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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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  • $\begingroup$ @DouglasS.Stones: The gaps are between consecutive primes. $\endgroup$
    – Harald Hanche-Olsen
    Apr 12, 2013 at 19:21
  • $\begingroup$ @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. $\endgroup$
    – Calvin Lin
    Apr 12, 2013 at 19:22

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

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  • $\begingroup$ 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. $\endgroup$
    – Calvin Lin
    May 15, 2013 at 14:11
  • $\begingroup$ @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. $\endgroup$
    – Zander
    May 16, 2013 at 0:57
  • $\begingroup$ 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. $\endgroup$
    – Calvin Lin
    May 16, 2013 at 15:55
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See OEIS sequence A000230 and references there.

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  • $\begingroup$ 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? $\endgroup$
    – Calvin Lin
    Apr 12, 2013 at 20:31
  • $\begingroup$ "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 $e^{3510}$. $\endgroup$
    – John Nicholson
    Mar 5, 2018 at 5:50

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