4
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Do you have a source? I would like to read more about it if it is an open problem. $\endgroup$ – feralin Jan 31 '16 at 5:40
  • $\begingroup$ I think that it is an unsolved problem known as Polignac's Conjecture. $\endgroup$ – user 170039 Jan 31 '16 at 5:46
  • $\begingroup$ @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. $\endgroup$ – feralin Jan 31 '16 at 5:48
5
$\begingroup$

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

$\endgroup$
0
$\begingroup$

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 ?

$\endgroup$
  • 1
    $\begingroup$ 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. $\endgroup$ – Ross Millikan Apr 7 '18 at 5:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.