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Claim. With $\log_k$ the $k$-fold iterated logarithm,

$$\limsup_{n\to\infty}\frac{p_{n+1}-p_n}{\log p_n\log_2p_n\log_4p_n\,(\log_3p_n)^{-2}}=\infty .$$

This is Theorem 1 of J. Maynard, Large gaps between primes, Ann. of Math. (2) 183 (2016), no. 3, 915–933, first posted as arXiv:1408.5110 on 21 August 2014 and described on its library card. Since $\log p_n\sim\log n$, the theorem says exactly that for every $C>0$ there are infinitely many $n$ with $p_{n+1}-p_n>C\log n\log_2n\log_4n/(\log_3n)^2$, the question of Problem 4, answered yes. The proof follows the Erdős–Rankin construction and changes only its final stage: the survivors $mp$ with $m$ smooth and $p$ a large prime are covered using the sieve weights developed for small gaps between primes, which is what lets the constant grow without bound, and the few smooth survivors are covered one at a time. The paper notes that Ford, Green, Konyagin and Tao obtained the same conclusion independently by a different route, posted one day earlier; their result has its own claim page. A remark in the paper announces a quantitative improvement of Rankin's bound; the five-author paper of 2018 records that Maynard obtained it, $G(X)\gg\log X\log_2X/\log_3X$, in unpublished work, and improves it further by a factor $\log_4X$ (see its claim page).

Acceptance. The paper is a refereed journal publication, the refereed evidence. The site's curator, Thomas Bloom, labels the problem proved and credits its solution to this paper and to that of Ford, Green, Konyagin and Tao, the reviewed evidence. The page is dated by the preprint's first posting; the journal issue appeared in May 2016.

Depends on. Nothing beyond the cited paper.