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I cannot find a proof of this theorem. May anyone assist?

$p_{n+1}-p_n\gg\frac{\log \log \log p_n}{\log \log \log \log p_n} \log{p_n}$

GH from MO
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Cameron Barbeau
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    I'm voting to close this question because one cannot prove a false assertion. – Stefan Kohl Sep 11 '15 at 09:08
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    Probably OP means that this happens infinitely often. – Fedor Petrov Sep 11 '15 at 12:49
  • I am voting to close as Googling "long gaps between primes" yields the state of the art. – Boris Bukh Sep 11 '15 at 13:09
  • The lower bound of Westzynthius involves sifting an interval $[R, R+ p_n\xi]$ with the first n primes in three stages: cross out multiples of all of the (k+1)st through lth primes, then choose residues to maximally sieve the remaining with the first k primes. This leaves much fewer than n-l holes in the interval to be covered by the remaining primes. $\xi$ in the paper is "like" a constant times $\frac{\log \log p_n}{\log \log \log p_n}$. I intend to post a review of the lower bound argument eventually. Gerhard "Still Playing With Upper Bound" Paseman, 2015.09.11 – Gerhard Paseman Sep 11 '15 at 21:07

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The bound as stated is false, because not all prime gaps are that large. In fact we know since the work of Yitang Zhang (2013) that there are infinitely many bounded prime gaps.

The state of the art regarding (occasional) large prime gaps is contained in the work of Ford-Green-Konyagin-Maynard-Tao. Consulting the references in this paper should lead you to earlier but still rather strong results, e.g. the work of Rankin.

GH from MO
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    The bound is reminiscent of Westzynthius (1931). A reference for this work can be found at http://mathoverflow.net/questions/37679 . Gerhard "And A Few Other Places" Paseman, 2015.09.10 – Gerhard Paseman Sep 10 '15 at 23:56