# Sequence of consecutive gaps in prime numbers

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 $k=2$.

• For $k=1$, $a!+2, a!+3, ..., a!+a$ make an arbitrary large gap, when $a$ is an arbitrary large number. – Arash Ahadi Dec 4 '16 at 3:48
• 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. – Gerhard Paseman Dec 4 '16 at 3:59
• 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. – Fan Zheng Dec 4 '16 at 4:07
• I am sure that there was a good intention but to me, this question does not make any sense whatsoever. – Włodzimierz Holsztyński Dec 5 '16 at 7:27
• @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. – GH from MO Dec 5 '16 at 11:33