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?
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2$\begingroup$ This is an eccentric use of the term "bounded" which, after all, has a universally accepted meaning. Why not use another term? $\endgroup$– luluJun 24, 2022 at 22:37
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$\begingroup$ @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. $\endgroup$– ArthurJun 24, 2022 at 22:38
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1$\begingroup$ @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. $\endgroup$– luluJun 24, 2022 at 22:39
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4$\begingroup$ 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 $\endgroup$– Mark BennetJun 24, 2022 at 22:42
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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.
