# Large gaps between consecutive prime numbers

@article{Ford2014LargeGB,
title={Large gaps between consecutive prime numbers},
author={Kevin Ford and Ben Green and Sergei Konyagin and Terence Tao},
journal={arXiv: Number Theory},
year={2014}
}
• Published 20 August 2014
• Mathematics
• arXiv: Number Theory
Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdos, we show that $$G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2},$$ where $f(X)$ is a function tending to infinity with $X$. Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the existence and distribution of long arithmetic progressions…

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## References

SHOWING 1-10 OF 60 REFERENCES
Large gaps between primes
We show that there exists pairs of consecutive primes less than $x$ whose difference is larger than $t(1+o(1))(\log{x})(\log\log{x})(\log\log\log\log{x})(\log\log\log{x})^{-2}$ for any fixed $t$. Our
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• Mathematics, Computer Science
• 2014
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Unusually large gaps between consecutive primes
• Mathematics
• 1990
Let G(x) denote the largest gap between consecutive primes below x. In a series of papers from 1935 to 1963, Erdos, Rankin, and Schonhage showed that G(x) > (c + o(1)) logx loglogx
Very Large Gaps between Consecutive Primes
Abstract LetG(X) denote the largest gap between consecutive primes belowX. Improving earlier results of Erdős, Rankin, Schonhage, and Maier-Pomerance, we prove G(X)⩾(2e γ +o(1)) log Xlog 2 Xlog 4
The Difference between Consecutive Prime Numbers V
Let p n denote the n th prime and let e be any positive number. In 1938 (3) Ishowed that, for an infinity of values of n , where, for k ≧1, log k +1 x = log (log k x ) and log 1 x = log x . In a
The primes contain arbitrarily long arithmetic progressions
• Mathematics
• 2004
We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi�s theorem, which asserts that any subset of the integers of
Small gaps between primes
We introduce a renement of the GPY sieve method for studying prime k-tuples and small gaps between primes. This renement avoids previous limitations of the method and allows us to show that for each
On the difference between consecutive primes
Update: This work reproduces an earlier result of Peck, which the author was initially unaware of. The method of the proof is essentially the same as the original work of Peck. There are no new
On the number of positive integers . . .
• Mathematics
• 1966
1. Introduction Let P(x,y) denote the number of integers specified in the title. A number of estimates and asymptotic formulae for this function have been given (cf. [1] and the literature mentioned