Related papers: On difference sets with small $\lambda$

In 1977, Hugh Williams studied Lucas pseudoprimes to all Lucas sequences of a fixed discriminant. These are composite numbers analogous to Carmichael numbers and they satisfy a Korselt-like criterion: $n$ must be a product of distinct…

We prove an explicit analogue of Legendre's conjecture for almost primes. Namely, for every integer $n \geq 1$, the interval $(n^2,(n+1)^2)$ contains an integer having at most $3$ prime factors, counted with multiplicity. This improves the…

We develop a lower bound sieve for primes under the (unlikely) assumption of infinitely many exceptional characters. Compared with the illusory sieve due to Friedlander and Iwaniec which produces asymptotic formulas, we show that less…

In this paper we study combinatorial aspects of permutations of $\{1,\ldots,n\}$ and related topics. In particular, we prove that there is a unique permutation $\pi$ of $\{1,\ldots,n\}$ such that all the numbers $k+\pi(k)$ ($k=1,\ldots,n$)…

The Goldbach conjecture states that every even integer greater than 2 can be expressed as the sum of two prime numbers. This conjecture was first proposed by German mathematician Christian Goldbach in 1742 and, despite being obviously true,…

A subset of the integers larger than 1 is $primitive$ if no member divides another. Erdos proved in 1935 that the sum of $1/(a\log a)$ for $a$ running over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he…

The basic theme of this paper is the fact that if $A$ is a finite set of integers, then the sum and product sets cannot both be small. A precise formulation of this fact is Conjecture 1 below due to Erd\H os-Szemer\'edi [E-S]. (see also…

Let $\Gamma\subset \bar{\mathbb Q}^{\times}$ be a finitely generated multiplicative group of algebraic numbers. Let $\alpha_1,\ldots,\alpha_r\in\bar{\mathbb Q}^\times$ be algebraic numbers which are $\mathbb{Q}$-linearly independent and let…

Let $(G, +)$ be an abelian group. In 2004, Eliahou and Kervaire found an explicit formula for the smallest possible cardinality of the sumset $A+A$, where $A \subseteq G$ has fixed cardinality $r$. We consider instead the smallest possible…

We prove new results on additive properties of finite sets $A$ with small multiplicative doubling $|AA|\leq M|A|$ in the category of real/complex sets as well as multiplicative subgroups in the prime residue field. The improvements are…

We estimate the asymptotic density of the set $\bar{A}$ of primes $p$ satisfying the constraint that $p+1$ and $p-1$ have only one prime divisor larger than $3$. We also estimate the density of a maximal subset $\bar{B} \subset \bar{A}$…

Let G be a finite abelian group with |G|>1. Let a_1,...,a_k be k distinct elements of G and let b_1,...,b_k be (not necessarily distinct) elements of G, where k is a positive integer smaller than the least prime divisor of |G|. We show that…

In the first paper under this title (1977), the first author utilized a duality identity between the largest and smallest prime factors involving the Moebius function, to establish the following result as a consequence of the Prime Number…

In this paper, when the order of $\theta$ is even, we prove that there exists no central difference sets in $A_2(m,\theta)$ and establish some non-existence results of central partial difference sets in $A_p(m,\theta)$ with $p>2$. When the…

In 1737 Leonard Euler gave what we often now think of as a new proof, based on infinite series, of Euclid's theorem that there are infinitely many prime numbers. Our short paper uses a simple modification of Euler's argument to obtain new…

The arithmetic properties of the ordinary partition function $p(n)$ have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form $p(\ell n+\beta)\equiv 0\pmod\ell$ for the…

A finite set $S \subset \mathbb{Z}$ is a Sidon set if its pairwise differences are distinct. Recall that a perfect difference set (PDS) of order $n$ is a set $B \subset \mathbb{Z}_v$ ($v = n^2 - n + 1$) of size $n$ such that every nonzero…

Skew Hadamard difference sets are an interesting topic of study for over seventy years. For a long time, it had been conjectured the classical Paley difference sets (the set of nonzero quadratic residues in $\mathbb{F}_q$ where $q \equiv 3…

For a finite set of non-zero natural numbers that contains at least one element different from 1 and the least common multiple of any of its subsets, there exists a subset of at least half of its members which has a common divisor larger…

We show that, in contrast to the integers setting, almost all even order abelian groups $G$ have exponentially fewer maximal sum-free sets than $2^{\mu(G)/2}$, where $\mu(G)$ denotes the size of a largest sum-free set in $G$. This confirms…