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Thursday, February 5, 2015

**Abstract:** We show how to improve quantitatively the best known bounds for large gaps between consecutive prime numbers by a factor of $\log\log\log x$. More precisely, if $G(x)$ is the largest gap between consecutive prime numbers which are at most $x$, then we show that $$G(x) \ge c \log x \frac{\log\log x \log\log\log\log x}{\log\log\log x}$$ for some positive $c$ and large enough $x$. This improves upon the recent bounds of the speaker (in joint work with B. Green, S. Konyagin and T. Tao) and J. Maynard, who independently solved a long-standing conjecture of Erdos on the growth of $G(x)$. The new work, which is joint with Ben Green, Sergei Konyagin, James Maynard and Terence Tao, combines ideas from both of the aforementioned papers and also introduces some new ideas from probabilistic graph theory (hypergraph packing, Rodl nibble).