Large gaps between consecutive prime numbers

2016/04/18 by Kevin Ford, Ben Green, Sergeĭ Konyagin +1 · 1 citation
Mathematics · #Analytic Number Theory Research #Limits and Structures in Graph Theory #Finite Group Theory Research #Mathematics #Prime (order theory) #Combinatorics

paper · doi:10.4007/annals.2016.183.3.4

openalex publication_date 2016/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/12

Abstract

Let G(X) denote the size of the largest gap between consecutive primes below X. Answering a question of Erd os, we show that G(X)> f(X) logX log logX log log log logX (log log logX) 2 ; where f(X) is a function tending to innity 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.

Citations

• Gaps between primes, and the pair correlation of zeros of the zeta-function
• The Difference Between Consecutive Primes
• Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4⋅10¹⁸
• Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes
• The Difference Between Consecutive Primes, II
• The primes contain arbitrarily long arithmetic progressions
• Small gaps between primes
• On the order of magnitude of the difference between consecutive prime numbers

Cited by

• List of multiple discoveries [wikipedia]

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