Problem #4: Is it true that, for any $C>0$, there are infinitely many...

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Progress

The peculiar quantitative form of Erdős' question was motivated by an old result of Rankin [Ra38], who proved there exists some constant $C>0$ such that the claim holds. Solved by Maynard [Ma16] and Ford, Green, Konyagin, and Tao [FGKT16]. The best bound available, due to all five authors [FGKMT18], is that there are infinitely many $n$ such that\[p_{n+1}-p_n\gg \frac{\log\log n\log\log\log\log n}{\log\log \log n}\log n.\]The likely truth is a lower bound like $\gg(\log n)^2$. In [Er97c] Erdős revised the value of this problem to \$5000 and reserved the \$10000 for a lower bound of $>(\log n)^{1+c}$ for some $c>0$.

The best known upper bound is\[p_{n+1}-p_n \ll n^{0.525+o(1)},\]proved by Baker, Harman, and Pintz [BHP01].

See also [687].

This is discussed in problem A8 of Guy's collection [Gu04].