# 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}
}
• Published 16 December 2014
• Mathematics, Computer Science
• arXiv: Number Theory
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.
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 $$G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log Limit points and long gaps between primes • Mathematics • 2015 Let d_n = p_{n+1} - p_n, where p_n denotes the nth smallest prime, and let R(T) = \log T \log_2 T\log_4 T/(\log_3 T)^2 (the "Erd{\H o}s--Rankin" function). We consider the sequence Long gaps in sieved sets • Mathematics • 2018 For each prime p, let I_p \subset \mathbb{Z}/p\mathbb{Z} denote a collection of residue classes modulo p such that the cardinalities |I_p| are bounded and about 1 on average. We show that Large Gaps between Primes in Arithmetic Progressions For (M,a)=1, put \begin{equation*} G(X;M,a)=\sup_{p^\prime_n\leq X}(p^\prime_{n+1}-p^\prime_n), \end{equation*} where p^\prime_n denotes the n-th prime that is congruent to a\pmod{M}. We show On the distribution of maximal gaps between primes in residue classes. Let q>r\ge1 be coprime positive integers. We empirically study the maximal gaps G_{q,r}(x) between primes p=qn+r\le x, n\in{\mathbb N}. Extensive computations suggest that almost always A lower bound for the least prime in an arithmetic progression • Mathematics • 2016 Fix k a positive integer, and let \ell be coprime to k. Let p(k,\ell) denote the smallest prime equivalent to \ell \pmod{k}, and set P(k) to be the maximum of all the p(k,\ell). We seek CYCLOTOMIC POLYNOMIALS WITH PRESCRIBED HEIGHT AND PRIME NUMBER THEORY • Mathematics • 2019 Given any positive integer n, let A(n) denote the height of the n^{\text{th}} cyclotomic polynomial, that is its maximum coefficient in absolute value. It is well known that A(n) is Bounded gaps between primes in short intervals • Mathematics • 2017 Baker, Harman, and Pintz showed that a weak form of the Prime Number Theorem holds in intervals of the form$$[x-x^{0.525},x]$$[x-x0.525,x] for large x. In this paper, we extend a result of Maynard On the gaps between consecutive primes • Mathematics, Computer Science Forum Mathematicum • 2022 The result concerning the least primes in arithmetic progressions concerning then-th prime is obtained and a related result about the small primes is obtained. ## References SHOWING 1-10 OF 77 REFERENCES Large gaps between consecutive prime numbers • Mathematics • 2014 Let G(X) denote the size of the largest gap between consecutive primes below X. Answering a question of Erdos, we show that$$G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log
Large gaps between 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
• 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
• J. Maynard
• Mathematics, Computer Science
Compositio 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
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