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Let $p_i$ be $i^{th}$ prime number (for example $p_4=7$). Is there $m=m(n, k)$ such that $n \leq \min \{p_{m+i}-p_{m+i-1} : 1 \leq i \leq k\}$ for every $n$ and $k$? Even, consider the problem for $k=2$.

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  • $\begingroup$ For $k=1$, $a!+2, a!+3, ..., a!+a$ make an arbitrary large gap, when $a$ is an arbitrary large number. $\endgroup$ – Arash Ahadi Dec 4 '16 at 3:48
  • $\begingroup$ This is a consequence of the prime k-tuples conjecture. Based on recent work involving k=2 (Tao, Pintz, Maynard and others) it is likely that m will be exponential in n and k. Gerhard "A Prime Time For Mathematics" Paseman, 2016.12.03. $\endgroup$ – Gerhard Paseman Dec 4 '16 at 3:59
  • $\begingroup$ AFAIK, the OP is asking whether there are arbitrarily long run of arbitrarily large prime gaps, which is not exactly the question the above comments are answering. $\endgroup$ – Fan Zheng Dec 4 '16 at 4:07
  • $\begingroup$ I am sure that there was a good intention but to me, this question does not make any sense whatsoever. $\endgroup$ – Włodzimierz Holsztyński Dec 5 '16 at 7:27
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    $\begingroup$ @WłodzimierzHolsztyński: The question makes full sense. As Fan Zheng remarked, it can be reformulated in plain English as: are there arbitrarily long runs of arbitrarily large prime gaps? Then answer is yes, see my response. $\endgroup$ – GH from MO Dec 5 '16 at 11:33
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Yes, this was proved in a strong quantitative form by Maier in his paper Chains of large gaps between consecutive primes, Adv. in Math. 39 (1981), 257-269. Recently, Pintz strengthened this result with Maynard's method, see Theorem 4' in this arXiv preprint.

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