Gaps between pairs of twin primes. I'm sorry if this has been answered before but I've not been able to find much info on it; so my question is: Say I have a pair of twin prime numbers $P_n$ and ($P_n+2$) , and the next largest pair of twin prime numbers $P_{n+1}$ , and ($P_{n+1}+2$). Can it be proven that $P_{n+1}-(P_n+2)-1$ > $1$ for any $P_{n+1}$ and $P_n$>$3$? I hope I've worded that in a way that makes sense, basically besides the pairs {3,5} and {5,7}, I don't believe that there are any pairs of twin primes with only one whole number or less between them, and I'm wondering how or if I could prove that. P.S. There seem to be very many pairs separated by only 3 whole numbers though, such as {5,7} and {11,13} or {2081,2083} and {2087,2089} which is pretty cool.

Yes, it can be proven as among $2m+1, 2m+3, 2m+5$ there is a multiple of $3$, which is not prime unless there it is equal to three.

On the post script: yes, there a conjecture to be infinitely many primes $p$ such that $p+2$, $p+6$, $p+8$ are all prime too. This is called a prime quadruplet.