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?
In fact it is expected that every even number occurs as a prime gap infinitely often. See Polignac's conjecture.
See OEIS sequence A000230 and references there.