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© Copyright 1999, Jim Loy
It may surprise you to know that all primes greater than 3 are either one more than a multiple of 6 or one less than a multiple of 6. Is that mysterious, or what? Well, it is not very mysterious at all. All whole numbers can be represented in one of these forms: 6n, 6n+1, 6n+2, 6n+3, 6n+4, or 6n+5. That is obvious from Modular Arithmetic, by the way. Of those six forms, 6n is a multiple of 6, 6n+3 is a multiple of 3 (and is prime when n=0), and 6n+2 and 6n+4 are multiples of 2 (and 6n+2 is prime when n=0). That only leaves 6n+1 and 6n+5 (6n+5 is also one less than a multiple of 6) as the only possible primes, after the first few. So, there is nothing mysterious there. It is similar to the observation that all primes greater than 2 are odd. It is the same reasoning, actually.
The above also means that (after the first few primes) twin primes (pairs of primes that have a difference of 2, like 11 and 13) have a multiple of 6 immediately between them. Unfortunately, this does not give any clue to how many twin primes there are. Despite the fact that there are infinitely many primes, no one knows whether there are finitely or infinitely many twin primes.
Also see The Infinitude Of Primes and Sieve Of Eratosthenes.