Related papers: On an inverse problem in additive number theory

We improve some results on the size of the greatest prime factor of integers of the form ab+1, where a and b belong to finite sets of integers with rather large density.

Let G be an additive abelian group whose finite subgroups are all cyclic. Let A_1,...,A_n (n>1) be finite subsets of G with cardinality k>0, and let b_1,...,b_n be pairwise distinct elements of G with odd order. We show that for every…

We investigate the occurrence of additive and multiplicative structures in random subsets of the natural numbers. Specifically, for a Bernoulli random subset of $\mathbb{N}$ where each integer is included independently with probability…

We study solutions to the equation $a+b=c$, where $a,b,c$ form a triple of coprime natural numbers. The $abc$ conjecture asserts that, for any $\epsilon>0$, such triples satisfy $\mathrm{rad}(abc) \ge c^{1-\epsilon}$ with finitely many…

Let $\mathcal{P}$ denote the set of all primes. $P_{1},P_{2},P_{3}$ are three subsets of $\mathcal{P}$. Let $\underline{\delta}(P_{i})$ $(i=1,2,3)$ denote the lower density of $P_{i}$ in $\mathcal{P}$, respectively. It is proved that if…

Let G be any additive abelian group with cyclic torsion subgroup, and let A, B and C be finite subsets of G with cardinality n>0. We show that there is a numbering {a_i}_{i=1}^n of the elements of A, a numbering {b_i}_{i=1}^n of the…

For any integer $x$, let $T_x$ denote the triangular number $\frac{x(x+1)}{2}$. In this paper we give a complete characterization of all the triples of positive integers $(\alpha, \beta, \gamma)$ for which the ternary sums $\alpha x^2…

Does a given system of linear equations with nonnegative constraints have an integer solution? This is a fundamental question in many areas. In statistics this problem arises in data security problems for contingency table data and also is…

Let $G$ be an additive finite abelian group with exponent $\exp(G)$. For $L\subseteq \mathbb N$, let $\mathsf{s}_{L}(G)$ be the smallest integer $\ell$ such that every sequence $S$ over $G$ of length $\ell$ has a zero-sum subsequence $T$ of…

We describe a method for bounding the set of exceptional integers not represented by a given additive form in terms of the exceptional set corresponding to a subform. Illustrating our ideas with examples stemming from Waring's problem for…

In this paper, we prove the conjecture that if there is an odd perfect number, then there are infinitely many of them.

We prove that finite sets of real numbers satisfying $|AA| \leq |A|^{1+\epsilon}$ with sufficiently small $\epsilon > 0$ cannot have small additive bases nor can they be written as a set of sums $B+C$ with $|B|, |C| \geq 2$. The result can…

In the several contexts such as combinatorial number theory, families of sets of positive integers closed under taking subsets have been investigated. Then it is sometimes useful to give bijections between the set of the one-sided infinite…

We introduce a general class $F_0$ of additive functions $f$ such that $f(p) = 1$ and prove a tight bound for exponential sums of the form $\sum_{n \le x} f(n) e(\alpha n)$ where $f \in F_0$ and $e(\theta) = \exp(2\pi i \theta)$. Both…

A route to evaluate exact sums represented by Dirichlet eta and beta functions, both of which are alternating and divergent at negative integer arguments, is advocated. It rests on precise polynomial extrapolations and stands as a…

For a given subset $A\subseteq \mathbb F_q^*$, we study the problem of finding a large packing set $B$ of $A$, that is, a set $B \subseteq \mathbb F_q^*$ such that $|AB|=|A||B|$. We prove the existence of such a $B$ of size $|B|\ge…

We prove the following theorem: for all positive integers $b$ there exists a positive integer $k$, such that for every finite set $A$ of integers with cardinality $|A| > 1$, we have either $$ |A + ... + A| \geq |A|^b$$ or $$ |A \cdot ...…

For an $n$-bit positive integer $a$ written in binary as $$ a = \sum_{j=0}^{n-1} \varepsilon_{j}(a) \,2^j $$ where, $\varepsilon_j(a) \in \{0,1\}$, $j\in\{0, \ldots, n-1\}$, $\varepsilon_{n-1}(a)=1$, let us define $$ \overleftarrow{a} =…

Erd\H{o}s asked whether there are infinitely many finite sets of distinct primes $p_1<\cdots<p_k$ and positive integers $m$ such that \begin{equation}\label{eq:erdos-original} \frac1{p_1}+\cdots+\frac1{p_k}=1-\frac1m. \end{equation} This is…

Limit theorems for non-additive probabilities or non-linear expectations are challenging issues which have raised progressive interest recently. The purpose of this paper is to study the strong law of large numbers and the law of the…