# Large gaps between primes

@article{Maynard2014LargeGB, title={Large gaps between primes}, author={James Maynard}, journal={arXiv: Number Theory}, year={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 proof works by incorporating recent progress in sieve methods related to small gaps between primes into the Erdos-Rankin construction. This answers a well-known question of Erdos.

## 26 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…

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.

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…

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
- 2018

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.

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

- Mathematics
- 2015

Note On The Maximal Primes Gaps

- Mathematics
- 2013

This note presents a result on the maximal prime gap of the form p_(n+1) - p_n 0 is a constant, for any arbitrarily small real number e > 0, and all sufficiently large integer n > n_0. Equivalently,…

On the Ratio of Consecutive Gaps Between Primes

- Mathematics
- 2015

In the present work we prove a common generalization of Maynard–Tao’s recent result about consecutive bounded gaps between primes and of the Erdős–Rankin bound about large gaps between consecutive…

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…

On the distribution of gaps between consecutive primes

- Mathematics
- 2014

Erd\"os conjectured that the set J of limit points of d_n/logn contains all nonnegative numbers, where d_n denotes the nth primegap. The author proved a year ago (arXiv: 1305.6289) that J contains an…

## References

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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…

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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…

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Abstract LetG(X) denote the largest gap between consecutive primes belowX. Improving earlier results of Erdős, Rankin, Schonhage, and Maier-Pomerance, we prove G(X)⩾(2e γ +o(1)) log Xlog 2 Xlog 4…

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We introduce a renement of the GPY sieve method for studying prime k-tuples and small gaps between primes. This renement avoids previous limitations of the method and allows us to show that for each…

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- Mathematics
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is greater than (c2/2)n. A simple calculation now shows that the primes satisfying (4) also satisfy the first inequality of (3) iÎ€ = e(ci) is chosen small enough. The second inequality of (3) is…

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Let p n denote the n th prime and let e be any positive number. In 1938 (3) Ishowed that, for an infinity of values of n , where, for k ≧1, log k +1 x = log (log k x ) and log 1 x = log x . In a…

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- Mathematics
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I = lim inf v n-a* log n The purpose of this paper is to combine the methods used in two earlier papers' in order to prove the following theorem. THEOREM. (1) 1 5 c(1 + 4e)/5, where c 1 -c, so that…

On the difference between consecutive prime numbers

- Mathematics
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/ = lim inf ̂ 11-tl . »->» log pn The purpose of this paper is to combine the methods used in two earlier papers1 in order to prove the following theorem. Theorem. (1) / = c(l + 40)/5, where c<…

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

- Mathematics
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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…

Linear equations in primes

- Mathematics
- 2006

Consider a system ψ of nonconstant affine-linear forms ψ 1 , ... , ψ t : ℤ d → ℤ, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [-N, N] d be convex. A generalisation…