P. Moree, B. Sury
Given positive integers a,b,c and d such that c and d are coprime we show that the primes p=c(mod d)dividing a^k+b^k for some k>=1 have a natural density and explicitly compute this density. We demonstrate our results by considering some claims of Fermat that he made in a 1641 letter to Mersenne.
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We describe first-degree prime ideals of biquadratic extensions in terms of first-degree prime ideals of two underlying quadratic fields. The identification of the prime divisors is given by numerical conditions involving their ideal norms. Interestingly, the correspondence between these ideals in the larger ring and those in the smaller ones extends to the divisibility of principal ideals in their respective rings, with some exceptions that we explicitly provide. Finally, we hint at possible applications of this correspondence.
Let a,f and g be integers, with a and f coprime. Under the generalized Riemann hypothesis it follows from work of Hooley and Lenstra that the set of primes p such that p=a(mod f) and g is primitive root mod p has a natural density. In this note we explicitly evaluate this density and give some applications of this result.
Arithmetic progression in a finite field with prescribed norms
本文研究了在有限域 $\mathbb{F}_{q^{n}}$ 中，由本原元构成且至少包含一个具有指定范数的正规元的算术级数的存在性，并证明了在特定参数下仅有10种可能例外。
Let a and f be coprime positive integers. Let g be an integer. Under the Generalized Riemann Hypothesis (GRH) it follows by a result of H.W. Lenstra that the set of primes p such that p=a(mod f) and g is a primitive root modulo p has a natural density. In this note this density is explicitly evaluated with an Euler product as result.
We investigate the question of which growth rates are possible for the number of periodic points of a compact group automorphism. Our arguments involve a modification of Linnik's Theorem, concerning small prime numbers in arithmetic progressions which lie in intervals.
A reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in this paper. Results are both, on the one hand, from a metrical and a non-metrical point of view and, on the other hand, from an asymptotic and also a uniform point of view. The principal novelty is a Khintchine type theorem for uniform approximation in this context. Some applications of this theory are also discussed.
Hausdorff dimension of sets of numbers whose continued fractions contain arbitrarily long arithmetic progressions
本文研究了偏商序列严格递增且包含任意长等差数列的无理数的豪斯多夫维数。
Counting newforms with prescribed ramified supercuspidal components
本文给出了$S_k^{\rm new}(N)$中在素数集合$T$处具有指定分歧超尖点分量$π_p$的新形式数量公式，该公式基于Atkin--Lehner算子的迹，且在$T$为单个素数或$N$所有素因子时完全显式。
Various authors, including McNew, Nathanson and O'Bryant, have recently studied the maximal asymptotic density of a geometric progression free sequence of positive integers. In this paper we prove the existence of geometric progression free sequences with small gaps, partially answering a question posed originally by Beiglböck et al. Using probabilistic methods we prove the existence of a sequence $T$ not containing any $6$-term geometric progressions such that for any $x\geq1$ and $\varepsilon>0$ the interval $[x,x+C_{\varepsilon}\exp((C+\varepsilon)\log x/\log\log x)]$ contains an element of…
Define B(n) to be the largest height of a polynomial in $\mathbb{Z}[x]$ dividing $x^n-1$. We formulate a number of conjectures related to the value of B(n) when $n$ is of a prescribed form. Additionally, we prove a lower bound for $B(p^aq^b)$ where $p,q$ are distinct primes.
Let E/F be a CM field split above a finite place v of F, let l be a rational prime number which is prime to v, and let S be the set of places of E dividing lv. If E_S denotes a maximal algebraic extension of E unramified outside S, and if u is a place of E dividing v, we show that any field embedding E_S \to \bar{E_u} has a dense image. The "unramified outside S" number fields we use are cut out from the l-adic cohomology of the "simple" Shimura varieties studied by Kottwitz and Harris-Taylor. The main ingredients of the proof are then the local Langlands correspondence for GL_n, the main glob…
Given any positive integer $n,$ let $A(n)$ denote the height of the $n^{\text{th}}$ cyclotomic polynomial, that is its maximum coefficient in absolute value. It is well known that $A(n)$ is unbounded. We conjecture that every natural number can arise as value of $A(n)$ and prove this assuming that for every pair of consecutive primes $p$ and $p'$ with $p\ge 127$ we have $p'-p<\sqrt{p}+1.$ We also conjecture that every natural number occurs as maximum coefficient of some cyclotomic polynomial and show that this is true if Andrica's conjecture that always $\sqrt{p'}-\sqrt{p}<1$ holds. This is th…
Thresholds and Fluctuations for Colorful Arithmetic Progressions in Sparse Random Colorings
本文确定了稀疏随机着色中彩色等差数列出现的完整阈值区域，并刻画了其数量在可满足区域内的渐近正态性及阈值曲面上的三种泊松型波动状态。
The distribution of the multiplicative index of algebraic numbers over residue classes
在广义黎曼假设下，研究数域中乘性指数属于指定集合且具有指定弗罗贝尼乌斯元的素理想的自然密度，并重点分析等差数列情形下密度的正性。
On the number of elements with prescribed norm and trace
本文研究有限域扩张中范数和迹均为给定值的元素个数，通过Artin-Schreier曲线上的有理点计数改进估计，并应用于环面Calabi-Yau簇及不可约多项式计数。
Magyar has shown that if $B \subset \mathbb{Z}^d$ has positive upper density $(d \geq 5)$, then the set of squared distances $\{ \|b_1-b_2 \|^2 \text{ }: \text{ } b_1,b_2 \in B \}$ contains an infinitely long arithmetic progression, whose period depends only on the upper density of $B$. We extend this result by showing that $B$ contains locally isometrically embedded copies of every tree with edge lengths in some given arithmetic progression (whose period depends only on the upper density of $B$ and the number of vertices of the sought tree). In particular, $B$ contains all chains of elements …
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 …
We show how to construct discretely ordered principal ideal subrings of $\mathbb Q[x]$ with various types of prime behaviour. Given any set $\mathcal D$ consisting of finite strictly increasing sequences $(d_1,d_2,\dots, d_l)$ of positive integers such that, for each prime integer $p$, the set $\{p\mathbb Z, d_1+p\mathbb Z,\dots, d_l+p\mathbb Z\}$ does not contain all the cosets modulo $p$, we can stipulate to have, for each $(d_1,\dots, d_l)\in \mathcal D$, a cofinal set of progressions $(f, f+d_1, \dots, f+d_l)$ of prime elements in our principal ideal domain $R_τ$. Moreover, we can simultan…
In this paper, we introduce and study a variant of Kummer's notion of (ir)regularity of primes which we call G-irregularity. It is based on Genocchi numbers $G_n$, rather than Bernoulli number $B_n.$ We say that an odd prime $p$ is G-irregular if it divides at least one of the integers $G_2,G_4,\ldots, G_{p-3}$, and G-regular otherwise. We show that, as in Kummer's case, G-irregularity is related to the divisibility of some class number. Furthermore, we obtain some results on the distribution of G-irregular primes. In particular, we show that each primitive residue class contains infinitely ma…
In this paper we construct a cover {a_s(mod n_s)}_{s=1}^k of Z with odd moduli such that there are distinct primes p_1,...,p_k dividing 2^{n_1}-1,...,2^{n_k}-1 respectively. Using this cover we show that for any positive integer m divisible by none of 3, 5, 7, 11, 13 there exists an infinite arithmetic progression of positive odd integers the m-th powers of whose terms are never of the form $2^n\pm p^a$ with p a prime and a,n in {0,1,2,...}. We also construct another cover of Z with odd moduli and use it to prove that $x^2-F_{3n}/2$ has at least two distinct prime factors whenever n is a nonne…
Simultaneous Residue-Class Selection in Prescribed-Difference Packings
本文证明素数差等差数列打包问题的两个渐近最优公式，达到已知下界并解决相关猜想，方法结合循环相位选择、格覆盖与素数规则块分解。
$α$-monogeneity of pure number fields: criterion and density
本文通过Dedekind指数定理证明了纯数域中$α$-monogeneity的准则，并给出了自然密度的显式表达式及渐进行为。
Rational dynamics of a prime-representing map
本文研究了素数表示常数递归构造中出现的映射$\mathcal{T}(x)=\lfloor x floor(1+\{x\})$的有理动力学，定义了有理数的阶，并证明了有限阶有理数的自然密度为1，同时给出了特定分母下的显式分数家族。
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