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Claim. Write $G(X)$ for the largest gap between consecutive primes below $X$ and $\log_k$ for the $k$-fold iterated logarithm. There is a function $f(X)$ tending to infinity with $X$ such that

$$G(X)\ge f(X)\,\frac{\log X\log_2X\log_4X}{(\log_3X)^2}$$

for all sufficiently large $X$; equivalently, for every $C>0$ the right side with $C$ in place of $f(X)$ is a lower bound for $G(X)$ once $X$ is large enough. This is the main theorem of K. Ford, B. Green, S. Konyagin and T. Tao, Large gaps between consecutive prime numbers, Ann. of Math. (2) 183 (2016), no. 3, 935–974, first posted as arXiv:1408.4505 on 20 August 2014. Since $\log p_n\sim\log n$, the theorem gives, for every $C>0$, infinitely many $n$ with $p_{n+1}-p_n>C\log n\log_2n\log_4n/(\log_3n)^2$, which is the question of Problem 4 answered in the affirmative: the constant in Rankin's 1938 bound can be taken arbitrarily large. The statement is recorded from the arXiv abstract and from the historical paragraph of the authors' later joint paper with Maynard (see its library card), which states that the 2014 papers of these authors and of Maynard answered Erdős's conjecture. The proof keeps the Erdős–Rankin sieve and replaces its last stage by a random covering of the surviving primes by arithmetic progressions, drawing on recent results on the existence and distribution of long arithmetic progressions of primes. Maynard reached the same conclusion independently one day later by a sieve route; Maynard's result has its own 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 Maynard's, 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.