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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.

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    $\begingroup$ I'm not sure I got your question correctly, but you are aware of the fact that, for any integer $n\ge 1$, at least one of the numbers $2n+1, 2n+3, 2n+5$ is divisible by $3$? $\endgroup$
    – Thomas
    Commented Jan 16, 2016 at 16:10
  • $\begingroup$ @Thomas oh I wasn't aware of that, haha Thanks! So this means I can't have 4 consecutive odd numbers that are prime right? $\endgroup$
    – T. Jones
    Commented Jan 16, 2016 at 16:35
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    $\begingroup$ @T.Jones Apart from $3, 5, 7$ we can't even have $3$ consecutive odd numbers that are prime. $\endgroup$
    – Arthur
    Commented Jan 16, 2016 at 16:35

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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.

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