# 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?

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@DouglasS.Stones: The gaps are between consecutive primes. –  Harald Hanche-Olsen 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. –  Calvin Lin 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. –  Calvin Lin 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? –  Calvin Lin Apr 12 '13 at 20:31