what is Prime Gaps relationship with number 6? Out of the 78499 prime number under 1 million. There are 32821 prime gaps (difference between two consecutive prime numbers) of a multiple 6. A bar chart of differences and frequency of occurrence shows a local maximum at each multiple of 6. Why is 6 so special? 


take any integer $n> 3$, and divide it by $6$.  That is, write
$n = 6q + r$
where $q$ is a non-negative integer and the remainder $r$ is one of $0$, $1$, $2$, $3$, $4$, or $5$.
If the remainder is $0$, $2$ or $4$, then the number $n$ is divisible by $2$, and can not be prime.
If the remainder is $3$, then the number $n$ is divisible by $3$, and can not be prime.
So if $n$ is prime, then the remainder $r$ is either


*

*$1$ (and $n = 6q + 1$   is one more than a multiple of six), or

*$5$ (and $n = 6q + 5 = 6(q+1) - 1$   is one less than a multiple of six).

