Describe Prime number gaps pattern by Logistic mapping
Wang Liang
基于符号动力学方法，我们证明了素数空位模式可以用Logistic映射X（k+1）=1-uX（k）^2，u=1.5437的混沌轨道来描述.如果是这样，就会有任意多个孪生素数。
Based on symbolic dynamics methods, we show the primes gap pattern could be described by the chaos orbit of Logistic mapping X(k+1)=1-uX(k)^2, u=1.5437. If so, there will be arbitrarily many twin primes.
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Primes in Tuples IV: Density of small gaps between consecutive primes
我们表明，一个积极的比例连续素数之间的所有差距是小差距。我们提供了几个定量的结果，一些无条件和一些有条件的，在这种味道。
We discuss recent advances on weak forms of the Prime $k$-tuple Conjecture, and its role in proving new estimates for the existence of small gaps between primes and the existence of large gaps between primes.
Tables of record gaps between prime constellations
我们目前的表记录（最大）之间的差距denominated总理星座，或k-元组。表中包含了素数k元组之间的所有最大间隔，最大间隔为10^15，对于每个k<=7。
Positive Proportion of Small Gaps Between Consecutive Primes
我们证明了一个积极的比例连续素数之间的差距是短的差距长度小于任何固定分数的平均间距之间的素数。
On small gaps between primes and almost prime powers
在最近与D.A. Goldston和C.Y. Yildirim我们刚刚错过了一个证明素数之间有界间隙无限频繁出现的证据。在目前的工作表明，添加到素数一个更薄的一套，所谓的几乎素数权力，工会的一套素数和几乎素数权力已经包含无限多有界的差距。更确切地说，它表明，如果我们添加到一组素数要么几乎素平方正好有两个，几乎相等的素因子，或者如果我们添加到一组素数几乎素立方体正好有三个，几乎相等的素因子，那么所产生的集合已经包含无限多个有界间隙。
We implement the Maynard-Tao method of detecting primes in tuples to investigate small gaps between primes in arithmetic progressions, with bounds that are uniform over a range of moduli.
In this paper, we study the gaps between primes in Beatty sequences following the methods in the recent breakthrough of J. Maynard.
Bounded gaps and perfect power gaps in sequences of consecutive primes
本文研究连续素数间隙中是否存在较大间隙或完美幂间隙，并探讨相关数学问题。
Expected gaps between prime numbers
我们直接通过Eratosthenes筛法来研究连续素数之间的间隔。使用初等方法，我们确定了这些差距和特定序列的连续差距，称为星座的递归关系。使用这种递归，我们可以估计一个素数和它的平方之间出现的间隙或星座的数量。这种递归也有明确的含义开放的问题之间的差距素数，包括三个问题提出的埃尔德什和图兰。
The Path to Recent Progress on Small Gaps Between Primes
本文介绍了一些想法，我们的工作中使用的小差距素数的发展。
We prove a modification as well as an improvement of a result of K. Ford, D. R. Heath-Brown and S. Konyagin concerning prime avoidance of square-free numbers and perfect powers of prime numbers.
On the ratio of consecutive gaps between primes
在目前的工作中，我们证明了一个共同的推广梅纳德-陶最近的结果连续有界的差距之间的素数和Erdens-Rankin界之间的连续素数的大差距。这项工作的答案在一个强有力的形式60岁的问题埃尔德什，问是否比两个连续primegaps可以无限经常任意小，和任意大，分别。
Large gaps between primes in arithmetic progressions -- an empirical approach
本文概述了算术级数中素数间间隙的全新穷举计算结果，并给出了异常大的最小素数的新数值结果。
For any positive integer $k$, we show that infinitely often, perfect $k$-th powers appear inside very long gaps between consecutive prime numbers, that is, gaps of size $$ c_k \frac{\log p \log_2 p \log_4 p}{(\log_3 p)^2}, $$ where $p$ is the smaller of the two primes.
Small Gaps Between Primes I
我们使用短除数和来近似素数元组和素数在短间隔的时刻。通过将这些结果与经典矩问题相联系，我们能够证明连续素数的正比例在素数之间的平均间距的四分之一以内。
In this paper, using the well known fact that the series of reciprocals of primes diverges, we obtain a general inequality for gaps of consecutive primes that holds for infinitely many primes. As it is shown the key ingredient for this direct approach is a consequence of the the Kummer's characterization of summable sequences of positive terms. Some interesting consequences are then presented. In particular, we show how the twin-prime conjecture is related to our main result.
Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim
这是一篇关于最近Goldston、Pintz和Yildirim关于素数之间的小间隔的奇妙定理的简要文章。
We derive heuristically formula for the $k$--moments $M_k(x)$ of the gaps between consecutive primes$<x $ represented directly by $x$$π(x)$ --- the number of primes up to: $M_k(x)= Γ(k+1)x^k/π^{k-1}(x)+\mathcal{O}(x)$, We illustrate obtained results by computer data.
We survey some past conditional results on the distribution of large differences between consecutive primes and examine how the Hardy-Littlewood prime k-tuples conjecture can be applied to this question.
Let $r \ge 2$ be an integer. We adapt the Maynard-Tao sieve to produce the asymptotically best-known bounded gaps between products of $r$ distinct primes. Our result applies to positive-density subsets of the primes that satisfy certain equidistribution conditions. This improves on the work of Thorne and Sono.
We show that the existence of arithmetic progressions with few primes, with a quantitative bound on "few", implies the existence of larger gaps between primes less than x than is currently known unconditionally. In particular, we derive this conclusion if there are certain types of exceptional zeros of Dirichlet L-functions.
Prime Gaps In The Gaussian Integers
本文定义了高斯整数中的素数间隙，并利用数值方法证明了其渐近上界为 O(log^2|p_n|)。
Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture
在本工作中，我们证明了一些令人惊讶的结果之间的差距连续素数和等差数列的广义孪生素数序列，不可能被证明没有最近的神奇成就张一堂连续素数之间存在有界的差距。这些结果中的大多数在十年前都属于科幻小说的范畴。然而，所提出的结果是远远没有直接后果张的著名定理：他们需要各种新的想法，其他重要的性质的应用筛功能和更密切的分析方法的Goldstone-Pintz-Yildirim，绿色陶，和张，分别。
A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which sequences are known as constellations. By studying this recursion on the cycles of gaps across stages of Eratosthenes sieve, we are able to provide evidence on a number of open problems regarding gaps between prime numbers. The basic counts of short constellations in the cycles of gaps provide evidence toward the twin prime conjecture and toward resolving a series of q…
In this paper, we apply the method of Maynard and Tao to the set of products of two distinct primes (E2-numbers). We obtain several results on the distribution of E2-numbers and primes. Among others, the result of Goldston, Pintz, Yildirim and Graham on small gaps between m consecutive E2-numbers is improved.
A poor man's improvement on Zhang's result: there are infinitely many prime gaps less than 60 million
采取简单的选择，我证明了一个可以降低张的总理差距从7000万至6000万。一个穷人的进步！
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.
On Maximal Prime Gaps
本文研究了素数间隙的问题，证明了素数与其后继素数之间的间隙不超过2log²p_n，并由此推导出对于足够大的数，存在特定类型的区间内必然存在素数。
A note on bounded gaps between primes
作为张义堂最近工作的一个改进，我们证明了：当k至少为181000时，任意可容许的k-整数元组在每一个组成部分中无限频繁地包含至少两个素数和几乎素数.这意味着在连续素数之间存在无穷多个间隔，其大小至多为2530338，并且Andrew W.萨瑟兰呼叫2326476
For each $m\geq 1$, there exist infinitely many primes $p_1<p_2<\ldots<p_{m+1}$ such that $p_{m+1}-p_1=O(m^4e^{8m})$ and $p_j+2$ has at most $\frac{16m}{\log 2}+\frac{5\log m}{\log 2}+37$ prime divisors for each $j$.
Using Duke's large sieve inequality for Hecke Gr{ö}ssencharaktere and the new sieve methods of Maynard and Tao, we prove a general result on gaps between primes in the context of multidimensional Hecke equidistribution. As an application, for any fixed $0<ε<\frac{1}{2}$, we prove the existence of infinitely many bounded gaps between primes of the form $p=a^2+b^2$ such that $|a|<ε\sqrt{p}$. Furthermore, for certain diagonal curves $\mathcal{C}:ax^α+by^β=c$, we obtain infinitely many bounded gaps between the primes $p$ such that $|p+1-\#\mathcal{C}(\mathbb{F}_p)|<ε\sqrt{p}$.
Bounded gaps between primes in special sequences
利用Maynard方法证明了序列$\{\lfloor nα\rfloor\}$中素数之间存在有界间隙，其中$α$是有限型的无理数.另外，给出了一个满足Leitmann性质的超线性函数$f$，证明了对于所有的$m$，存在无穷多个包含$m$素数和至少一个$\lfloor f(q)\rfloor$形式的整数的有界区间，其中$q$是正整数.
We study the first occurrences of gaps between primes in the arithmetic progression (P): $r$, $r+q$, $r+2q$, $r+3q,\ldots,$ where $q$ and $r$ are coprime integers, $q>r\ge1$. The growth trend and distribution of the first-occurrence gap sizes are similar to those of maximal gaps between primes in (P). The histograms of first-occurrence gap sizes, after appropriate rescaling, are well approximated by the Gumbel extreme value distribution. Computations suggest that first-occurrence gaps are much more numerous than maximal gaps: there are $O(\log^2 x)$ first-occurrence gaps between primes in (P) …
Bounded prime gaps in short intervals
本文推广了Zhang和Pintz关于有界素缺的最新结果，给出了短区间$[x,x+x (\log x)^{-A}]$中有界素对个数为6* 10 ^7的下界.我们的结果只通过分析张的定理1的证明，但我们也解释了如何张的定理2的一个尖锐的变体将意味着相同的结果为更短的时间间隔。
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]$ for large $x$. In this paper, we extend a result of Maynard and Tao concerning small gaps between primes to intervals of this length. More precisely, we prove that for any $δ\in [0.525,1]$ there exist positive integers $k,d$ such that for sufficiently large $x$, the interval $[x-x^δ,x]$ contains $\gg_{k} \frac{x^δ}{(\log x)^k}$ pairs of consecutive primes differing by at most $d$. This confirms a speculation of Maynard that results on small gaps between primes can be re…
Let $k\geq 2$ be a fixed natural number. We establish the existence of infinitely many pairs of consecutive primes $p_n$, $p_{n+1}$ satisfying $$ p_{n+1}-p_n\geq c\:\frac{\log p_n\: \log_2 p_n\: \log_4 p_n}{\log_3 p_n}\:,$$ with $c$ being a fixed positive constant, for which the interval $(p_n, p_{n+1})$ contains the $k$-th power of a prime number.
We derive heuristically approximate formulas for the negative $k$--moments $M_{-k}(x)$ of the gaps between consecutive primes$<x $ represented directly by $π(x)$ --- the number of primes up to $x$. In particular we propose an analytical formula for the sum of reciprocals of gaps between consecutive primes $<x : ~ M_{-1}(x)\sim \frac{π^2(x)}{x-2π(x)}\log\Big(\frac{x}{2π(x)}\Big) \sim x \log \log(x)/\log^2(x)$. We illustrate obtained results by the enormous computer data up to $x=4\times 10^{18}$.
On the frequency of small gaps between the primes
本文针对连续素数间隙倒数和发散所需间隙大小的问题，在Friedlander依赖深层k元组猜想的结果之外，对其弱化形式给出了无条件解答。
Suppose that $1<c<9/8$. For any $m\geq 1$, there exist infinitely many $n$ such that $$ \{[n^c],\ [(n+1)^c],\ \ldots,\ [(n+k_0)^c]\} $$ contains at least $m+1$ primes, if $k_0$ is sufficiently large (only depending on $m$).
Limit points and long gaps between primes
本文通过结合Maynard和Tao的工作，证明归一化素数间隔序列的极限点集包含至少25%的非负实数，并探讨了素数间长间隔的性质。
Question 10208b (1992) of the American Mathematical Monthly asked: does there exist an increasing sequence $\{a_k\}$ of positive integers and a constant $B > 0$ having the property that $\{ a_k + n\}$ contains no more than $B$ primes for every integer $n$? A positive answer to this question became known as Golomb's conjecture. In this note we give a negative answer, making use of recent progress in prime number theory.
Small Prime Gaps in Abelian Number Fields
我们证明了一个类似的结果Goldston，Pintz和Yildirim的素数之间的小差距，完全分裂在一个阿贝尔数域。我们证明了一个条件的结果假设的Elliott-Halbertmann猜想，和一个无条件的结果。在类数1的二次扩张的特殊情况下，我们也给出了相同结果的另一个证明，它依赖于二次数域上的Bennieri-Vinogradov定理的推广。
A new and exciting breakthrough due to Maynard establishes that there exist infinitely many pairs of distinct primes $p_1,p_2$ with $|p_1-p_2|\leq 600$ as a consequence of the Bombieri-Vinogradov Theorem. In this paper, we apply his general method to the setting of Chebotarev sets of primes. We study applications of these bounded gaps with an emphasis on ranks of prime quadratic twists of elliptic curves over $\mathbb{Q}$, congruence properties of the Fourier coefficients of normalized Hecke eigenforms, and representations of primes by binary quadratic forms.
We introduce a refinement of the GPY sieve method for studying prime $k$-tuples and small gaps between primes. This refinement avoids previous limitations of the method, and allows us to show that for each $k$, the prime $k$-tuples conjecture holds for a positive proportion of admissible $k$-tuples. In particular, $\liminf_{n}(p_{n+m}-p_n)<\infty$ for any integer $m$. We also show that $\liminf(p_{n+1}-p_n)\le 600$, and, if we assume the Elliott-Halberstam conjecture, that $\liminf_n(p_{n+1}-p_n)\le 12$ and $\liminf_n (p_{n+2}-p_n)\le 600$.
The existence of small prime gaps in subsets of the integers
我们考虑在不同的整数集合$\mathcal{C}$中寻找小素数缺口的问题。在Goldston-Pintz-Yildirim的工作之后，我们将考虑在等差数列中得到良好控制的自然数的集合。令$q_n$表示$\mathcal{C}$中的第$n$个素数，我们将建立对于任何小常数$ε>0$，集合$\left\{q_n| q_{n+1}-q_n \leq ε\log n \right\}$构成所有素数的正比例。使用的技术开发的梅纳德和陶，我们也将证明，$\mathcal{C}$有界素差距。具体的例子，如$\mathcal{C}$的情况下，是一个等差数列已经研究过，所以本文的目的是提出一般类的结果集。
A positive integer is called an $E_j$-number if it is the product of $j$ distinct primes. We prove that there are infinitely many triples of $E_2$-numbers within a gap size of $32$ and infinitely many triples of $E_3$-numbers within a gap size of $15$. Assuming the Elliot-Halberstam conjecture for primes and $E_2$-numbers, we can improve these gaps to $12$ and $5$, respectively. We can obtain even smaller gaps for almost primes, almost prime powers, or integers having the same exponent pattern in the their prime factorizations. In particular, if $d(x)$ denotes the number of divisors of $x$, we…
Real exponential sums over primes and prime gaps
研究证明在短区间内存在素数，解决了长期存在的问题，并验证了关于素数分布的旧猜想。
Small gaps between primes or almost primes
令$p_n$表示$n^{th}$素数。Goldston，Pintz和Yildirim最近证明了$ \liminf_{n\to \infty} \frac{(p_{n+1}-p_n)}{\log p_n} =0.$。我们给出了这个结果的另一种证明。我们还证明了一些相应的结果，数有两个素因子。令$q_n$表示恰好是两个不同素数的乘积的$n^{th}$数。我们证明了$\liminf_{n\to \infty} (q_{n+1}-q_n) \le 26.$如果适当的推广的Elliott-Halberten猜想是真的，那么上述界可以提高到6。
Let $α, β\geq 0$ and $α+ β< 1$. In this short note, we show that $\liminf_{n \to \infty} p_n^β(p_{n+1}^α- p_n^α) = 0$, where $p_n$ is the $n$th prime. This notes an improvement over results of Sándor and gives additional evidence towards a conjecture of Andrica. This follows directly from recent results on prime pairs from Maynard, Tao, Zhang.
Most prime gaps results have been proven using tools from analytic or algebraic number theory in the last few centuries. In this paper, we would like to present some probabilistic way of proving many essential results. A major component of the proof is a probabilistic approach to the sieve method. In addition, we discuss their connections with recent work by Zhang and Maynard on small and large gaps in prime numbers.
Note On Prime Gaps And Very Short Intervals
本文在黎曼假设的前提下，讨论了一个新的初等论证，它似乎证明了一个有限序列的素数p_1，p_2，.的最大素数间隔，p_n <= x，满足max {p_（n+1）- p_n：p_n <= x} <= c1（（logx）^2）/loglogx，c1 > 0常数。等价地，它表明非常短的区间（x，x + y）包含素数，对于所有y > c2（（logx）^2）/loglogx，c2 > 0常数，并且x足够大> 0。
Let $p_n$ denote the $n$th prime and $g_n:=p_{n+1}-p_n$ the $n$th prime gap. We demonstrate the existence of infinitely many values of $n$ for which $g_n>g_{n+1}>\cdots>g_{n+m}$ with $m\gg \log\log\log n$ and similarly for the reversed inequalities. In doing so we settle a conjecture of Erdös for the case $m=2$.
On smooth gaps between primes using the Maynard-Tao sieve
本文利用 Zhang、Maynard 和 Polymath8b 在孪生素数猜想上的进展及 Maynard-Tao 筛法，将素数间光滑间隙的界限降低至 47，并证明存在无穷多个间隙均为 $y_m$-光滑的 $m$-元组素数。
In the framework of Cramer's probabilistic model of primes, we explore the exact and asymptotic distributions of maximal prime gaps. We show that the Gumbel extreme value distribution exp(-exp(-x)) is the limit law for maximal gaps between Cramer's random primes. The result can be derived from a general theorem about intervals between discrete random events occurring with slowly varying probability monotonically decreasing to zero. A straightforward generalization extends the Gumbel limit law to maximal gaps between prime constellations in Cramer's model.
Erdős considered the second moment of the gap-counting function of prime divisors in 1946 and proved an upper bound that is not of the right order of magnitude. We prove asymptotics for all moments. Furthermore, we prove a generalisation stating that the gaps between primes $p$ for which there is no $\mathbb{Q}_p$-point on a random variety are Poisson distributed.
Bounded gaps between primes in number fields and function fields
Hardy-Littlewood素数$k$ -元组猜想一直被认为是用现有方法完全不可接近的。虽然这仍然是事实，但张，梅纳德和陶的惊人突破仍然在这个问题上取得了重大进展。本文将Maynard-Tao方法推广到数域和函数域$\mathbb{F}_q(t)$。
Density-based structural frameworks for prime numbers, prime gaps, and Euler products
本文提出基于密度的结构框架，用于分析素数、素数间隙和欧拉乘积，揭示了Hardy-Littlewood、Cramer和PNT预测之间的张力，并通过截断欧拉乘积分析ζ函数的临界线与零点分布。
A famous conjecture of Artin states that there are infinitely many prime numbers for which a fixed integer $g$ is a primitive root, provided $g \neq -1$ and $g$ is not a perfect square. Thanks to work of Hooley, we know that this conjecture is true, conditional on the truth of the Generalized Riemann Hypothesis. Using a combination of Hooley's analysis and the techniques of Maynard-Tao used to prove the existence of bounded gaps between primes, Pollack has shown that (conditional on GRH) there are bounded gaps between primes with a prescribed primitive root. In the present article, we provide …
Let $q>r\ge1$ be coprime integers. Let ${\mathbb P}_c={\mathbb P}_c(q,r,{\cal H})$ be an increasing sequence of primes $p$ satisfying two conditions: (i) $p\equiv r$ (mod $q$) and (ii) $p$ starts a prime $k$-tuple with a given pattern ${\cal H}$. Let $π_c(x)$ be the number of primes in ${\mathbb P}_c$ not exceeding $x$. We heuristically derive formulas predicting the growth trend of the maximal gap $G_c(x)=\max_{p'\le x}(p'-p)$ between successive primes $p,p'\in{\mathbb P}_c$. Extensive computations for primes up to $10^{14}$ show that a simple trend formula $$G_c(x) \sim {x\overπ_c(x)}\cdot(\…
We propose the formula for the number of pairs of consecutive primes $p_n, p_{n+1}<x$ separated by gap $d=p_{n+1}-p_n$ expressed directly by the number of all primes $<x$, i.e. by $π(x)$. As the application of this formula we formulate 7 conjectures, among others for the maximal gap between two consecutive primes smaller than $x$, for the generalized Brun's constants and the first occurrence of a given gap $d$. Also the leading term $\log \log(x)$ in the prime harmonic sum is reproduced from our guesses correctly. These conjectures are supported by the computer data.
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