# Long gaps between primes

@article{Ford2014LongGB, title={Long gaps between primes}, author={Kevin Ford and Ben Green and Sergei Konyagin and James Maynard and Terence Tao}, journal={arXiv: Number Theory}, year={2014} }

Let $p_n$ denotes the $n$-th prime. We prove that $$\max_{p_{n+1} \leq X} (p_{n+1}-p_n) \gg \frac{\log X \log \log X\log\log\log\log X}{\log \log \log X}$$ for sufficiently large $X$, improving upon recent bounds of the first three and fifth authors and of the fourth author. Our main new ingredient is a generalization of a hypergraph covering theorem of Pippenger and Spencer, proven using the R\"odl nibble method.

## 60 Citations

Large gaps between consecutive prime numbers

- Mathematics
- 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…

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

- Mathematics
- 2015

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…

On the distribution of maximal gaps between primes in residue classes.

- Mathematics
- 2016

Let $q>r\ge1$ be coprime positive integers. We empirically study the maximal gaps $G_{q,r}(x)$ between primes $p=qn+r\le x$, $n\in{\mathbb N}$. Extensive computations suggest that almost always…

A lower bound for the least prime in an arithmetic progression

- Mathematics
- 2016

Fix $k$ a positive integer, and let $\ell$ be coprime to $k$. Let $p(k,\ell)$ denote the smallest prime equivalent to $\ell \pmod{k}$, and set $P(k)$ to be the maximum of all the $p(k,\ell)$. We seek…

CYCLOTOMIC POLYNOMIALS WITH PRESCRIBED HEIGHT AND PRIME NUMBER THEORY

- Mathematics
- 2019

Given any positive integer $n,$ let $A(n)$ denote the height of the $n^{\text{th}}$ cyclotomic polynomial, that is its maximum coefficient in absolute value. It is well known that $A(n)$ is…

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…

On the gaps between consecutive primes

- Mathematics, Computer ScienceForum Mathematicum
- 2022

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.

## References

SHOWING 1-10 OF 77 REFERENCES

Large gaps between consecutive prime numbers

- Mathematics
- 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…

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

- Mathematics
- 2015

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…

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…

Dense clusters of primes in subsets

- Mathematics, Computer ScienceCompositio Mathematica
- 2016

It is proved that any subset of the primes which is ‘well distributed’ in arithmetic progressions contains many primesWhich are close together, and bounds hold with some uniformity in the parameters.

A lower bound for the least prime in an arithmetic progression

- Mathematics
- 2016

Fix $k$ a positive integer, and let $\ell$ be coprime to $k$. Let $p(k,\ell)$ denote the smallest prime equivalent to $\ell \pmod{k}$, and set $P(k)$ to be the maximum of all the $p(k,\ell)$. We seek…

Large gaps between consecutive prime numbers containing perfect powers

- Mathematics
- 2015

For any positive integer k, we show that infinitely often, perfect kth powers appear inside very long gaps between consecutive prime numbers, that is, gaps of size
$$\displaystyle{c_{k}\frac{\log…

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…

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…

An inverse theorem for the Gowers U^{s+1}[N]-norm

- Mathematics
- 2010

We prove the inverse conjecture for the Gowers U^{s+1}[N]-norm for all s >= 3; this is new for s > 3, and the cases s [-1,1] is a function with || f ||_{U^{s+1}[N]} > \delta then there is a…