Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 4

claims/: The 7 claim pages of Problem 4, one per claimant's result; the problem's standing derives from them.

Statement. Is it true that, for any $C>0$ , there are infinitely many $n$ such that

p_{n+1}-p_n> C\frac{\log\log n\log\log\log\log n}{(\log\log \log n)^2}\log n?

Status. Proved, the site's label. The accepted claims are the 2014 theorem of Ford, Green, Konyagin and Tao and Maynard's independent 2014 theorem, both refereed in the Annals and credited by the site's curator with the solution; the stronger 2018 bound of Ford, Green, Konyagin, Maynard and Tao, refereed in the Journal of the American Mathematical Society and named by the curator as the best bound before 2026; and the stronger 2026 bound of the AI-written manuscript posted by DottedCalculator, which the curator credits in the commentary and expounds on the site. Rankin's 1938 bound, refereed in the Journal of the London Mathematical Society, is an accepted partial result for every $C$ below $1/3$ . Two claims are pending: a further strengthening by OpenAI and a Lean proof in Boris Alexeev's lean-proofs repository.

Source. erdosproblems.com/4, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #4, https://www.erdosproblems.com/4.

References.

[BHP01] Baker, R. C. and Harman, G. and Pintz, J., The difference between consecutive primes. II. Proc. London Math. Soc. (3) (2001), 532-562.
[Er97c] Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I, Algorithms Combin. 13, Springer (1997), 47--67; the prime-gap passage with display (2.21), printed p. 58: the statement's conjecture carries the smaller of two prizes there, the larger having moved to $d_n>(\log n)^{1+\epsilon}$ . Library home: erdos_1997_some_my_favorite_problems_results; paged at display_2_21.
[FGKMT18] Ford, Kevin and Green, Ben and Konyagin, Sergei and Maynard, James and Tao, Terence, Long gaps between primes. J. Amer. Math. Soc. (2018), 65-105.
[FGKT16] Ford, Kevin and Green, Ben and Konyagin, Sergei and Tao, Terence, Large gaps between consecutive prime numbers. Ann. of Math. (2) (2016), 935-974.
[Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer (2004), xviii+437 pp. Section A8 "Gaps between primes. Twin primes.", printed p. 31: Rankin's bound for infinitely many $n$ "and Erdős offers $5,000 for a proof or disproof that the constant $c$ can be taken arbitrarily large". Library home: guy_2004_unsolved_problems_number_theory.
[Ma16] Maynard, James, Large gaps between primes. Ann. of Math. (2) (2016), 915-933.
[Ra38] Rankin, R. A., The Difference between Consecutive Prime Numbers. J. London Math. Soc. (1938), 242-247.

Formalization. Statement in formal-conjectures, at the linked commit: a statement with sorry marked research solved, with Rankin's bound as a variant; the site's label carries no Lean qualification. In Boris Alexeev's lean-proofs repository, Erdos4.lean and Erdos4b.lean prove the statement for every $C>0$ and the 2018 five-author bound, recorded on their claim page, and the module formalizing the 2026 manuscript is recorded on that claim page; OpenAI's repository formalizing its own 2026 note is recorded on its claim page. This corpus has built none of them.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (24)

Rankin: prime gaps at the question's scale with a fixed constant Ford, Green, Konyagin and Tao: Rankin's constant can be arbitrarily large Maynard: large gaps between primes with an arbitrarily large constant Ford, Green, Konyagin, Maynard and Tao: a further log log log X over Rankin's bound A Lean proof of the question and of the 2018 bound DottedCalculator: the tilted residue sieve and a stronger prime-gap bound OpenAI: a further log log n factor in the largest prime gap problems/primes/E0004/claims Primes integer_sequences/ford_2018_long_gaps_between_primes Theorem 1: G(X) ≫ log X log_2 X log_4 X / log_3 X Sieving intervals and Siegel zeros Corollary 2 (p. 4): Siegel zeros with 1 - beta < (log q)^{-B} give prime gaps >> log p_n (log log p_n)^{B-1}number_theory/guy_2004_unsolved_problems_number_theory Primes primes/baker_2001_difference_between_consecutive_primes Theorem 1 (p. 532): for all x > x_0 the interval [x - x^{0.525}, x] contains primes primes/maynard_2016_large_gaps_between_primes Proposition 5: the primes of R_m are covered by one residue class for each prime of a short interval in [x/2, x]Theorem 1: the prime gap divided by Rankin's scale has infinite limit superior ramsey_theory/erdos_1980_noen_mindre_kjente_problemer_i_kombinatorisk Problem (p. 163, unnumbered): can c be taken arbitrarily large in Rankin's bound for large prime gaps?ramsey_theory/erdos_1997_some_my_favorite_problems_results Display (2.21): d_n > (log n)^{1+ε} infinitely often, with the Erdős–Rankin conjecture

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