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. 
View PDF on arXiv
60 Citations
Highly Influential Citations
10
Background Citations
31
Methods Citations
6
Results Citations
3

60 Citations

Large gaps between consecutive prime numbers
Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdos, we show that $$G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log…
Large gaps between consecutive prime numbers containing perfect $k$-th powers of prime numbers
Limit points and long gaps between primes
Let $d_n = p_{n+1} - p_n$, where $p_n$ denotes the $n$th smallest prime, and let $R(T) = \log T \log_2 T\log_4 T/(\log_3 T)^2$ (the "Erd{\H o}s--Rankin" function). We consider the sequence…
Long gaps in sieved sets
For each prime $p$, let $I_p \subset \mathbb{Z}/p\mathbb{Z}$ denote a collection of residue classes modulo $p$ such that the cardinalities $|I_p|$ are bounded and about $1$ on average. We show that…
Large Gaps between Primes in Arithmetic Progressions
For $(M,a)=1$, put \begin{equation*} G(X;M,a)=\sup_{p^\prime_n\leq X}(p^\prime_{n+1}-p^\prime_n), \end{equation*} where $p^\prime_n$ denotes the $n$-th prime that is congruent to $a\pmod{M}$. We show…
On the distribution of maximal gaps between primes in residue classes.
Let $q>r\ge1$ be coprime positive integers. We empirically study the maximal gaps $G_{q,r}(x)$ between primes $p=qn+r\le x$, $n\in{\mathbb N}$. Extensive computations suggest that almost always…
A lower bound for the least prime in an arithmetic progression
Fix $k$ a positive integer, and let $\ell$ be coprime to $k$. Let $p(k,\ell)$ denote the smallest prime equivalent to $\ell \pmod{k}$, and set $P(k)$ to be the maximum of all the $p(k,\ell)$. We seek…
CYCLOTOMIC POLYNOMIALS WITH PRESCRIBED HEIGHT AND PRIME NUMBER THEORY
Given any positive integer $n,$ let $A(n)$ denote the height of the $n^{\text{th}}$ cyclotomic polynomial, that is its maximum coefficient in absolute value. It is well known that $A(n)$ is…
Bounded gaps between primes in short intervals
Baker, Harman, and Pintz showed that a weak form of the Prime Number Theorem holds in intervals of the form $$[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
TLDR
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
Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdos, we show that $$G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log…
Large gaps between consecutive prime numbers containing perfect $k$-th powers of prime numbers
Large gaps between primes
We show that there exists pairs of consecutive primes less than $x$ whose difference is larger than $t(1+o(1))(\log{x})(\log\log{x})(\log\log\log\log{x})(\log\log\log{x})^{-2}$ for any fixed $t$. Our…
Limit points and long gaps between primes
Let $d_n = p_{n+1} - p_n$, where $p_n$ denotes the $n$th smallest prime, and let $R(T) = \log T \log_2 T\log_4 T/(\log_3 T)^2$ (the "Erd{\H o}s--Rankin" function). We consider the sequence…
Dense clusters of primes in subsets
  • J. Maynard
  • Mathematics, Computer Science
    Compositio Mathematica
  • 2016
TLDR
It is proved that any subset of the primes which is ‘well distributed’ in arithmetic progressions contains many primesWhich are close together, and bounds hold with some uniformity in the parameters.
A lower bound for the least prime in an arithmetic progression
Fix $k$ a positive integer, and let $\ell$ be coprime to $k$. Let $p(k,\ell)$ denote the smallest prime equivalent to $\ell \pmod{k}$, and set $P(k)$ to be the maximum of all the $p(k,\ell)$. We seek…
Large gaps between consecutive prime numbers containing perfect powers
For any positive integer k, we show that infinitely often, perfect kth powers appear inside very long gaps between consecutive prime numbers, that is, gaps of size $$\displaystyle{c_{k}\frac{\log…
On the difference between consecutive primes
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
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
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…
...
...