# Is this generalization of prime gaps also bounded?

I know that it has been proven that prime gaps are bounded. Meaning, no matter how far you go along the number line, you will keep finding consecutive primes less than a fixed distance, which I believe was proven to be at most 70 million or so. However, I want to generalize this, and ask whether the gap between $$p_n$$ and $$p_{n+2}$$, where $$p_n$$ is the nth prime number, is also bounded. In fact, I want to know, is it true that for any positive integer $$k$$, the gap between $$p_n$$ and $$p_{n+k}$$ is also bounded? Or, is there a smallest positive integer $$k$$ where the gap is no longer bounded?

• This is an eccentric use of the term "bounded" which, after all, has a universally accepted meaning. Why not use another term?
– lulu
Commented Jun 24, 2022 at 22:37
• @lulu "Has finite liminf" perhaps? It doesn't flow so nicely, though, and I don't know that there is a more elegant phrasing that sounds better. Commented Jun 24, 2022 at 22:38
• @Arthur I agree, but that's not a reason to use a word with a universally accepted meaning to mean something entirely different. Better to call such sequences "good" or whatever.
– lulu
Commented Jun 24, 2022 at 22:39
• I think you will find that if you explore the Polymath project on prime gaps the figure of $70$ million has been considerably reduced ($246$? and a contingent $6$) and the gaps for two, three four and more primes have been explored, and bounded explicitly for low values - asone.ai/polymath/index.php?title=Bounded_gaps_between_primes Commented Jun 24, 2022 at 22:42

It is known that for every $$k \in \mathbb{N}$$ there is an integer $$H_k < cke^{4k}$$ (for some constant $$c$$) such that there are infinitely many $$x \in \mathbb{N}$$ for which there at least $$k$$ primes in the interval $$[x, x + H_k]$$. For a paper on this, see here.