D. R. Heath-Brown, Xiannan Li
We show that there are infinitely many primes $p$ such that not only does $p + 2$ have at most two prime factors, but $p + 6$ also has a bounded number of prime divisors. This refines the well known result of Chen.
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We give a more comrehensive treatment of Chen's double sieve and improve related constants in Goldbach's conjecture and the twin prime problem.
In this paper we show that if $A$ is a subset of Chen primes with positive relative density $α$, then $A+A$ must have positive upper density at least $cαe^{-c^\prime\log(1/α)^{2/3}(\log\log(1/α))^{1/3}}$ in the natural numbers.
We find a lower bound for the number of Chen primes in the arithmetic progression $a \bmod q$, where $(a,q)=(a+2,q)=1$. Our estimate is uniform for $q \leq \log^M x$, where $M>0$ is fixed.
We prove that there exists a k_0>0 such that every sufficiently large odd integer n with 3\mid n can be represented as p_1+p_2+p_3, where p_1,p_2 are Chen's primes and p_3 is a prime with p_3+2 has at most k_0 prime factors.
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).
The Exceptional Set in Goldbach's Problem with two Chen Primes
该研究针对满足$n\equiv4\pmod{6}$的自然数，通过结合Bombieri–Vinogradov定理的幂节省变体的高效筛策略，改进了哥德巴赫问题中两个陈素数和的相关结论，确定了其例外集的幂节省性质。
We show that every even number $>\exp\exp 36$ can be represented as the sum of a prime and a product of at most two primes.
For a fixed prime $p$, let $e_p(n!)$ denote the order of $p$ in the prime factorization of $n!$. Chen and Liu (2007) asked whether for any fixed $m$, one has $\{e_p(n^2!) \bmod m:\; n\in\mathbb{Z}\}=\mathbb{Z}_m$ and $\{e_p(q!) \bmod m:\; q {prime}\}=\mathbb{Z}_m$. We answer these two questions and show asymptotic formulas for $# \{n<x: n \equiv a \bmod d,\; e_p(n^2!)\equiv r \bmod m\}$ and $# \{q<x: q {prime}, q \equiv a \bmod d,\; e_p(q!)\equiv r \bmod m\}$. Furthermore, we show that for each $h\geq 3$, we have $\{n<x: n \equiv a \bmod d,\; e_p(n^h!)\equiv r \bmod m\} \gg x^{4/(3h+1)}$.
Representation of even Gaussian integer à la Chen
该研究类比有理数域的陈景润定理，证明了范数充分大的偶高斯整数可表示为一个高斯素数与至多含两个高斯素因子的高斯整数之和。
In 1973, J.-R. Chen showed that every large even integer is a sum of a prime and a product of at most two primes. In this paper, the author indicates and fixes the issues in a simplified proof of this result given by Pan et al.
We prove that assuming the Generalized Riemann Hypothesis every even integer larger than $\exp(\exp(15.85))$ can be written as the sum of a prime number and a number that has at most two prime factors.
On Chen's theorem, Goldbach's conjecture and almost prime twins II
本文证明了关于Goldbach猜想和孪生素数问题的筛法结果，将素数表示的下界渐近常数提升至1.9728，接近Hardy-Littlewood猜想的渐近常数2，结合了加权筛法、Chen转换原理等多种筛法技术。
An explicit version of Chen's theorem and the linear sieve
受Nathanson和Yamada工作启发，利用线性筛法证明大于$\exp(\exp(32.7))$的偶数均可表示为一个素数与至多两个素数乘积之和，给出了Chen定理的显式版本。
Let $\mathcal{P}$ be the set of all primes and $π(x)$ be the number of primes up to $x$. For any $n\ge 2$, let $P^+(n)$ be the largest prime factor of $n$. For $0<c<1$, let $$T_c(x)=\#\{p\le x:p\in \mathcal{P},P^+(p-1)\ge p^c\}.$$ In this note, we proved that there exists some $c<1$ such that $$\limsup_{x\rightarrow\infty}\frac{T_c(x)}{π(x)}<\frac12,$$ which disproves a conjecture of Chen and Chen.
We obtain the triple correlations for a truncated divisor sum related to primes. We also obtain the mixed correlations for this divisor sum when it is summed over the primes, and give some applications to primes in short intervals.
Intersective polynomials are polynomials in $\Z[x]$ having roots every modulus. For example, $P_1(n)=n^2$ and $P_2(n)=n^2-1$ are intersective polynomials, but $P_3(n)=n^2+1$ is not. The purpose of this note is to deduce, using results of Green-Tao \cite{gt-chen} and Lucier \cite{lucier}, that for any intersective polynomial $h$, inside any subset of positive relative density of the primes, we can find distinct primes $p_1, p_2$ such that $p_1-p_2=h(n)$ for some integer $n$. Such a conclusion also holds in the Chen primes (where by a Chen prime we mean a prime number $p$ such that $p+2$ is the …
On the Weakly Prime-Additive Numbers with Length 4
本文在广义黎曼假设下证明了对任意正整数m，存在无穷多个能被m整除的长度为4的弱素数可加数，并给出了一个与长度为3情形类似的相关结果。
Let $p>5$ be a fixed prime. We obtain an asymptotic formula related to small solutions of quadratic congruences of the form $x_1^2+x_2^2\equiv x_3^2\bmod{p^n}$ where $\max\{|x_1|,|x_2|,|x_3|\}\le p^{νn}$ with $ν>1/2$.
On the upper and lower bound orders of almost prime triples
本文研究形如$(p, P_{a}, P_{b})$的Hardy-Littlewood三元组，改进了其上下界阶，推广并改进了已有结果。
A new proof of Chen's theorem for Markoff graphs
本研究针对模$p$马尔可夫图相关的陈氏定理的马尔可夫推论，给出了一种替代证明，该推论与Baragar关于模$p$马尔可夫图对绝大多数素数连通的猜想相关。
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.
On a conjecture on shifted primes with large prime factors, II
本文研究移位素数 $p-1$ 具有大素因子的密度问题，证明当 $8/9\le c<1$ 时相应比例的上极限不超过 $8(1/c-1)$，并在 $c\to1$ 时趋于零。
The Density of Primes in the Eigensurface of ${\bf S}_3$
本文研究了${\bf S}_3$特征面上素数三元组的密度，证明其大于3ζ(3)/log N，表明该表面更频繁地包含素数。
The abc conjecture implies infinitely many non-Wieferich places for fixed bases in number fields
本文在abc猜想假设下，将非Wieferich素数增长结果推广至数域，证明Ding的结果在虚二次域整数环中除31个例外外对所有基成立。
The abc conjecture, one of the most famous open problems in number theory, claims that three positive integers satisfying a+b=c cannot simultaneously have significant repetition among their prime factors; in particular, the product of the distinct primes dividing the three integers should never be much less than c. Triples of numbers satisfying a+b=c are called abc triples if the product of their distinct prime divisors is strictly less than c. We catalog what is known about abc triples, both numerical examples found through computation and infinite familes of examples established theoreticall…
On Chen's theorem over Piatetski-Shapiro type primes and almost-primes
本文在Piatetski-Shapiro序列上建立新的Bombieri-Vinogradov型均值定理，推广并改进了先前关于该序列上素数、殆素数的陈氏定理相关结果。
Triple divisor type functions in arithmetic progressions to prime power moduli
本文研究非尖点GL_3自守形式Hecke特征值在素数幂模算术级数中的等分布，利用Kowalski-Lin-Michel和Milićević技术证明指数1/2+7/458-ε可取。
Let $K$ be a global field which contains a primitive $p$-th root of unity, where $p$ is a prime number. M. J. Hopkins and K. G. Wickelgren showed that for $p=2$, any triple Massey product over $K$ with respect to $\mathbb{F}_p$, contains 0 whenever it is defined. We show that this is true for all primes $p$.
For the ternary quadratic form Q(x) = x^2 + y^2 - z^2 and a non-zero Pythagorean triple x_0 in Z^3 lying on the cone Q(x) = 0, we consider an orbit O = x_0 Gamma of a finitely generated subgroup Gamma < SO_Q(Z) with critical exponent exceeding 1/2. We find infinitely many Pythagorean triples in O whose hypotenuse, area, and product of side lengths have few prime factors, where "few" is explicitly quantified. We also compute the asymptotic of the number of such Pythagorean triples of norm at most T, up to bounded constants.
A remark on weighted average multiplicities in prime factorisation
本文推广ABC三元组的质量定义，引入加权平均重数（WAM），证明其实部小于1时无界、大于1时全纯有界，并揭示其与最大素因子重数及ABC猜想的联系。
Triple convolution sums of the generalised divisor functions and related sums over primes
本文研究广义除数函数的三重卷积和及其在素数上的应用，通过多变量Dirichlet级数理论和概率论方法推导出渐进行为及下界。
It is well-known that pythagorean triples can be represented by points of the unit circle with rational coordinates. These points form an abelian group, and we describe its structure. This structural description yields, almost immediately, an enumeration of the normalized pythagorean triples with a given hypotenuse, and also to an effective method for producing all such triples. This effective method seems to be new. This paper is intended for the general mathematical audience, including undergraduate mathematics students, and therefore it contains plenty of background material, some history a…
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…
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…
Let $(a, b, c)$ be a primitive Pythagorean triple parameterized as $a=u^2-v^2,\ b=2uv,\ c=u^2+v^2$,\ where $u>v>0$ are co-prime and not of the same parity. In 1956, L. Je{ś}manowicz conjectured that for any positive integer $n$, the Diophantine equation $(an)^x+(bn)^y=(cn)^z$ has only the positive integer solution $(x,y,z)=(2,2,2)$. In this connection we call a positive integer solution $(x,y,z)\ne (2,2,2)$ with $n>1$ exceptional. In 1999 M.-H. Le gave necessary conditions for the existence of exceptional solutions which were refined recently by H. Yang and R.-Q. Fu. In this paper we give a un…
Let $L/F$ be a quadratic extension of totally real number fields. For any prime $p$ unramified in $L$, we construct a $p$-adic $L$-function interpolating the central values of the twisted triple product $L$-functions attached to a $p$-nearly ordinary family of unitary cuspidal automorphic representations of $\text{Res}_{L\times F/F}(\text{GL}_{2})$. Furthermore, when $L/\mathbb{Q}$ is a real quadratic number field and $p$ is a split prime, we prove a $p$-adic Gross-Zagier formula relating the value of the $p$-adic $L$-function outside the range of interpolation to the syntomic Abel-Jacobi imag…
Exact Frequencies of Consecutive Quadratic Residue and Nonresidue Patterns Modulo a Prime
研究模奇素数\(p\)时长度为二和三的连续二次剩余与非剩余模式的精确频率，利用有限域特征和基本性质推导计数公式，对的频率由\(p \pmod 4\)分类，三元组频率依赖\(p \pmod 8\)及雅可比和\(T_p\)。
For A,epsilon>0 and any sufficiently large odd n we show that for almost all k up to n^{1/5-epsilon} there exists a representation n=p1+p2+p3 with primes in residue classes b1,b2,b3 mod k for almost all admissible triplets b1,b2,b3 of reduced residues mod k.
The fibonomial triangle has been shown by Chen and Sagan to have a fractal nature mod 2 and 3. Both these primes have the property that the Fibonacci entry point of $p$ is $p+1$. We study the fibonomial triangle mod 5, showing with a theorem of Knuth and Wilf that the triangle has a recurring structure under divisibility by five. While this result is not new, our method of proof is new and suggests a conjecture for the divisibility of a fibonomial coefficient by a general prime $p$. We give necessary conditions for such primes, namely that the Fibonacci entry point must be greater than or equa…
Let $m$ be a fixed square-free positive integer, then equivalence classes of solutions of Diophantine equation $x^2+m\cdot y^2=z^2$ form an infinitely generated abelian group under the operation induced by the complex multiplication. A basis of this group is constructed here using prime ideals and the ideal class group of the field $\mathbb Q (\sqrt{-m})$.
An average Brun-Titchmarsh theorem and shifted primes with a large prime factor
该研究利用Maynard的三重卷积估计突破，改进了平均Brun-Titchmarsh定理，并应用于移位素数，证明无穷多个素数$p$满足$p-1$的最大素因子大于$p^{0.679}$。
We establish results of Bombieri-Vinogradov type for the von Mangoldt function $Λ(n)$ twisted by a nilsequence. In particular, we obtain Bombieri-Vinogradov type results for the von Mangoldt function twisted by any polynomial phase $e(P(n))$; the results obtained are as strong as the ones previously known in the case of linear exponential twists. We derive a number of applications of these results. Firstly, we show that the primes $p$ obeying a "nil-Bohr set" condition, such as $\|αp^k\|<\varepsilon$, exhibit bounded gaps. Secondly, we show that the Chen primes are well-distributed in nil-Bohr…
On Triple Quadratic Residue Symbols in Real Quadratic Fields
针对窄类群平凡的实二次域，引入三次二次剩余符号，通过模2算术三重Milnor不变量定义，并刻画其在8次二面体扩张中的分解律，推广了Rédei的三次符号。
Let $F=x^2+y^2-z^2$, $x_0 \in \mathbb{Z}^3$ primitive with $F(x_0)=0$, and $Γ\leq SO_F(\mathbb{Z})$ be a finitely generated thin subgroup. We consider the resulting thin orbits of Pythagorean triples $x_0 \cdot Γ$ - specifically which hypotenuses, areas, and products of all three coordinates arise. We produce infinitely many $R$-almost primes in these three cases whenever $Γ$ has exponent $δ_Γ>δ_0(R)$ for explicit $R$, $δ_0$.
Divisibility Estimates for Gaussian Norms and Almost-Prime Orders of a CM Elliptic Curve
该研究针对椭圆曲线E:y²=x³-x，结合高斯CM筛框架与交叉项，证明了特定素数p下#E(F_p)/8的无平方因子性及素因子个数限制，为相关数论问题提供了可除性估计与分布结果。
We introduce mod 3 triple Milnor invariants and triple cubic residue symbols for certain primes of the Eisenstein number field $\mathbb{Q}(\sqrt{-3})$, following the analogies between knots and primes. Our triple symbol generalizes both the cubic residue symbol and Rédei's triple symbol, and describes the decomposition law of a prime in a mod 3 Heisenberg extension of degree 27 over $\mathbb{Q}(\sqrt{-3})$ with restricted ramification, which we construct concretely in the form similar to Rédei's dihedral extension over $\mathbb{Q}$. We also give a cohomological interpretation of our symbols by…
Record values are determined for the order $|\Sha|$ of the Tate--Shafarevich group of an elliptic curve $E$, computed analytically by the Birch--Swinnerton-Dyer conjecture, and for the Goldfeld--Szpiro ratio $G=|\Sha|/\sqrt{N}$, where $N$ is the conductor of $E$. The curves have rank zero and are isogenous to quadratic twists of Frey curves constructed from coprime positive integers $(a,b,c)$ with $a+b=c$ and $c>r^{1.4}$, where the radical $r$ is the product of the primes dividing $abc$. Curves with $|\Sha|>250000^2$ and $G>12$ are found in 20 isogeny classes. Three curves have $G>150$. The la…
Markoff triples and Nielsen equivalence in $\text{SL}_2(\mathbb{F}_p)$
本文证明了关于SL₂(F_p)中马尔科夫三元组的Q-分类猜想，通过矩阵可逆性检查解决了相关等价类问题。
On $abc$ triples of the form $(1,c-1,c)$
本文研究形如 $(1,c-1,c)$ 的 $abc$ 三元组，推导出两个一般性结论，可复原现有文献中首项为1的 $abc$ 三元组序列，为 $abc$ 猜想相关研究提供支撑。
Given relatively prime integers $a_1, \dotsc, a_n$, the Frobenius number $g(a_1, \dotsc, a_n)$ is defined as the largest integer which cannot be expressed as $x_1 a_1 + \dotsb + x_n a_n$ with $x_i$ nonnegative integers. In this article, we give the Frobenius number of primitive Pythagorean triples. That is, \[ g(m^2-n^2, 2mn, m^2+n^2) = (m-1)(m^2-n^2) + (m-1)(2mn) - (m^2 + n^2). \]
Let $p$ be a prime number, $F$ a field containing a root of unity of order $p$, and $G_F$ the absolute Galois group. Extending results of Hopkins, Wickelgren, Minac and Tan, we prove that the triple Massey product $H^1(G_F)^3\to H^2(G_F)$ contains $0$ whenever it is nonempty. This gives a new restriction on the possible profinite group structure of $G_F$.
Consider three normalised cuspidal eigenforms of weight $2$ and prime level $p$. Under the assumption that the global root number of the associated triple product $L$-function is $+1$, we prove that the complex Abel-Jacobi image of the modified diagonal cycle of Gross-Kudla-Schoen on the triple product of the modular curve $X_0(p)$ is torsion in the corresponding Hecke isotypic component of the Griffiths intermediate Jacobian. The same result holds with the complex Abel-Jacobi map replaced by its étale counterpart. As an application, we deduce torsion properties of Chow-Heegner points associat…
On the Burgess bound and Pythagorean triples involving primitive roots
该研究结合Pierce和Xu的高维Burgess界与经典一维Burgess界，证明对任意足够大素数p，存在满足特定条件的毕达哥拉斯三元组，证实了孙智伟关于原根的相关猜想。
Uniqueness theorem of generalized Markov numbers that are prime powers
该研究将经典马尔可夫方程推广为带参数k的广义形式，拓展Cohn三元组概念，证明了素数幂形式的广义马尔可夫数的唯一性定理。
Split-Cubic Curves from D(n)-Triples: A Bounded Finite-Field Census
对与D(n)-三元组相关的分裂三次椭圆曲线在有限域上进行确定性普查，记录了1640条约化记录并审计点计数，为后续研究提供数据基础。
In paper on a classification of Lehmer triples, Juricevic conjectured that there are infinitely many primes of special form. We disprove one of his conjectures and consider the other one.
Triples and quadruples of consecutive squares or non-squares in a finite field
本文研究有限域中连续平方数或非平方数的三元组与四元组计数，利用 Jacobsthal 和求值给出显式公式，推广了 Monzingo 的结果，并处理了 5 的幂阶情形。
The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an L^2-L^p restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime k-tuples. Let a_1,...,a_k and b_1,...,b_k be positive integers. For t on the unit circle write h(t) := \sum_{n \in X} e(nt)$, where X is the set of all n <= N such that the numbers a_1n + b_1,..., a_kn + b_k are all prime. We obtain upper bounds for the L^p norm of h, p > 2, which are (conditionally on the prime tuple conjecture) of the corre…
We prove results pertaining to strong approximation for Markoff triples in the case of prime moduli.
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