Coincidence? Why are Primes Clustered Around Multiples of 6?

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Solution:

Why are the prime numbers clustered around multiples of 6?

 

Let's consider what happens when we divide an integer by 6.  There are 6 possible remainders, including the remainder of 0, which we get when the number is a multiple of 6:

 

Remainder when we divide by 6	Can the number be prime?
0	Must be a multiple of 2 and 3.
1	May be prime, such as 13.  But not always, such as 25.
2	Must be a multiple of 2.
3	Must be a multiple of 3.
4	Must be a multiple of 2.
5	May be prime, such as 29.  But not always, such as 35.

 

Looking at the chart, we can see that numbers leaving a remainder of 0, 2, or 4 must be a multiple of 2, and numbers leaving a remainder of 0 or 3 must be a multiple of 3.  Other than 2 and 3 themselves, these numbers are all composite.  Therefore all primes greater than 3 must have a remainder of 1 or 5 when we divide by 6.  In other words, they must be one more or one less than a multiple of 6.