Problem 4

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?

Source: erdosproblems.com/4

Number theory
Primes

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

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.

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: Erdős 1997; paged at Erdős 1997, 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.
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.

Problem 4

Status

Proof claims

Discussion