
2 * 3 = 6: 5 and 7 are prime.
2 * 3 * 2 = 12: 11 and 13 are prime
2 * 3 * 3 = 18: 17 and 19 are prime
2 * 3 * 5 = 30: 11 and 13 are prime
2 * 3 * 7 = 42: 41 and 43 are prime.
However, 2 * 3 * 101 = 606 but 605 is not prime.
But,
2 * 3 * 5 * 5 = 150 and 149 and 151 are prime.
2 * 3 * 2 * 3 * 5 = 180 and 179 and 181 are prime.
Re:Prime number patterns
by jjaredsimpson on Tuesday July 17, 2012 @06:57PM
(#4258371)
I don't see where the insight is. twin primes must be of the form (6k-1, 6k+1). So of course there will be a 2,3 at least.
Smaller numbers have multiples that are more densely distributed among the integers.
The point is twin primes (6k-1, 6k+1) are more likely for a large k when k is a composite number than a prime AND the more factors of k the higher chance for twin primes.
EX: K = (6 * 2 * 3 * 5 * 7 * 11 * 13 ) gives a twin prime.