# 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} }

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…

## Tables from this paper

## 47 Citations

Large gaps between primes

- Mathematics
- 2014

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…

Long gaps between primes

- Mathematics, Computer Science
- 2014

The main new ingredient is a generalization of a hypergraph covering theorem of Pippenger and Spencer, proven using the R\"odl nibble method.

On the gaps between consecutive primes

- Mathematics, Computer Science
- 2018

The result concerning the least primes in arithmetic progressions concerning then-th prime is obtained and a related result about the small primes is obtained.

Large gaps between consecutive prime numbers containing perfect $k$-th powers of prime numbers

- Mathematics
- 2015

Bounded gaps between primes in short intervals

- Mathematics
- 2017

Baker, Harman, and Pintz showed that a weak form of the Prime Number Theorem holds in intervals of the form $$[x-x^{0.525},x]$$[x-x0.525,x] for large x. In this paper, we extend a result of Maynard…

An Algorithm to Generate Random Factored Smooth Integers.

- Mathematics, Computer Science
- 2020

This work presents an algorithm that will generate an integer n at random, with known prime factorization, such that every prime divisor of $n$ is $\le y$ and presents other running times based on differing sets of assumptions and heuristics.

Chains of large gaps between primes

- Mathematics
- 2018

Let pn denote the n-th prime, and for any \(k \geqslant 1\) and sufficiently large X, define the quantity
$$\displaystyle G_k(X) := \max _{p_{n+k} \leqslant X} \min ( p_{n+1}-p_n, \dots ,…

Limit points and long gaps between primes

- Mathematics
- 2015

Let $d_n = p_{n+1} - p_n$, where $p_n$ denotes the $n$th smallest prime, and let $R(T) = \log T \log_2 T\log_4 T/(\log_3 T)^2$ (the "Erd{\H o}s--Rankin" function). We consider the sequence…

Long gaps in sieved sets

- Mathematics
- 2018

For each prime $p$, let $I_p \subset \mathbb{Z}/p\mathbb{Z}$ denote a collection of residue classes modulo $p$ such that the cardinalities $|I_p|$ are bounded and about $1$ on average. We show that…

Large Gaps between Primes in Arithmetic Progressions

- Mathematics
- 2018

For $(M,a)=1$, put \begin{equation*} G(X;M,a)=\sup_{p^\prime_n\leq X}(p^\prime_{n+1}-p^\prime_n), \end{equation*} where $p^\prime_n$ denotes the $n$-th prime that is congruent to $a\pmod{M}$. We show…

## References

SHOWING 1-10 OF 60 REFERENCES

Large gaps between primes

- Mathematics
- 2014

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…

Long gaps between primes

- Mathematics, Computer Science
- 2014

The main new ingredient is a generalization of a hypergraph covering theorem of Pippenger and Spencer, proven using the R\"odl nibble method.

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

- Mathematics
- 1997

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

- Mathematics
- 1963

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

- Mathematics
- 2013

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

- Mathematics
- 2012

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…