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The nth prime gap is defined as $p_{n+1} - p_n $, [sequence A001223 in OEIX] ( 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
up vote 2 down vote accepted

In fact it is expected that every even number occurs as a prime gap infinitely often. See Polignac's conjecture.

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There seems to be recent proof by Zhang Yitang, that infinitely many prime gaps do not exceed 70 million. If this holds true, then the answer would be no. – Calvin Lin May 15 '13 at 14:11
@CalvinLin Zhang's result doesn't contradict Polignac's conjecture at all. It just says that there are infinitely many "small" gaps. But as you said in the question there are arbitrarily large gaps, and a result of Rankin says that there are infinitely many arbitrarily large gaps as well. – Zander May 16 '13 at 0:57
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

See OEIS sequence A000230 and references there.

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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

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