# Are there prime gaps of every size?

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.

• Do you have a source? I would like to read more about it if it is an open problem. – feralin Jan 31 '16 at 5:40
• I think that it is an unsolved problem known as Polignac's Conjecture. – user 170039 Jan 31 '16 at 5:46
• @user170039 my question is distinct from Polignac's Conjecture. I ask if there is at least one prime gap of every size, not infinitely many prime gaps of every size. – feralin Jan 31 '16 at 5:48

## 2 Answers

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

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 ?

• 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 7 '18 at 5:25