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



> 
> $p\_{n+1}-p\_n\gg\frac{\log \log \log p\_n}{\log \log \log \log p\_n} \log{p\_n}$
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> 
> 



Proof that $p_{n+1}-p_n\gg\frac{\log \log \log p_n}{\log \log \log \log p_n} \log {p_n}$
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](http://de.arxiv.org/abs/1412.5029). Consulting the references in this paper should lead you to earlier but still rather strong results, e.g. the work of Rankin.


