Gaps between consecutive pairs of twin primes.

Question

One can have a gap of $4$ between consecutive pairs of twin primes as in $17-13=4$ for twins $(11,13)$ and $(17,19)$. There is also a gap of $10$ between $19$ and $29$ in twins $(17,19)$ and $(29,31)$ were it not for $23; 19$ and $29$ are not consecutive primes. Are there such gaps greater than $4$ for consecutive primes that are members of pairs of twin primes?

Answer 1

Yes: $419,421,431,433$ are consecutive primes who are also twin primes, there are probably smaller counterexamples.

Edit: Yep, $137,139,149,151$ is the smallest counterexample.

Answer 2

All primes greater than $3$ have the form $6m\pm 1$ (which is NOT to say that all numbers of the form $6m\pm 1$ are primes), so any twin prime (other than $3,5$) will consist of numbers $6m-1,\ 6m+1$ for some $m$. We can then see that the separation between any two twin prime pairs will be $(6k-1)-(6j+1)=6(k-j)-2$.

The first thing this tells us is that the separation between twin prime pairs will be $-2\equiv 4 \bmod 6$, or numbers such as $4,10,16,22, \dots$

You identified separations of $4$ which are true gaps, having no intervening primes (you can prove to yourself that a separation of $4$, when it occurs, must be a gap without any primes in it). 'I was suspended for talking' identified gaps of $10$. Gaps of $16$ are not hard to find: $(1931,1933),(1949,1951)$ and $(2111,2113),(2129,2131)$.

As an oddity, you might be interested in $(4217,4219),(4229,4231),(4241,4243)$ which features two consecutive gaps of $10$.

I suspect that there is no reason that examples of larger gaps cannot be found, although the effort of looking for them may not be entirely worthwhile, unless you have some futher application in mind.