Are there two consecutive gaps of size $4$ between prime numbers?

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?

Solution 1:

One of $\{p,p+4,p+8\}$ will be divisible by $3$, so if they are all to be primes, one of them must be $3$ itself. Then we have $3,7,11$, which are all prime, but unfortunately 3 to 7 is not a gap, so no, there cannot be two consecutive prime gaps of length 4.

Solution 2:

Just to elaborate on @Henning Makholm's answer:

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