Equal Consecutive Prime Number Gaps are Multiples of Six

Theorem
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.

Proof
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 $$6$$.