Prime Number Gaps Multiples of Six
If you list the gaps between consecutive primes > 5 ( 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, .....) you will notice that consecutive gaps that are equal are of the form $6x$. This is always the case.
Suppose there were two consecutive gaps between 3 consecutive prime numbers that were equal, but not divisible by $6$.
Then the difference is $2k$ where $k$ is not divisible by $3$, and so the (supposed) prime numbers will be $p, p+2k, p+4k$.
But then $p+4k$ is congruent modulo 3 to $p+k$.
That makes the three numbers congruent to $p, p+k, p+2k$.
One of those is divisible by $3$ and so can not be prime.
So two consecutive gaps must be divisible by $3$ and therefore (as they have to be even) by $6$.