# 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}
}
• Published 20 August 2014
• Mathematics
• arXiv: Number Theory
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…
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
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.
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
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
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