Some Collatz quadratic prime sequences

关于 $\mathbb{F}_2[x]$ 中 Collatz 映射的停时

On the stopping time of the Collatz map in $\mathbb{F}_2[x]$

本文研究了有限域多项式环 $\mathbb{F}_2[x]$ 中 Collatz 映射的停时，将其上界改进至 $O({\rm deg} (f)^{1.5})$，并证明了特定多项式序列停时中存在无界长度的等差数列。

广义Collatz转移关系在Büchi算术中的逻辑不可定义性

Logical Undefinability of the Generalized Collatz Transition Relation in Büchi Arithmetic

证明广义Collatz映射的任意步转移关系在基2 Büchi算术中不可一阶定义，通过反证法利用指数集非半线性与Cobham-Semënov定理矛盾。

Prime-universal diagonal quadratic forms

A (positive definite and integral) quadratic form is said to be $\textit{prime-universal}$ if it represents all primes. Recently, Doyle and Williams in [2] classified all prime-universal diagonal ternary quadratic forms, and all prime-universal diagonal quaternary quadratic forms under two conjectures proposed by themselves. In this article, we classify all prime-universal diagonal quadratic forms regardless of ranks. Furthermore, we prove, so called, $67$-Theorem for a diagonal quadratic form to be prime-universal.

Statistics of small prime quadratic non-residues

We prove that the average of the $k$-th smallest prime quadratic non-residue modulo a prime approximates the $2k$-th smallest prime.

Primes in quadratic fields

This paper presents algorithms for calculating the quadratic character and the norms of prime ideals in the ring of integers of any quadratic field. The norms of prime ideals are obtained by means of a sieve algorithm using the quadratic character for the field considered. A quadratic field, and its ring of integers, can be represented naturally in a plane. Using such a representation, the prime numbers - which generate the principal prime ideals in the ring - are displayed in a given bounded region of the plane.

Primes represented by quadratic polynomials via exceptional characters

We estimate the number of primes represented by a general quadratic polynomial with discriminant $Δ$, assuming that the corresponding real character is exceptional.

The largest cycles consist by the quadratic residues and Fermat primes

This paper studies the largest cycles consisted by the quadratic residues modulo prime numbers. We give some formulae about the maximum length of the cycles. Especially, the formula for modulo Fermat primes is given.

A characterization of almost universal ternary quadratic polynomials with odd prime power conductor

An integral quadratic polynomial (with positive definite quadratic part) is called almost universal if it represents all but finitely many positive integers. In this paper, we introduce the conductor of a quadratic polynomial, and give an effective characterization of almost universal ternary quadratic polynomials with odd prime power conductor.

Primes in Quadratic Progressions on Average

In this paper, we establish a theorem on the distribution of primes in quadratic progressions on average.

Quadratic Frobenius probable prime tests costing two selfridges

By an elementary observation about the computation of the difference of squares for large in- tegers, deterministic quadratic Frobenius probable prime tests are given with running times of approximately 2 selfridges.

素数模二次Dirichlet特征和矩的界

Bounds for moments of quadratic Dirichlet character sums of prime moduli

本文在广义黎曼假设下，先对模8p的二次Dirichlet L-函数平移矩建立精确上界，再据此证明素数模二次Dirichlet特征和矩的界。

Quadratic polynomials at prime arguments

For a fixed quadratic irreducible polynomial $f$ with no fixed prime factors at prime arguments, we prove that there exist infinitely many primes $p$ such that $f(p)$ has at most 4 prime factors, improving a classical result of Richert who requires 5 in place of 4. Denoting by $P^+(n)$ the greatest prime factor of $n$, it is also proved that $P^+(f(p))>p^{0.847}$ infinitely often.

On the Representation of Almost Primes by Sets of Quadratic Forms

We investigate the problem of r almost-primes represented by sets of quadratic forms and give upper bounds for r. Our results extend work of Diamond and Halberstam in which they investigated the corresponding problem for polynomials.

Local Densities of Diagonal Integral Ternary Quadratic Forms at Odd Primes

We give formulas for local densities of diagonal integral ternary quadratic forms at odd primes. Exponential sums and quadratic Gauss sums are used to obtain these formulas. These formulas (along with 2-adic densities and Siegel's mass formula) can be used to compute the representation numbers of certain ternary quadratic forms.

The quadratic form in 9 prime variables

Let $f(x_1,\ldots,x_n)$ be a regular indefinite integral quadratic form with $n\ge 9$, and let $t$ be an integer. It is established that $f(x_1,\ldots,x_n)=t$ has solutions in prime variables if there are no local obstructions.

Binary quadratic forms that represent almost the same primes

Jagy and Kaplansky exhibited a table of 68 pairs of positive definite binary quadratic forms that represent the same odd primes and conjectured that this list is complete outside of "trivial" pairs. In this article, we find all pairs of such forms that represent the same primes outside of a finite set.

On Primes Represented by Quadratic Polynomials

This is a survey article on the Hardy-Littlewood conjecture about primes in quadratic progressions. We recount the history and quote some results approximating this hitherto unresolved conjecture.

On the Representation of Primes by Binary Quadratic Forms, and Elliptic Curves

It is shown that, under some mild technical conditions, representations of prime numbers by binary quadratic forms can be computed in polynomial complexity by exploiting Schoof's algorithm, which counts the number of $\mathbb F_q$-points of an elliptic curve over a finite field $\mathbb F_q$. Further, a method is described which computes representations of primes from reduced quadratic forms by means of the integral roots of polynomials over $\mathbb Z$. Lastly, some progress is made on the still-unsettled general problem of deciding which primes are represented by which classes of quadratic f…

Non-vanishing of central values of quadratic Hecke $L$-functions of prime moduli in the Gaussian field

We study the first and second mollified moments of central values of a quadratic family of Hecke $L$-functions of prime moduli to show that more than nine percent of the members of this family do not vanish at the central values.

关于实二次域中素理想分布的一个注记

A note on the distribution of prime ideals in real quadratic fields

本文总结了虚二次域素理想分布的研究，并基于相同方法建立了实二次域的类似结果。

Non-vanishing of quadratic twists of modular $L$-functions of prime-related moduli

In this paper, we study central values of the family of quadratic twists of modular $L$-functions of moduli $8p$, with $p$ ranging over odd primes. Assuming the truth of the generalized Riemann hypothesis, we establish a positive proportion non-vanishing result for the corresponding $L$-values.

On Primes in Quadratic Progressions

We verify the Hardy-Littlewood conjecture on primes in quadratic progressions on average. The results in the present paper significantly improve those of a previous paper of the authors(arXiv:math.NT/0605563).

Catalan's Equation For Even Primes over Quadratic Fields

This article lists all the solution of the Catalan equation $x^p-y^q=1$ for $x,y \in \mathbb{Z}[i]$, when one of the primes $p and q$ is even.

与素数相关模数的二次狄利克雷 $L$-函数的大值

Large values of quadratic Dirichlet $L$-functions of prime-related moduli

本文在广义黎曼假设下，应用长谐振子方法，展示了与素数相关模数的二次狄利克雷 $L$-函数族在中心点处的大值。

Almost-Primes Represented by Quadratic Polynomials

In his paper Almost-Primes Represented by Quadratic Polynomials, Iwaniec proved that the polynomial n^2 + 1 takes on values with at most two prime factors (counted with multiplicity) infinitely often. He states that "in order to avoid technical complications, we shall restrict our proof to the polynomial n^2 + 1.". In this exposition, we follow Iwaniec's proof and show that for any irreducible quadratic polynomial G(n) (satisfying some obviously necessary hypotheses), G(n) has at most two prime factors for infinitely many values of n.

Mersenne Primes in Real Quadratic Fields

The concept of Mersenne primes is studied in real quadratic fields of class number 1. Computational results are given. The field $Q(\sqrt{2})$ is studied in detail with a focus on representing Mersenne primes in the form $x^{2}+7y^{2}$. It is also proved that $x$ is divisible by 8 and $y\equiv \pm3\pmod{8}$ generalizing the result of F Lemmermeyer, first proved in \cite{LS} using Artin's Reciprocity law.

On prime values of binary quadratic forms with a thin variable

In this paper we generalize the result of Fouvry and Iwaniec dealing with prime values of the quadratic form $x^2 + y^2$ with one input restricted to a thin subset of the integers. We prove the same result with an arbitrary primitive positive definite binary quadratic form. In particular, for any positive definite binary quadratic form $F$ and binary linear form $G$, there exist infinitely many $\ell, m\in\mathbb{Z}$ such that both $F(\ell, m)$ and $G(\ell, m)$ are primes as long as there are no local obstructions.

高斯域中与素数相关模的二次Hecke $L$-函数的大值

Large values of quadratic Hecke $L$-functions with prime-related moduli of the Gaussian field

本文在广义黎曼假设下，利用共振方法证明高斯域中与素数相关模的二次Hecke $L$-函数在中心点可取到任意大的值。

Representing Primes as the Form $x^2+ny^2$ in Some Imaginary Quadratic Fields

We give criteria of the solvability of the diophantine equation $p=x^2+ny^2$ over some imaginary quadratic fields where $p$ is a prime element. The criteria becomes quite simple in special cases.

Explicit Result on Equivalence of Rational Quadratic Forms Avoiding Primes

Given a pair of regular quadratic forms over $\mathbb{Q}$ which are in the same genus and a finite set of primes $P$, we show that there is an effective way to determine a rational equivalence between these two quadratic forms which are integral over every prime in $P$. This answers one of the principal questions posed by Conway and Sloane in their book {\em Sphere packings, lattices and groups}, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol 290, Springer-Verlag, New York, 1999; page 402.

On the average distribution of primes represented by binary quadratic forms

We investigate the average distribution of primes represented by positive definite integral binary quadratic forms, the average being taken over negative fundamental discriminants in long ranges. In particular, we prove corresponding results of Bombieri-Vinogradov type and of Barban-Davenport-Halberstam type, although with shorter ranges than in the original theorems for primes in arithmetic progressions: The results imply that, for all $a>0$, the least prime that can be represented by any given positive definite binary quadratic form of discriminant $q$ is smaller than $|q|^{7+a}$ for all for…

Quadratic residues and non-residues for infinitely many Piatetski-Shapiro primes

In this paper, we prove a quantitative version of the statement that every nonempty finite subset of $\mathbb{N}^+$ is a set of quadratic residues for infinitely many primes of the form $[n^c]$ with $1\leqslant c\leqslant243/205$. Correspondingly, we can obtain a similar result for the case of quadratic non-residues under reasonable assumptions. These results generalize the previous ones obtained by S. Wright in certain aspects.

The number of imaginary quadratic fields with prime discriminant and class number up to $H$

In this paper, we obtain an asymptotic formula for the number of imaginary quadratic fields with prime discriminant and class number up to $H$, as $H\to \infty$. Previously, such an asymptotic was only known under the assumption of the Generalized Riemann Hypothesis, by the recent work of Holmin, Jones, Kurlberg, McLeman and Petersen.

虚二次域中素理想的渐近估计与切比雪夫偏差

The asymptotic estimation of prime ideals in imaginary quadratic fields and Chebyshev's bias

该研究聚焦虚二次域中满足特定条件的素理想的渐近估计，探讨其分布的切比雪夫偏差现象，丰富了代数数论中素理想分布的相关研究。

函数域上素二次Dirichlet L-函数的混合二阶矩

The mixed second moment of prime quadratic Dirichlet $L$-functions over function fields

该研究在函数域背景下，针对首一不可约多项式对应的二次Dirichlet L-函数的μ阶和ν阶导数，计算出其混合二阶矩的渐近公式，为相关L-函数导数矩的研究提供了结果。

具有素数相关模的二次狄利克雷\(L\)函数的极值

Extreme values of quadratic Dirichlet $L$-functions with prime-related moduli

研究具有素数相关模的二次狄利克雷\(L\)函数极值，通过给出新下界推广前人工作并改进近期成果。

Bounds for moments of quadratic Dirichlet $L$-functions of prime-related moduli

In this paper, we study the $k$-th moment of central values of the family of quadratic Dirichlet $L$-functions of moduli $8p$, with $p$ ranging over odd primes. Assuming the truth of the generlized Riemann hypothesis, we establish sharp upper and lower bounds for the $k$-th power moment of these $L$-values for all real $k \geq 0$.

素数幂模下二元二次型的特征和

Character sums to prime power moduli evaluated at binary quadratic forms

研究素数幂模下二元二次型的特征和估计，采用p进分析方法，补充Heath-Brown对平方自由模的估计。

素数导体二次狄利克雷L函数的混合二阶矩

The Mixed Second Moment of Quadratic Dirichlet $L$-functions with Prime Conductors

本文在函数域语境下，针对首一不可约多项式平均的情况，建立了含完备L函数二阶导数的混合二阶矩的渐近公式。

Quadratic Forms in Prime Variables with small off-diagonal ranks

The main goal of this note is to establish the limits of L. Zhao's techniques for counting solutions to quadratic forms in prime variables. Zhao considered forms with rank at least 9, and showed that these equations have solutions in primes provided there are no local obstructions. We consider in detail the degenerate cases of off-diagonal rank 1 and 2, and improve the rank lower bounds to at least 6 and at least 8 respectively. These results complement a recent breakthrough of Green on the non-degenerate rank 8 case.

Prime polynomial values of quadratic functions in short intervals

In this paper we establish the function field analogue of Bateman-Horn conjecture in short interval in the limit of a large finite field. Hence we start with counting prime polynomials generated by primitive quadratic functions in short intervals. To this end we further work out on function field analogs of cancellation of Mobius sums and its correlations(Chowla type sums) and confirm that square root cancellation in Mobius sums is equivalent to square root cancellation in Chowla type sums.

Threefield identities and simultaneous representations of primes by binary quadratic forms

Kaplansky [2003] proved a theorem on the simultaneous representation of a prime $p$ by two different principal binary quadratic forms. Later, Brink found five more like theorems and claimed that there were no others. By putting Kaplansky-like theorems into the context of threefield identities after Andrews, Dyson, and Hickerson, we find that there are at least two similar results not on Brink's list. We also show how such theorems are related to results of Muskat on binary quadratic forms

One level density of low-lying zeros of quadratic Hecke $L$-functions to prime moduli

In this paper, we study the one level density of low-lying zeros of a family of quadratic Hecke $L$-functions to prime moduli over the Gaussian field under the generalized Riemann hypothesis (GRH) and the ratios conjecture. As a corollary, we deduce that at least $75 \%$ of the members of this family do not vanish at the central point under GRH.

Translation invariant quadratic forms and dense sets of primes

Let $f(x_1,\ldots,x_s)$ be a translation invariant indefinite quadratic form of integer coefficients with $s\ge 10$. Let $\mathcal{A}\subseteq \mathcal{P}\cap \{1,2,\ldots,X\}$. Let $X$ be sufficiently large. Subject to a rank condition, we prove that there exist distinct primes $p_1,\ldots,p_s\in \mathcal{A}$ such that $f(p_1,\ldots,p_s)=0$ as soon as $|\mathcal{A}|\ge \frac{X}{\log X} (\log\log X)^{-\frac{1}{80}}.$

Bounded gaps between product of two primes in imaginary quadratic number fields

We study the gaps between products of two primes in imaginary quadratic number fields using a combination of the methods of Goldston-Graham-Pintz-Yildirim \cite{GGPY}, and Maynard \cite{MAY}. An important consequence of our main theorem is existence of infinitely many pairs $α_1, α_2$ which are product of two primes in the imaginary quadratic field $K$ such that $|σ(α_1-α_2)|\leq 2$ for all embedding $σ$ of $K$ if the class number of $K$ is one and $|σ(α_1-α_2)|\leq 8$ for all embedding $σ$ of $K$ if the class number of $K$ is two.

Sign changes of Fourier coefficients of cusp forms of half-integral weight over split and inert primes in quadratic number fields

In this paper, we investigate sign changes of Fourier coefficients of half-integral weight cusp forms. In a fixed square class $t\mathbb{Z}^2$, we investigate the sign changes in the $tp^2$-th coefficient as $p$ runs through the split or inert primes over the ring of integers in a quadratic extension of the rationals. We show that sign changes occur in both sets of primes when there exists a prime dividing the discriminant of the field which does not divide the level of the cusp form and find an explicit condition that determines whether sign changes occur when every prime dividing the discrim…

椭圆曲线殆素数二次扭曲的中心极端 $L$-值

Extreme central $L$-values of almost prime quadratic twists of elliptic curves

该研究证明椭圆曲线殆素数二次扭曲的中心 $L$-函数极值，并在 BSD 猜想下推出其二次扭曲族中 Tate--Shafarevich 群的极值。

Explicit isogenies of prime degree over quadratic fields

Let $K$ be a quadratic field which is not an imaginary quadratic field of class number one. We describe an algorithm to compute the primes $p$ for which there exists an elliptic curve over $K$ admitting a $K$-rational $p$-isogeny. This builds on work of David, Larson-Vaintrob, and Momose. Combining this algorithm with work of Bruin-Najman, Özman-Siksek, and most recently Box, we determine the above set of primes for the three quadratic fields $\mathbb{Q}(\sqrt{-10})$, $\mathbb{Q}(\sqrt{5})$, and $\mathbb{Q}(\sqrt{7})$, providing the first such examples after Mazur's 1978 determination for $K =…

Simultaneous Cubic and Quadratic Diagonal Equations In 12 Prime Variables

The system of equations \[ u_1p_1^2 + \ldots + u_sp_s^2 = 0 \] \[ v_1p_1^3 + \ldots + v_sp_s^3 = 0 \] has prime solutions $(p_1, \ldots, p_s)$ for $s \geq 12$, assuming that the system has solutions modulo each prime $p$. This is proved via the Hardy-Littlewood circle method, building on Wooley's work on the corresponding system over the integers and recent results on Vinogradov's mean value theorem. Additionally, a set of sufficient conditions for local solvability is given: If both equations are solvable modulo 2, the quadratic equation is solvable modulo 3, and for each prime $p$ at least 7…

关于具有判别式1-4p的二元二次型数量的论文

On the number of binary quadratic forms having discriminant $1-4p$, $p$ prime

本文研究了具有判别式1-4p的二元二次型数量，并提出了一种随机欧拉乘积模型来解释Hurwitz类数的分布。

The least prime number represented by a binary quadratic form

Let $D<0$ be a fundamental discriminant and $h(D)$ be the class number of $\mathbb{Q}(\sqrt{D})$. Let $R(X,D)$ be the number of classes of the binary quadratic forms of discriminant $D$ which represent a prime number in the interval $[X,2X]$. Moreover, assume that $π_{D}(X)$ is the number of primes, which split in $\mathbb{Q}(\sqrt{D})$ with norm in the interval $[X,2X].$ We prove that $$ \Big(\frac{π_D(X)}{π(X)}\Big)^2 \ll \frac{R(X,D)}{h(D)}\Big(1+\frac{h(D)}{π(X)}\Big), $$ where $π(X)$ is the number of primes in the interval $[X,2X]$ and the implicit constant in $\ll$ is independent of $D$ …

GRH下二次数域中素数分母的丢番图逼近

Diophantine approximation with prime denominator in quadratic number fields under GRH

本文在广义黎曼猜想对Hecke $L$-函数成立的条件下，将Matomäki关于实数无理数素数分母丢番图逼近的结果推广至所有二次数域。

Walks on Primes in Imaginary Quadratic Fields

We generalize the Gaussian moat problem to the finitely many imaginary quadratic fields exhibiting unique factorization in their integers. We map this problem to the Euclidean minimum spanning tree problem and use Delaunay triangulation and Kruskal's algorithm to find moats in these fields in $O(|\mathbf{P}| \log |\mathbf{P}|)$ time, where $\mathbf{P}$ denotes the set of primes we are searching through. Our algorithm uses symmetry and boundaries to correctly and efficiently identify all moats, in order of increasing length, along with the farthest prime reached for each moat, in an underlying …

Quaternary quadratic forms with prime discriminant

Let $Q$ be a positive-definite quaternary quadratic form with prime discriminant. We give an explicit lower bound on the number of representations of a positive integer $n$ by $Q$. This problem is connected with deriving an upper bound on the Petersson norm $\langle C, C \rangle$ of the cuspidal part of the theta series of $Q$. We derive an upper bound on $\langle C, C \rangle$ that depends on the smallest positive integer not represented by the dual form $Q^{*}$. In addition, we give a non-trivial upper bound on the sum of the integers $n$ excepted by $Q$.