Related papers: Multiplicative complements II

Let $0\le \alpha \le \beta\le 1$. For any finite set $B\subset\mathbb{N}$, we show that there exists a set $A\subset\mathbb{N}$ such that $\underline{d}(A+B) = \alpha$ and $\bar{d}(A+B) = \beta$, where $\underline{d}(A+ B)$ and…

Assuming the Generalized Riemann Hypothesis, the authors study when a character sum over all n <= x is o(x); they show that this holds if log x / log log q -> infinity and q -> infinity (q is the size of the finite field).

There is a standard "word length" metric canonically associated to any set of generators for a group. In particular, for any integers a and b greater than 1, the additive group of integers has generating sets {a^i}_{i=0}^{\infty} and…

We prove that if $B\subseteq A$ is an extension of finite dimensional algebras such that the projective dimension of $A/B$ as a $B$-bimodule is finite, if $A$ has finite finitistic dimension, then so does $B$. We exhibit examples…

A positive integer $n$ is called perfect if $ \sigma(n)=2n$, where $\sigma(n)$ denote the sum of divisors of $n$. In this paper we study the ratio $\frac{\sigma(n)}{n}$. We define the function Abundancy Index $I:\mathbb{N} \to \mathbb{Q}$…

It is proved that all sufficiently large integers $n$ can be represented as $$n=x_1^2+x_2^3+\cdots+x_{13}^{14},$$ where $x_1,\ldots,x_{13}$ are positive integers. This improves upon the current record with $14$ variables in place of $13$.

We investigate additive properties of sets $A,$ where $A=\{a_1,a_2,\ldots ,a_k\}$ is a monotone increasing set of real numbers, and the differences of consecutive elements are all distinct. It is known that $|A+B|\geq c|A||B|^{1/2}$ for any…

We show that for every fixed $\ell\in\mathbb{N}$, the set of $n$ with $n^\ell|\binom{2n}{n}$ has a positive asymptotic density $c_\ell$, and we give an asymptotic formula for $c_\ell$ as $\ell\to \infty$. We also show that $\# \{n\le x,…

Let $N(x,y)$ denote the number of integers $n\le x$ which are divisible by a shifted prime $p-1$ with $p>y$, $p$ prime. Improving upon recent bounds of McNew, Pollack and Pomerance, we establish the exact order of growth of $N(x,y)$ for all…

Le n be any positive integer. A hyperbinary expansion of n is are presentation of n as sum of powers of 2, each power being used at most twice. In this paper we study some properties of a suitable edge-coloured and vertex-weighted oriented…

Let $A$ be a finite multiset of integers. If $B$ be a multiset such that $A$ and $B$ are $t$-complementing multisets of integers, then $B$ is periodic. We obtain the Biro-type upper bound for the smallest such period of $B$: Let…

Let A be a positive operator in an infinite sigma-finite von Neumann factor M and let B_j be a sequence of positive elements in M. We give sufficient conditions for decomposing A into a sum of elements C_j equivalent to B_j for all j ( C…

We prove that assuming the Generalized Riemann Hypothesis every even integer larger than $\exp(\exp(15.85))$ can be written as the sum of a prime number and a number that has at most two prime factors.

Let $A\subset\mathbb{N}$, $\alpha\in(0,1)$, and for $x\in\mathbb{R}$ let $e(x):=e^{2\pi ix}$. We set $$S_{A}(\alpha,N):=\sum_{\substack{n\in A\n\leq N}}e(n\alpha).$$ Recently, Lambert A'Campo proposed the following question: is there an…

Some new results concerning the equation $\sigma(N)=aM, \sigma(M)=bN$ are proved. As a corollary, there are only finitely many odd superperfect numbers with a fixed number of distinct prime factors.

In this paper, we prove the number of countable models of a countable supersimple theory is either 1 or infinite. This result is an extension of Lachlan's theorem on a superstable theory.

Let $\Omega(n)$ denote the number of prime factors of a positive integer $n$ counted with multiplicities. We show that for any bounded functions $a,b\colon\mathbb{N}\to\mathbb{C}$, $$\frac{1}{\log{N}}\sum_{n=1}^N…

In this paper, we consider representations of integers as sums of generalized heptagonal numbers with a prescribed number of repeats of each heptagonal number appearing in the sum. In particular, we investigate the classification of such…

We prove dual theorems to theorems proved by author in \cite {5}. Beginning with Section 10, we introduce and study so-called "twin numbers of the second kind" and a postulate for them. We give two proofs of the infinity of these numbers…

A celebrated unresolved conjecture of Peter Frankl states that every finite union-closed collection of sets ($B$), with non-empty universe, admits an abundant element. The best result in the literature states that if $|B|=n$, then there…