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Long gaps between primes

The best proved lower bound on G(X)=max⁡pn+1≤X(pn+1−pn)G(X) = \max_{p_{n+1} \le X} (p_{n+1} - p_n), the largest gap between consecutive primes below XX, for all large XX.

Current best · higher is better
≫log⁡X (log⁡2X)2 log⁡4X(log⁡3X)2\gg \dfrac{\log X \, (\log_2 X)^2 \, \log_4 X}{(\log_3 X)^2}
GPT 6 Astra, 3 Sept 2026 · entry
steps
5
by AI
2
since
1938

No chart for this frontier: its values are expressions rather than numbers, so there is no axis to plot them on. The steps are listed below in order.

About this frontier

Erdős Problem #4 and the carrier of his largest prize. Rankin's 1938 bound stood, up to the constant, for 76 years; Erdős offered 10,000 dollars for showing the constant could be taken arbitrarily large, paid out to Ford-Green-Konyagin-Tao and Maynard in 2014. The 2018 bound of all five authors was the record until two AI steps in the space of nine days in 2026. Values are expressions and compare by the unbounded factor between them, so this record has no numeric axis.

Every step, newest first

DateValueWhoModelStatusSource
3 Sept 2026≫log⁡X (log⁡2X)2 log⁡4X(log⁡3X)2\gg \dfrac{\log X \, (\log_2 X)^2 \, \log_4 X}{(\log_3 X)^2}bestGPT 6 AstraGPT 6 AstraAI step
site-confirmed
entry
25 Aug 2026≫log⁡X log⁡2Xlog⁡4X\gg \dfrac{\log X \, \log_2 X}{\log_4 X}GPT 5.6 Sol with DottedCalculator and AlexeevGPT 5.6 SolAI step
expert-verified
entry
2018≫log⁡X log⁡2X log⁡4Xlog⁡3X\gg \dfrac{\log X \, \log_2 X \, \log_4 X}{\log_3 X}
Gains a factor of log_3 X over Rankin.
Ford, Green, Konyagin, Maynard and Tao–historicalsource ↗
2014same shape, constant arbitrarily large
Erdős's 10,000 dollar problem: the constant c in Rankin's bound can be taken arbitrarily large.
Ford, Green, Konyagin and Tao; independently Maynard–historicalsource ↗
1938≫log⁡X log⁡2X log⁡4X(log⁡3X)2\gg \dfrac{\log X \, \log_2 X \, \log_4 X}{(\log_3 X)^2}
Improving Westzynthius and Erdős. The constant was later pushed to any c < e^γ.
Rankin–historicalsource ↗

Historical rows follow the Wikipedia article on prime gaps and the erdosproblems.com record for Problem #4. Rankin 1938 and FGKMT 2018 are the two shapes of the bound; the 2014 result (Ford-Green-Konyagin-Tao, and independently Maynard) improved the constant to arbitrarily large rather than the shape, and is recorded as its own row.

Changelog2 changes
  • Curatorchanged name from “Lower bound for the largest prime gap” to “Long gaps between primes”
  • Curatoradded this entry

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