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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. For all sufficiently large $X$,

$$G(X)\gg\frac{\log X\log_2X\log_4X}{\log_3X},$$

with an effective implied constant. This is Theorem 1 of K. Ford, B. Green, S. Konyagin, J. Maynard and T. Tao, Long gaps between primes, J. Amer. Math. Soc. 31 (2018), no. 1, 65–105, first posted as arXiv:1412.5029 on 16 December 2014, with its library card paging the theorem at theorem_1. The bound exceeds the gap of Problem 4, $C\log n\log_2n\log_4n/(\log_3n)^2$, for every $C>0$ with a factor $\log_3X$ to spare ($\log p_n\sim\log n$ carries the comparison from $X$ to the index), so it answers the question in the affirmative and improves by that factor the 2014 theorems of Ford, Green, Konyagin and Tao and of Maynard, which first showed that the constant in Rankin's bound can be arbitrarily large; the paper's introduction records that Maynard had meanwhile obtained $G(X)\gg\log X\log_2X/\log_3X$ in unpublished work. The proof reduces, through the Chinese remainder theorem (Lemma 1.1), to the covering bound (1.2), $Y(x)\gg x\log x\log_3x/\log_2x$, for the longest initial interval covered by one residue class modulo each prime $p\le x$; that bound combines multidimensional sieve weights of Maynard type with a generalization of the Pippenger–Spencer hypergraph covering theorem proved by the Rödl nibble. The basis of this page is the statement and its reduction as the arXiv preprint (v3) gives them, not the proof of (1.2). The bound was improved in 2026 by the manuscript posted by DottedCalculator.

Acceptance. The paper is a refereed journal publication, the refereed evidence. The site's curator, Thomas Bloom, labels the problem proved and names this theorem in the problem's commentary as the best bound available before the 2026 improvement, the reviewed evidence. The page is dated by the preprint's first posting; the journal published the paper online on 23 February 2017.

Depends on. Nothing beyond the cited paper.