A remark on relatively prime sets

Combinatorial Identities Via Phi Functions and Relatively Prime Subsets

Let $n$ be a positive integer and let $A$ be nonempty finite set of positive integers. We say that $A$ is relatively prime if $\gcd(A) =1$ and that $A$ is relatively prime to $n$ if $\gcd(A,n)=1$. In this work we count the number of nonempty subsets of $A$ which are relatively prime and the number of nonempty subsets of $A$ which are relatively prime to $n$. Related formulas are also obtained for the number of such subsets having some fixed cardinality. This extends previous work for the cases where $A$ is an interval or a set in arithmetic progression. Applications include: a) An exact formul…

有限域上多元不可约多项式和互素多项式对的个数

Number of Irreducible Polynomials and Pairs of Relatively Prime Polynomials in Several Variables over Finite Fields

讨论了有限域上多元给定次数的不可约多项式和给定次数的互素多项式对的计数结果。考虑了度的两个概念，{ \em全次数}和{ \em向量度}。我们证明了不可约数可以递归地按度计算，并且互素对的数目可以用不可约数来表示。我们还得到了不可约数和互素对数的渐近公式。不可约数的渐近公式推广和改进了Carlitz，Cohen和Bodin的一些结果.

A remark on prime (non)congruent numbers

This paper discusses prime numbers that are (resp. are not) congruent numbers. Particularly the only case not fully covered by earlier results, namely primes of the form $p=8k+1$, receives attention.

Counting $r$-tuples of positive integers with $k$-wise relatively prime components

Let $r\ge k\ge 2$ be fixed positive integers. Let $\varrho_{r,k}$ denote the characteristic function of the set of $r$-tuples of positive integers with $k$-wise relatively prime components, that is any $k$ of them are relatively prime. We use the convolution method to establish an asymptotic formula for the sum $\sum_{n_1,\ldots,n_r\le x} \varrho_{r,k}(n_1,\ldots,n_r)$ by elementary arguments. Our result improves the error term obtained by J. Hu (2013).

Digital images unveil geometric structures in pairs of relatively prime numbers

We present a transformation, based on the Bézout's identity, which maps the set of pairs of relatively prime numbers $(p,q)$ with fixed $p$ and $0<q<p$, to pairs of relatively prime numbers in the $p\times p$ square in $\mathbb R^2$, in such a way that intriguing quadratic arcs show up. We exhibit parametrizations of quadratic curves which fit such quadratic arcs and we also justify algebraically the ensuing geometry.

On numbers $n$ relatively prime to the $n$th term of a linear recurrence

Let $(u_n)_{n \geq 0}$ be a nondegenerate linear recurrence of integers, and let $\mathcal{A}$ be the set of positive integers $n$ such that $u_n$ and $n$ are relatively prime. We prove that $\mathcal{A}$ has an asymptotic density, and that this density is positive unless $(u_n / n)_{n \geq 1}$ is a linear recurrence.

The Density of Tuples Restricted by Relatively r-Prime Conditions

In order to consider j-wise relative r-primality conditions that do not necessarily require all j-tuples of elements in a Dedekind domain to be relatively r-prime, we define the notion of j-wise relative r-primality with respect to a fixed j-uniform hypergraph H. This allows us to provide further generalisations to several results on natural densities not only for a ring of algebraic integers, but also for the ring F_q[x].

整数等差数列与其数字和互质的研究

Arithmetic progressions of integers that are relatively prime to their digital sums

本文研究了$b$-反尼文数等差数列的最大长度，探讨了其在不同基底下的性质和特征。

计数互素的回文数对

Counting relatively prime pairs of palindromes

本文针对任意基数下固定奇数长度的回文数对，给出了互素回文数对数量的渐近公式，解决了 Banks 和 Shparlinski 提出的开放问题。

Piatetski-Shapiro序列中的互素对

Relatively prime pairs in the Piatetski-Shapiro sequences

本文证明了任意大阶Piatetski-Shapiro序列中互素对数量的渐近公式，改进了先前阶数限制的结果，关键方法包括平均技巧和推广Deshouillers的分布结果。

法里序列、斯特恩序列及带互素乘数的弗罗贝尼乌斯问题中的分母

Denominators in the Farey Sequence, the Stern Sequence, and the Frobenius Problem with Relatively Prime Multipliers

该研究针对带系数互素限制的弗罗贝尼乌斯开放问题，证明此类问题仍存在无法表示的最大整数，并探究其与斯特恩序列等的关联及相关性质。

Some remarks in elementary prime number theory

We give an outline of a generalization of the Gelfond-Schnirelmann method in elementary number theory. It is related to an integral of Selberg (1944) generalizing the Euler beta integral. The result we explain was obtained by Nair and Chudnovsky independently in the early eighties.

关于素计数函数的一个不等式的注记

A remark on an inequality for the prime counting function

我们注意到不等式$0.92 \frac{x}{\log(x)} <π(x)< 1.11 \frac{x}{\log(x)}$并不对所有的$x\ge 30$都成立，与某些文献相反。这些估计$π(x)$最近出现在论文代数数论。

关于孪生素数分布的几点注记

Some Remarks on the Distribution of twin Primes

本文介绍了连续双胞胎之间的间隙的计算机数据，最高可达$2^{44}\approx 1.76\times 10^{13}$。P.F.Kelly和T.Pilling \cite{Kelly1}、\cite{Kelly2}最近的论文中所载的描述计算机结果的启发式公式的简单推导，并与“实验”值作了比较。

A remark on the conditional estimate for the sum of a prime and a square

Hardy and Littlewood conjectured that every sufficiently large integer is either a square or the sum of a prime and a square. Let $E(x)$ be the number of positive integers up to $x\ge4$ which does not satisfy this condition. We prove $E(x)\ll x^{1/2}(\log x)^A(\log\log x)^4$with $A=3/2$ under the Generalized Riemann Hypothesis. This is a small improvement of the previous remarks of Mikawa (1993) and Perelli-Zaccagnini (1995) which claims $A=4,3$ respectively.

Uniform upper bounds of the distribution of relatively r-prime lattice points

We estimate the distribution of relatively $r$-prime lattice points in number fields $K$ with their components having a norm less than $x$. In the previous paper we obtained uniform upper bounds as $K$ runs through all number fields under assuming the Lindelöf hypothesis. And we also showed unconditional results for abelian extensions with a degree less than or equal to $6$. In this paper we remove all assumption about number fields and improve uniform upper bounds. Throughout this paper we consider estimates for distribution of ideals of the ring of integer $\mathcal{O}_K$ and obtain uniform …

The probability that random positive integers are k-wise relatively prime

An s-tuple of positive integers are k-wise relatively prime if any k of them are relatively prime. Exact formula is obtained for the probability that s positive integers are k-wise relatively prime.

Norm forms represent few integers but relatively many primes

Norm forms, examples of which include $x^2 + y^2$, $x^2 + x y - 57 y^2$, and $x^3 + 2 y^3 + 4 z^3 - 6 x y z$, are integral forms arising from norms on number fields. We prove that the natural density of the set of integers represented by a norm form is zero, while the relative natural density of the set of prime numbers represented by a norm form exists and is positive. These results require tools from class field theory, including the Artin-Chebotarev density theorem and the Hilbert class field. We introduce these tools as we need them in the course of the main arguments. This article is expo…

Remarks on the distribution of the primitive roots of a prime

Let $\mathbb{F}_p$ be a finite field of size $p$ where $p$ is an odd prime. Let $f(x)\in\mathbb{F}_p[x]$ be a polynomial of positive degree $k$ that is not a $d$-th power in $\mathbb{F}_p[x]$ for all $d\mid p-1$. Furthermore, we require that $f(x)$ and $x$ are coprime. The main purpose of this paper is to give an estimate of the number of pairs $(ξ,ξ^αf(ξ))$ such that both $ξ$ and $ξ^αf(ξ)$ are primitive roots of $p$ where $α$ is a given integer. This answers a question of Han and Zhang.

关于素数分解中加权平均重数的一个注记

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。

关于特殊类型素数模一的 $\sqrt{p}$ 分布的一个注记

A remark on the distribution of $\sqrt{p}$ modulo one involving primes of special type II

研究特殊类型素数模一的平方根分布问题，证明不等式有无穷多素数解，改进了已有结果。

A Remark on the Piatetski-Shapiro-Hua Theorem

In this paper, we prove that for any fixed $205/243<γ\leqslant1$, every sufficiently large $N$ satisfying $N\equiv 5 \pmod {24}$ can be represented as five squares of primes with one prime in $\mathcal{P}_γ$, which improves the previous result of Zhang and Zhai.

A remark on Greenberg's generalized conjecture for imaginary $S_3$-extensions of $\mathbb{Q}$

Let $K/ \mathbb{Q}$ be an imaginary $S_3$-extension, and $p$ a prime number which splits into exactly three primes in $K$. We give a sufficient condition for the validity of Greenberg's generalized conjecture for $K$ and $p$.

Relatively Prime Sets, Divisor Sums, and Partial Sums

For a nonempty finite set $A$ of positive integers, let $\gcd\left(A\right)$ denote the greatest common divisor of the elements of $A$. Let $f\left(n\right)$ and $Φ\left(n\right)$ denote, respectively, the number of subsets $A$ of $\left\{1, 2, \ldots, n\right\}$ such that $\gcd\left(A\right) = 1$ and the number of subsets $A$ of $\left\{1, 2, \ldots, n\right\}$ such that $\gcd\left(A\cup\left\{n\right\}\right) =1$. Let $D\left(n\right)$ be the divisor sum of $f\left(n\right)$. In this article, we obtain partial sums of $f\left(n\right)$, $Φ\left(n\right)$ and $D\left(n\right)$. We also obtain…

Boundary slopes for the Markov ordering on relatively prime pairs

Following McShane, we employ the stable norm on the homology of the modular torus to investigate the Markov ordering on the set of relatively prime integer pairs $(q,p)$ with $q\ge p\ge0$. Our main theorem is a characterization of slopes along which the Markov ordering is monotone with respect to $q$, confirming conjectures of Lee-Li-Rabideau-Schiffler that refine conjectures of Aigner. The main tool is an explicit computation of the slopes at the corners of the stable norm ball for the modular torus.

On some properties of the function of the number of relatively prime subsets of $\{1, 2, ..., n\}$

In the paper we solve few problems proposed by Prapanpong Pongsriiam. Let $f(n)$ denote the number of relatively prime subsets of $\{1, 2, 3, \dots, n\}$ and $g(n)$ denote the number of subsets $A$ of $\{1, 2, 3, \dots, n\}$ such that gcd$(A)>1$ and gcd$(A, n+1)=1$ . We show that $f_n^2-f_{n-k}f_{n+k}>0$ for $n\geq k+1\quad (k\geq2)$. We also show $\frac{g(6n-2)}{g(6n-4)}>\frac{g(6n)}{g(6n-2)}>\frac{g(6n+2)}{g(6n)}<\frac{g(6n+4)}{g(6n+2)}$ for large $n$.

关于超奇异素数处带复乘的椭圆曲线的反分圆Selmer群特征元的一个注记

A remark on the characteristic elements of anticyclotomic Selmer groups of elliptic curves with complex multiplication at supersingular primes

本文针对超奇异素数处带复乘的椭圆曲线，基于Burungale--Kobayashi--Ota的局部点结构研究突破，建立了反分圆扩张下余挠带符号Selmer群与精细Selmer群的特征理想关联，补充了Agboola--Howard的相关结果。

Principality of prime ideals of algebraic number fields

We discuss principality of prime ideals of finite algebraic number fields $L=K(θ)$ over an algebraic number field $K ([K:\mathbb{Q}]<\infty)$ defined by irreducible polynomials $f(x)\in \mathfrak{O}_{K}[x]$ and $f(θ)=0$. Our main Theorem says that if a principal prime ideal $(π)\subset \mathfrak{O}_{K}$ is relatively prime to conductor $\mathfrak{F} =\{α\in \mathfrak{O}_{L}|$ a principal ideal $(α)$ of $\mathfrak{O}_{L}\subset \mathfrak{O}_{K}[θ]\}$ and splits completely over $L$: $(π)\mathfrak{O}_{L}=\prod \mathfrak{p}_{i}$, then $\mathfrak{p}_{i}$ is a principal ideal of $\mathfrak{O}_{L}$ f…

Dirichlet's theorem and Jacobsthal's function

If $a$ and $d$ are relatively prime, we refer to the set of integers congruent to $a$ mod $d$ as an `eligible' arithmetic progression. A theorem of Dirichlet says that every eligible arithmetic progression contains infinitely many primes; the theorem follows from the assertion that every eligible arithmetic progression contains at least one prime. The Jacobsthal function $g(n)$ is defined as the smallest positive integer such that every sequence of $g(n)$ consecutive integers contains an integer relatively prime to $n$. In this paper, we show by a combinatorial argument that every eligible ari…

关于素数与循环数的猜想

Conjectures about Primes and Cyclic Numbers

本文定义了循环数，回顾并提出了素数与循环数的类比猜想，利用小于$10^8$的循环数进行测试，推翻了Hardy和Littlewood素数第二猜想的循环类比。

连续素数中的有界间隙与完美幂间隙

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。

广义哥德巴赫划分中最小素数大小排序的预测

Predicting the size ranking of minimal primes in the generalised Goldbach partitions

本文研究广义哥德巴赫划分中最小素数大小的排序，提出仅依赖m2和m1素因子的预测函数，与实验数据高度相关，并给出数值结果。

Solving $a x^p + b y^p = c z^p$ with $abc$ containing an arbitrary number of prime factors

In this paper we prove new cases of the asymptotic Fermat equation with coefficients. This is done by solving remarkable $S$-units equations and applying a method of Frey-Mazur.

回文殆素数个数的无穷性

Infinitude of palindromic almost-prime numbers

本文证明了在任意基数下，存在无穷多个至多含六个较大素因子的回文数，通过等分布估计和矩上界实现。

Malle's Conjecture for $S_n\times A$ for $n = 3,4,5$

We propose a framework to prove Malle's conjecture for the compositum of two number fields based on proven results of Malle's conjecture and good uniformity estimates. Using this method we can prove Malle's conjecture for $S_n\times A$ over any number field $k$ for $n=3$ with $A$ an abelian group of order relatively prime to 2, for $n= 4$ with $A$ an abelian group of order relatively prime to 6 and for $n=5$ with $A$ an abelian group of order relatively prime to 30. As a consequence, we prove that Malle's conjecture is true for $C_3\wr C_2$ in its $S_9$ representation, whereas its $S_6$ repres…

关于自然数加法表示的若干注记

Remarks on additive representations of natural numbers

本文研究互素无平方因子整数下形如 $a p+b P_{2}$ 的自然数表示，给出表示个数的新下界，并推广和改进了短区间、算术级数及哥德巴赫型上界等特殊情形中的已有结果。

关于Kurepa左阶乘的几点注记

Some remarks on Kurepa's left factorial

建立了Kurepa K（n）的次阶乘函数S（n）与左阶乘函数之间的联系.描述了这两个函数的一些基本性质和同余式。最后，我们计算了K（n）小于10000的素数的素数分布。

Splitting Behavior of $S_n$-Polynomials

We analyze the probability that, for a fixed finite set of primes S, a random, monic, degree n polynomial f(x) with integer coefficients in a box of side B around 0 satisfies: (i) f(x) is irreducible over the rationals, with splitting field over the rationals having Galois group $S_n$; (ii) the polynomial discriminant Disc(f) is relatively prime to all primes in S; (iii) f(x) has a prescribed splitting type at each prime p in S. The limit probabilities as $B \to \infty$ are described in terms of values of a one-parameter family of measures on $S_n$, called splitting measures, with parameter $z…

A remark on primality testing and decimal expansions

We show that for any fixed base $a$, a positive proportion of primes have the property that they become composite after altering any one of their digits in the base $a$ expansion; the case $a=2$ was already established by Cohen-Selfridge and Sun, using some covering congruence ideas of Erdős. Our method is slightly different, using a partially covering set of congruences followed by an application of the Selberg sieve upper bound. As a consequence, it is not always possible to test whether a number is prime from its base $a$ expansion without reading all of its digits. We also present some sli…

Generating Bézout trees for Pythagorean pairs

Relatively prime pairs of integers can be represented as nodes in three way branching trees. We construct trees of Bézout coefficients which correspond to the relatively prime pairs in the aforementioned trees. As one application, we compare the Bézout coefficients in these trees to those returned by the gcd function in Matlab. As another application, we use these trees to decrease the computation time required to create computer generated hyperbolic wallpaper designs.

On Ramanujan expansions and primes in arithmetic progressions

A celebrated theorem of Delange gives a sufficient condition for an arithmetic function to be the sum of the associated Ramanujan expansion with the coefficients provided by a previous result of Wintner. By applying the Delange theorem to the correlation of the von Mangoldt function with its incomplete form, we deduce an inequality involving the counting function of the prime numbers in arithmetic progressions. A remarkable aspect is that such an inequality is equivalent to the famous conjectural formula by Hardy and Littlewood for the twin primes.

不同素数乘积模下的整数幂和的同余关系

A Congruence for Sums of Integer Powers Modulo Products of Distinct Primes

本文证明了在特定条件下，不同素数乘积模下的整数幂和满足一种同余关系。

A remark on partial sums involving the Mobius function

Let $<¶> \subset \N$ be a multiplicative subsemigroup of the natural numbers $\N = \{1,2,3,...\}$ generated by an arbitrary set $¶$ of primes (finite or infinite). We given an elementary proof that the partial sums $\sum_{n \in < ¶>: n \leq x} \frac{μ(n)}{n}$ are bounded in magnitude by 1. With the aid of the prime number theorem, we also show that these sums converge to $\prod_{p \in ¶} (1 - \frac{1}{p})$ (the case when $¶$ is all the primes is a well-known observation of Landau). Interestingly, this convergence holds even in the presence of non-trivial zeroes and poles of the associated zeta…

Factorization Theorems for Relatively Prime Divisor Sums, GCD Sums and Generalized Ramanujan Sums

We generalize recent matrix-based factorization theorems for Lambert series generating functions generating the coefficients $(f \ast 1)(n)$ for some arithmetic function $f$. Our new factorization theorems provide analogs to these established expansions generating sums of the form $\sum_{d: (d,n)=1} f(d)$ (type I) and the Anderson-Apostol sums $\sum_{d|(m,n)} f(d) g(n/d)$ (type II) for any arithmetic functions $f$ and $g$. Our treatment of the type II sums includes a matrix-based factorization method relating the partition function $p(n)$ to arbitrary arithmetic functions $f$. We also conclude…

On Primorial Numbers

Prime numbers have attracted the attention of mathematiciansand enthusiasts for millenniums due to their simple definition and remarkable properties. In this paper, we study primorial numbers (the product of the first prime numbers) to define primorial sets, primorial intervals, primorial tables, and primorial totative numbers. We establish relationships between prime numbers and primorial totative numbers and between admissible k-tuples of prime numbers and admissible k-tuples of primorial totative. Finally, we study the Goldbach conjecture and derive four Goldbach conjectures using primordia…

Some remarks related to Maeda's conjecture

In this article we deal with the problem of counting the number of pairs of normalized eigenforms $ (f,g) $ of weight $k$ and level $N$ such that $ a_p (f) = a_p (g) $ where $a_p (f) $ denotes the $p-$th Fourier coefficient of $f$. Here $p$ is a fixed prime.

计数使得若干$k$元多项式的值互素的正整数$k$元组

Counting $k$-tuples of positive integers such that the values of several polynomials of $k$ variables are relatively prime

本文针对满足一般性质的多项式系统，建立了关于其最大公因数函数的有界强乘性函数和的渐近公式，并由此推导出$k$元组互素密度的Ekedahl-Poonen公式的误差项版本。

Remarks about the Arithmetic of Graphs

The arithmetic of N, Z, Q, R can be extended to a graph arithmetic where N is the semiring of finite simple graphs and where Z and Q are integral domains, culminating in a Banach algebra R. A single network completes to the Wiener algebra. We illustrate the compatibility with topology and spectral theory. Multiplicative linear functionals like Euler characteristic, the Poincare polynomial or the zeta functions can be extended naturally. These functionals can also help with number theoretical questions. The story of primes is a bit different as the integers are not a unique factorization domain…

On the Lehmer's problem involving Euler's totient function

The Euler's totient function $ φ(n) $ counts the positive integers up to a given integer $ n$ that are relatively prime to $ n $. We solve a problem due to Lehmer that there is no composite number $ n $ such that $ φ(n)\mid n-1 $. We also note that there are infinitely many primes that are not Sophie Germain primes.

Pairwise Relative Primality of Positive Integers

Given a graph $G=(V,E)$ with $V=\{1,2,...,k\}$, the $k$ positive integers $a_1,a_2, ...,a_k$ are $G$-wise relatively prime if $(a_i, a_j)=1$ for $\{i,j\} \in E$. In this note we consider the problem of finding the probability $A_G$ that k positive integers are $G$-wise relatively prime. As an application of our results, we solve the problems of finding probabilities that k positive integers have exact (or at least) r relatively prime pairs, which was proposed by P. Moree.

Simple proof of the integrality of the Somos-5 sequence

In this paper, a simple proof that shows that the Somos-5 sequence produces only natural numbers is given. We also give simple proof that each $a_n$ in the Somos-5 sequence is relatively prime to its four predecessors. The manner in which the integrality is proven is taken from the 1992 proof for the integrality of the Somos-4 sequence by Janice. L. Malouf. The manner in which it is proven that each $a_n$ is relatively prime to its four predecessors is taken from a paper from Michael J. Crabb, who proved that each $a_n$ in the Somos-5 sequence is relatively prime to its two predecessors.

大Zsigmondy素数

Large Zsigmondy Primes

该数论研究明确了互素正整数对$(a,b)$与正整数$n$对应的大Zsigmondy素数的定义，完成了所有不存在大Zsigmondy素数的整数三元组$(a,b,n)$的分类工作。

关于函数域上Catalan方程的注记

Remarks on Catalan's equation over function fields

本文研究特征为素数的整体函数域上Catalan方程的解，证明在特定条件下当指数均大于1时不存在非常数解。

分拆函数及相关函数的闭式公式

Closed-Form Formula for the Partition Function and Related Functions

本文为分拆函数及 Andrews 的 spt 函数推广提出了新的闭式算术与递归公式，利用容斥原理推导了非互质分拆函数公式，并通过分拆的最大公约数定理将分拆与素数联系，推导出互质函数公式及多个数论函数的新恒等式。