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 $n$.
We say that n ideals of algebraic integers in a fixed number ring are k-wise relatively r-prime if any k of them are relatively r-prime. In this article, we provide an exact formula for the probability that n nonzero ideals of algebraic integers in a fixed number ring are k-wise relatively r-prime.
The paper solves the problems of determining the asymptotics of the number of primes and the sums of functions of primes in a subset of the natural series that satisfies the conditions that the asymptotic density of the number of primes in this subset is constant and not equal to zero.
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].
Arithmetic progressions of integers that are relatively prime to their digital sums
The College of Idaho（爱达荷学院）；Cedar Crest College（雪松crest学院）；Bryn Mawr College（布林茅尔学院）；University of Michigan（密歇根大学）；Kutztown University of Pennsylvania（宾夕法尼亚库兹敦大学）
Benford's Law describes the prevalence of small numbers as the leading digits of numbers in many sets of integers. We prove a variant of Benford's law for many positive-density subsets of the primes. This follows from a more general result over number fields.
We introduce a wide class of deterministic subsets of primes of zero relative density and we prove Roth's Theorem in these sets, namely, we show that any subset of them with positive relative upper density contains infinitely many non-trivial three-term arithmetic progressions. We also prove that the Hardy--Littlewood majorant property holds for these subsets of primes. Notably, our considerations recover the results for the Piatetski--Shapiro primes for exponents close to $1$, which are primes of the form $\lfloor n^c\rfloor$ for a fixed $c>1$.
In a recent work \cite{key-11}, A. Fish proved that if $E_{1}$ and $E_{2}$ are two subsets of $\mathbb{Z}$ of positive upper Banach density, then there exists $k\in\mathbb{Z}$ such that $k\cdot\mathbb{Z}\subset\left(E_{1}-E_{1}\right)\cdot\left(E_{2}-E_{2}\right).$ In this article we will show that a similar result is true for the set of primes $\mathbb{P}$ (which has density $0$). We will prove that there exists $k\in\mathbb{N}$ such that $k\cdot\mathbb{N}\subset\left(\mathbb{P}-\mathbb{P}\right)\cdot\left(\mathbb{P}-\mathbb{P}\right),$ where $\mathbb{P}-\mathbb{P}=\left\{ p-q:p>q\,\text{and}…
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.
Norm forms represent few integers but relatively many primes
范数形式，其例子包括$x^2 + y^2$、$x^2 + x y - 57 y^2$和$x^3 + 2 y^3 + 4 z^3 - 6 x y z$，是由数域上的范数产生的整数形式。证明了由范数形式表示的整数集合的自然密度为零，而由范数形式表示的素数集合的相对自然密度存在且为正.这些结果需要类域理论的工具，包括Artin-Chebotarev密度定理和Hilbert类域。我们介绍这些工具，因为我们需要他们在过程中的主要论点。这篇文章是暂时的性质，并假定只有第一当然代数数论。
In this paper, a simple explanation for the Goldbach Conjecture is given. We have shown that the probability of violating the conjecture not only for the prime numbers, but also for any subset of natural numbers whose distribution is similar to the prime numbers is negligible. This result makes it possible to generalize the conjecture to any subset of natural numbers whose distribution is similar to the prime numbers. Additionally, we selected several new subsets whose distribution amongst the natural numbers are similar to the prime numbers by randomly addition of +1 and -1 to the prime numbe…
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…
如果$a$和$d$是互素，我们把全等于$a$ mod $d$的整数集合称为“合格”等差数列。狄利克雷的一个定理说，每一个合格的等差数列包含无穷多个素数;该定理从断言，每一个合格的等差数列包含至少一个素数。Jacobsthal函数$g(n)$被定义为最小正整数，使得$g(n)$连续整数的每个序列包含与$n$互素的整数。在本文中，我们通过一个组合的论点，每一个合格的算术级数与$d\le76$包含至少一个素数，我们表明，某些合理的界限上的Jacobsthal函数的素数将意味着，每一个合格的等差数列包含至少一个素数。也就是说，Jacobsthal函数的某些合理的界限将导致狄利克雷定理的初等证明。
