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

Let $A$ be a finite set of relatively prime positive integers, and let $S(A)$ be the set of all nonnegative integral linear combinations of elements of $A$. The set $S(A)$ is a semigroup that contains all sufficiently large integers. The…

Kurepa's conjecture states that there is no odd prime $p$ that divides $!p=0!+1!+\cdots+(p-1)!$. We search for a counterexample to this conjecture for all $p<2^{34}$. We introduce new optimization techniques and perform the computation…

1. There is no existing any quadratic interval $\eta_{n}:=(n^{2},(n+1)^{2}],$ which contains less than 2 prime numbers. The number of prime numbers within $\eta_{n}$ goes averagely linear with n to infinity. 2. The exact law of the number…

We give a solution stated in the title to problem 3 of part 1 of the problems listed in the book of Eklof and Mekler [EM],(p.453). There, in pp. 241-242, this is discussed and proved in some cases. The existence of strongly lambda-free ones…

Assuming that Siegel zeros exist, we prove a hybrid version of the Chowla and Hardy--Littlewood prime tuples conjectures. Thus, for an infinite sequence of natural numbers $x$, and any distinct integers $h_1,\dots,h_k,h'_1,\dots,h'_\ell$,…

Let $\pi$ be a set of primes such that $|\pi|\geqslant 2$ and $\pi$ differs from the set of all primes. Denote by $r$ the smallest prime which does not belong to $\pi$ and set $m=r$ if $r=2,3$ and $m=r-1$ if $r\geqslant 5$. We study the…

A $(v,k,\lambda, \mu)$-partial difference set (PDS) is a subset $D$ of a group $G$ such that $|G| = v$, $|D| = k$, and every nonidentity element $x$ of $G$ can be written in either $\lambda$ or $\mu$ different ways as a product $gh^{-1}$,…

A well-known conjecture asserts that, for any given positive real number $\lambda$ and nonnegative integer $m$, the proportion of positive integers $n \le x$ for which the interval $(n,n + \lambda\log n]$ contains exactly $m$ primes is…

For a subset $A$ of $\{1,2,\ldots,N\}^2$ of size $\alpha N^2$ we show existence of $(m,n)\neq(0,0)$ such that the set $A$ contains at least $(\alpha^3 - o(1))N^2$ triples of points of the form $(a,b)$, $(a+m,b+n)$, $(a-n,b+m)$. This answers…

A classic theorem in combinatorial design theory is Fisher's inequality, which states that a family $\mathcal F$ of subsets of $[n]$ with all pairwise intersections of size $\lambda$ can have at most $n$ non-empty sets. One may weaken the…

Let $\{ B(t) \colon 0\leq t\leq 1\}$ be a linear Brownian motion and let $\dim$ denote the Hausdorff dimension. Let $\alpha>\frac12$ and $1\leq \beta \leq 2$. We prove that, almost surely, there exists no set $A\subset[0,1]$ such that $\dim…

Let $G$ be an additive group of order $v$. A $k$-element subset $D$ of $G$ is called a $(v, k, \lambda, t)$-almost difference set if the expressions $gh^{-1}$, for $g$ and $h$ in $D$, represent $t$ of the non-identity elements in $G$…

This paper provides a survey of results on the greatest prime factor, the number of distinct prime factors, the greatest squarefree factor and the greatest m-th powerfree part of a block of consecutive integers, both without any assumption…

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…

We state a general purpose algorithm for quickly finding primes in evenly divided sub-intervals. Legendre's conjecture claims that for every positive integer $n$, there exists a prime between $n^2$ and $(n+1)^2$. Oppermann's conjecture…

In 2000 Deaconescu raised a question whether there exists a composite $n$ for which $S_2(n)|\phi(n)-1$, where $\phi(n)$ is Euler's function and $S_2(n)$ is Schemmel's totient function. In this paper we prove that any such $n$ is odd,…

In the present paper we prove several results concerning the existence of low-discrepancy point sets with respect to an arbitrary non-uniform measure $\mu$ on the $d$-dimensional unit cube. We improve a theorem of Beck, by showing that for…

We study the distribution of prime numbers under the unlikely assumption that Siegel zeros exist. In particular we prove for \[ \sum_{n \leq X} \Lambda(n) \Lambda(\pm n+h) \] an asymptotic formula which holds uniformly for $h = O(X)$. Such…

It is conjectured that Hadamard matrices exist for all orders $4t$ ($t>0$). However, despite a sustained effort over more than five decades, the strongest overall existence results are asymptotic results of the form: for all odd natural…

Let $a\geq 1, b\geq 0$ and $k\geq 2$ be any given integers. It has been proven that there exist infinitely many natural numbers $m$ such that sum of divisors of $m$ is a perfect $k$th power. We try to generalize this result when the values…