Prime Number Gaps Multiples of Six

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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 $6$.