TITLE: Are there two consecutive gaps of size $4$ between prime numbers?
QUESTION [10 upvotes]: Are there consecutive gaps(difference) of 4 between prime numbers? 
Seeing at the first few gaps
$1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6,.....$.
(for example 6 is repeated consecutively at some places).
Are there consecutive gaps of 4 between prime numbers? 
If there is no twin gap of 4, then how to prove it, what is the proof of that?

REPLY [2 votes]: Just to elaborate on @Henning Makholm's answer:

$p\equiv0\pmod3 \implies p=3 \implies p+4\neq5\text{ which is the next prime}$
$p\equiv1\pmod3 \implies p+8\equiv9\equiv0\pmod3 \implies p+8\text{ is not prime}$
$p\equiv2\pmod3 \implies p+4\equiv6\equiv0\pmod3 \implies p+4\text{ is not prime}$