Alex V. Kontorovich
Selberg identified the "parity" barrior, that sieves alone cannot distinguish between integers having an even or odd number of factors. We give here a short and self-contained demonstration of parity breaking using bilinear forms, modeled on the Twin Primes Conjecture.
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We study a special set of constellations of primes generated by twin primes.
A classification of twin primes implies special twin primes. When applied to triplets, it yields exceptional prime number triplets. These generalize yielding exceptional prime number multiplets.
Using the Rowland idea, we find two infinite sets of generators of primes. We also pose some conjectures concerning twin primes.
We discuss various recent advances on weak forms of the Twin Prime Conjecture.
We resolve the function field analogue of the conjecture concerning distribution of twin primes in arithmetic progression.
The Legendre type relation for the counting function of ordinary twin primes is reworked in terms of the inverse of the Riemann zeta function. Its analysis sheds light on the distribution of the zeros of the Riemann zeta function in the critical strip and their links to primes and the twin prime problem.
We give an account of the arguments that lead from the assumption of the existence of exceptional characters to the asymptotics in related ranges for the counting function of twin primes.
A new explicit formula is proved for the contribution of the major arcs in the Goldbach and Generalized Twin Prime Problem, in which the level of the major arcs can be chosen very high. This will have many applications in the approximations of the mentioned problems.
For earlier considered our sequence A166944 in [4] we prove three statements of its connection with twin primes. We also give a sufficient condition for the infinity of twin primes and pose several new conjectures; among them we propose a very simple conjectural algorithm of constructing a pair $(p,\enskip p+2)$ of twin primes over arbitrary given integer $m\geq4$ such that $p+2\geq m.$
We give a more comrehensive treatment of Chen's double sieve and improve related constants in Goldbach's conjecture and the twin prime problem.
We adopt an empirical approach to the characterization of the distribution of twin primes within the set of primes, rather than in the set of all natural numbers. The occurrences of twin primes in any finite sequence of primes are like fixed probability random events. As the sequence of primes grows, the probability decreases as the reciprocal of the count of primes to that point. The manner of the decrease is consistent with the Hardy--Littlewood Conjecture, the Prime Number Theorem, and the Twin Prime Conjecture. Furthermore, our probabilistic model, is simply parameterized. We discuss a sim…
Classifications of twin primes are established and then applied to triplets that generalize to all higher multiplets. Mersenne and Fermat twins and triplets are treated in this framework. Regular prime number multiplets are related to quadratic and cubic prime number generating polynomials.
An Equivalent form of Twin Prime Conjecture connected with a sequence of arithmetic progressions
本文提出了一种与算术进展序列对称性质相关的 twin素数猜想等价形式。
A serious error has been found in the paper, specifically, Lemma 8 is incorrect.
Based on Golomb's arithmetic formulas, Dirichlet series for two classes of twin primes are constructed and related to the roots of the Riemann zeta function in the critical strip.
We bring to bear an empirical model of the distribution of twin primes and produce two distinct results. The first is that we can make a quantitative probabilistic prediction of the occurrence of gaps in the sequence of twins within the primes. The second is that the ``high jumper'' i.e., the separation with greatest likelihood (in terms of primes) is always expected to be zero.
Twin primes and the $3$-sphere
本文通过研究素数模下 $3$-球面的点群及其与已知群、Chebyshev 多项式的联系，构造了一个收敛性等价于孪生素数对有限性的无穷级数，从而将证明孪生素数猜想转化为证明该级数发散。
A Positive Lower Bound for the Sum of Log-Reciprocal Twin Prime Products via Weighted Sieve
本文通过改进加权筛法，证明了孪生素数乘积对数倒数之和的正下界，从而推导出孪生素数对的无限性。
Numerical evidence suggests that for only about $2\%$ of pairs $p,p+2$ of twin primes, $p+2$ has more primitive roots than does $p$. If this occurs, we say that $p$ is exceptional (there are only two exceptional pairs with $5 \leq p \leq 10{,}000$). Assuming the Bateman-Horn conjecture, we prove that at least $0.47\%$ of twin prime pairs are exceptional and at least $65.13\%$ are not exceptional. We also conjecture a precise formula for the proportion of exceptional twin primes.
We use the methods developed in our papers on moments and divisor correlations to derive heuristically the conjectural ratios formula for two zetas over two zetas.
We improve Bombieri's asymptotic sieve to localise the variables. As a consequence, we prove, under a Elliott-Halberstam conjecture, that there exists an infinity of twins almost prime. Those are prime numbers $p$ such that for all $\varepsilon>0$, $p-2$ is either a prime number or can be written as $p_1 p_2$ where $p_1$ and $p_2$ are prime and $p_1<X^{\varepsilon}$, and we give the explicit asymptotic.
We present the geometry lying behind counting twin prime polynomials in $\mathbb{F}_q[T]$ in general. We compute cohomology and explicitly count points by means of a twisted Lefschetz trace formula applied to these parametrizing varieties for cubic twin prime polynomials. The elliptic curve $X^3 = Y(Y-1)$ occurs in the geometry, and thus counting cubic twin prime polynomials involves the associated modular form. In theory, this approach can be extended to higher degree twin primes, but the computations become harder. The formula we get in degree $3$ is compatible with the Hardy-Littlewood heur…
We show that if besides the primes some other sequences (involving the Liouville function and the primes) have a common distribution level exceeding 0.7231 then for any positive even integer $h$ there are arbitrarily long arithmetic progressions of primes $p$ such that $p+h$ is also prime for each element of the progression. In case of $h=2$ this means that under some plausible unproved hypotheses about regular distribution of the primes and other sequences in arithmetic progressions we really have arbitrarily long arithmetic progressions in the sequence of twin primes.
For every even integer N, denote by D_{1,2}(N) the number of representations of N as a sum of a prime and an integer having at most two prime factors. In this paper, we give a new lower bound for D_{1,2}(N).
Sieves are constructed for twin primes in class I, which are of the form 2m+/-D, D>=3 odd. They are characterized by their twin-D-I rank m. They have no parity problem. Non-rank numbers are identified and counted using odd primes p>=5. Twin-D-I ranks and non-ranks make up the set of positive integers. Regularities of non-ranks allow obtaining the number of twin-D-I ranks. It involves considerable cancellations so that the asymptotic form of its main term collapses to the expected form, but its coefficient depends on D.
A probabilistic approach to the twin prime and cousin prime conjectures
本文从概率角度探讨孪生素数与表兄弟素数对是否无限多的问题，通过划分连续素数平方区间并结合Merten第三定理与二项分布性质证明包含概率趋于1，进而提出更严格的新猜想并证明二者具有相同的渐近分布。
In this paper we prove two results concerning Vinogradov's three primes theorem with primes that can be called almost twin primes. First, for any $m$, every sufficiently large odd integer $N$ can be written as a sum of three primes $p_1, p_2$ and $p_3$ such that, for each $i \in \{1,2,3\}$, the interval $[p_i, p_i + H]$ contains at least $m$ primes, for some $H = H(m)$. Second, every sufficiently large integer $N \equiv 3 \pmod{6}$ can be written as a sum of three primes $p_1, p_2$ and $p_3$ such that, for each $i \in \{1,2,3\}$, $p_i + 2$ has at most two prime factors.
Arithmetic closed forms count the Mersenne primes, the Fermat primes and the twin-prime pairs
本文提出通过算术闭式生成和计数梅森素数、费马素数、孪生素数对及索菲·热尔曼素数的方法，结合威尔逊定理和特定测试技术。
An intermediate conjecture between Goldbach and Dubner: every even number is the sum of a prime and a twin prime
该研究提出介于哥德巴赫与杜布纳猜想间的中间猜想，用罗曼诺夫方法结合塞尔伯格筛法给出条件性结果，验证了10^14以内偶数均满足该猜想。
Jing Run Chen proved in 1966 that $p+2$ has at most two prime factors for infinitely many primes $p$. However, due to the parity problem we do not know whether $p+2$ has an odd (or even) number of prime factors infinitely often. In the present work it is proved that $p+d$ has an odd number of prime factors for at least one value of d=2,4,...16.
The computer data up to $2^{44}\approx 1.76\times 10^{13}$ on the gaps between consecutive twins is presented. The simple derivation of the heuristic formula describing computer results contained in the recent papers by P.F.Kelly and T.Pilling \cite{Kelly1}, \cite{Kelly2} is provided and compared with the ``experimental'' values.
A Generalized Elliott-Halberstam Conjecture Implying the Twin Prime Hypothesis
本研究提出关乎算术级数素数对分布的移位卷积广义Elliott-Halberstam猜想（GEH-2），给出von Mangoldt函数相关性的分布水平，证明其蕴含孪生素数猜想并讨论相关数论意义。
We present an algorithm analogous to the sieve of Eratosthenes that produces the list of twin primes. Next, we count the number of twin primes resulting from the construction with two different heuristic arguments. The first method is essentially the same as the one in Hardy and Wright. However, the second method is novel. It results in the same asymptotic formula but it uses a simpler correction factor than theirs. Though we have no theory for the accuracy of our estimates, we compute them both without and with the correction factor and they turn out to be close to the actual counts up to 800…
This is an exposition of recent developments in the theory of bounded differences between primes. Readers are expected to be beginners of analytic number theory. The present text is a substantially improved and augmented version of the one that I had prepared for my talk which I delivered at the Annual Meeting of the Mathematical Society of Japan on 15 March 2014.
Let E be an elliptic curve over Q. In 1988, Koblitz conjectured a precise asymptotic for the number of primes p up to x such that the order of the group of points of E over the finite field F_p is prime. This is an analogue of the Hardy and Littlewood twin prime conjecture in the case of elliptic curves. Koblitz's conjecture is still widely open. In this paper we prove that Koblitz's conjecture is true on average over a two-parameter family of elliptic curves. One of the key ingredients in the proof is a short average distribution result in the style of Barban-Davenport-Halberstam, where the a…
The main result of the paper is that assuming that the level $θ$ of distribution of primes exceeds 1/2, then there exists a positive $d\leq C(θ)$ such that there are arbitrarily long arithmetic progressions with the property that $p'=p+d$ is the next prime for each element of the progression. If $θ>0.971$, then the above holds for some $d\leq 16$.
Infinitely many twin prime polynomials of odd degree
本文在有限域框架下，构造了一类新的无穷多奇数次孪生素数多项式组，并探讨其与Wieferich素数存在性及Bell数算术性质的关联，拓展了有限域上孪生素数问题的构造性研究。
The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that if $x \ge R_n$, then the interval $(x/2,x]$ contains at least $n$ primes. We sharpen Laishram's theorem that $R_n < p_{3n}$ by proving that the maximum of $R_n/p_{3n}$ is $R_5/p_{15} = 41/47$. We give statistics on the length of the longest run of Ramanujan primes among all primes $p<10^n$, for $n\le9$. We prove that if an upper twin prime is Ramanujan, then so is the lower; a table gives the number of twin primes below $10^n$ of three types. Finally, we relate runs of Ramanujan primes to prime gaps. Along the way we st…
In his Classical approximation to the Twin prime problem, Selberg proved that for $x$ sufficiently large, there is an $n \in (x,2x)$ such that $2^{Ω(n)}+2^{Ω(n+2)} \leq λ$ with $λ=14$, where $Ω(n)$ is the number of prime factors of $n$ counted with multiplicity. This enabled him to show that for infinitely many $n$, $n(n+2)$ has atmost $5$ prime factors, with one having atmost $2$ and the other having atmost $3$ prime factors. By adopting Selberg's approach and using a refinement suggested by Selberg, we improve this value of $λ$ to about $λ=12.59$.
Garcia, Kahoro, and Luca showed that the Bateman-Horn conjecture implies $ϕ(p-1) \geq ϕ(p+1)$ for a majority of twin-primes pairs $p,p+2$ and that the reverse inequality holds for a small positive proportion of the twin primes. That is, $p$ tends to have more primitive roots than does $p+2$. We prove that Dickson's conjecture, which is much weaker than Bateman-Horn, implies that the quotients $\frac{ϕ(p+1)}{ϕ(p-1)}$, as $p,p+2$ range over the twin primes, are dense in the positive reals. We also establish several Schinzel-type theorems, some of them unconditional, about the behavior of $\frac{…
Recently we have introduced a novel characterisation of the distribution of twin primes that consists of three essential elements. These are: that the twins are most naturally viewed as a subsequence of the primes themselves, that the likelihood of a particular prime in sequence being the first element of a twin is akin to a fixed-probability random event, and that this probability varies with $π_{1}$, the count of primes up to this number, in a simple way. Our initial studies made use of two unproven assumptions: that it was consistent to model this fundamentally discrete system with a contin…
We consider the exceptional set in the binary Goldbach problem for sums of two almost twin primes. Our main result is a power-saving bound for the exceptional set in the problem of representing $m=p_1+p_2$ where $p_1+2$ has at most $2$ prime divisors and $p_2+2$ has at most $3$ prime divisors. There are three main ingredients in the proof: a new transference principle like approach for sieves, a combination of the level of distribution estimates of Bombieri--Friedlander--Iwaniec and Maynard with ideas of Drappeau to produce power savings, and a generalisation of the circle method arguments of …
We study the number of ways to decompose a monic polynomial in F_q[t] of degree n as a sum of two monic irreducible polynomials in F_q[t]. Our principal result is an asymptotic formula for the number of such representations in the case when q is large compared to n. In its range of validity, this formula agrees with what is suggested by heuristic arguments from the rational setting. We also present similar results towards an analogue of the twin prime conjecture.
In Wilson's Theorem the primality of a number hinges on a congruence. We present a similar test where the primality of a number m hinges, instead, on the indivisibility of 4(m-5)! by m. One implication of this theorem is a necessary and sufficient condition for two numbers to be twin primes, a result reminiscent of Clement's theorem but involving indivisibility. MSC: 11A41 and 11A51.
This is an extension and background to a talk I gave on 9 October 2013 to the Brown Graduate Student Seminar, called `A friendly intro to sieves with a look towards recent progress on the twin primes conjecture.' During the talk, I mention several sieves, some with a lot of detail and some with very little detail. I also discuss several results and built upon many sources. I'll provide missing details and/or sources for additional reading here.
We say that a prime number $p$ is an $\textit{Artin prime}$ for $g$ if $g$ mod $p$ generates the group $(\mathbb{Z}/p\mathbb{Z})^{\times}$. For appropriately chosen integers $d$ and $g$, we present a conjecture for the asymptotic number $π_{d,g}(x)$ of primes $p \leq x$ such that both $p$ and $p+d$ are Artin primes for $g$. In particular, we identify a class of pairs $(d,g)$ for which $π_{d,g}(x) =0$. Our results suggest that the distribution of Artin prime pairs, amongst the ordinary prime pairs, is largely governed by a Poisson binomial distribution.
A hypothesis is put forward regarding the function $π_2(x)$ which describes the distribution of twin primes in the set of natural numbers. The function $π_2(x)$ is tested by evaluation and an empirical $π_2^{\ast}(x)$ is arrived at, which is shown to be highly accurate. Several other questions are also addressed.
We study the distribution of prime numbers under the unlikely assumption that Siegel zeros exist. In particular we prove for \[ \sum_{n \leq X} Λ(n) Λ(\pm n+h) \] an asymptotic formula which holds uniformly for $h = O(X)$. Such an asymptotic formula has been previously obtained only for fixed $h$ in which case our result quantitatively improves those of Heath-Brown (1983) and Tao and Teräväinen (2021). Since our main theorems work also for large $h$ we can derive new results concerning connections between Siegel zeros and the Goldbach conjecture and between Siegel zeros and primes in almost al…
The transference principle of Green and Tao enabled various authors to transfer Szemerédi's theorem on long arithmetic progressions in dense sets to various sparse sets of integers, mostly sparse sets of primes. In this paper, we provide a transference principle which applies to general affine-linear configurations of finite complexity. We illustrate the broad applicability of our transference principle with the case of almost twin primes, by which we mean either Chen primes or "bounded gap primes", as well as with the case of primes of the form $x^2+y^2+1$. Thus, we show that in these sets of…
The results of the computer investigation of the sign changes of the difference between the number of twin primes $π_2(x)$ and the Hardy--Littlewood conjecture $c_2\Li_2(x)$ are reported. It turns out that $π_2(x) - c_2\Li_2(x)$ changes the sign at unexpectedly low values of $x$ and for $x<2^{42}$ there are over 90000 sign changes of this difference. It is conjectured that the number of sign changes of $π_2(x) - c_2\Li_2(x)$ for $x\in (1, T)$ is given by $\sqrt T/\log(T)$.
This article is a collected information from some books and papers, and in most cases the original sentences is reserved about twin prime conjecture.
Using geometric methods, we improve on the function field version of the Burgess bound, and show that, when restricted to certain special subspaces, the Möbius function over $\mathbb F_q[T]$ can be mimicked by Dirichlet characters. Combining these, we obtain a level of distribution close to $1$ for the Möbius function in arithmetic progressions, and resolve Chowla's $k$-point correlation conjecture with large uniformity in the shifts. Using a function field variant of a result by Fouvry-Michel on exponential sums involving the Möbius function, we obtain a level of distribution beyond $1/2$ for…
Let $p, q$ be twin prime numbers with $q-p=2$ . Consider the elliptic curves E=E_σ: y^2 = x (x+σp)(x+σq) . (σ=\pm 1). E=E_σis also denoted as E_+ or E_- when σ= +1or $-1.Here the Mordell-Weil group and the rank of the elliptic curve E over the Gauss field K=Q(\sqrt -1) (and over the rational field Q is determined in several cases; and results on solutions of related Diophantine equations and simultaneous Pellian equations will be given. The arithmetic constructs over Q of the elliptic curve E have been studied in the last paper1, the Selmer groups are determined, results on Mordell-Weil group,…
Computer experiments reveal that twin primes tend to center on nonsquarefree multiples of 6 more often than on squarefree multiples of 6 compared to what should be expected from the ratio of the number of nonsquarefree multiples of 6 to the number of squarefree multiples of 6 equal $π^2/3-1$, or ca 2.290. For multiples of 6 surrounded by twin primes, this ratio is 2.427, a relative difference of ca $6.0\%$ measured against the expected value. A deviation from the expected value of this ratio, ca $1.9\%$, exists also for isolated primes. This shows that the distribution of primes is biased towa…
Prime isogenous discriminant ideal twins
本文研究了素数等价判别理想双生的分类，证明在数域中，当p∈{2,3,5,7,13}时，存在有限个这样的双生。
A sieve is constructed for twin primes at distance 4, which are of the form 3(2m+1)+/-2, and are characterized by their twin-4 rank 2m+1. It has no parity problem. Non-ranks are identified as all other odd numbers and counted using odd primes p>=5. Twin-4 ranks and non-ranks make up the set of odd numbers. Regularities of non-ranks allow gathering information on them to obtain a Legendre-type sum for the number of twin-4 ranks. Due to considerable cancellations in it, the asymptotic law of its main term has the expected form and magnitude of its coefficient.
A modification of the linear sieve, and the count of twin primes
本文提出一种满足强分解性质的线性筛法修改，将素数分布水平提升至 $x^{10/17}$，超越以往工作，并据此获得孪生素数计数新上界，实现自1986年以来最大改进。
Average twin prime conjecture for elliptic curves in arithmetic progressions
该研究将Koblitz关于椭圆曲线约化素数阶的平均孪生素数猜想推广到算术级数中的素数，得到了一致成立的渐近公式，且平均常数与理论预测一致。
The results of the computer investigation of the sign changes of the difference between the number of twin primes $π_2(x)$ and the Hardy--Littlewood conjecture $C_2\Li_2(x)$ are reported. It turns out that $d_2(x)=π_2(x) - C_2\Li_2(x)$ changes the sign at unexpectedly low values of $x$ and for $x<2^{48}=2.81\...\times10^{14}$ there are 477118 sign changes of this difference. It is conjectured that the number of sign changes of $d_2(x)$ for $x\in (1, T)$ is given by $\sqrt T/\log(T)$. The running logarithmic densities of the sets for which $d_2(x)>0$ and $d_2(x)<0$ are plotted for $x$ up to $2^…
We consider the representation of primes as a sum of a prime and twice a triangular number. We prove that a subset of the primes having density 1 is expressible in this form. We conjecture that every odd prime number is expressible as a sum of a twin prime and twice a triangular number. We show that this conjecture implies the existence of infinitely many twin primes.
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