The largest gap between consecutive primes below X grows faster than log X log_2 X log_4 X/(log_3 X)^2

You need to know: Prime numbers.

Background: Let $p_n$ denotes the n-th prime. For $X>2$ , let $G(X)=\sup\limits_{p_n \leq X}(p_{n+1}-p_n)$ be the largest gap between consecutive primes below X. For $X>16$ , let $f(X)=\frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2}$ . With notation $\log_r (X)$ for the r-th fold logarithm, this simplifies to $f(X)=\frac{\log X \log_2 X \log_4 X}{(\log_3 X)^2}$ .

The Theorem: On 20th August 2014, Kevin Ford, Ben Green, Sergei Konyagin, and Terence Tao submitted to arxiv a paper in which they proved that $\lim\limits_{X\to\infty} \frac{G(X)}{f(X)}=\infty$ .

Short context: How fast does the largest gap $G(X)$ between consecutive primes below X grow with X? It is conjectured that $G(X)$ grows like a constant times $\log^2 X$ , but this is wide open. In 1931, Westzynthius proved that $G(X) \geq c\frac{\log X \log_3 X}{\log_4 X}$ for some constant $c>0$ . This was improved to $G(X) \geq c\frac{\log X \log_2 X}{(\log_3 X)^2}$ by Erdős in 1935, and to $G(X) \geq c f(X)$ by Rankin in 1938. This strange-looking bound, surprisingly, standed for almost 80 years, except of improvements by a constant factor. At last, the Theorem proves that $G(X)$ grows faster than $f(X)$ . The same result was proved independently by Maynard. See here for even better lower bound proved in a later work, and here for an upper bound on $G(x)$ .