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Large gaps between consecutive prime numbers

Kevin Ford, Ben Green, Sergei Konyagin, Terence Tao

Abstract

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.

Large gaps between consecutive prime numbers

Abstract

Let denote the size of the largest gap between consecutive primes below . Answering a question of Erdos, we show that where is a function tending to infinity with . 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.

Paper Structure

This paper contains 14 sections, 18 theorems, 152 equations, 1 table.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Notational conventions
  4. On arithmetic progressions consisting of primes
  5. Main construction
  6. Probability estimates
  7. Proof of Lemma \ref{['QQ1_normal']}
  8. A preliminary lemma
  9. Proof of Lemma \ref{['sieveunited']}
  10. Linear equations in primes with large shifts
  11. Proof of Lemma \ref{['first']}(i)
  12. Proof of Lemma \ref{['first']}(ii)
  13. Further comments and speculations
  14. Linear equations in primes

Key Result

Theorem 1

Let $R>0$. Then for any sufficiently large $X$, there are at least consecutive composite natural numbers not exceeding $X$.

Theorems & Definitions (20)

  • Theorem 1
  • Definition 1
  • Lemma 1.1
  • Theorem 2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 3: First reduction
  • Theorem 4: Second reduction
  • ...and 10 more