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Is it true that for every even natural number $k$ there exists some $n \in \mathbb{N}$ such that $g_n = p_{n+1} - p_n = k$?

I don't know how to approach the problem at all, and in fact I don't even know enough about prime gaps to even form a conjecture as to the answer. I feel like the answer is "yes", but only because that would be "nicer" than having some even integers never appear in sequence of prime gaps.

I hope it's not an unsolved problem!

Edit: My question is distinct from Polignac's Conjecture, since I ask if there is at least one prime gap, instead of infinitely many prime gaps, for every size.

feralin
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2 Answers2

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It appears to be open if every even number is the difference of two primes, let alone consecutive primes. Here is a m.se question mentioning that and an mo question here

Community
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For $n$ a positive integer, the numbers $$(n+1)!+2,(n+1)!+3,..,(n+1)!+n+1$$ are $n$ consecutive composite integers. Does this help ?

user385459
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    No, this doesn't help. OP asked about some pair of primes that have a given difference. Finding a long stretch of non-primes does not speak to this at all. – Ross Millikan Apr 07 '18 at 05:25
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    I think it's unfair to say that this "does not speak to this at all". This result is clearly related, for instance, anytime (n+1)!-1 and (n+1)! + (n+2) are prime this would give a solution for k = n+3. – Adam Boocher Nov 13 '21 at 15:27