Large gaps between consecutive prime numbers

By Kevin Ford, Ben Joseph Green, Sergei V. Konyagin, Terence Tao. 2014

80 citations recorded by the source corpus.

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 consisting entirely of primes.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/593d7ada5d6013cd63a13b567a41b7beee57e95e

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

Citations in the corpus

Long gaps between primes (2014, methodology): “The arguments in [12] used recent results [18, 19, 20] on the number of solutions to linear equations in primes, whereas the arguments in [29] instead relied on multidimensional prime-detecting sieves introduced in [27].”