# 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.