Ford 2018 long gaps between primes

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equation_1_2: The best held lower bound for the longest initial interval covered by one residue class modulo each prime up to x, the covering form of the 2018 prime-gap theorem.

lemma_1_1: The Chinese-remainder transfer from a residue covering of an initial interval to a prime gap, and the identity between the covering function and Jacobsthal's function at the primorial.

theorem_1: The 2018 lower bound for the largest gap between consecutive primes below X, deduced from the covering bound (1.2) through Lemma 1.1.

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Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, Terence Tao, Long gaps between primes. Journal of the American Mathematical Society 31 (2018). arXiv:1412.5029, doi:10.1090/jams/876. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1412.5029), every other right reserved.

The copy read for this card is arXiv:1412.5029v3 (14 July 2016, 40 pages; the arXiv stamp is on p. 1), with a complete text layer. The paper appeared in J. Amer. Math. Soc. 31 (2018), no. 1, 65--105 (published online 23 February 2017; the Crossref record of DOI 10.1090/jams/876 and the arXiv listing's journal reference were read). The journal text was not compared; the page numbers and labels below are the preprint's. Read status: claims checked for Definition 1, Lemma 1.1, displays (1.2) and (1.3), Theorem 1 and Corollary 1 (pp. 2--4), each read clause by clause on the page images and in the text layer on 2026-09-18, with the proof of Lemma 1.1 read in full; the proof of (1.2) (Sections 3--8, pp. 8--39) was not read. The rest of the digest records an earlier reading that was not repeated.

Theorem 1 shows that G(X) = max_{p_{n+1} <= X} (p_{n+1} - p_n) is at least of order log X log_2 X log_4 X / log_3 X for large X, with an effective constant, improving Rankin's classical bound and the authors' earlier resolutions of Erdos's conjecture that the Rankin constant can be arbitrarily large. The proof reduces (Lemma 1.1, Definition 1) to the covering function Y(x), the largest y such that residue classes a_p mod p, one per prime p <= x, cover [1,y]; the bound established is Y(x) >> x log x log_3 x / log_2 x, and Y(x) = j(P(x)) - 1 links it to Jacobsthal's function. The new ingredient is a generalization of the Pippenger-Spencer hypergraph covering theorem proved by the Rodl nibble, combined with multidimensional prime-detecting sieve weights of Maynard type to sieve a set of primes efficiently. Corollary 1 (p. 4) transfers the bound to M(k), the largest least prime in a progression mod k, when k has no prime factor up to x for some sufficiently large x <= log k. For problem 688 this is the state-of-the-art constructive interval covering with one residue class per prime, and the machinery a lower-bound argument would have to adapt to a truncated prime window.

Source: https://arxiv.org/abs/1412.5029.

Bears on. #688: context, the covering machinery over every prime up to x, not the truncated window of that problem. #687: display (1.2) is the best lower bound for that problem's Y(x) in a refereed source, Lemma 1.1 its transfer to prime gaps, (1.3) its identity with Jacobsthal's function, and p. 4 one of the two library sources for Iwaniec's Y(x) << x^2 and the only library source for the Maier-Pomerance conjecture. Iwaniec's own paper is filed as iwaniec_1978_problem_jacobsthal; its Corollary C(r) << r^2 log^2 r on printed p. 226, read there clause by clause on the page image and paged on corollary, gives Y(x) << x^2 at r = pi(x) in a one-line step recorded on that page, the paper never stating the bound in that form. #970: (1.2) with (1.3) gives j(P(x)) >> x log x log_3 x / log_2 x, a lower bound for Jacobsthal's function of a number with pi(x) prime factors; the translation to that problem's h(k) is made on its page. #929: that problem's S(k) is the least x with Y(x) >= k, so (1.2) inverts to an upper bound for S(k) on its page. #4: Theorem 1 (p. 2 of v3, page image and text layer), G(X) >> log X log_2 X log_4 X / log_3 X for large X, exceeds the problem's gap C log n log_2 n log_4 n / (log_3 n)^2 for every C (a factor log_3 X to spare, with log p_n ~ log n); the historical paragraph on p. 2 records that the authors' 2014 paper and Maynard's, the site's [FGKT16] and [Ma16], already showed the Rankin constant can be arbitrarily large, "answering in the affirmative a long-standing conjecture of Erdős", the problem's question (theorem_1).

Results to transcribe.

• Theorem 1 (p. 2): G(X) >> log X log_2 X log_4 X / log_3 X for sufficiently large X, with an effective implied constant.
• Equation (1.2) (p. 3): Y(x) >> x log x log_3 x / log_2 x, where Y(x) is the longest interval coverable by one residue class per prime p <= x; this implies Theorem 1. The page also records p. 4's comparison with Rankin's and Maynard's bounds and its attestation of Iwaniec's upper bound and the Maier-Pomerance conjecture.
• Lemma 1.1 and (1.3) (pp. 3-4): G(P(x)+Y(x)+x) >= Y(x) via the Chinese remainder theorem, and Y(x) = j(P(x)) - 1 for Jacobsthal's function j.
• Corollary 1 (p. 4): If k has no prime factor <= x for some x <= log k, and x is sufficiently large (so that log_2 x and log_3 x are positive), then the largest least prime M(k) in a progression mod k satisfies M(k) >> k x log x log_3 x / log_2 x.
• Theorem 3 (Probabilistic covering, Section 4.2, p. 12; proved in Section 5) and Corollary 3 (Generalized Pippenger-Spencer, p. 14): A generalization of the Pippenger-Spencer hypergraph covering theorem, proved by the Rodl nibble, of independent interest.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.