Related papers: Carmichael Numbers in All Possible Arithmetic Prog…

We consider a tuple $\Phi = (\phi_1,\ldots,\phi_m)$ of commuting maps on a finitary matroid $X$. We show that if $\Phi$ satisfies certain conditions, then for any finite set $A\subseteq X$, the rank of $\{\phi_1^{r_1}\cdots\phi_m^{r_m}(a):a…

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…

We introduce the concept of an almost prime number generalizing a prime number. It turns out that a composite almost prime number must be a Carmichael number, in case it exists. We prove several properties of almost prime numbers and…

Assuming the generalized Riemann hypothesis, we give asymptotic bounds on the size of intervals that contain primes from a given arithmetic progression using the approach developed by Carneiro, Milinovich and Soundararajan [Comment. Math.…

We consider various arithmetic questions for the Piatetski-Shapiro sequences $\fl{n^c}$ ($n=1,2,3,...$) with $c>1$, $c\not\in\N$. We exhibit a positive function $\theta(c)$ with the property that the largest prime factor of $\fl{n^c}$…

Based on the Goldbach conjecture and arithmetic fundamental theorem, the Goldbach conjecture was extended to more general situations, i.e., any positive integer can be written as summation of some specific prime numbers, which depends on…

Let $K$ be a finitely generated field of characteristic zero. We study, for fixed $m \geq 2$, the rational functions $\phi$ defined over $K$ that have a $K$-orbit containing infinitely many distinct $m$th powers. For $m \geq 5$ we show the…

In his book \emph{Topics in Analytic Number Theory}, Hans Rademacher conjectured that the limits of certain sequences of coefficients that arise in the ordinary partial fraction decomposition of the generating function for partitions of…

We obtain results for the following question where $m\ge 1$ and $n\ge 2$ are integers. {\bf Question.} For which continuous functions $f\colon [0,\infty)\to [0,\infty)$ does there exist a continuous function $\phi\colon (0,1)\to (0,\infty)$…

We prove that in every ring of generalised power series with non-positive real exponents and coefficients in a field of characteristic zero, every series admits a factorisation into finitely many irreducibles of infinite support, the number…

Iannucci considered the positive divisors of a natural number $n$ that do not exceed $\sqrt{n}$ and found all forms of numbers whose such divisors are in arithmetic progression. In this paper, we generalize Iannucci's result by excluding…

We introduce the generic Lah polynomials $L_{n,k}(\phi)$, which enumerate unordered forests of increasing ordered trees with a weight $\phi_i$ for each vertex with $i$ children. We show that, if the weight sequence $\phi$ is…

A space $G(M, \varPhi)$ of infinitely differentiable functions in ${\mathbb R}^n$ constructed with a help of a family $\varPhi=\{\varphi_m\}_{m=1}^{\infty}$ of real-valued functions $\varphi_m \in~C({\mathbb R}^n)$ and a logarithmically…

A positive integer $n$ is said to be a Zumkeller number or an integer-perfect number if the set of its positive divisors can be partitioned into two subsets of equal sums. In this paper, we prove several results regarding Zumkeller numbers.…

Define {\em the Liouville function for $A$}, a subset of the primes $P$, by $\lambda_{A}(n) =(-1)^{\Omega_A(n)}$ where $\Omega_A(n)$ is the number of prime factors of $n$ coming from $A$ counting multiplicity. For the traditional Liouville…

While solving a special case of a question of Erd\H{o}s and Graham Steinerberger asks for all integers $n$ with $\phi(n)=\frac{2}{3} \cdot (n+1)$. He discovered the solutions $n\in\{5, 5 \cdot 7, 5\cdot 7\cdot 37, 5\cdot 7\cdot 37\cdot…

We find upper bounds that are sharp for the number of $k$th powers inside arbitrary arithmetic progressions whose step has $O(1)$ many divisors.

Let $c$ be a fixed integer such that $c \in \{0,2\}.$ Let $n$ be a positive integer such that either $n\geq 2$ or $2n+1 \neq 3^u$ for any integer $u\geq 2$ according as $c = 0$ or not. Let $\phi(x)$ belonging to $\mathbb{Z}[x]$ be a monic…

Let $C({\mathbb R}^n)$ denote the set of real valued continuous functions defined on ${\mathbb R}^n$. We prove that for every $n\ge 2$ there are positive numbers $\lambda _1 , \ldots , \lambda _n$ and continuous functions $\phi_1 ,\ldots ,…

In this paper we will give a scheme-theoretic discussion on the unramified extensions of an arithmetic function field in several variables. The notion of unramified discussed here is parallel to that in algebraic number theory and for the…