Possible values of prime gaps

The nth prime gap is defined as $p_{n+1} - p_n$, [sequence A001223 in OEIX] (http://oeis.org/A001223). What values can occur as a prime gap?

Clearly with the exception of $1 = 3 - 2$, all the prime gaps must be even. We also know that this sequence must contain infinitely large numbers, since there are no primes between $n!+2$ and $n! + n$.

Is it true that every even number occurs as a prime gap?

• @DouglasS.Stones: The gaps are between consecutive primes. Apr 12 '13 at 19:21
• @DouglasS.Stones Your point about 2 arbitrary primes is something I want to consider too (perhaps as a separate question). However, we can show that $2k+1$ is a difference of primes if and only if $2k+3$ is prime, since one of the primes in the difference must be 2. Apr 12 '13 at 19:22

• Ah, I misread the article. Sorry for the confusion. Yes, we can easily show that there are infinitely many arbitrarily large gaps using $n!+2 - n!+n$. I was wondering. May 16 '13 at 15:55
• Thanks. It seems to me that we don't know if the number $2n$ can appear as a prime gap, since the largest known prime gap with identified proven primes as gap ends has length 337446. I can't seem to find any existence results, like can 12345678 be a prime gap? Apr 12 '13 at 20:31
• "can 12345678 be a prime gap?" As far as we know, yes, but the prime number of its first occurrence is going to be very big prime compared to the currently know first occurrences. Near $e^{3510}$. Mar 5 '18 at 5:50